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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a scheme, locally of finite type over a locally noetherian scheme \(S\), and let \({ F}:={Hilb}_{X/S}\) be Grothendieck's Hilbert functor, i.e. the set \({\mathcal F}(T)\) consists of closed schemes of \( Z \subseteq X \times_S T\) which are flat and of proper support over \(T\). If \(X\) is quasiprojective (resp. separable) over \(S\), then the Hilbert funcor \(\text{Hilb}_{X/S}\) is representable by a quasiprojective scheme (resp. an algebraic space) due to fundamental theorems of \textit{A. Grothendieck} [Sem. Bourbaki 13(1960/61), No. 221 (1961; Zbl 0236.14003)] and \textit{A. Artin} [in: Global Analysis, Papers in Honor of K. Kodaira, 21--71 (1969; Zbl 0205.50402)]. One of Artin's necessary conditions for e.g. \({\mathcal F}\) to be representable, is that formal deformations are effective, i.e. that the map \({\mathcal F}(A) \to {\displaystyle\lim_{\longleftarrow}}{\mathcal F}(A/m^n)\) is surjective for any complete local ring \((A,m)\). The authors of this paper show that if \(X\) is not separated over \(S\), then the Hilbert functor \(\text{Hilb}_{X/S}^1\) of one point has non-effective formal deformations. Thus \(\text{Hilb}_{X/S}^1\) is not representable by a scheme or an algebraic space. Indeed \(\text{Hilb}_{X/S}^1 \simeq {\Hom}_S(-,X)\) if and only if \(X\to S\) is separated. Hilbert functor; representability; separated scheme; algebraic space Lundkvist C. and Skjelnes R., Non-effective deformations of Grothendieck's Hilbert functor, Math. Z. 258 (2008), no. 3, 513-519. Parametrization (Chow and Hilbert schemes), Local deformation theory, Artin approximation, etc., Generalizations (algebraic spaces, stacks) Non-effective deformations of Grothendieck's Hilbert functor | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{F. C. Kirwan} [Cohomology of quotients in symplectic and algebraic geometry. Princeton, New Jersey: Princeton University Press (1984; Zbl 0553.14020)] studied the map from the equivariant cohomology of a Hamiltonian group action to the cohomology of the symplectic quotient. The paper under review deals with the quantum version of this situation. Let \(X\) be a smooth projectively embedded variety with a connected reductive group action such that the stable locus is equal to the semistable locus.
Let \(QH(X/\!/G)\) be the quantum cohomology of the GIT quotient \(X/\!/G\), and let \(QH_G(X)= H_G(X)\otimes \Lambda_X^G\) denote the equivariant quantum cohomology where \( \Lambda_X^G \subset \Hom(H_2^G(X,\mathbb{Z}))\) is the Novikov field. \textit{A. B. Givental} [Int. Math. Res. Not. 1996, No. 13, 613--663 (1996; Zbl 0881.55006)] equipped the latter with a product structure arising from the equivariant Gromov-Witten theory. \textit{E. Gonzalez} and the author of the paper under review [``Quantum Witten localization and abelianization for qde solutions'', \url{arXiv:0811.3358}] had proved that, under some conditions, the moduli space of stable gauged maps from a smooth connected projective curve is a proper Deligne-Mumford stack equipped with a perfect obstruction theory which leads to the definition of the gauged Gromov-Witten invariants.
The first result of the paper under review is the construction of the ``Quantum Kirwan Morphism''
\[
\kappa_X^G:QH_G(X)\to QH(X/\!/G)
\]
by means of the virtual integration over a compactified stack of affine gauged maps. \(\kappa_X^G\) is a morphism of CohFT algebras, which can be considered as a non-linear generalization of an algebra homomorphism.
The second main result of the paper under review relates in the large area limit the graph potentials via the quantum Kirwan morphism.
More precisely, suppose that \(C\) is a smooth projective curve such that all the semistable gauged maps from \(C\) to \(X\) are stable for sufficiently large stability parameters. Then
\[
\tau_{X/\!/G}\circ\kappa^G_X=\lim_{\rho\to \infty}\tau_X^G
\]
where \(\rho\) is the stability parameter, \(\tau_{X/\!/G}\) is the genus zero graph potential for the GIT quotient \(X/\!/G\), and \(\tau^G_X\) is the gauged Gromov-Witten potential. The paper under review also proves the localized version of this theorem that arises as the fixed point contributions for a circle acting on the domain. This localization in the case of Gromov-Witten invariants gives rise to a solution for a version of the Picard-Fuchs quantum differential equation for \(X/\!/G\).
Some of the results have overlaps and connections with the works of Givental, Lian-Liu-Yau, Iritani, Ciocan-Fontanine-Kim-Maulik, and Coates-Corti-Iritani-Tseng who have taken different approaches. quantum Kirwan morphism; gauged Gromov-Witten invariants C. Woodward, Quantum Kirwan morphism and Gromov-Witten invariants of quotients I, II, III, Transform. Groups 20 (2015), 507-556, 881-920, 1155-1193. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum Kirwan morphism and Gromov-Witten invariants of quotients. I | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors prove that Lusztig's Frobenius map (for quantum groups at roots of unity) can be, after dualizing, viewed as a characteristic zero lift of the geometric Frobenius splitting of \(G/B\) (in char \(p>0\)) introduced by \textit{V. B. Mehta} and \textit{A. Ramanathan} [Ann. Math. (2) 122, 27-40 (1985; Zbl 0601.14043)]. Frobenius map; Schubert variety; quantum group; geometric Frobenius splitting Kumar, S., Littelmann, P.: Frobenius splitting in characteristic zero and the quantum Frobenius map. J. Pure Appl. Algebra 152, 201--216 (2000) Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized function algebras) and their representations, Cohomology theory for linear algebraic groups Frobenius splitting in characteristic zero and the quantum Frobenius map | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials M. Auslander initiated the use of noncommutative algebras of finite dimension by studying the representation theory of Cohen-Macaulay rings, that is, more generally \textit{orders}. Two classes of noncommutative algebras were singled out, the Auslander algebras and the non-singular orders. The representation theory of finite-dimensional representations is encoded in the structure of their Auslander algebras. All Cohen-Macaulay modules are projective, so that the study of the non-singular orders are more basic.
The study of such algebras gives applications in algebraic geometry. A first example is Van den Bergh's definition of \textit{Noncommutative Crepant Resolution}, NCCR: Let \(R\) be a commutative noetherian normal domain. Then a reflexive \(R\)-module \(M\) is said to give a NCCR of \(\text{Spec}R\) if \(\Lambda=\text{End}_R(M)\) is a nonsingular \(R\)-order, which means that \(\Lambda_{\mathfrak p}=\text{End}_{R_{\mathfrak p}}(M_{\mathfrak p},M_{\mathfrak p})\) is a maximal Cohen-Macaulay \(R_{\mathfrak p}\)-module for each \(\mathfrak p\in\text{Spec}R\).
This article considers the slightly simpler concept of NCR, that is \textit{Noncommutative resolution}: A finitely generated module \(M\) over a commutative noetherian ring \(R\) is called a NCR of \(\text{Spec}(R)\) if \(M\) is faithful and \(\text{End}_R(M)\) has finite global dimension. NCRs exist when \(R\) is artinian, or reduced and one-dimensional. This article treats the question on which rings \(R\) that have an NCR. The conditions are given in the terms of the Grothendieck group of the category of finitely generated \(R\)-modules, and its subcategories. The existence of an NCR forces strong constraints on the singularities of \(R\). The formulations of the conditions on \(R\) in terms of the Grothendieck group leads to influence and use of results from algebraic K-theory. This leads to one of the main results stating that for surface singularities over an algebraically closed field, the existence of a NCR characterise rational singularities. Thus the rationality of a surface singularity can be tested on the existence of a NCR.
The main results of the article, more or less verbatim, is as follows:
{Theorem 2.5.} Let \(R\) be a semilocal ring and assume that \(M\) gives a NCR of \(R\). Let \(\mathcal C_M\) be the full subcategory of mod\(R\) consisting of \(X\) satisfying \(\text{supp}X\subset\text{NG}(M).\) Then \(K_0(R)/\langle\mathcal C_M\rangle\) is a finitely generated abelian group. (\(K_0(R)\) denotes the Grothendieck group of \(R\), \(\text{NG}(M)\) is the nongenerating locus of \(M\).)
{Theorem 3.11.} Let \(R\) be a normal, Cohen-Macaulay standard graded algebra over a subfield \(k\) of \(\mathbb C\). Let \(\mathfrak m\) be the irrelevant ideal of \(R\). Suppose that \(\text{Spec}R\setminus\{\mathfrak m\}\) has only rational singularities. Suppose moreover that there exists an \(R\)-module \(M\) giving a NCR. Then \(\text{Spec}R\) has only rational singularities.
A few relevant, explicit examples are given, and all in all this article gives nice results from the noncommutative algebraic geometry to the commutative. Also, a really nice historical survey is given in the introduction, and a good list of references ends the article. non-commutative resolution; NCR; non-commutative crepant resolution; NCCR; non-generating locus; semilocal ring Dao, H., Iyama, O., Takahashi, R., Vial, C.: Non-commutative resolutions and Grothendieck groups. J. Noncommut. Geom. \textbf{9}(1), 21-34 (2015) Grothendieck groups, \(K\)-theory and commutative rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Representations of orders, lattices, algebras over commutative rings, Homological dimension (category-theoretic aspects) Non-commutative resolutions and Grothendieck groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors consider the Riemann-Hilbert correspondence in the context of Riemann surfaces of genus \(g\) with \(s\) holes and \(n\) bordered cusps. They study the Poisson structures on representation spaces induced by the Fock-Rosly bracket and prove that the quotients by unipotent Borel subgroups giving rise to decorated character varieties are Poisson reductions. monodromy; Riemann-Hilbert correspondence; Lie-Poisson structure Families, moduli of curves (analytic), Algebraic moduli problems, moduli of vector bundles, Quantum groups (quantized enveloping algebras) and related deformations, Structure of families (Picard-Lefschetz, monodromy, etc.) Algebras of quantum monodromy data and character varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{F. C. Kirwan} [Cohomology of quotients in symplectic and algebraic geometry. Princeton, New Jersey: Princeton University Press (1984; Zbl 0553.14020)] studied the map from the equivariant cohomology of a Hamiltonian group action to the cohomology of the symplectic quotient. The paper under review deals with the quantum version of this situation.
Let \(X\) be a smooth projectively embedded variety with a connected reductive group action such that stable locus is equal to the semistable locus. Let \(QH(X/\!/G)\) be the quantum cohomology of the GIT quotient \(X/\!/G\), and let \(QH_G(X)= H_G(X)\otimes \Lambda_X^G\) denote the equivariant quantum cohomology where \( \Lambda_X^G \subset\mathrm{Hom}(H_2^G(X,\mathbb{Z}))\) is Novikov field. \textit{A. B. Givental} [Int. Math. Res. Not. 1996, No. 13, 613--663 (1996; Zbl 0881.55006)] equipped the latter with a product structure arising from the equivariant Gromov-Witten theory. \textit{E. Gonzalez} and and the author of the paper under review [Math. Z. 273, No. 1--2, 485--514 (2013; Zbl 1258.53092)] proved that, under some conditions, the moduli space of stable gauged maps from a smooth connected projective curve is a proper Deligne-Mumford stack equipped with a perfect obstruction theory which leads to the definition of the Gauged Gromov-Witten invariants.
The paper under review constructs virtual fundamental classes on the moduli spaces used in the construction of the quantum Kirwan map and the gauged Gromov-Witten potential. This is the second paper in a sequence of papers aiming at constructing a quantum version of the Kirwan map.
The first result of the series of papers [\textit{C. T. Woodward}, Transform. Groups 20, No. 2, 507--556 (2015; Zbl 1326.14134)] is the construction of ``Quantum Kirwan Morphism''
\[
\kappa_X^G:QH_G(X)\to QH(X/\!/G)
\]
by means of the virtual integration over a compactified stack of affine gauged maps. \(\kappa_X^G\) is a morphism of CohFT algebras, which can be considered as a non-linear generalization of an algebra homomorphism.
The second main result of the series of the papers relates in the large area limit the graph potentials via the quantum Kirwan morphism.
More precisely, suppose that \(C\) is a smooth projective curve such that all the semistable gauged maps from \(C\) to \(X\) are stable for sufficiently large stability parameters. Then
\[
\tau_{X/\!/G}\circ\kappa^G_X=\lim_{\rho\to \infty}\tau_X^G
\]
where \(\rho\) is the stability parameter, \(\tau_{X/\!/G}\) genus zero graph potential for the GIT quotient \(X/\!/G\), and \(\tau^G_X\) is the gauged Gromov-Witten potential. Moreover, the localized version of this theorem that arises as the fixed point contributions for a circle acting on the domain. This localization in the case of Gromov-Witten invariants gives rise to a solution for a version of the Picard-Fuchs quantum differential equation for \(X/\!/G\).
Some of these results have overlaps and connections with the works of Givental, Lian-Liu-Yau, Iritani, Ciocan-Fontanine-Kim-Maulik, and Coates-Corti-Iritani-Tseng who have taken different approaches. Gromov-Witten invariants; quantum cohomology C. Woodward, \textit{Quantum Kirwan morphism and Gromov-Witten invariants of quotients II}, preprint. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum Kirwan morphism and Gromov-Witten invariants of quotients. II | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(K\) be a field, \(G_K\) the absolute Galois group of \(K\), \(X\) a hyperbolic curve over \(K\) and \(\pi_1(X)\) the étale fundamental group of \(X\). The absolute Grothendieck conjecture in anabelian geometry asks the following question: Is it possible to recover \(X\) group-theoretically, solely from \(\pi_1(X)\) (not \(\pi_1(X)\twoheadrightarrow G_K\))? When \(K\) is a \(p\)-adic field (i.e., a finite extension of \(\mathbb{Q}_p\)), this conjecture (called the \(p\)-adic absolute Grothendieck conjecture) is unsolved. To approach this problem, we introduce a certain \(p\)-adic analytic invariant defined by Serre (which we call \(i\)-invariant). Then the absolute \(p\)-adic Grothendieck conjecture can be reduced to the following problems: (A) determining whether a proper hyperbolic curve admits a rational point from the data of the \(i\)-invariants of the sets of rational points of the curve and its coverings; (B) recovering the \(i\)-invariant of the set of rational points of a proper hyperbolic curve group-theoretically. The main results of the present paper give a complete affirmative answer to (A) and a partial affirmative answer to (B). anabelian geometry; Grothendieck conjecture; \(p\)-adic analytic manifold; rational point Coverings of curves, fundamental group, Galois theory, Rational points, Local ground fields in algebraic geometry A \(p\)-adic analytic approach to the absolute Grothendieck conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a quantum version of the Danilov-Jurkiewicz presentation of the cohomology of a compact toric orbifold with projective coarse moduli space. More precisely, we construct a canonical isomorphism from a formal version of the Batyrev ring from [\textit{V. V. Batyrev}, in: Journées de géométrie algébrique d'Orsay, France, juillet 20-26, 1992. Paris: Société Mathématique de France. 9--34 (1993; Zbl 0806.14041)] to the quantum orbifold cohomology at a canonical bulk deformation. This isomorphism generalizes results
of \textit{A. B. Givental} [Int. Math. Res. Not. 1996, No. 13, 613--663 (1996; Zbl 0881.55006)], \textit{H. Iritani} [J. Reine Angew. Math. 610, 29--69 (2007; Zbl 1160.14044)] and \textit{K. Fukaya} et al. [Lagrangian Floer theory and mirror symmetry on compact toric manifolds. Paris: Société Mathématique de France (SMF) (2016; Zbl 1344.53001)] for toric manifolds and \textit{T. Coates} et al. [Acta Math. 202, No. 2, 139--193 (2009; Zbl 1213.53106)] for weighted projective spaces. The proof uses a quantum version of Kirwan surjectivity (Theorem 2.6 below) and an equality of dimensions (Theorem 4.19 below) deduced using a toric minimal model program (tmmp). We show that there is a natural decomposition of the quantum cohomology where summands correspond to singularities in the tmmp, each of which gives rise to a collection of Hamiltonian non-displaceable Lagrangian tori. quantum cohomology; toric varieties Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Toric varieties, Newton polyhedra, Okounkov bodies, Geometric invariant theory, Quantization in field theory; cohomological methods Quantum cohomology and toric minimal model programs | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials From the authors' abstract: We show that for any minuscule or cominuscule homogeneous space \(X\), the Gromov-Witten varieties of degree \(d\) curves passing through three general points of \(X\) are rational or empty for any \(d\). Applying techniques of A. Buch and L. Mihalcea to constructions of the authors together with L. Manivel, we deduce that the equivariant \(K\)-theoretic three points Gromov-Witten invariants are equal to classical equivariant \(K\)-theoretic invariants on auxiliary spaces. quantum \(K\)-theory; homogeneous spaces; quantum to classical principle Chaput, P.-E.; Perrin, N., Rationality of some Gromov-Witten varieties and application to quantum \(K\)-theory, Commun. Contemp. Math., 13, 1, 67-90, (2011) Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Rationality of some Gromov -- Witten varieties and application to quantum \(K\)-theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials ``Consider the system of equations \(P_1=\dots= P_n=0\) in \((\mathbb{C}\setminus 0)^n\), where \(P_1,\dots, P_n\) are Laurent polynomials with the Newtonian polyhedrons \(\Delta_1,\dots, \Delta_n\). Let us associate each Laurent polynomial \(Q\) with the \(n\)-form
\[
\omega= Q\Biggl/ \Biggl( P\frac{dz_1}{z_1} \wedge\dots\wedge \frac{dz_n}{z_n} \Biggr),
\]
where \(z_1,\dots, z_n\) are independent variables and \(P= P_1\cdot \dots\cdot P_n\).
For general sets of the polyhedrons \(\Delta_1,\dots, \Delta_n\), the sum of the Grothendieck residues of the form \(\omega\) over all roots of the system of equations is evaluated''. Newton polyhedra; Grothendieck residues; Laurent polynomials Gel'fond, O. A.; Khovanskii, A. G.: Newtonian polytopes and Grothendieck residues. Dokl. math. 54, 700-702 (1996) Residues for several complex variables, Toric varieties, Newton polyhedra, Okounkov bodies Newtonian polyhedra and Grothendieck residues | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Grothendieck duality goes back to 1958, to the talk at the ICM in Edinburgh [\textit{A. Grothendieck}, ``The cohomology theory of abstract algebraic varieties'', in: Proc. Int. Congr. Math. 1958. New York: Cambridge Univ. Press. 103--118 (1960; Zbl 0119.36902)] announcing the result. Hochschild homology is even older, its roots can be traced back to the 1945 article [\textit{G. Hochschild}, Ann. Math. (2) 46, 58--67 (1945; Zbl 0063.02029)]. The fact that the two might be related is relatively recent. The first hint of a relationship came in 1987 in \textit{J. Lipman} [Contemp. Math. 61 (1987; Zbl 0606.14015)], and another was found in 1997 in \textit{M. Van den Bergh} [J. Algebra 195, No. 2, 662--679 (1997; Zbl 0894.16020)]. Each of these discoveries was interesting and had an impact, Lipman's mostly by giving another approach to the computations and van den Bergh's especially on the development of non-commutative versions of the subject. However in this survey we will almost entirely focus on a third, much more recent connection, discovered in 2008 by \textit{L. L. Avramov} and \textit{S. B. Iyengar} [Mich. Math. J. 57, 17--35 (2008; Zbl 1245.13011)] and later developed and extended in several papers, see for example [\textit{L. L. Avramov} et al., Adv. Math. 223, No. 2, 735--772 (2010; Zbl 1183.13021); \textit{S. B. Iyengar} et al., Compos. Math. 151, No. 4, 735--764 (2015; Zbl 1348.13022)].
There are two classical paths to the foundations of Grothendieck duality, one following Grothendieck and Hartshorne [\textit{R. Hartshorne}, Residues and duality. Berlin-Heidelberg-New York: Springer (1966; Zbl 0212.26101)] and (much later) \textit{B. Conrad} [Grothendieck duality and base change. Berlin: Springer (2000; Zbl 0992.14001)], and the other following \textit{P. Deligne} [``Cohomology à support propre en construction du foncteur \(f^!\)'', Lect. Notes Math. 20, 404--421 (1966)], \textit{J.-L. Verdier} [Algebr. Geom., Bombay Colloq. 1968, 393--408 (1969; Zbl 0202.19902)] and (much later) \textit{J. Lipman} [``Notes on derived functors and Grothendieck duality'', Lect. Notes Math. 1960, 1--259 (2009)]. The accepted view is that each of these has its drawbacks: the first approach (of Grothendieck, Hartshorne and Conrad) is complicated and messy to set up, while the second (of Deligne, Verdier and Lipman) might be cleaner to present but leads to a theory where it's not obvious how to compute anything.
The point of this article is that the recently-discovered connection with Hochschild homology and cohomology (the one due to Avramov and Iyengar) changes this. It renders clearly superior the highbrow approach to the subject, the one due to Deligne, Verdier and Lipman. Not only is it (relatively) easy to set up the machinery, the computations also become transparent. And in the process we learn that Grothendieck duality is not really about residues of meromorphic differential forms, it is about the local cohomology of the Hochschild homology. By a fortuitous accident, if \(f:X\to Y\) is a smooth map then the top Hochschild homology happens to be isomorphic to the relative canonical bundle, and its top local cohomology is represented by meromorphic differential forms. This is the reason that, as long as we stick to smooth maps, what comes up is residues of meromorphic forms. For non-smooth, flat maps it's Hochschild homology and maps from it that we need to study. Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Homotopical algebra, Quillen model categories, derivators, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) The relation between Grothendieck duality and Hochschild homology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper applies the general techniques of abstract homotopical algebra to describe derived categories of Grothendieck categories.
Model category structures are defined on the category of chain complexes over a Grothendieck abelian category and their behaviour with respect to tensor product and stabilization is studied. So are given convenient tools to construct and understand triangulated categories of motives.
Examples and illustrations show how this work is useful and gives a promising way. model category; Grothendieck category; derived functor; motivic complexes Cisinski, Denis-Charles; Déglise, Frédéric, Triangulated categories of mixed motives, (2009), preprint Nonabelian homotopical algebra, Chain complexes (category-theoretic aspects), dg categories, Grothendieck categories, Motivic cohomology; motivic homotopy theory Local and stable homological algebra in Grothendieck abelian categories | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this article, the author gives some reminiscences about quantum cohomology. He lists the intuitive interpretation of the Gromov-Witten invariant as an integral over the moduli space of stable \(J\)-holomorphic maps, then mentions the first intersection theory approach and the later virtual neighborhood technique. After presenting axioms for Gromov-Witten invariants and explaining how they can be packaged into a quantum product, he discusses glueing theory and the future research questions that he thinks will be important. quantum cohomology; Gromov-Witten invariant; \(J\)-holomorphic maps; glueing theory Y. Ruan, ''Quantum cohomology and its application,'' in Proc. Internat. Congress of Mathematicians, Vol. II, 1998, pp. 411-420. Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Quantization in field theory; cohomological methods Quantum cohomology and its application | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathbb Z}[Var_{k}]\) be the free abelian group having the isomorphism classes of \(k\)-varieties as generators. The additive structure of the Grothendieck ring \(K_{0}(Var_{k})\) is given as the quotient of \({\mathbb Z}[Var_{k}]\) by the relations \([X]=[Z]+[X\backslash {Z}]\) whenever \(X\) is a \(k\)-variety and \(Z\) its closed subvariety. The multiplication on \(K_{0}(Var_{k})\) is given by the formula: \([X] \cdot [Y] :=[(X {\times}_{k} Y)_{red}]\). Not much is known about this ring. It is clear that \([X]=[Y]\) if \(X\) and \(Y\) are piecewise isomorphic, i.e. \(X\) and \(Y\) admit finite partitions into locally closed subvarieties \(X_{1},\dots ,X_{n}\) and \(Y_{1},\dots ,Y_{n}\) respectively such that \(X_{i}\) is isomorphic to \(Y_{i}\) for \(1\leq i \leq n .\) The converse question was asked by \textit{M. Larsen} and \textit{V. Lunts} [Mosc. Math. J. 3, No. 1, 85--95 (2003; Zbl 1056.14015)]. The authors give an affirmative answer when \(k\) is algebraically closed of characteristic zero for the following cases: \(\dim X\leq 1\) or \(X\) is a smooth projective surface or \(X\) contains finitely many rational curves. Grothendieck ring; birational geometry; piecewise isomorphism André, Y.: Une introduction aux motifs (motifs purs, motifs mixtes, périodes). Panoramas et Synthèses 17, SMF Paris (2004) Varieties and morphisms, Rational and birational maps, Arcs and motivic integration The Grothendieck ring of varieties and piecewise isomorphisms | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the ``Euler characteristic integral'' of a certain cohomotopy class over its scheme of fixed points. When the base is a field and the fixed points are étale, we compute this integral in terms of Morel's identification of the ring of endomorphisms of the motivic sphere spectrum with the Grothendieck-Witt ring. In particular, we show that the Euler characteristic of an étale algebra corresponds to the class of its trace form in the Grothendieck-Witt ring. motivic homotopy theory; Grothendieck-Witt group; trace formula Hoyois, Marc, A quadratic refinement of the {G}rothendieck-{L}efschetz-{V}erdier trace formula, Algebr. Geom. Topol.. Algebraic \& Geometric Topology, 14, 3603-3658, (2014) Motivic cohomology; motivic homotopy theory, Algebraic theory of quadratic forms; Witt groups and rings, Fixed points and coincidences in algebraic topology A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We reformulate the algebraic structure of Zwiebach's quantum open-closed string field theory in terms of homotopy algebras. We call it the quantum open-closed homotopy algebra (QOCHA) which is the generalization of the open-closed homotopy algebra (OCHA) of Kajiura and Stasheff. The homotopy formulation reveals new insights about deformations of open string field theory by closed string backgrounds. In particular, deformations by Maurer Cartan elements of the quantum closed homotopy algebra define consistent quantum open string field theories. K. Münster and I. Sachs, Quantum open-closed homotopy algebra and string field theory, \textit{Comm. Math. Phys.}, 321 (2013), no. 3, 769--801.Zbl 1270.81185 MR 3070036 String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Quantum open-closed homotopy algebra and string field theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The purpose of this paper is to present a mathematical theory of the half-twisted \((0,2)\) gauged linear sigma model and its correlation functions that agrees with and extends results from physics. The theory is associated to a smooth projective toric variety \(X\) and a deformation \(\mathcal{E}\) of its tangent bundle \(T_X\). It gives a quantum deformation of the cohomology ring of the exterior algebra of \(\mathcal{E}^\ast\). We prove that in the general case, the correlation functions are independent of ``nonlinear'' deformations. We derive quantum sheaf cohomology relations that correctly specialize to the ordinary quantum cohomology relations described by Batyrev in the special case \(\mathcal{E} = T_X\). quantum cohomology; quantum sheaf cohomology; toric varieties; primitive collection; gauged linear sigma model R. Marsh and K. Rietsch, \textit{The B-model connection and mirror symmetry for Grassmannians}, arXiv:1307.1085. Sheaves and cohomology of sections of holomorphic vector bundles, general results, Quantum field theory on curved space or space-time backgrounds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) A mathematical theory of quantum sheaf cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0619.00007.]
Let \(A=\mathbb{C}[[ X_1,\ldots,X_ n]]\) and let \(G\) be a finite group acting linearly on \(A\) with ring of invariants \(R=A^ G\). It has been proved by Auslander and Reiten that \(R\) has a finitely generated Grothendieck group \(G(R)\) and that the reduced Grothendieck group \(\widetilde G(R)\) is finite, at least when \(G\) acts freely on the linear space \(L = \sum_{i=1} \mathbb{C} X_ i\) of 1-forms. In the present note, which is essentially completely self-contained, the authors show that \(\widetilde G(R)\) is finite if \(G\) is just assumed to be abelian. Moreover, the reduced Grothendieck group of all simple hypersurface singularities is explicitly calculated.
For the second part of the paper (the explicit calculation of \(\widetilde G(R)\)), the authors make use of the fact that for any two-dimensional normal local domain \((R,m,k)\) with \([k]=0\), \(\widetilde G(R) = Cl(R)\), the class group of \(R\). Indeed, this result implies that if \(G\) acts faithfully and linearly on \(A=\mathbb{C}[[ X,Y]]\), then \(\widetilde G(A^ G)=(G/H)^*\), where \(H\) is the subgroup of \(G\), which is generated by the pseudoreflections on \(G\). The explicit description of \(\widetilde G(R)\) referred to above, makes use of this as well as some results of Knörrer, which say that (i) if \(f\in \mathbb{C}[[ X_1,\ldots,X_ n]]\), then the stable \(AR\)-quivers of \(\mathbb{C}[[ X_1,\ldots,X_ n]]/(f)\) and \(\mathbb{C}[[ X_1,\ldots,X_ n,Y,Z]]/(f+Y^ 2+Z^ 2)\) coincide, and (ii) simple hypersurface singularities are of finite Cohen-Macaulay representation type. Indeed, in htis case the stable \(AR\)-quivers of \(M(R)\) determine \(G(R)\), so one only has to consider the Grothendieck groups of simple hypersurface singularities in dimension 1 and 2.
Note: As pointed out by the authors, Auslander and Reiten have proved independently that \(\widetilde G(R)\) is finite for any action of a finite, not necessarily abelian group. Moreover, the \(AR\)-quivers of simple hypersurface singularities have been calculated by Dieterich and Wiedemann in dimension~1 and by Auslander in dimension~2. ring of invariants; reduced Grothendieck group; simple hypersurface singularities; stable AR-quivers; exponential-type operators; uniform approximation Jürgen Herzog and Herbert Sanders, The Grothendieck group of invariant rings and of simple hypersurface singularities, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 134 -- 149. Singularities of surfaces or higher-dimensional varieties, Geometric invariant theory, Grothendieck groups (category-theoretic aspects), Grothendieck groups, \(K\)-theory and commutative rings, Representation theory of associative rings and algebras, Group actions on varieties or schemes (quotients) The Grothendieck group of invariant rings and of simple hypersurface singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a short new computation of the quantum cohomology of an arbitrary smooth (semiprojective) toric variety \(X\), by showing directly that the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono [\textit{K. Fukaya} et al., Lagrangian Floer theory and mirror symmetry on compact toric manifolds. Paris: Société Mathématique de France (SMF) (2016; Zbl 1344.53001)] defines an isomorphism onto a suitable Jacobian ring. In contrast to previous results of this kind, \(X\) need not be compact. The proof is based on the purely algebraic fact that a class of \textit{generalized Jacobian rings} associated to \(X\) are free as modules over the Novikov ring. When \(X\) is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure. Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Mirror symmetry (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology and closed-string mirror symmetry for toric varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Quantum states are represented by positive semidefinite Hermitian operators with unit trace, known as density matrices. An important subset of quantum states is that of separable states, the complement of which is the subset of \textit{entangled} states. We show that the problem of deciding whether a quantum state is entangled can be seen as a moment problem in real analysis. Only a small number of such moments are accessible experimentally, and so in practice the question of quantum entanglement of a many-body system (e.g, a system consisting of several atoms) can be reduced to a truncated moment problem. By considering quantum entanglement of \(n\) identical atoms we arrive at the truncated moment problem defined for symmetric measures over a product of \(n\) copies of unit balls in \(\mathbb{R}^d\). We work with moments up to degree 2 only, since these are most readily available experimentally. We derive necessary and sufficient conditions for belonging to the moment cone, which can be expressed via a linear matrix inequality of size at most \(2d+2\), which is independent of \(n\). The linear matrix inequalities can be converted into a set of explicit semialgebraic inequalities giving necessary and sufficient conditions for membership in the moment cone, and show that the two conditions approach each other in the limit of large \(n\). The inequalities are derived via considering the dual cone of nonnegative polynomials, and its sum-of-squares relaxation. We show that the sum-of-squares relaxation of the dual cone is asymptotically exact, and using symmetry reduction techniques [\textit{G. Blekherman} and \textit{C. Riener}, ``Symmetric nonnegative forms and sums of squares'', Preprint, \url{arXiv:1205.3102}; \textit{K. Gatermann} and \textit{P. A. Parrilo}, J. Pure Appl. Algebra 192, No. 1--3, 95--128 (2004; Zbl 1108.13021)], it can be written as a small linear matrix inequality of size at most \(2d+2\), which is independent of \(n\). For the cone of symmetric nonnegative polynomials with the relevant support we also prove an analogue of the half-degree principle for globally nonnegative symmetric polynomials [\textit{C. Riener}, J. Pure Appl. Algebra 216, No. 4, 850--856 (2012; Zbl 1242.05272); \textit{V. Timofte}, J. Math. Anal. Appl. 284, No. 1, 174--190 (2003; Zbl 1031.05130)]. Quantum coherence, entanglement, quantum correlations, Quantum state spaces, operational and probabilistic concepts, Moment problems, Real algebraic and real-analytic geometry, Entanglement measures, concurrencies, separability criteria, Linear inequalities of matrices Quantum entanglement, symmetric nonnegative quadratic polynomials and moment problems | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Denote by \(W_{\vec{a}}\) Schubert varieties associated to \(\vec{a}\) and by \(\sigma_{\vec{a}}\in\text{H}^{2|\vec{a}|}(G,\mathbb{C})\) the corresponding elements in cohomology where \(G\) is the Grassmannian. The symbol \(W_a\) stands for a special Schubert variety associated to \((a,0,\dots,0)\). Choose general points \(p_1,\dots,p_N\in\mathbb{P}^1\) and general translates of the \(W_{\vec{a}}\). The Gromov-Witten intersection number \(\langle W_{\vec{a}_1},\dots,W_{\vec{a}_N}\rangle_d\) is, by a naive definition, the number of holomorphic maps \(f:\mathbb{P}^1\to G\) of degree \(d\) with the property that \(f(p_i)\in W_{\vec{a}_i}\) for all \(i=1,\dots,N\). For \(d=0\) one gets the original intersection number. The small quantum ring is the vector space \(\text{H}^\ast(G,\mathbb{C})[q]\) over \(C[q]\) with an associative product which obeys
\[
\sigma_{\vec{a}_1}\ast\dots\ast\sigma_{\vec{a}_N}= \sum_{d\geq 0}q^d\left(\sum_{\vec{a}} \langle W_{\vec{a}},W_{\vec{a}_1},\dots,W_{\vec{a}_N}\rangle_d \sigma_{\vec{a}}\right).
\]
The paper generalizes Giambelli's formula and Pieri's formula to the small quantum ring. In the quantum Giambelli formula that reads \(\sigma_{\vec{a}}=\Delta_{\vec{a}}(\sigma_\ast)\) no higher terms in \(q\) arise. The Giambelli determinant in cohomology classes corresponding to special Schubert varieties is evaluated in \(\text{H}^\ast(G,\mathbb{C})[q]\). On the other hand, the quantum Pieri formula has a correction term,
\[
\sigma_a\ast\sigma_{\vec{a}}= p_{a,\vec{a}}(\sigma_{\vec\ast})+ q\left(\sum_{\vec c}\sigma_{\vec c}\right),
\]
with an appropriate range of \(\vec c\). Before giving the proofs the Gromov-Witten number is defined rigorously by considering intersections of Schubert varieties on the moduli space \(\mathcal M_d\) of holomorphic maps of degree \(d\) from \(\mathbb{P}^1\) to \(G\) with \(\mathcal M_d\) being an open subscheme in the Grothendieck quoted scheme. As a corollary of the quantum Giambelli's formula the author also shows a Vafa and Intriligator formula for the Gromov-Witten intersection number of special Schubert varieties. Gromov-Witten intersection number; small quantum ring; Giambelli's formula; Pieri's formula A. Bertram. ''Quantum Schubert calculus''. Adv. Math. 128 (1997), pp. 289--305.DOI. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Grassmannians, Schubert varieties, flag manifolds, Relationships between surfaces, higher-dimensional varieties, and physics, Quantum field theory on curved space or space-time backgrounds Quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We define and classify the analogues of the affine Kac-Moody Lie algebras for the ring of functions on the complex projective line minus three points. The classification is given in terms of Grothendieck's dessins d'enfants. We also study the question of conjugacy of Cartan subalgebras for these algebras. reductive group scheme; dessins d'enfants; torsor; loop algebra; affine and three-point affine Lie algebras; Cartan subalgebra Chernousov, V., Gille, P. and Pianzola, A., Three-point Lie algebras and Grothendieck's dessins d'enfants. arXiv:1311.7097. Infinite-dimensional Lie (super)algebras, Galois cohomology of linear algebraic groups, Dessins d'enfants theory, Group actions on varieties or schemes (quotients), Cohomology theory for linear algebraic groups, Linear algebraic groups over adèles and other rings and schemes Three-point Lie algebras and Grothendieck's dessins d'enfants | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(S_{0}({\mathcal V}_{\mathbb C})\) be the Grothendieck semiring of complex quasi-projective varieties. The generators of the additive semigroup are isomorphism classes \([X]\) of such varieties subject to the relation \([X]=[X-Y]+[Y]\) if \(Y\subset X\) is a closed subvariety of \(X\). Multiplication is given by the cartesian product. A power structure over a semiring \(R\) is a map
\[
(1+T\cdot R[[T]]) \times R \rightarrow 1+T\cdot R[[T]] : (A(T), m) \rightarrow (A(T))^m,
\]
where \(A(T)= 1+ a_{1}T + a_{2}T^2 +\cdots , a_{i}\in R, m\in R, \) satisfying all the usual properties of the exponential function. The main result of the paper is the computation of the generating series (in \(S_{0}({\mathcal V}_{\mathbb C})\)) of Hilbert schemes of zero-dimensional subschemes on a smooth quasi-projective variety of dimension \(d\) as an exponent of such series for the complex affine space \({\mathbb A}^d.\) Grothendieck semiring; Hilbert scheme; generating series Gusein-Zade, SM; Luengo, I; Melle-Hernández, A, Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points, Mich. Math. J., 54, 353-359, (2006) Parametrization (Chow and Hilbert schemes), Grothendieck groups (category-theoretic aspects) Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be a number field and let \(\overline k\) be its algebraic closure. A smooth algebraic curve \(X\) of genus \(g\) is said to be hyperbolic if \(2g-2+n>0\) where \(n\) denotes the number of punctures. Let \(\pi_1(X\otimes\overline k)\) be the algebraic (étale) fundamental group of \(X\). The Grothendieck conjecture asserts that \(\pi_1(X\otimes\overline k)\) determines the curve \(X\) itself. The conjecture was solved in the affirmative by H. Nakamura, A. Tamagawa, and S. Mochizuki. Nakamura proved the conjecture in the cases \(g=0, 1\) and \(n\geq 3\) using weight filtration, Kummer theory, and diophantine equations. Then Tamagawa proved the cases \(n>0\) (for affine curves) using class field theory of algebraic curves over finite fields. Finally, Mochizuki completed the proof in the remaining cases \(n=0\) based on \(p\)-adic Hodge theory. The three mathematicians developed their own methods; Mochizuki's approach based on \(p\)-adic interpolation of the problem might be regarded as the most powerful. This is a short (rather personal) report by Ihara on the work of the three mathematicians on the occasion of their receipt of the Autumn Prize of the Mathematical Society of Japan (1997). fundamental groups of algebraic curves; hyperbolic curves; Grothendieck's conjecture; H. Nakamura; A. Tamagawa; S. Mochizuki Coverings of curves, fundamental group, History of mathematics in the 20th century, History of algebraic geometry The solution of the Grothendieck conjecture on fundamental groups of algebraic curves. The work of Hiroaki Nakamura, Akio Tamagawa, and Shinichi Mochizuki | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review is a survey of the authors' paper [J. Am. Math. Soc. 14, No. 2, 429-469 (2001; Zbl 1040.14010)], devoted to an extension of the theory of motivic integration, called arithmetic motivic integration, and its applications to the uniform dependence for large \(p\) of \(p\)-adic integrals of a very general type. The motivic integration, a \(\mathbb{C}[[t]]\)-analogue of usual \(p\)-adic integration, was first introduced by Kontsevich to prove the birational invariance of Hodge numbers for projective Calabi-Yau manifolds, and developed further by Batyrev and the authors.
To illustrate the main application of the arithmetic motivic integration, consider the following special case of a much more general result of the authors on \(p\)-adic integrals. Let \(X\) be an algebraic variety over \(\mathbb Z\), and for each prime \(p\) and natural number \(n\), let \(N_{p, n}\) denote the cardinality of the image of the projection \(X(\mathbb{Z}_p) \to X(\mathbb{Z}/p^{n + 1})\), where \(\mathbb{Z}_p\) is the ring of \(p\)-adic integers. The rationality of the formal power series \(P_p(T) = \sum_n {N_{p, n} T^n}\), conjectured by Serre and Oesterlé, was proved in 1983 by Denef using Macintyre's quantifier elimination for \(p\)-adic fields. For studying the dependence of the rational power series \(P_p(T)\) on \(p\), the authors consider the motivic Grothendieck ring \(K_0^{\text{mot}}(\text{Var}_{\mathbb Q})\) of algebraic varieties over \(\mathbb Q\), as well as the Grothendieck ring \(K_0(PFF_{\mathbb Q})\) of the theory of pseudo-finite fields (infinite models of the elementary theory of finite fields) of characteristic \(0\), constructing a canonical morphism \(K_0(PPF_{\mathbb Q}) \to K_0^{\text{mot}}(\text{Var}_{\mathbb Q}) \otimes {\mathbb Q}\). There exists a canonically defined rational power series \(P(T)\) over \(K_0^{\text{mot}}(\text{Var}_{\mathbb Q}) \otimes {\mathbb Q}\) such that , for \(p >> 0, P_p(T)\) is obtained from \(P(T)\) by applying to each coefficient of \(P(T)\) some operator \(N_p\), induced by associating to an algebraic variety over \(\mathbb Q\) its number of rational points over the prime field \(\mathbb{F}_p\), for \(p >>0\). motivic integration; \(p\)-adic integration; quantifier elimination, pseudo-finite field; Poincaré series J. Denef and F. Loeser, Motivic integration and the Grothendieck group of pseudo-finite fields, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 13 -- 23. Arcs and motivic integration, Finite ground fields in algebraic geometry, Varieties over finite and local fields, Zeta functions and \(L\)-functions, Quantifier elimination, model completeness, and related topics, Applications of model theory, Field arithmetic, Model theory of fields, Local ground fields in algebraic geometry, Global ground fields in algebraic geometry, Ultraproducts and field theory, Étale and other Grothendieck topologies and (co)homologies, Birational geometry Motivic integration and the Grothendieck group of pseudo-finite fields | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article is in two parts. The first part (paragraphs 1-3) presents a detailed introduction into the concept of Gröbner bases. The second part presents some results showing how to use the Gröbner bases to study polynomial mappings and four examples of calculation. Especially, example 4 shows how to use Gröbner bases to verify the Jacobian conjecture. Gröbner bases; polynomial mappings; Jacobian conjecture Polynomial rings and ideals; rings of integer-valued polynomials, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Automorphisms of curves Application of Gröbner bases in the theory of polynomial mappings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this brief note, the author announces the main results of his doctoral dissertation entitled ``A formula for the Gromov-Witten invariants of toric varieties'' (Univ. Strasbourg 1999). In the meantime, this thesis has appeared: \textit{H. Spielberg}, Républication Math. Inst. Recherche Univ. Strasbourg (1999; Zbl 0964.14045).
The main result of the author's thesis, and of this research announcement, consists in an explicit formula for all genus-zero Gromov-Witten invariants of an arbitrary smooth projective toric variety. This formula, based upon a localization argument due to \textit{T. Graber} and \textit{R. Pandharipande} [Invent. Math. 135, No. 2, 487-518 (1999; Zbl 0953.14035)] is then applied to the study of some special toric varieties. In particular, the author gives here an example of a (non-Fano) toric manifold and its Gromov-Witten invariants, whose explicit computation follows from his general approach. Being in accordance with the definition of the quantum cohomology ring of a projective manifold à la Behrend-Fantechi, on the one hand, but differing from V. Batyrev's definition of the quantum cohomology ring of a symplectic toric manifold, this example shows that these two approaches to quantum cohomology rings do not coincide. Gromov-Witten invariants; toric variety; quantum cohomology rings Holger Spielberg, The Gromov-Witten invariants of symplectic toric manifolds, and their quantum cohomology ring, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 8, 699 -- 704 (English, with English and French summaries). Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Fano varieties, Toric varieties, Newton polyhedra, Okounkov bodies The Gromov-Witten invariants of symplectic toric manifolds, and their quantum cohomology ring | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\theta\in\mathbb{R}\backslash\mathbb{Q}\) be an irrational parameter. Then a quantum torus \(A_\theta\) is a transformation group \(C^\ast\)-algebra \(C^\ast(\theta\mathbb{Z},\mathbb{R}/\mathbb{Z})\) for the group action of \(\theta\mathbb{Z}\) on \(\mathbb{R}/\mathbb{Z}.\) It is known that \(A_\theta\) is a universal \(C^\ast\)-algebra generated by two unitaries \(U,V\in A_\theta\) satisfying the relation \(UV=\exp(2\pi i x)VU.\) \(A_\theta\) is called a quantum torus with real multiplication if \(\theta \) is an irrational root of a quadratic equation with rational coefficients (a real quadratic irrationality).
For real quadratic fields \(k\), it is proposed to use quantum tori with real multiplication \(A_\theta\) as geometric objects associated to \(k\). This is compared to using elliptic curves with complex multiplication \(E_\tau\), \(\tau\in k^\prime\backslash\mathbb{Q}\) as associated geometric object to complex quadratic fields \(k^\prime\).
The elliptic curve \(E_\tau\) is said to have complex multiplication if \(\operatorname{End}(E_\tau)\) is larger that \(\mathbb{Z}.\) This is fulfilled if and only if \(\tau\) is a complex quadratic number. Real multiplication of quantum tori has similar interpretation when morphisms are considered in the sense of noncommutative geometry. This is thoroughly treated in the introduction to the present article.
The author constructs the graded ring of quantum theta functions \(R=\bigoplus_{n\geq 0}R_n\) associated to \(A_\theta\). This graded ring is a (sort of) subalgebra of the tensor algebra \(E\) of arithmetically constructed \(A_\theta - A_\theta\)-imprimitivity bimodules \(E_n\). The author defines and analyzes \(A - B\)-imprimitivity bimodules for general \(C^\ast\)-algebras in connection with strong Morita equivalence in section 2, which is quite necessary to define the associated graded ring \(R\). It is also proved that for \(n\geq 1\), the elements of \(R_n\) are quantum theta functions. The author recalls the definition of such.
The ring \(R\) have elements that are formally elements of quantum tori \(A_\theta,\) but the multiplication law is different from the one in \(A_\theta\). Also, \(R\) is isomorphic to a kind of Segre square of \(E\).
Two standard examples of strong Morita equivalence and imprimitivity modules needed in the construction of \(R\) are given. A lot of partial results is given and explicitly proved in this setting. This leads, among other results, to the fact that \(A_\theta\) is strongly Morita equivalent to \(A_{g\theta}\) of any \(g\) in \(\text{GL}_2(\mathbb{Z}).\) The last section contains the construction and main results of the ring \(R\). Explicit descriptions, including dimensions of each homogeneous part are given. The last result also proves that \(\theta\) and \(g\) can be chosen such that the ring \(R(g,\theta)\) is Koszul. quantum theta functions; noncommutative torus; imprimitivity module; strong Morita equivalence; real multiplication Vlasenko, M.: The graded ring of quantum theta functions for noncommutative torus with real multiplication. Int. Math. Res. Not. 15825 (2006) Algebraic groups, Noncommutative algebraic geometry The graded ring of quantum theta functions for noncommutative torus with real multiplication | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We review mirror symmetry for the quantum cohomology D-module of a compact weak-Fano toric manifold. We also discuss the relationship to the GKZ system, the Stanley-Reisner ring, the Mellin-Barnes integrals, and the \(\widetilde{\Gamma}\)-integral structure. Mirror symmetry (algebro-geometric aspects), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Toric varieties, Newton polyhedra, Okounkov bodies, Research exposition (monographs, survey articles) pertaining to algebraic geometry Quantum D-modules of toric varieties and oscillatory integrals | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We begin with some general remarks on ordinary cohomology theory, which is a basic concept in modern geometry with many applications. We review briefly its origin and purpose, and the way in which it is used by researchers. Quantum cohomology is much more recent and less developed, but it already has deep relations with physics, differential geometry, and algebraic geometry, as well as topology. We give an informal definition of quantum cohomology, with some concrete examples, and conclude with a discussion of the quantum differential equations which are responsible for this deeper structure. Gromov-Witten invariants; Quantum cohomology; quantum differential equations Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Research exposition (monographs, survey articles) pertaining to differential geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Homology and cohomology theories in algebraic topology Introduction to quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a semi-simple algebraic group and \(R\) its root system. Denote by \(\Theta\) (resp. \(\theta\)) the highest root (resp. the highest short root) of \(R\). A dominant weight is said to be adjoint if it is equal to \(\Theta\) and coadjoint if it is equal to \(\theta\). Similarly, if \(P\) is the parabolic subgroup of \(G\) associated to the adjoint (resp. coadjoint) weight, then \(G/P\) is called an adjoint (resp. coadjoint) homogeneous space. Examples are the orthogonal and symplectic Grassmannians of lines, odd-dimensional quadrics and projective spaces, the two-step flag variety \(F(1,n;n+1)\) as well as six exceptional homogeneous spaces.
The paper under review studies various aspects of the (small) quantum cohomology of (co)adjoint varieties, in line with previous work on minuscule homogeneous spaces by the same authors and \textit{L. Manivel} [Transform. Groups 13, No. 1, 47--89 (2008; Zbl 1147.14023)]. Let \(q\) denote the quantum parameter of a (co)adjoint variety \(X=G/P\). The main results are a simplified formula for the quantum multiplication by the hyperplane class of \(X\), a bound on the possible degrees in \(q\) of the quantum product of two Schubert classes of \(X\), and a presentation of the quantum cohomology ring for the six exceptional (co)adjoint varieties -- such a presentation being already known for the classical (co)adjoint varieties from [\textit{A.S. Buch}, \textit{A. Kresch} and \textit{H. Tamvakis}, Invent. Math. 178, No. 2, 345--405 (2009; Zbl 1193.14071)].
As a consequence of the presentation of the quantum cohomology ring, the authors are able to tell whether the (small) quantum cohomology ring of (co)adjoint varieties is semi-simple. Semi-simplicity of the quantum cohomology is an important problem. For instance, it appears in a conjecture of \textit{B. Dubrovin} [Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 315--326 (1998 ; Zbl 0916.32018)] on Fano varieties, in relation with properties of their derived categories. Another consequence of semi-simplicity for homogeneous spaces \(G/P\) is a phenomenon known as strange duality, i.e. the existence of an involution of the localised quantum cohomology ring \(\text{QH}^*(G/P)_{q\neq 0}\). Strange duality for (co)minuscule homogeneous spaces has been studied in [\textit{P. E. Chaput}, \textit{L. Manivel} and \textit{N. Perrin}, Canadian J. Math. 62, No. 6, 1246--1263 (2010; Zbl 1219.14060)]. Here the authors prove that the quantum cohomology ring of coadjoint varieties is almost never semi-simple, while it always is for adjoint non-coadjoint varieties. In the latter case they give a partial description of strange duality. quantum cohomology; adjoint homogeneous spaces; Schubert calculus; strange duality Chaput, P.; Perrin, N., On the quantum cohomology of adjoint varieties, Proc. Lond. Math. Soc., 103, 294-330, (2011) Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) On the quantum cohomology of adjoint varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that the quantum Frobenius morphism constructed by Lusztig in the setting of the quantum enveloping algebra \(U_{\mathcal{B}}\) specialized at a root of unity admits a (nonunital) multiplicative splitting. We construct the splitting using a basis of the toral part of the small quantum algebra consisting of pairwise orthogonal idempotents summing up to 1, and likewise in the modular case of the algebra of distributions for a semisimple algebraic group. Gros, Michel; Kaneda, Masaharu, Un scindage du morphisme de Frobenius quantique, Ark. Mat., 53, 2, 271-301, (2015) Quantum groups (quantized enveloping algebras) and related deformations, Linear algebraic groups over arbitrary fields, Homogeneous spaces and generalizations A splitting of the quantum Frobenius morphism | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author shows that Grothendieck's standard conjectures over a field of characteristic zero follow from either of two other motivic conjectures.
The first conjecture is the one on the existence of a motivic \(t\)-structure. Let \(k\) be any field, and \(DM_k\) be the triangulated category of geometric motives with \({\mathbb Q}\)-coefficients over \(k\). It is an idempotently complete triangulated rigid tensor \({\mathbb Q}\)-category. Denote by \(\mathrm{Vec}_{{\mathbb Q}}\) and \(\mathrm{Vec}_{{\mathbb Q}_{\ell}}\) the categories of finite-dimensional \({\mathbb Q}\)- and \({\mathbb Q}_{\ell}\)-vector spaces, respectively. For \(\ell\) a prime different from the characteristic of \(k\), there is the \(\ell\)-adic realization functor \(r_{\ell}: DM_k \to D^b(\mathrm{Vec}_{{\mathbb Q}_{\ell}})\). For \(k\) of characteristic zero, each embedding \(\iota:k \hookrightarrow {\mathbb C}\) yields the Betti realization functor \(r_{\iota}:DM_k \to D^b(\mathrm{Vec}_{\mathbb Q})\). Let \(r\) be either of these realization functors. For a \(t\)-structure \(\mu\) on \(DM_k\), let \(\mathcal{M}_k\) be its heart and \(^{\mu}H: DM_k \to \mathcal{M}_k\) be the cohomology functor. The \(t\)-structure \(\mu\) is called \textit{motivic} if it is non-degenerate and compatible with \(\otimes\) and \(r\). It is a conjecture that a motivic \(t\)-structure exists. The author shows that the existence of a motivic \(\mu\) would imply the Lefschetz and Künneth type standard conjectures.
The second conjecture is a weak version of Suslin's homology conjecture. For a smooth projective complex variety \(X\), let \(C_r(X)\) be the topological monoid of effective \(r\)-cycles on \(X\) and \(C_r(X)^+\) be its group completion. The Lawson homology groups of \(X\) are defined as \(L_rH_{2r+i}(X, {\mathbb Z}):=\pi_i(C_r(X)^+)\).
In connection with results of \textit{E. M. Friedlander} [Ann. Sci. ENS 28, No. 3, 317--343 (1995; Zbl 0854.14006)], the author proves the surprising fact that Grothendieck's standard conjectures over fields of characteristic zero are equivalent to the surjectivity of the natural map from Lawson to singular homology
\[
L_rH_a(X) \to H_a(X, {\mathbb Z})
\]
for all smooth projective complex varieties \(X\) and all \(a \geq \mathrm{dim}\,X+r\). In fact, Suslin conjectures that this map is even an isomorphism. motives; standard conjectures; Suslin conjecture Beilinson, A., Remarks on Grothendieck's standard conjectures, (Regulators. Regulators, Contemp. Math., vol. 571, (2012), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 25-32 (Equivariant) Chow groups and rings; motives, Motivic cohomology; motivic homotopy theory, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Remarks on Grothendieck's standard conjectures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper focuses on a triple central extension of the translation algebra in 4-dimensions, here taken to be the phase space of a particle moving in 2-dimensional configuration space. As such, this algebra represents a non-commutative quantum system with non-commuting coordinates and momenta and the standard commutation relations between coordinates and momenta. It is thus a deformation of the Heisenberg algebra of a 2-dimenional quantum system. The physical prototype is a particle moving in a 2-dimensional plane with a magnetic field perpendicular to the plane.
The unitary irreducible representations of this algebra is then constructed using the method of coadjoint orbits. Explicit realizations of the algebra as differential operators acting on the smooth square integrable functions are constructed. From this a class of gauge equivalent gauge potentials is identified that contains the standard symmetric and Landau gauges.
Next non-commutative 4-Tori are considered. These are essentially the \(C^*\) algebras generated by the Weyl-algebra associated with the triply extended algebra of translations in 4-dimensions. These algebras are then explicitly classified in terms of the anti-symmetric \(4\times 4\) matrix that codes the Weyl algebra. The associated star product is also explicitly constructed.
Finally, projective modules over the 4-tori and connections of constant curvature on them are constructed. translation algebra; central extensions; Weyl algebra; coadjoint orbits; unitary representations Noncommutative geometry in quantum theory, Toric varieties, Newton polyhedra, Okounkov bodies, Coadjoint orbits; nilpotent varieties, Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics On the plethora of representations arising in noncommutative quantum mechanics and an explicit construction of noncommutative 4-tori | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The articles of this volume will be reviewed individually. Grothendieck theory; Dessins d'enfants L. Schneps, ed., \textit{The Grothendieck Theory of Dessin d'Enfants}, Cambridge Univ. Press, Cambridge, (1994). Collections of articles of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to algebraic geometry The Grothendieck theory of dessins d'enfants | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(P_n= K[x_1, \dots, x_n]\) be the polynomial algebra over a field \(K\) of zero characteristic and \(p\in P_n\). Analogously to \textit{A. A. Zolotykh} and \textit{A. A. Mikhalev} [in Russ. Acad. Sci. Dokl., Math. 49, No. 1, 189-193 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 334, No. 6, 690-693 (1994; Zbl 0839.17002)] and \textit{U. U. Umirbaev} [see Fundam. Prikl. Mat. 2, No. 1, 313-315 (1996; Zbl 0899.20010)], the author considers the outer rank of \(p\) (i.e., the minimal number of generators \(x_i\), on which an automorphic image of \(p\) can depend) in connection with the ideal \(I_{d(p)}\) of \(P_n\), generated by partial derivatives of \(p\). In general, one has that outer rank\((p)\geq \text{rank} (I_{d(p)})\), where \(\text{rank} (I_{d(p)})\) is the minimal number of generators of this ideal. The author investigates the possibility for \(p\) to be included in a basis of the algebra \(P_n\), consisting of \(n\) elements. In this case, \(p\) must have the outer rank 1 and it is necessary that the row \(\left( {\partial p\over \partial x_1}, \dots, {\partial p\over \partial x_n} \right)\) can be transformed to \((1,0, \dots,0)\) using only elementary transformations. The last condition is also sufficient in the case \(n=2\), where the mentioned elementary transformations are chosen by means of an appropriate Buchberger \(S\)-construction used for obtaining the Gröbner basis for \(I_{d(p)}\). A result related to the Jacobian conjecture [\textit{H. Bass}, \textit{E. H. Connell} and \textit{D. Wright}, Bull. Am. Math. Soc., New Ser. 7, 287-330 (1982; Zbl 0539.13012)] is obtained, as well. polynomial algebra; Gröbner basis; Jacobian conjecture Vladimir Shpilrain and Jie-Tai Yu, Polynomial automorphisms and Gröbner reductions, J. Algebra 197 (1997), no. 2, 546 -- 558. Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomial rings and ideals; rings of integer-valued polynomials, Polynomials over commutative rings Polynomial automorphisms and Gröbner reductions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Bukhvostov-Lipatov model is an exactly soluble model of two interacting Dirac fermions in \(1 + 1\) dimensions. The model describes weakly interacting instantons and anti-instantons in the \(\operatorname{O}(3)\) non-linear sigma model. In our previous work [the authors, ibid. 911, 863--889 (2016; Zbl 1346.81064)] we have proposed an exact formula for the vacuum energy of the Bukhvostov-Lipatov model in terms of special solutions of the classical sinh-Gordon equation, which can be viewed as an example of a remarkable duality between integrable quantum field theories and integrable classical field theories in two dimensions. Here we present a complete derivation of this duality based on the classical inverse scattering transform method, traditional Bethe ansatz techniques and analytic theory of ordinary differential equations. In particular, we show that the Bethe ansatz equations defining the vacuum state of the quantum theory also define connection coefficients of an auxiliary linear problem for the classical sinh-Gordon equation. Moreover, we also present details of the derivation of the non-linear integral equations determining the vacuum energy and other spectral characteristics of the model in the case when the vacuum state is filled by 2-string solutions of the Bethe ansatz equations. Bazhanov, V. V.; Lukyanov, S. L.; Runov, B. A., Nucl. Phys. B, 927, 468, (2018) Model quantum field theories, Exactly and quasi-solvable systems arising in quantum theory, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Inverse scattering problems in quantum theory, Exactly solvable models; Bethe ansatz, Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory Bukhvostov-Lipatov model and quantum-classical duality | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the quantum Witten-Kontsevich series introduced by \textit{A. Buryak} et al. [Int. Math. Res. Not. 2020, No. 24, 10381--10446 (2020; Zbl 1464.37071)] as the logarithm of a quantum tau function for the quantum KdV hierarchy. This series depends on a genus parameter \(\epsilon\) and a quantum parameter \(\hbar\). When \(\hbar=0\), this series restricts to the Witten-Kontsevich generating series for intersection numbers of psi classes on moduli spaces of stable curves.
We establish a link between the \(\epsilon=0\) part of the quantum Witten-Kontsevich series and one-part double Hurwitz numbers. These numbers count the number of nonequivalent holomorphic maps from a Riemann surface of genus \(g\) to \(\mathbb{P}^1\) with a complete ramification over \(0\), a prescribed ramification profile over \(\infty\) and a given number of simple ramifications elsewhere. \textit{I. P. Goulden} et al. [Adv. Math. 198, No. 1, 43--92 (2005; Zbl 1086.14022)] proved that these numbers have the property of being polynomial in the orders of ramification over \(\infty\). We prove that the coefficients of these polynomials are the coefficients of the quantum Witten-Kontsevich series. We also present some partial results about the full quantum Witten-Kontsevich power series. moduli space of curves; double ramification cycle; quantum KdV; quantum tau function; Hurwitz numbers Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Deformation quantization, star products, Families, moduli of curves (algebraic) The quantum Witten-Kontsevich series and one-part double Hurwitz numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this short note, we give an elementary proof for the fact that the Grothendieck group of complete toric Deligne-Mumford stack is torsion free. Grothendieck group; \(K\)-theory; toric stacks; Stanley-Reisner ring Generalizations (algebraic spaces, stacks), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies On the Grothendieck groups of toric stacks | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials According to Givental's mirror theorem genus zero Gromov-Witten potential (\(I\)-series) of complete intersections in projective toric varieties can be found from a solution (\(J\)-series) to an explicit differential equation, the Riemann-Roch equation. In this note the author generalizes the mirror theorem to complete intersections in weighted projective spaces and singular toric varieties. As a consequence, he verifies that the remaining \(3\) families of Fano threefolds with Picard group \(\mathbb{Z}\) satisfy the Golyshev conjecture, i.e. solutions to their so-called counting equations are modular. For the other \(14\) families this follows from an old result of Beauville. V. V. Przyjalkowski, Quantum cohomology of smooth complete intersections in weighted projective spaces and in singular toric varieties. \textit{Rossiïskaya Akademiya} \textit{Nauk. Matematicheskiï Sbornik }198 (2007), 107--122. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Fano varieties, Toric varieties, Newton polyhedra, Okounkov bodies Quantum cohomology of smooth complete intersections in weighted projective spaces and in singular toric varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider Fock representations of the \(Q\)-deformed commutation relations \(\partial_s \partial_t^{\dagger} = Q(s, t) \partial_t^{\dagger} \partial_s + \delta(s, t)\) for \(s, t \in T\). Here \(T : = \mathbb{R}^d\) (or more generally \(T\) is a locally compact Polish space), the function \(Q : T^2 \rightarrow \mathbb{C}\) satisfies \(| Q(s, t) | \leq 1\) and \(Q(s, t) = \overline{Q(t, s)}\), and \(\int_{T^2} h(s) g(t) \delta(s, t) \sigma(d s) \sigma(d t) : = \int_T h(t) g(t) \sigma(d t)\), \(\sigma\) being a fixed reference measure on \(T\). In the case, where \(| Q(s, t) | \equiv 1\), the \(Q\)-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev. These generalized statistics contain anyon statistics as a special case (with \(T = \mathbb{R}^2\) and a special choice of the function \(Q\)). The related \(Q\)-deformed Fock space \(\mathcal{F}(\mathcal{H})\) over \(\mathcal{H} : = L^2(T \rightarrow \mathbb{C}, \sigma)\) is constructed. An explicit form of the orthogonal projection of \(\mathcal{H}^{\otimes n}\) onto the \(n\)-particle space \(\mathcal{F}_n(\mathcal{H})\) is derived. A scalar product in \(\mathcal{F}_n(\mathcal{H})\) is given by an operator \(\mathcal{P}_n \geq 0\) in \(\mathcal{H}^{\otimes n}\) which is strictly positive on \(\mathcal{F}_n(\mathcal{H})\). We realize the smeared operators \(\partial_t^{\dagger}\) and \(\partial_t\) as creation and annihilation operators in \(\mathcal{F}(\mathcal{H})\), respectively. Additional \(Q\)-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form \(\partial_s^{\dagger} \partial_t^{\dagger} = Q(t, s) \partial_t^{\dagger} \partial_s^{\dagger}\), \(\partial_s \partial_t = Q(t, s) \partial_t \partial_s\), valid for those \(s, t \in T\) for which \(|Q(s,t)| = 1\).{
\copyright 2017 American Institute of Physics} Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Commutation relations and statistics as related to quantum mechanics (general), Bergman spaces and Fock spaces, Formal methods and deformations in algebraic geometry, Anyons Fock representations of \(Q\)-deformed commutation relations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(z=\{z_{ij}\}_{1\leq i,j\leq n}\) be variables and \(v,w\in S_n\) permutations. Let \(Z^{(v)}\) be the specialized generic matrix, i.e. with entries \(z_{ij}\) but specialized to \(z_{n-v(i)+1,i}=1\), \(z_{n-v(i)+1,a}=0\;,\;z_ {b,i}=0\) if \(a>i, b>n-v(i)+1\). Let \(z^{(v)}\subset z\) be the remaining variables. Let \(r^w_{i,j}=\#\{k\;|\;w(k)\geq n-i+1, k\leq j\}\) and denote by \(Z^{(v)}_{ab}\) the southwest \(a\times b\) submatrix of \(Z^{(v)}\). The Kazhdan--Lusztig ideal \(I_{v,w}\subseteq\mathbb{C}[z^{(v)}]\) is the ideal generated by all minors of size \(1+r^w_{ij}\) of \(Z^{(v)}_{ij}\) for all \(i,j\) (the so--called defining minors). Consider the following monomial ordering:
\[
z_{ij}<z_{kl}, \text{ if } j<l\;\text{ or if } j=l\text{ and } i<k\;.
\]
It is proved that the defining minors are a Gröbner basis of \(I_{v,w}\).
Kazhdan--Lusztig ideals provide an explicite choice of coordinates and equations encoding a neighborhood of a torus--fixed point of a Schubert variety on a type \(A\) flag variety.
The Gröbner basis turns out to be a tool to explain several combinatorical formulas. Gröbner basis; Kazhdan-Lusztig ideal; Schubert variety Woo, Alexander; Yong, Alexander, A Gröbner basis for Kazhdan-Lusztig ideals, Amer. J. Math., 134, 4, 1089-1137, (2012) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Determinantal varieties A Gröbner basis for Kazhdan-Lusztig ideals | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the present paper, we discuss the Grothendieck conjecture for hyperbolic curves over Kummer-faithful fields. In particular, we prove that every point-theoretic and Galois-preserving outer isomorphism between the étale/tame fundamental groups of affine hyperbolic curves over Kummer-faithful fields arises from a uniquely determined isomorphism between the original hyperbolic curves. This result generalizes results of \textit{A. Tamagawa} [Compos. Math. 109, No. 2, 135--194 (1997; Zbl 0899.14007)], i.e., our main result in the case where the basefields are either finite fields or mixed-characteristic local fields. Grothendieck conjecture; hyperbolic curve; Kummer-faithful field Coverings of curves, fundamental group, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) On the Grothendieck conjecture for affine hyperbolic curves over Kummer-faithful fields | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using the formalism of quantized quadratic Hamiltonians [\textit{A.~B.~Givental}, Mosc. Math. J. 1, No. 4, 551--568 (2001; Zbl 1008.53072)], the authors are able to prove quantum versions of three classical theorems in algebraic geometry; namely, the Riemann-Roch theorem, Serre duality, and the Lefschetz hyperplane section theorem. The key ingredient consists in introducing a notion of twisted Gromov-Witten invariants of a compact projective complex manifold \(X\); the quantum version of the aforementioned theorems can then be seen as relations between the twisted and the nontwisted Gromov-Witten theory of \(X\).
More precisely, let \(X_{g,n,d}\) be the moduli space of genus \(g\), \(n\)-pointed stable maps to \(X\) of degree \(d\), where \(d\) is an element in \(H_2(X;\mathbb{Z})\), and let \(E\) be a holomorphic vector bundle on \(X\). Since a point in \(X_{g,n,d}\) is represented by a pair \((\Sigma,f)\), where \(\Sigma\) is a complex curve and \(f:\Sigma\to X\) a holomorphic map, one can use \(f\) to pull back \(E\) on \(\Sigma\) and then consider the \(K\)-theory Euler character of \(f^*E\), i.e., the virtual vector space \(H^0(\Sigma,f^*E)\ominus H^1(\Sigma,f^*E)\), as the fiber over \([(\Sigma,f)]\) of a virtual vector bundle \(E_{g,n,d}\) over \(X_{g,n,d}\). This intuitive construction is made completely rigorous by considering \(K\)-theory push-pull \(K^0(X)\to K^0(X_{g,n,d})\) along the diagram
\[
\begin{tikzcd} X_{g,n+1,d}\rar["\mathrm{ev}_{n+1}"]\dar["\pi" '] &X\\ X_{g,n,d}\end{tikzcd}
\]
A rational invertible multiplicative characteristic class of a complex vector bundle is an expression of the form
\[
\mathbf{c}(\cdot)=\exp\left(\sum_{k=0}^\infty s_k \text{ch}_k(\cdot)\right),
\]
where \(\text{ch}_k\) are the components of the Chern character, and the \(s_k\) are arbitrary parameters. These data determine a cohomology class \(\mathbf{c}(E_{g,n,d})\) (actually, a formal family of cohomology classes parametrized by the \(s_k\)) in \(H^*(X_{g,n,d};\mathbb{Q})\), and one can define the total \((\mathbf{c},E)\)-twisted descendant potential \(\mathcal{D}_{\mathbf{c},E}^g\) as
\[
\mathcal{D}_{\mathbf{c},E}(t_0,t_1,\dots)=\exp\left(\sum_{g\geq 0}\hbar^{g-1}\mathcal{F}^g_{\mathbf{c},E}(t_0,t_1,\dots)\right),
\]
where
\[
\mathcal{F}_{\mathbf{c},E}^g(t_0,t_1,\dots)=\sum_{n,d}\frac{Q^d}{n!}\int_{[X_{g,n,d}]}\mathbf{c}(E_{g,n,d}) (\sum_{k_1=0}^\infty(\text{ev}_1^*t_k)\psi_1^{k_1}) \cdots (\sum_{k_1=0}^\infty(\text{ev}_n^*t_k)\psi_n^{k_n}).
\]
Here \(Q^d\) is the representative of \(d\) in the semigroup ring of degrees of holomorphic curves in \(X\), \(t_0,t_1,\dots\) are rational cohomology classes on \(X\), and \(\psi_i\) is the first Chern class of the universal cotangent bundle over \(X_{g,n,d}\) corresponding to the \(i\)-th marked point of \(X\). For \(E\) the zero element in \(K^0(X)\), the twisted potential \(\mathcal{D}_{\mathbf{c},E}^g\) reduces to \(\mathcal{D}_X\), the total descendant potential of \(X\).
At this point the formalism of quantized quadratic hamiltonians enters the picture. One considers the symplectic space \(\mathcal{H}=H^*(X;\mathbb{Q})((z^{-1}))\) of Laurent polynomials in \(z^{-1}\) with coefficients in the cohomology of \(X\), endowed with the symplectic form
\[
\Omega(\mathbf{f},\mathbf{g})=\frac{1}{2\pi i}\oint \left(\int_X\mathbf{f}(-z)\mathbf{g}(z)\right)\,dz.
\]
The subspace \(\mathcal{H}_+=H^*(X;\mathbb{Q})[z]\) is a Lagrangian subspace, and \((\mathcal{H},\Omega)\) is identified with the canonical symplectic structure on \(T^*\mathcal{H}_+\). Finally, given an infinitesimal symplectic transformation \(T\) of \(\mathcal{H}\), one can consider the differential operator \(\hat{T}\) of order \(\leq 2\) on functions on \(\mathcal{H}_+\), which is associated by quantization to the quadratic Hamiltonian \(\Omega(T\mathbf{f},\mathbf{f})/2\) on \(\mathcal{H}\). By the inclusion \(\mathcal{H}_+\hookrightarrow H^*(X;\mathbb{Q})[[z]]\), the operator \(\hat{T}\) acts on asymptotic elements of the Fock space, i.e., on functions of the formal variable \(\mathbf{q}(z)=q_0+q_1z+q_2z^2+\cdots\) in \(H^*(X;\mathbb{Q})[[z]]\). By the dilaton shift, i.e., setting \(\mathbf{q}(z)=\mathbf{t}(z)-z\), with \(\mathbf{t}(z)=t_0+t_1z+t_2z^2+\cdots\), the operator \(\hat{T}\) acts on any function of \(t_0,t_1,\dots\), notably on the descendant potentials.
Having introduced this formalism, the authors are able to express the relation between twisted and untwisted Gromov-Witten invariants in an extremely elegant way: up to a scalar factor,
\[
\mathcal{D}_{\mathbf{c},E}=\hat{\Delta}\mathcal{D}_X,
\]
where \(\Delta:\mathcal{H}\to \mathcal{H}\) is the linear symplectic transformation defined by the asymptotic expansion of
\[
\sqrt{\mathbf{c}(E)}\prod_{m=1}^\infty \mathbf{c}(E\otimes L^{-m})
\]
under the identification of the variable \(z\) with the first Chern class of the universal line bundle \(L\). This is the quantum Riemann-Roch theorem; it explicitly determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invariants. The result is a consequence of Mumford's Grothendieck-Riemann-Roch theorem applied to the universal family \(\pi:X_{g,n+1,d}\to X_{g,n,d}\). If \(E=\mathbb{C}\) is the trivial line bundle, then \(E_{g,n,d}=\mathbb{C}\ominus \mathbf{E}_g^*\), where \(\mathbf{E}_g\) is the Hodge bundle, and one recovers from quantum Riemann-Roch results of \textit{D. Mumford} [Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271--328 (1983; Zbl 0554.14008)] and \textit{C. Faber, R. Pandharipande} [Invent. Math. 139, No.1, 173--199 (2000; Zbl 0960.14031)] on Hodge integrals.
If \(\mathbf{c}^*\) is the multiplicative characteristic class
\[
\mathbf{c}^*(\cdot)=\exp\left(\sum_{k=0}^\infty(-1)^{k+1} s_k\mathrm{ch}_k(\cdot)\right),
\]
then \(\mathbf{c}^*(E^*)=1/\mathbf{c}(E)\), and one the following quantum version of Serre duality:
\[
\mathcal{D}_{\mathbf{c}^*,E^*}(\mathbf{t}^*)=(\mathrm{sdet}\,\mathbf{c}(E))^{-\frac{1}{24}}\mathcal{D}_{\mathbf{c},E}(\mathbf{t}),
\]
where \(\mathbf{t}^*(z)=\mathbf{c}(E)\mathbf{t}(z)+(1-\mathbf{c}(E))z\).
Finally, if \(E\) is a convex vector bundle and a submanifold \(Y\subset X\) is defined by a global section of \(E\), then the genus zero Gromov-Witten invariants of \(Y\) can be expresssed in terms of the invariants of \(X\) twisted by the Euler class of \(E\). These are in turn related to the untwisted Gromov-Witten invariants of \(X\) by the quantum Riemann-Roch theorem, so the authors end up with a quantum Lefschetz hyperplane section principle, expressing genus zero Gromov-Witten invariants of a complete intersection \(Y\) in terms of those of \(X\). This extends earlier results [\textit{V.~V.~Batyrev, I.~Ciocan-Fontanine, B.~Kim} and \textit{D.~van Straten}, Acta Math. 184, No. 1, 1--39 (2000; Zbl 1022.14014); \textit{A.~Bertram}, Invent. Math. 142, No. 3, 487--512 (2000; Zbl 1031.14027); \textit{A.~Gathmann}, Math. Ann. 325, No. 2, 393--412 (2003; Zbl 1043.14016); \textit{B.~Kim}, Acta Math. 183, No. 1, 71--99 (1999; Zbl 1023.14028); \textit{Y.-P.~Lee}, Invent. Math. 145, No. 1, 121--149 (2001; Zbl 1082.14056)], and yields most of the known mirror formulas for toric complete intersections. The idea of deriving mirror formulas by applying the Grothendieck-Riemann-Roch theorem to universal stable maps is not new: according to the authors it can be traced back at least to Kontsevich's investigations in the early 1990s, and to Faber's and Pandharipande's work on Hodge integrals. Gromov-Witten invariants; mirror symmetry; Grothendieck-Riemann-Roch theorem; Lefschetz hyperplane section principle; Serre duality T. Coates, A. Givental, Quantum Riemann-Roch, Lefschetz and Serre. \textit{Ann. of Math}. (2) \textbf{165} (2007), 15-53. MR2276766 Zbl 1189.14063 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Riemann-Roch theorems, Mirror symmetry (algebro-geometric aspects) Quantum Riemann-Roch, Lefschetz and Serre | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth projective variety. Let \(\bar{M}_{0,n}=\bar{M}_{0,n}(X,\beta)\) be the moduli stack of genus \(0\) stable maps to \(X\) with \(n\) marked points and of class \(\beta\in H_2(X;\mathbb{Z})\). The modified psi classes are defined by \(\bar{\psi}_i=\pi_i^* \psi_i\), where \(\pi_i: \bar{M}_{0,n} \to \bar{M}_{0,1}\) is the projection forgetting all the marked points but the \(i\)-th, and \(\psi_i\) is the usual psi class on \(\bar{M}_{0,1}\) [\textit{T. Graber}, \textit{J. Kock} and \textit{R. Pandharipande}, Am. J. Math. 124, 611--647 (2002; Zbl 1055.14056)].
These modified psi classes are used to construct a new quantum product, called the tangency quantum product, on \(H^*(X;\mathbb{Q})\otimes_{\mathbb{Q}} \Lambda[[x,y]]\), where \(\Lambda\) is the usual Novikov ring and \(x,y\) are variables which parametrize an element in \(H^{\text{even}}(X;\mathbb{Q})\). Associativity is proven by means of a WDVV type equation for the tangency quantum potential. This product recovers the usual quantum product upon resetting \(x=y=0\).
As an example, if \(X\) is a homogeneous variety, this structure encodes the characteristic numbers of rational curves in \(X\). For the projective plane, the product is equivalent to that of the contact cohomology of \textit{L. Ernström} and \textit{G. Kennedy} [Am. J. Math. 121, 73--96 (1999; Zbl 0933.14011)]. quantum cohomology; WDVV equations; Gromov-Witten invariants DOI: 10.1112/S0010437X03000101 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Tangency quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A result of D.~Peterson identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of \(\text{GL}_n\) is proven. The methods developed are used to give a new proof of a formula of C.~Vafa, K.~Intriligator, and A.~Bertram for the structure constants (Gromov-Witten invariants). Certain inequalities for Schur polynomials at roots of unity are proven. Gromov-Witten invariants; quantum cohomology rings; Grassmannians; reduced coordinate rings; Schur polynomials K. Rietsch, Quantum cohomology rings of Grassmannians and total positivity, Duke Math. J. 110 (2001), no. 3, 523--553. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over the reals, the complexes, the quaternions Quantum cohomology rings of Grassmannians and total positivity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Mirror manifolds to toric Calabi-Yau threefolds are encoded in algebraic curves. The quantization of these curves leads naturally to quantum-mechanical operators on the real line. We show that, for a large number of local del Pezzo Calabi-Yau threefolds, these operators are of trace class. In some simple geometries, like local \(\mathbb{P}^2\), we calculate the integral kernel of the corresponding operators in terms of Faddeev's quantum dilogarithm. Their spectral traces are expressed in terms of multi-dimensional integrals, similar to the state-integrals appearing in three-manifold topology, and we show that they can be evaluated explicitly in some cases. Our results provide further verifications of a recent conjecture which gives an explicit expression for the Fredholm determinant of these operators, in terms of enumerative invariants of the underlying Calabi-Yau threefolds. Kashaev, R.; Mariño, M., Operators from mirror curves and the quantum dilogarithm, Commun. Math. Phys., 346, 967, (2016) Quantization in field theory; cohomological methods, Calabi-Yau manifolds (algebro-geometric aspects), Calabi-Yau theory (complex-analytic aspects), Supersymmetric field theories in quantum mechanics, Groups and algebras in quantum theory and relations with integrable systems, Selfadjoint operator theory in quantum theory, including spectral analysis, Commutation relations and statistics as related to quantum mechanics (general), Mirror symmetry (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Relationships between algebraic curves and integrable systems, Relationships between algebraic curves and physics, Polylogarithms and relations with \(K\)-theory Operators from mirror curves and the quantum dilogarithm | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that, for smooth quasi-projective varieties over a field, the \(K\)-theory \(K(X)\) of vector bundles is the universal cohomology theory where \(c_1(L \otimes \overline{L}) = c_1(L) + c_1(\overline{L}) - c_1(L) c_1(\overline{L})\). Then, we show that Grothendieck's Riemann-Roch theorem is a direct consequence of this universal property, as well as the universal property of the graded \(K\)-theory \(GK^\bullet(X) \otimes \mathbb{Q}\). Riemann-Roch; \(K\)-theory; cohomology theories Navarro, A., On Grothendieck's Riemann-Roch theorem, Expo. Math., 35, 3, 326-342, (2017) Riemann-Roch theorems, Relations of \(K\)-theory with cohomology theories, Applications of methods of algebraic \(K\)-theory in algebraic geometry On Grothendieck's Riemann-Roch theorem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A Gröbner basis for the small quantum cohomology of Grassmannian \(G_{k,n}\) is constructed and used to obtain new recurrence relations for Kostka numbers and inverse Kostka numbers. Using these relations it is shown how to determine inverse Kostka numbers which are related to the mod-\(p\) Wu formula. quantum cohomology; Gröbner basis; Grassmannian; (inverse) Kostka numbers Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Characteristic classes and numbers in differential topology Recurrence formulas for Kostka and inverse Kostka numbers via quantum cohomology of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from a geometric point of view. This is done in the perspective of a joint result of the author with \textit{A. Uribe} [Commun. Math. Phys. 233, No. 3, 493--512 (2003; Zbl 1027.53113)] which relates the quantum group and the Weyl quantizations of the moduli space of flat SU\((2)\)-connections on the torus. Two results are emphasized: the reconstruction from Weyl quantization of the restriction to the torus of the modular functor, and a description of a basis of the space of quantum observables on the torus in terms of colored curves, which answers a question related to quantum computing. Witten--Reshetikhin--Turaev invariants; theta functions; Weyl quantization; modular functor Răzvan Gelca, On the holomorphic point of view in the theory of quantum knot invariants, J. Geom. Phys. 56 (2006), no. 10, 2163 -- 2176. Topological field theories in quantum mechanics, Invariants of knots and \(3\)-manifolds, Quantum groups and related algebraic methods applied to problems in quantum theory, Geometry and quantization, symplectic methods, Geometric quantization, Theta functions and curves; Schottky problem, Quantum computation On the holomorphic point of view in the theory of quantum knot invariants | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathbb P}^{d}\) denote projective space over an algebraically closed field \(k\). Let \(K_{0}( {\mathbb P}^{d} )\) denote the Grothendieck group of locally free sheaves on \({\mathbb P}^{d}\). For \(x \in K_{0}( {\mathbb P}^{d} )\) with Chern classes \(c_{i}(x)\) the Chern polynomial is defined by
\[
C_{x}(t) = 1 + c_{1}(x)t + c_{2}(x)t^{2} + \dots + c_{d}(x)t^{d}
\]
lying in \({\mathbb Z}[t]/(t^{d+1})\). On the other hand a graded, finitely generated module \(M\) over the graded ring \(k[x_{0}, \dots , x_{d}]\) gives a coherent sheaf on \({\mathbb P}^{d}\) and a class in the Grothendieck group \(G_{0}( {\mathbb P}^{d} ) \cong K_{0}( {\mathbb P}^{d} )\). The Hilbert polynomial \(P_{M}(t)\) is related to \(C_{M}(t)\) by the Hirzebruch-Riemann-Roch theorem [see \textit{D. Eisenbud}, ``Commutative algebra. With a view towards algebraic geometry'', Grad. Texts Math. 150 (1995; Zbl 0819.13001)].
The author shows that the homomorphism
\[
\xi : K_{0}( {\mathbb P}^{d} ) \rightarrow ({\mathbb Z}[t]/(t^{d+1}))^{*} \times {\mathbb Z}
\]
given by \(\xi(M) = (C_{M}(t) , \text{rank}M))\) is injective. As an application she shows that the classes of \(M\) and \(N\) in \(K_{0}({\mathbb P}^{d})\) are equal if and only if \(C_{M}(t) = C_{N}(t)\) and \( \text{rank}M) = \text{rank}N)\) or if and only if \(P_{M}(t) = P_{N}(t)\). The paper concludes with a section detailing the precise relation between Chern and Hilbert polynomials, which appears in [\textit{D. Eisenbud}, loc. cit., Exercise 19.18]. Chan, C. -Y.J.: A correspondence between Hilbert polynomials and Chern polynomials over projective spaces, Illinois J. Math. 48, No. 2, 451-462 (2004) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry A correspondence between {H}ilbert polynomials and {C}hern polynomials over projective spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth complex projective variety and \(f: X \relbar \to X'\) an ordinary \({\mathbb P}^r\) flop. One of the main results established in this paper is that the big quantum cohomology ring is invariant under simple ordinary flops after analytic continuation over the extended Kähler moduli space (Theorem 0.2). The authors also prove that if \(f\) is a Mukai flop, then \(X\) and \(X'\) are diffeomorphic, hence they have isomorphic Hodge structures and Gromov-Witten theory (Theorem 0.3).
Y. Ruan and C.-L. Wang independently conjectured that \(K\)-equivalent smooth varieties have canonically isomorphic quantum cohomology rings over the extended Kähler moduli spaces (Conjecture 0.4). This conjecture was first established for threefolds by A. Li and Y. Ruan, and the paper under review establishes the analogous result in higher dimension, thus providing further evidence for the Ruan--Wang conjecture. quantum cohomology; Gromov-Witten invariants; ordinary flops; analytic continuation Lee, Y-P; Lin, H-W; Wang, C-L, Flops, motives, and invariance of quantum rings, Ann. Math., 172, 243-290, (2010) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), (Equivariant) Chow groups and rings; motives, Minimal model program (Mori theory, extremal rays) Flops, motives, and invariance of quantum rings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper proposes a generalization of the classical \(l\)-modular polynomial for elliptic curves, the \textit{set of Hilbert modular polynomials}, for principally polarized abelian varieties of dimension \(g\)\, with maximal real multiplication for a totally real field \(K_0\). It also provides an algorithm to compute them.
The \(l\)-modular polynomial for elliptic curves parametrises \(l\)-isogenies. Analogously the set of Hilbert modular polynomials are related with isogenies of cyclic kernel, the \(\mu\)-isogenies (Definition 2.2), with \(\mu\in K_0\) a totally positive prime.
Section 1 gives an summary of the proposal and states the main result (Theorem 1.5, proved in Section 6), concerning the existence of Hilbert modular polynomials. Section 2 summarizes the notions of maximal real multiplication and Hilbert modular forms.
Section 3 studies the existence of RM isomorphism invariants (Proposition 3.1). Section 4 gives an \textit{if and only if} condition for the existence of a \(\mu\)-isogeny (Proposition 4.5). Section 5 considers the computation of RM isomorphism invariants in the case \(g=2\).
Section 6 gives an algorithm to compute a set of Hilbert polynomials (Algorithm 6.3) and provides the proof of Theorem 1.5. In the case of surfaces (\(g=2\)) Section 7 presents improvements of the algorithm (Algorithm 7.8) and details of an implementation in MAGMA for \(K_0= Q(\sqrt 5)\) and \(K_0= Q(\sqrt 2)\).
Finally Section 8 shows three possible applications: point counting for curves of genus two with maximal real multiplication, see [\textit{S. Ballentine} et al., Assoc. Women Math. Ser. 9, 63--94 (2017; Zbl 1414.11076)], walking on isogeny graphs for genus 2 curves, see [\textit{A. Dudeanu} et al., ``Cyclic isogenies for abelian varieties with real multiplication'', Preprint, \url{arXiv:1710.05147}] and finally computing Hilbert class polynomials to genus 2, generalizing a method of \textit{A. V. Sutherland} [Math. Comput. 80, No. 273, 501--538 (2011; Zbl 1231.11144)]. Hilbert modular polynomials; cyclic isogenies; abelian varieties; genus two; maximal real multiplication Number-theoretic algorithms; complexity, Abelian varieties of dimension \(> 1\), Complex multiplication and moduli of abelian varieties, Isogeny Hilbert modular polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials To each polynomial \(P\) with integral non-negative coefficients and constant term equal to 1, of degree \(d\), we associate a pair of elements \((y,w)\) in the symmetric group \(S_n\), where \(n=1+d+P(1)\), for which we prove that the Kazhdan-Lusztig polynomial \(P_{y,w}\) equals \(P\). This pair satisfies \(\ell(w)-\ell(y)=2d+P(1)-1\), where \(\ell(w)\) denotes the number of inversions of \(w\). -- For details see the author's paper: \textit{P. Polo}, Represent. Theory 3, No. 4, 90-104 (1999; Zbl 0968.14029). Kazhdan-Lusztig polynomials; Schubert varieties; intersection cohomology Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Representations of finite symmetric groups, Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Construction of arbitrary Kazhdan-Lusztig polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a vector space with a non-degenerate symmetric form and OG be the orthogonal Grassmannian which parametrizes maximal isotropic subspaces in \(V\). The authors give a presentation for the small quantum cohomology ring \(QH^*(\text{OG})\) and show that its product structure is determined by the ring of \(\tilde P\)-polynomials of \textit{P.~Pragacz} and \textit{J.~Ratajski} [Compos. Math. 107, 11--87 (1997; Zbl 0916.14026)]. A ``quantum Schubert calculus'' is formulated, which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing Gromov-Witten invariants. As an application, it is shown that the table of three-point, genus zero Gromov-Witten invariants for OG coincides with that for a corresponding Lagrangian Grassmannian LG, up to an involution.
In a companion paper to this one [\textit{A.~Kresch} and \textit{H.~Tamvakis}, J.~Algebr. Geom., 12, No.~4, 777--810 (2003; Zbl 1051.53070)], the authors provide an analogous analysis for the Lagrangian Grassmannian. The situation in the orthogonal case is similar, but with significant differences, both in the results and in their proofs. quot schemes; Schubert calculus Kresch A., Tamvakis H.: Quantum cohomology of orthogonal Grassmannians. Composit. Math. 140, 482--500 (2004) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Quantum cohomology of orthogonal Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This monograph grew out of several lecture courses given by the author at the Max-Planck-Institut für Mathematik in Bonn, Germany, between 1994 and 1998, and many shorter lecture series delivered at various summer schools and conferences. Large parts of the material presented here have been circulating, for quite a while, in preprint form under the title ``Frobenius manifolds, quantum cohomology, and moduli spaces. I, II, III'' [cf.: MPI preprint 96-113, Max-Planck-Institut für Mathematik (Bonn 1996)]. The present book is the final and complete version of the author's lectures on these subjects. Its main goal is to summarize, in a coherent and systematic way, some of the pioneering mathematical developments that took place in the past ten years and that aimed at establishing a mathematical version of quantum cohomology.
The author, being one of the great pioneers, ultimate experts, leading inspirators and most active researchers in the field, has contributed a great deal to these developments, whether by his own work or by the joint work with his collaborators \textit{K. Behrend, C. Hertling, R. Kaufmann, M. Kontsevich}, and others. In view of this fact, it is a matter of course that the approach to quantum cohomology described in this book is closely related to the author's original work and/or his research colleagues: The text consists of six chapters, each of which is divided into several sections, and a foregoing introduction dedicated to the general motivation for mathematical quantum cohomology. The contents of the single chapters are as follows:
Chapter 0: ``Introduction: What is quantum cohomology?'': This introduction gives a rather detailed overview of the two central themes of the book: quantum cohomology and Frobenius manifolds. The author explains the (preliminary) definitions underlying these concepts, gives some illustrations by important examples, and derives from this motivating discussion the strategic plan of the book. Typically for the author's well-known style of writing, already the introduction is pointed, concise, directing and highly enlightening.
Chapter I: ``Introduction to Frobenius manifolds'': This chapter is based on \textit{B. Dubrovin}'s innovating work on Frobenius (super-)manifolds [in: Integrable systems and quantum groups, Montecatini 1993, Lect. Notes Math. 1620, 120-348 (1996; Zbl 0841.58065)] and provides, together with some important enhancements by the author himself, a systematic exposition of the fundaments of this theory. This includes the definition of Frobenius manifolds, Dubrovin's structure connection, Euler fields, the extended structure connection, semi-simple Frobenius manifolds, examples of Frobenius manifolds and a first encounter with quantum cohomology in this context, weak Frobenius manifolds, and relations to Poisson structures.
Chapter II: ``Frobenius manifolds and isomonodromic deformations'': In this chapter, the author continues the study of Frobenius (super-)manifolds from the deformation-theoretic viewpoint. The main topics treated here are the so-called second structure connection on Frobenius manifolds, the formal Laplace transform, isomonodromic deformations of connections, versal deformations, Schlesinger equations and their Hamiltonian structure, semisimple Frobenius manifolds as special solutions to the Schlesinger equations, and applications to the quantum cohomology ring of a projective space. The concluding section of this chapter discusses, in greater detail, the three-dimensional semisimple case of Frobenius manifolds and its connection with a special family of nonlinear ordinary differential equations, the so-called family ``Painlevé VI''. Again, much of the material presented here originates from Dubrovin's fundamental work cited above.
Chapter III: ``Frobenius manifolds and moduli spaces of curves'': This chapter turns to the more algebraic aspects of Frobenius manifolds in their supergeometric setting. The author introduces formal Frobenius manifolds, \(\text{Comm}_\infty\)-algebras, abstract (polynomial) correlation functions, and the Euler operator in the formal case. Then he discusses prestable pointed algebraic curves, together with their associated graphs and their moduli spaces, stratifications of moduli spaces of stable pointed curves, the particular structure of these moduli spaces in the case of genus zero and, in particular, the recent work of \textit{S. Keel} [Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992; Zbl 0768.14002)]. The fourth section of this chapter establishes the link between formal Frobenius manifolds and their abstract correlation functions, on the one hand, and cohomological field theories (with so-called tree level structure) and their natural correlation functions, on the other hand. Then, in section 5, Gromov-Witten invariants and the quantum cohomology ring of a projective manifold are described via the axiomatic approach by the author and M. Kontsevich [cf. \textit{M. Kontsevich} and \textit{Yu. Manin}, Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020)]. This framework is then tied up with the structural theory of formal Frobenius supermanifolds introduced at the beginning of this chapter. In this context, the author also discusses cohomological field theories and formal Frobenius supermanifolds of rank one in connection with Weil-Petersson volume forms on moduli spaces of stable \(n\)-pointed rational curves and Mumford classes. -- The last four sections of this chapter are devoted to R. Kaufmann's study of tensor products in the categories of local and global Frobenius manifolds [cf. \textit{R. M. Kaufmann}, ``The geometry of moduli spaces of pointed curves, the tensor product in the theory of Frobenius manifolds and the explicit Künneth formula in quantum cohomology'' (Bonn 1998; Zbl 0918.14011)], K. Saito's framework families over Frobenius manifolds, Gepner's Frobenius manifolds, Gerstenhaber-Batalin-Vilkovyski supercommutative algebras and their Maurer-Cartan equations, relations to symplectic manifolds, and special Frobenius manifolds associated to Calabi-Yau manifolds and symplectic Lefschetz manifolds.
Chapter IV: ``Operads, graphs, and perturbation series'': This chapter serves as a concise introduction to the more technical framework of operads and generating functions for moduli spaces of curves and quantum cohomology rings. The author reviews the classical linear operads and their occurrence in the homology theory of moduli spaces of stable \(n\)-pointed rational curves, sketches then the general functorial interpretation of operads, proves subsequently some formal identities for certain infinite sums taken over graphs of various topological types, and concludes this chapter by calculating several types of generating functions related to moduli spaces of curves and quantum cohomology rings. The latter topic is discussed along the lines of the work of \textit{W. Fulton} and \textit{R. MacPherson} [Ann. Math., II. Ser. 139, No. 1, 183-225 (1994; Zbl 0820.14037)] and the material of the whole chapter is closely related to the work of E. Getzler and M. Kapranov on cyclic and modular operads (1995-1998).
Chapter V: ``Stable maps, stacks, and Chow groups'': Although quantum cohomology, the main subject of the book, has been invoked in several places in the first four chapters, whether in the form of illustrating examples in chapter II or as an axiomatic framework in chapter III, its proof of existence as well as its systematic treatment had to be postponed until the final chapter VI. This is due to the fact that either construction of a mathematical quantum cohomology structure on the cohomology ring of a projective manifold requires a tremendous amount of advanced algebro-geometric techniques. Chapter V provides an overview of these methods and results needed, in addition, for the author's construction of quantum cohomology: prestable curves and prestable maps, flat families of these objects, groupoids and moduli groupoids, algebraic stacks à la Artin and Deligne-Mumford, homological Chow groups of schemes, homological Chow groups of stacks, operational Chow groups of schemes and stacks, and the related intersection and deformation theory of schemes and stacks.
Whereas chapters I--IV are reasonably self-contained and offer complete proofs of the main results, this chapter V is comparatively sketchy and survey-like. As the author points out in the preface of the book, this chapter and the following chapter VI are meant as an introduction to the wealth of original papers on the subjects discussed here and cannot replace the study of those.
Chapter VI: ``Algebraic geometric introduction to the gravitational quantum cohomology'': This concluding chapter focuses on the algebro-geometric construction of explicit Gromov-Witten-type invariants. The approach described here was formulated by the author himself and K. Behrend in their 1996 paper [\textit{K. Behrend} and \textit{Yu. Manin}, Duke Math. J. 85, No. 1, 1-60 (1996; Zbl 0872.14019)]. Later on this programme has been rigorously worked out by \textit{K. Behrend} [Invent. Math. 127, No. 3, 601-617 (1997; Zbl 0909.14007) and by \textit{K. Behrend} and \textit{B. Fantechi}, Invent. Math. 128, No. 1, 45-88 (1997; Zbl 0909.14006)]. The Manin-Behrend-Fantechi theory of algebro-geometric Gromov-Witten invariants is heavily based on the technical framework reviewed in chapter V. The author discusses this construction procedure in all of its crucial steps, turns then to the resulting quantum cohomology structure via gravitational descendants and Virasoro constraints, and shows how to calculate the occurring correlator functions on the rational cohomology of the underlying projective manifold (phase space) via the established Gromov-Witten correspondences and the intersection theory of the involved moduli spaces. Moreover, it is explained how the quantum cohomology constructed here can be interpreted as a Frobenius manifold, and why this viewpoint might be useful with regard to further generalizations of the Martin-Behrend-Fantechi approach to more general Frobenius manifolds.
In the course of the text, the author frequently points to other available approaches to Gromov-Witten theories and quantum cohomology models. However, although the celebrated ``mirror conjecture'' for Calabi-Yau manifolds initially provided the main stimulus for the development of mathematical quantum cohomology theories, it is not treated in this book. Fortunately, there are at least two recent (complementary) monographs which stress this particular aspect, together with the alternate approaches to Gromov-Witten invariants and quantum cohomology. The booklet ``Mirror, symmetry'' by \textit{C. Voisin} [cf. SMF/AMS Texts Monogr. 1 (1999; Zbl 0945.14021)] gives a nice panoramic overview of the recent developments concerning the mirror conjecture and the related quantum cohomology theory, whereas the voluminous book ``Mirror symmetry and algebraic geometry'' by \textit{D. Cox} and \textit{S. Katz} [cf. Math. Surveys Monogr. 68 (1999; 951.14026)] provides a nearly encyclopaedic exposition of these topics. Together with these two books, the monograph under review represents the current standard literature in textbook form on these subjects, without any doubt.
However, as has been stated before, Yu. I. Manin's book is not entirely self-contained. The exposition of the material is rather concise and condensed, nevertheless coherent, comprehensible and educating. The reader is required to have quite a bit of expertise in algebraic geometry, complex differential geometry, category theory, non-commutative algebra, Hamiltonian systems, and modern quantum physics. On the other hand, the wealth of both mathematical information and inspiration provided by the text is absolutely immense, and in this vein, the book is an excellent source for experts and beginning researchers in the field. quantum cohomology; Frobenius manifolds; isomonodromic deformations of connections; versal deformations; Calabi-Yau manifolds; algebraic stacks; homological Chow groups; Gromov-Witten-type invariants; Painlevé VI Yu. I. Manin, \textit{Frobenius manifolds, Quantum Cohomology, and Moduli Spaces} (Faktorial, Moscow, 2002) [in Russian]. Calabi-Yau manifolds (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to quantum theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Frobenius manifolds, quantum cohomology, and moduli spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper deals with the classification of a certain class of quadratic regular algebras of global dimension four, i.e. those having the same Hilbert series as the polynomial ring in four indeterminates, and which map onto a twisted homogeneous coordinate ring of a nonsingular quadric in \(\mathbb{P}^3\) via a graded morphism (of degree zero). quadratic regular algebras; global dimension; Hilbert series; twisted homogeneous coordinate rings; nonsingular quadrics Vancliff, M.; Van Rompay, K., Embedding a quantum nonsingular quadric in a quantum \(\mathbb{P}^3\), J. algebra, 195, 93-129, (1997) Twisted and skew group rings, crossed products, Noncommutative algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations, Homological dimension in associative algebras, Elliptic curves, Graded rings and modules (associative rings and algebras), Low codimension problems in algebraic geometry Embedding a quantum nonsingular quadric in a quantum \(\mathbb{P}^3\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author focuses on quantum differential equations, which can be regarded as examples of equations with certain universal properties. The author uses the language of D-modules which is a proper tool to investigate the relation between quantum cohomology and the theory of integrable systems. This work also contains some explanations on how quantum cohomology is related to other parts of mathematics. Quantum differential equations; Quantum cohomology; D- modules Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Groups and algebras in quantum theory and relations with integrable systems, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Schrödinger operator, Schrödinger equation Differential equations aspects of quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this article is to supply simpler proofs of the main theorems about the (small) quantum cohomology ring of a Grassmann variety. This includes \textit{A. Bertram's} quantum version of the Pieri and Giambelli formulas [Adv. Math. 128, 289--305 (1997; Zbl 0945.14031)]. In contrast to Bertram's proofs, which require the use of quot schemes, the presented proofs in this article stay with more elementary algebraic geometric methods and do not use any moduli space techniques. Essentially everything is based only on the definition of the Gromov-Witten invariants.
The author shows that the quantum Pieri formula is a consequence of the classical Pieri formula. Furthermore, he shows that the quantum Giambelli formula follows immediately from the quantum Pieri formula together with the classical Giambelli formula and associativity of the quantum cohomology. Also a proof is given of the Grassmannian case of a formula of Fulton and Woodward for the minimal \(q\)-power which appears in a quantum product of two Schubert classes. A proof of a simple version of the rim-hook algorithm is supplied. Finally, the presentation in terms of generators and relations of the quantum cohomology of Grassmannians is obtained.
The idea of the author is to start from the simple fact that if a rational curve of degree \(d\) passes through a Schubert variety in the Grassmannian \(\text{Gr}(l,\mathbb C^n)\), then the linear span of the points of this curve gives rise to a point in \(\text{Gr}(l+d,\mathbb C^n)\) which must lie in a related Schubert variety. This idea can be used to conclude in many cases that no curves pass trough three Schubert varieties in general position because the intersection of the related Schubert varieties is empty. In particular, the quantum Giambelli formula can be deduced by knowing that certain Gromov-Witten invariants are zero, which follows because the codimensions of the related Schubert varieties add up to more than the dimension of \(\text{Gr}(l+d,\mathbb C^n)\). quantum cohomology; Gromov-Witten invariants; Grassmann varieties; Pieri formula; Giambelli formula A. Buch. ''Quantum cohomology of Grassmannians''. Comp. Math. 137 (2003), pp. 227--235. DOI. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Quantum cohomology of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author gives a survey of recent joint work with \textit{A. S. Buch} and \textit{H. Tamvakis} [J. Am. Math. Soc. 16, No. 4, 901--915 (2003; Zbl 1063.53090) and Adv. Math. 185, No. 1, 80--90 (2004; Zbl 1053.05121)] on the quantum cohomology of Grassmannians for classical Lie types. The starting point is a review of both classical and quantum formulae for \(\text{Gr}(k,n)\), including a description of the Gromov-Witten invariants as certain classical structure constants on a partial flag variety \(F(k-d;k+d;n)\).
By using a conjectural interpretation of the classical cohomology of a partial flag variety by \textit{A. Knutson, T.C. Tao} and \textit{C. T. Woodward} [J. Am. Math. Soc. 17, 19--48 (2004; Zbl 1043.05111)], the author derives a combinatorial description for genus 0 Gromov-Witten invariants. This conjectural quantum Littlewood-Richardson formula has been proved for \(k\leq 3\) and computer-checked for \(n\leq 16\).
Finally, Grassmannians of the other classical types \(B\), \(C\) and \(D\) are considered, and a quantum Pieri formula is announced. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum cohomology of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper is a survey of classical results of Grothendieck on vanishing cycles. Consider a regular, proper and flat curve \(X\) over a strictly local trait \(S=(S,s,\eta )\), whose generic fiber is smooth and whose reduced special fiber is a divisor with normal crossings. One can analyse the difference between the (étale) cohomology of the special fiber \(H^*(X_s)\) and that of the generic geometric fiber \(H^*(X_{\eta})\), the coefficients ring being \(\mathbb{Z}_{\ell}\), \(\ell\) a prime number invertible on \(S\). Assuming that the action of the inertia group \(I\) on \(H^*(X_{\bar{\eta}})\) is tame, one shows that the defect of the specialization map \(H^*(X_s)\rightarrow H^*(X_{\bar{\eta}})\) is an isomorphism controlled by the vanishing cycles groups. This construction leads to Grothendieck's local monodromy theorem and his monodromy pairing for abelian varieties over local fields. The author discusses related current developments and questions and includes the proof of an unpublished result of Gabber giving a refined bound for the exponent of unipotence of the local monodromy for torsion coefficients. étale cohomology; monodromy; Milnor fiber; nearby and vanishing cycles; alteration; hypercovering; semistable reduction; intersection complex; abelian scheme; Picard functor; Jacobian; Néron model; Picard-Lefschetz formula; \(\ell\)-adic sheaf Research exposition (monographs, survey articles) pertaining to algebraic geometry, History of algebraic geometry, Development of contemporary mathematics, Galois representations, Abelian varieties of dimension \(> 1\), Derived categories and commutative rings, Structure of families (Picard-Lefschetz, monodromy, etc.), Étale and other Grothendieck topologies and (co)homologies, Local ground fields in algebraic geometry, Picard schemes, higher Jacobians, Arithmetic ground fields for curves, Formal groups, \(p\)-divisible groups, Group schemes Grothendieck and vanishing cycles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the corresponding \(D\)-module when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system. In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko. \(D\)-module; quantum completely integrable systems; eigenvalue problem; differential Galois group; algebraic integrability Braverman A., Etingof P., Gaitsgory D.: Quantum integrable systems and differential Galois theory. Transform. Groups 2, 31--57 (1997) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Feynman integrals and graphs; applications of algebraic topology and algebraic geometry, Applications of dynamical systems, Curves in algebraic geometry, Differential algebra, Commutative rings of differential operators and their modules Quantum integrable systems and differential Galois theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we prove a certain geometric version of the Grothendieck Conjecture for tautological curves over Hurwitz stacks. This result generalizes a similar result obtained by Hoshi and Mochizuki in the case of tautological curves over moduli stacks of pointed smooth curves. In the process of studying this version of the Grothendieck Conjecture, we also examine various fundamental geometric properties of ``profiled log Hurwitz stacks'', i.e., log algebraic stacks that parametrize Hurwitz coverings for which the marked points are equipped with a certain ordering determined by combinatorial data which we refer to as a ``profile''. anabelian geometry; Grothendieck conjecture; Hurwitz stack; universal curve Coverings of curves, fundamental group, Families, moduli of curves (algebraic), Generalizations (algebraic spaces, stacks) Geometric version of the Grothendieck conjecture for universal curves over Hurwitz stacks | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a sequence \(E_{\bullet}:E_{0}\to E_{1}\to \cdots \to E_{n}\) of vector bundles and bundle maps over a nonsingular variety \(X\) one can obtain a quiver variety \(\Omega_{r}=\{x\in X\mid \text{rank} (E_{i} (x) \to E_{j}(x)) \leq r_{ij}\), \(i<j\} \) where \(r=\{r_{ij}\} \) is a collection of integers with \(0\leq i<j\leq n.\) (Clearly \(\Omega_{r}\) also depends on \(E_{\bullet}\) but the notation indicating such is suppressed here.) Then \(\Omega_{r}\) is a subscheme of \(X\). The maximum codimension of \( \Omega_{r}\) is \(d(r) :=\sum_{i<j}(r_{i,j-1}-r_{ij}) (r_{i+1,j}-r_{ij})\).
The objective in this paper is to find a formula for \(\mathcal{O}_{\Omega_{r}},\) the structure sheaf of \(\Omega_{r},\) in \(K^{\circ}X,\) the Grothendieck ring of algebraic vector bundles on \(X\). Denoting the class of \( \Omega_{r}\) in \(K^{\circ}X\) by \([ \mathcal{O}_{\Omega_{r}}] \) gives the following description of the formula:
\[
[ \mathcal{O}_{\Omega_{r}}] =\sum_{| \mu | \geq d(r)}c_{\mu}(r) G_{\mu_{i}}(E_{1}-E_{0}) \cdots G_{\mu_{n}}(E_{n}-E_{n-1})
\]
where \(c_{\mu}(r) \) are certain integers described combinatorially in the paper and \(G_{\mu_{i}}(E_{i}-E_{i-1}) \) are the stable Grothendieck polynomials (which are also defined in the paper). The sum is over a finite number of sequences of partitions \(\mu \) such that the weights sum to at least \(d(r) .\) This formula is analogous to the formula for the cohomology class of \(\Omega_{r}\) as presented in the author's previous work with Fulton, however in this situation one needs the codimension to be precisely \(d(r) .\)
The paper starts with a treatment of these Grothendieck polynomials, including their geometric significance. Following, the algorithm for generating the coefficients \(c_{\mu}(r) \) is presented. The author conjectures that \((-1)^{| \mu | -d(r)}c_{\mu}(r) \geq 0,\) that is that the signs of the coefficients alternate with the weight of \(\mu .\) Evidence to support this conjecture appears in the fifth section where it is shown that Grothendieck polynomials are special cases of the formula. Finally, a Gysin formula, which calculates \(K\)-theoretic pushforwards from a Grassmann bundle, is shown using a generalization of the Jacobi-Trudi formula for Schur functions. quiver varieties; Grothendieck classes; cohomology; Gysin formula \beginbarticle \bauthor\binitsA. S. \bsnmBuch, \batitleGrothendieck classes of quiver varieties, \bjtitleDuke Math. J. \bvolume115 (\byear2002), no. \bissue1, page 75-\blpage103. \endbarticle \OrigBibText Anders Skovsted Buch, Grothendieck classes of quiver varieties , Duke Math. J. 115 (2002), no. 1, 75-103. \endOrigBibText \bptokstructpyb \endbibitem Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Determinantal varieties, \(K\)-theory of schemes Grothendieck classes of quiver varieties. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is a clearly written survey on the Grothendieck ring of varieties. The Grothendieck group is defined in a general setting and its basic properties are discussed. The Grothendieck ring of varieties is then introduced, the principal known results are reviewed and thoroughly motivated, and open problems are formulated. Johannes Nicaise and Julien Sebag, The Grothendieck ring of varieties, Motivic integration and its interactions with model theory and non-Archimedean geometry. Volume I, London Math. Soc. Lecture Note Ser., vol. 383, Cambridge Univ. Press, Cambridge, 2011, pp. 145 -- 188. Applications of methods of algebraic \(K\)-theory in algebraic geometry The Grothendieck ring of varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A notion of affine stacks introduced by the author in [Sel. Math., New Ser. 12, No. 1, 39--134 (2006; Zbl 1108.14004)] is a homotopy version of the notion of affine schemes and gives several applications in the context of algebraic topology and algebraic geometry. Furthermore, a map \(\mathbb{Z}\to\mathbb{H}\) of presheaves is defined in the paper quoted above and Conjecture 2.3.6 states that the deduced map \(K(\mathbb{Z},n)\to K(\mathbb{H},n)\) of simplicial presheaves is affine.
The objective of this work is to reconsider the schematization problem of \url{https://grothendieck.umontpellier.fr/archives-grothendieck/}, with a particular focus on the global case over \(\mbox{Spec}(\mathbb{Z})\). For this, the author proves the conjecture above which gives a formula for the homotopy groups of the schematization of a simply connected homotopy type stated in Theorème 4.1 being the main result of this paper. Then, several results on the behaviour of the schematization functor over \(\mbox{Spec}(\mathbb{Z})\) is deduced, which are proposed as a solution to the schematization problem. presheaf; schematic homotopy type of schemata; schematization functor; stack \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories, Schemes and morphisms, Classification of homotopy type Grothendieck's schematization problem revisited | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Factorization statistics are functions defined on the set \(\text{Poly}_d(\mathbb F_q)\) of all monic degree {\textit d} polynomials with coefficients in \(F_q\) which only depend on the degrees of the irreducible factors of a polynomial. We show that the expected values of factorization statistics are determined by the representation theoretic structure of the cohomology of point configurations in \(\mathbb R^3\). This {\textit twisted Grothendieck-Lefschetz formula for} \(\text{Poly}_d(\mathbb F_q)\) is analogous to a result of Church, Ellenberg, and Farb for squarefree polynomials. Our proof uses formal power series methods which also lead to a new proof of the Church, Ellenberg, and Farb result circumventing algebraic geometry. arithmetic statistics; symmetric group representations; configuration space Combinatorial aspects of algebraic geometry, Configurations and arrangements of linear subspaces Factorization statistics and the twisted Grothendieck-Lefschetz formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct in an abstract fashion (without using Gromov-Witten invariants) the orbifold quantum cohomology of weighted projective space, starting from a certain differential operator. We obtain the product, grading, and intersection form by making use of the associated self-adjoint D-module and the Birkhoff factorization procedure. The method extends in principle to the more difficult case of Fano hypersurfaces in weighted projective space, where Gromov-Witten invariants have not yet been computed, and we illustrate this by means of an example originally studied by A. Corti. In contrast to the case of weighted projective space itself or the case of a Fano hypersurface in projective space, a ``small cell'' of the Birkhoff decomposition plays a role in the calculation. quantum cohomology; D-module; weighted projective space; Birkhoff decomposition Guest, M.A., Sakai, H.: Orbifold quantum D-modules associated to weighted projective spaces (2008). arXiv:0810.4236v1 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Orbifold quantum D-modules associated to weighted projective spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathcal C\) be a category with a final object \( pt\), on which a fiber product is defined together with a class of independent squares and confined maps [cf. \textit{W. Fulton} and \textit{R. MacPherson}, Matematika: Novoe v Zarubezhnoj Nauke, 33. Moskva: Izdatel'stvo ``Mir'' (1983; Zbl 0526.55007)]. A bivariant theory \(\mathbb B\) on the category \(\mathcal C\) with values in the category of abelian groups is a correspondence that assigns to each morphism in \(\mathcal C\)
\[
X @ >f>> Y
\]
an abelian group
\[
{\mathbb B}(X @>f>>Y)
\]
equipped with the product, pushforward and pullback operations. These operations have to satisfy natural compatibility conditions [loc. cit.]. For a bivariant theory \(B_{*}(X):={\mathbb B}(X\rightarrow pt)\) is a covariant functor for confined maps, and \(B^{*}(X):= {\mathbb B}(X\rightarrow X)\) is a contravariant functor. Let \(\mathcal C\) and \({\mathcal C}^{\prime}\) be two categories as above, \(-: {\mathcal C}\rightarrow {\mathcal C}^{\prime}\) a functor preserving the basic structures, and \(\mathbb B\) and \({\mathbb B}^{\prime}\) two corresponding bivariant theories on \(\mathcal C\) and \({\mathcal C}^{\prime}\) respectively. A Grothendieck transformation: \({\gamma}: {\mathbb B}\rightarrow {\mathbb B}^{\prime}\) is a collection of group homomorphisms
\[
{\gamma}_{f}: {\mathbb B}(X @>f>>Y)\longrightarrow {\mathbb B}^{\prime} ({\overline X} @>{\bar f}>>{\overline Y})
\]
preserving the three basic operations. The main examples of bivariant theories are: bivariant \textbf{K} theory introduced by Fulton and MacPherson [loc. cit.] and Fulton-MacPherson bivariant \(\mathbb F\) theory of constructible functions. One also has bivariant versions of homology theories with various coefficient groups. The Grothendieck transformation \({\gamma}: {\mathbb B}\rightarrow {\mathbb B}^{\prime}\) yields natural transformations of the associated covariant and contravariant functors. The authors study the reverse question concentrating on the covariant aspect. Namely, given a natural transformation \(\tau : {\mathbb B}_{*} \rightarrow {\mathbb B}_{*}^{\prime}\) of covariant functors coming from bivariant theories \({\mathbb B}\) and \({\mathbb B}^{\prime}\) does there exist a Grothendieck transformation \({\gamma}\) of bivariant theories whose covariant part is \({\tau}?\) A typical situation is given by MacPherson's Chern class transformation \(c_{*} : F\rightarrow H_{*}\) from the covariant functor of constructible functions to the Borel-Moore homology with integer coefficients. The authors give the existence and a kind of uniqueness theorem for a suitable general Grothendieck transformation \(\gamma\) associated to a natural transformation \(c_{*} : {\mathbb F}_{*}\rightarrow {\mathbb H}_{*}\) of bivariant theories \({\mathbb F}\) and \({\mathbb H}.\) Here \(c_{*}\) has to satisfy certain conditions, but it is not required to be necessarily the MacPherson's Chern class. Grothendieck transformation; bivariant theory; bivariant Chern class Brasselet, J.-P.; Schürmann, J.; Yokura, S., On Grothendieck transformation in fulton-Macpherson's bivariant theory, J. pure appl. algebra, 211, 665-684, (2007) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Riemann-Roch theorems, Other homology theories in algebraic topology On Grothendieck transformations in Fulton-MacPherson's bivariant theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe a method for counting maps of curves of given genus (and variable moduli) to \(\mathbb{P}^2\), essentially by splitting the \(\mathbb{P}^2\) in half; then specialising to the case of genus 0 we show that the method of quantum cohomology may be viewed as the ``mirror'' of the former method where one splits the \(\mathbb{P}^1\) rather than the \(\mathbb{P}^2\); finally we indicate an analogue of the former method where \(\mathbb{P}^2\) is replaced by a K3 quartic. quantum cohomology of the plane; K3 quartic; number of maps Z Ran, On the quantum cohomology of the plane, old and new, and a \(K3\) analogue, Collect. Math. 49 (1998) 519 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic), Enumerative problems (combinatorial problems) in algebraic geometry On the quantum cohomology of the plane, old and new, and a K3 analogue | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum cohomology ring \(qH^*(G/P)\) of a homogeneous space is obtained from the ordinary cohomology ring \(H^*(G/P)\) by tensoring with \(\mathbb{Z}[q]\) and then introducing a new multiplication based on the Gromov--Witten invariants. Tensoring with \(\mathbb{C}\) yields the complexified quantum cohomology ring \(qH^*_\mathbb{C}(G/P)\). Peterson's theorem asserts that the latter is isomorphic to the affine coordinate ring of a suitable affine variety. It is the purpose of this paper to present an elementary proof of this fact in the cases where \(G\) is, respectively, \(SO_{2n+1}(\mathbb{C})\), \(SO_{2n+n}(\mathbb{C})\), and \(Sp_{2n}(\mathbb{C})\), and \(P\) is a parabolic subgroup of a certain designated type. In general, to a group \(G\) with Langland's dual \(G^\vee\) and a parabolic subgroup \(P\), the author associates a certain affine variety \(V_P \subset G^\vee\). Then for each of the above cases he established an isomorphism from \(qH^*_\mathbb{C}(G/P)\) to the coordinate ring of \(V_P\). The proof proceeds by discovering and employing an explicit presentation of all the rings involved. homogeneous space; parabolic subgroup; Gromov-Witten Cheong, D, Quantum cohomology rings of Lagrangian and orthogonal Grassmannians and total positivity, Trans. Am. Math. Soc., 361, 5505-5537, (2009) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Homogeneous spaces and generalizations, Representation theory for linear algebraic groups Quantum cohomology rings of Lagrangian and orthogonal Grassmannians and total positivity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a finite-dimensional representation of a complex connected reductive algebraic group \(G\). Let \(X\) be the closure of a \(G\)-orbit in \(V\). Assume that \(G\) contains a one-dimensional torus acting on \(X\) by multiplication. Then the coordinate ring \(\mathbb{C}[X]\) has a natural grading coming from this action. Let \(K_0'(X)\) be the Grothendieck group of the category of graded finitely generated \(\mathbb{C}[X]\)-modules with rational \(G\)-action compatible with the module structure. In this paper, it is shown that if \(X\) admits a ``coherent desingularization'', then \(K_0'(X)\) is generated by the shifts in grading of the Euler characters of sheaves arising from global sections of line bundles on \(P/B\), \(P\) being a certain parabolic subgroup associated to the ``coherent desingularization''. categories of graded finitely generated modules; complex connected reductive algebraic groups; coordinate rings; Grothendieck groups; Euler characters of sheaves; global sections of line bundles Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients), Grothendieck groups, \(K\)-theory, etc., Linear algebraic groups over the reals, the complexes, the quaternions, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The Grothendieck group of \(G\)-equivariant modules over coordinate rings of \(G\)-orbits | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author gives an introductory survey of genus zero Gromov-Witten invariants and quantum cohomology [cf. \textit{W. Fulton} and \textit{R. Pandharipande}, Proc. Symp. Pure Math. 62, Part 2, 45--96 (1997; Zbl 0898.14018)] and its formalization in the framework of Frobenius manifolds [\textit{Yu. I. Manin}, ``Frobenius manifolds, quantum cohomology, and moduli spaces.''Coll. Publ. AMS 47 (1999; Zbl 0952.14032)]. This very clear account of the theory may serve well as a guide for the first steps into this rapidly growing field. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main objects of this paper are the Gromov-Witten invariants for symplectic manifolds defined by means of the moduli spaces of pseudo-holomorphic curves. It is well-known that these invariants can be used in topological quantum field theories to describe the correlation functions. In this case they must satisfy a fundamental axiom which is called composition law. The author gives a detailed description of these invariants in order to formulate precise statements about quantum multiplication and to calculate it in some examples.
In Section 1 he describes the Gromov-Witten invariants and demonstrates that they satisfy the composition law. The description of the moduli spaces of pseudo-holomorphic curves is given in Section 2. In Section 3 the author gives the definition of quantum multiplication and describes its properties. He finishes this section by pointing out some related questions and approaches such as the approach of Kontsevich-Manin, manifolds of Frobenius, mirror symmetry and Floer cohomology. Gromov-Witten invariants; symplectic manifolds; topological field theories; quantum multiplication Audin, M.: Cohomologie quantique. Astérisque 241, 29-58 (1997) Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Topological field theories in quantum mechanics, Symplectic aspects of Floer homology and cohomology, Calabi-Yau manifolds (algebro-geometric aspects), Moduli problems for topological structures, Spaces of embeddings and immersions, Moduli problems for differential geometric structures, Complex-analytic moduli problems, Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces Quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Quantum groups are non-cocommutative Hopf algebra deformations of the universal enveloping of a Lie algebra while quantum cohomology is a commutative deformation of the algebra structure on the cohomology of an algebraic variety that takes into account the enumerative geometry of rational curves in the variety. That is, quantum cohomology is a commutative associative deformation of ordinary multiplication in equivariant cohomology \(H^\bullet_{\mathsf{G}}(X)\) defined by \((\gamma_1 * \gamma_2, \gamma_3) = \sum_{\beta>0} q^\beta \langle \gamma_1,\gamma_2,\gamma_3 \rangle_\beta\), where \((\gamma_1,\gamma_2)=\int_X \gamma_1 \cup \gamma_2\) is the standard bilinear form on \(H^\bullet_{\mathsf{G}}(X)\), \(\beta\) ranges over the cone of effective classes in \(H_2(X,\mathbb{Z})\), \(q^\beta\) denotes the corresponding element of the semigroup algebra of the effective cone, and \(\langle \gamma_1,\gamma_2,\gamma_3 \rangle_\beta \in H^\bullet_{\mathsf{G}}(\mathsf{pt},\mathbb{Q})\) is the virtual count of rational curves of degree \(\beta\) meeting cycles Poincaré dual to \(\gamma_1\), \(\gamma_2\), and \(\gamma_3\).
The authors study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver \(Q\), connecting equivariant quantum cohomology of symplectic resolutions with their quantizations and derived autoequivalences. Using a geometric \(R\)-matrix formalism (Section 1.2.1, page 3), which is related to the reflection operator in Liouville field theory, they construct the Yangian \(\mathsf{Y}_Q\) of \(Q\) acting on the cohomology \(H(\mathsf{w}) = \bigoplus_{\mathsf{v}} H^\bullet_{\mathsf{G}}\left(\mathcal{M}_{\theta,\zeta}(\mathsf{v},\mathsf{w})\right)\) of Nakajima quiver varieties. When \(Q\) does not have a loop, this construction is related to \textit{M. Varagnolo}'s [Lett. Math. Phys. 53, No. 4, 273--283 (2000; Zbl 0972.17010)] and \textit{H. Nakajima}'s [J. Am. Math. Soc. 14, No. 1, 145--238 (2001; Zbl 0981.17016)], who construct a certain subalgebra of \(\mathsf{Y}_Q\) via generators and relations.
The authors prove a formula for quantum multiplication by divisors in terms of the Yangian action, where the structure constants of quantum multiplication are formal power series in \(q^\beta\). The quantum connection through the commuting difference connection can be identified with the trigonometric Casimir connection for \(\mathsf{Y}_Q\), which is a generalization of the rational Casimir connection. Equivalently, the divisor operators correspond to certain elements of Baxter subalgebras of \(\mathsf{Y}_Q\). A key role is played by geometric shift operators (Section 8.1.7, page 115) which can be identified with the quantum KZ difference connection.
In the second part of the book under review, the authors give an extended example of the general theory for moduli spaces of sheaves on \(\mathbb{C}^2\) framed at infinity. In Chapter 12 (page 141), the general framework of the quiver has been specialized to the one with one vertex and one loop. The Uhlenbeck compactification, which is the compactification of the moduli of framed instantons, as well as polarizations of Nakajima varieties to construct a Quot-scheme, Baranovsky operators, Fock spaces, and Virasoro algebras have been defined. The Yangian action is analyzed explicitly in terms of a free field realization, and they show that divisor operators generate the quantum ring, which is identified with the full Baxter subalgebras. As a corollary, an action of the \(W\)-algebra \(\mathcal{W}\big(\mathfrak{gl}(r)\big)\) on the equivariant cohomology of rank \(r\) moduli spaces is obtained. quantum group; quantum cohomology; Nakajima quiver varieties; Yangians; quantum connections; Hilbert schemes; Baxter algebras; Fock bosons; instanton moduli; Baranovsky operators; Virasoro algebra; Gamma functions Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Quantum groups (quantized enveloping algebras) and related deformations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum groups and quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author describes a formalism based on quantization of quadratic Hamiltonians and symplectic actions of loop groups which provides a convenient home for most of the known results and conjectures about Gromov-Witten invariants of compact symplectic manifolds. Gromov-Witten invariants; quantization; quadratic Hamiltonians A. B. Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians. \textit{Mosc. Math. J}. \textbf{1} (2001), 551-568, 645. MR1901075 Zbl 1008.53072 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Deformation quantization, star products Gromov-Witten invariants and quantization of quadratic Hamiltonians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\emptyset \ne \mathcal{A} \subset \mathbb{R}^n\) and \(\mathcal{B} \subset \mathbb{N}^n \setminus (2\mathbb{N})^{n}\) be finite sets. Let \( \mathbb{R}[\mathcal{A}, \mathcal{B}]\) be the set of all functions \(f \colon \mathbb{R}^n \to \mathbb{R} \cup \{\infty\}\) of the form \[f(\mathbf{x}) = \sum_{\alpha \in \mathcal{A}} c_\alpha| \mathbf{x}|^{\alpha} + \sum_{\beta \in \mathcal{B}} d_\beta \mathbf{x}^{\beta},\] where \(c_\alpha\) and \(d_\beta\) are real numbers. By definition, \(\mathbb{R}[\mathcal{A}, \mathcal{B}]\) is a vector space of dimension \(|\mathcal{A}| + |\mathcal{B}|.\)
Let \(f := \sum_{\alpha \in \mathcal{A}} c_\alpha| \mathbf{x}|^{\alpha} + \sum_{\beta \in \mathcal{B}} d_\beta \mathbf{x}^{\beta} \in \mathbb{R}[\mathcal{A}, \mathcal{B}].\) We say that \(f\) is
\begin{enumerate}
\item an {\em even AG function} if at most one of the \(c_\alpha\) is negative and all the \(d_\beta\) are zero; and
\item an {\em odd AG function} if all the \(c_\alpha\) are non-negative and at most one of the \(d_\beta\) is nonzero.
\end{enumerate}
The function \(f\) is called an {\em AG (arithmetic-geometric mean) function} if \(f\) is an even AG function or an odd AG function.
Let \(C_{\mathcal{S}}(\mathcal{A}, \mathcal{B})\) denote the conic hull of the set \begin{eqnarray*} \{f \in \mathbb{R}[\mathcal{A}, \mathcal{B}]) \ | \ f \textrm{ is an AG function and } f \ge 0 \textrm{ on } \mathbb{R}^n\}, \end{eqnarray*} which is a closed pointed convex cone. By definition, this cone generalizes and unifies sums of arithmetic-geometric mean exponentials (SAGE) and sums of non-negative circuit polynomials (SONC).
In the paper under review, the authors first show that every \(f \in C_{\mathcal{S}}(\mathcal{A}, \mathcal{B})\) can be decomposed into a sum of non-negative AG functions whose supports are contained in the support of \(f.\) Then the authors present a characterization of the dual cone of the cone \(C_{\mathcal{S}}(\mathcal{A}, \mathcal{B}).\) Consequently, the following facts are obtained:
\begin{enumerate}
\item a characterization of the extreme rays of the \(C_{\mathcal{S}}(\mathcal{A}, \mathcal{B}).\)
\item a characterization of a class of non-negative AG functions with simplex Newton polytopes.
\item an approximation theorem of nonnegative univariate polynomials in terms of SONC polynomials.
\end{enumerate}
The paper is clearly written and well organized. polynomials; sums of arithmetic-geometric mean exponentials; sums of non-negative circuit polynomials Real algebraic sets, Nonlinear programming, Convex sets in \(n\) dimensions (including convex hypersurfaces), Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) A unified framework of SAGE and SONC polynomials and its duality theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The \(p\)-loop amplitude of closed oriented bosonic strings in 26 dimensions is considered. The integration measure in this case is a measure on the moduli space \(\bar M_ p\) of Riemann surfaces of genus \(p\). It is proved that for \(p>1\) this measure is a squared modulus of a holomorphic function having a second order pole on degenerate surfaces, divided by \((\det N_ 1)^{13}\), \(N_ 1\) is a matrix of scalar products of holomorphic one-forms on a surface. This property fixes the measure uniquely; this can be derived either by a direct calculation, or using a theorem of Mumford. oriented bosonic strings in 26 dimensions A. Belavin and V. G. Knizhnik, ''Complex geometry and the theory of quantum strings'', Zh. Eksp. Teor. Fiz.,91, No. 2, 364--390 (1986). String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Moduli, classification: analytic theory; relations with modular forms Algebraic geometry and the geometry of quantum strings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For fields \(k\), the \emph{Brauer group} \(\operatorname{Br}(k)\) goes back to work of
Brauer, Albert and Noether.
Its elements are equivalence classes of \emph{central simple algebras} \(A\) of finite degree.
Addition and inverses come from tensor product and opposite algebras, and
the matrix algebras \(\operatorname{Mat}_n(k)\) are made trivial by the equivalence relation.
These algebras \(A\) can also be characterized as twisted forms
of matrix rings, and in this way generalize to algebras over arbitrary rings \(R\),
or sheaves of algebras over structure sheaves \(\mathscr{O}_X\). In this more general context, they are called \emph{Azumaya algebras}.
Brauer groups of fields, rings, or ringed spaces are truly foundational invariants that have striking applications in various
fields of mathematics. To mention a few: In group theory, they measure the obstruction to pass from a projective representation
to a linear representation. In algebraic number theory, they provide an elegant formulation of class field theory.
In algebraic geometry, they are directly related to twisted forms of projective \(n\)-spaces,
and frequently contain obstructions against existence of tautological objects for moduli problems.
In complex geometry, they describe the relation between algebraic and transcendental cycles.
In arithmetic geometry, they can be used to explain why certain schemes over numbers fields may or may not
contain rational points.
One of Grothendieck's many insights was that each Azumaya \(\mathscr{O}_X\)-algebra \(\mathscr{A}\) induces, via non-abelian cohomology, a class in \(H^2(X,\mathbb{G}_m)\),
which embeds the Brauer group into the cohomology group.
This is analogous to the interpretation of the Picard group, with two crucial differences:
For the Brauer group, one gets in general only an inclusion rather than an equality, and one
has to work with the étale topology.
This point of view, with its numerous ramifications and applications, was developed by Grothendieck in
a series of three highly influential papers [\textit{A. Grothendieck}, Adv. Stud. Pure Math. 3, 88--188 (1968; Zbl 0198.25901)].
Since then, several books appeared that treated certain aspects of the theory, for example [\textit{J. S. Milne}, Étale cohomology. Princeton, NJ: Princeton University Press (1980; Zbl 0433.14012)] and [\textit{P. Gille} and \textit{T. Szamuely}, Central simple algebras and Galois cohomology. Cambridge: Cambridge University Press (2006; Zbl 1137.12001)] and
[\textit{S. Gorchinskiy} and \textit{C. Shramov}, Unramified Brauer group and its applications. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1423.14002)].
The monograph of Colliot-Thelene and Skorobogatov is the first book entirely devoted to the subject ob Brauer groups.
It has a length about 450 pages and contains 16 chapters.
The first three chapters provide the necessary prerequisites from Galois cohomology and étale cohomology,
and introduce the Brauer group for schemes. The authors use the somewhat idiosyncratic notation
\(\operatorname{Br}(X)=H^2(X,\mathbb{G}_m)\), and call it the \emph{Brauer-Grothendieck group},
and write \(\operatorname{Br}_{\text{Az}}(X)\subset H^2(X,\mathbb{G}_m)\) for the group of classes of Azumaya algebras,
and here is referred to as the \emph{Brauer-Azumaya group}.
Section 4 discusses methods from \emph{stack theory}, which in the last two decades found some striking
applications to Brauer groups. Namely, one may view \(\alpha\in H^2(X,\mathbb{G}_m)\) as a gerbe \(\mathfrak{X}\) on \(X\),
interpret it as an Artin stack, and then study coherent sheaves \(\mathscr{E}\) on this Artin stack.
From this viewpoint, the Azumaya algebras on \(X\) representing \(\alpha\) are nothing but certain locally free
sheaves on \(\mathfrak{X}\). Using this approach, de Jong gave a proof of a result attributed to Gabber,
that the Brauer-Azumaya group is the torsion part in the Brauer-Grothendieck group, provided that
\(X\) is quasi-compact and admits an ample invertible sheaf. The book gives the first published
version of de Jong's arguments.
In Section 5, the authors concentrate on separated schemes \(X\) of finite type over a ground field \(k\),
which are called \emph{varieties}, and examine the interplay between
the geometric Brauer group \(\operatorname{Br}(X^s)\) obtained after passing to a separable closure,
the kernel \(\operatorname{Br}_1(X)\) for the base-change map \(\operatorname{Br}(X)\rightarrow\operatorname{Br}(X^s)\),
and the image \(\operatorname{Br}_0(X)\) for \(\operatorname{Br}(k)\rightarrow \operatorname{Br}(X)\).
The topic of Section 6 is birational invariance of Brauer groups,
and its relation to the \emph{ramified Brauer group} \(\operatorname{Br}_{\text{nr}}(K/k)\bigcap\operatorname{Br}(A)\),
where \(A\) runs through all discrete valuation rings with \(K=\operatorname{Frac}(A)\) and \(k\subset A\).
In Section 7 the relation to the Severi--Brauer varieties is discussed, which are twisted forms
of some \(\mathbb{P}^n\), whereas Section 8 discusses the contribution of singularities to the Brauer groups.
Section 9 discusses the results of Bogomolov and Saltman
about the unramified Brauer group of certain fields of invariants.
Section 10 contains various computations concerning schemes over local noetherian rings that are henselian.
In Section 11, the authors study more generally the Brauer groups arising
in connection with dominant morphisms \(f:X\rightarrow Y\) between integral schemes, and give a discussion
of the examples of \textit{M. Artin} and \textit{D. Mumford} [Proc. Lond. Math. Soc. (3) 25, 75--95 (1972; Zbl 0244.14017)] for unirational schemes that are not rational.
Such rationality questions are examined in more detail in Section 12, with a discussion
of the examples of \textit{B. Hassett} et al. [Acta Math. 220, No. 2, 341--365 (2018; Zbl 1420.14115)]
for families of fourfolds, where some members are rational, while others are not stably rational.
The sections 13--15 are devoted to a comprehensive presentation of the \emph{Brauer-Manin obstruction}
for algebraic varieties \(X\) over number fields \(k\).
They discuss in detail the set of adelic points \(X(\boldsymbol{A}_k)\) and the ensuing Brauer-Manin set \(X(\boldsymbol{A}_k)^{\operatorname{Br}}\),
which contains the closure of \(X(k)\). If the scheme \(X\) has empty Brauer-Manin set but admits adelic points
one says that there is a \emph{Brauer-Manin obstruction to the Hasse principle} for \(X\).
Density properties of \(X(k)\subset X(\boldsymbol{A}_k)^{\operatorname{Br}}\) lead to various notions
of weak approximation and strong approximations, coming with related obstructions, all of which is discussed in neat detail.
The role of Schinzel's Hypothesis is explained, and applications to rationally connected varieties and zero-cycles are studied.
The final Section 16 is devoted to the Tate Conjecture, mainly in the context of K3 surfaces and abelian varieties.
The volume closes with a comprehensive list of references, containing about 470 entries, and a short list of symbols.
There are no exercises.
The monograph is directed to researchers in algebraic and arithmetic geometry who use Brauer groups in one form or another,
and also to graduate students who want to learn about the topic and its applications.
The book gives a comprehensive, clear, up-to date
presentation of the theory, including most proofs. A particular strength is that it nicely
collects many results, examples and counterexamples from various areas of algebraic and arithmetic geometry
that are otherwise somewhat scattered in the literature.
Summing up, the book fills a wide gap and is a most welcome addition to the literature. Brauer groups Research exposition (monographs, survey articles) pertaining to algebraic geometry, Brauer groups of schemes The Brauer-Grothendieck group | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a compact quasi-smooth derived scheme \(M\) with \((- 1)\)-shifted cotangent bundle \(N\), there are at least two ways to localise the virtual cycle of \(N\) to \(M\) via torus and cosection localisations, introduced by \textit{Y. Jiang} and \textit{R. P. Thomas} [J. Algebr. Geom. 26, No. 2, 379--397 (2017; Zbl 1401.14221)]. We produce virtual cycles on both the projective completion \(\overline{N} : = \mathbb{P}(N \oplus \mathcal{O}_M)\) and projectivisation \(\mathbb{P}(N)\) and show the ones on \(\overline{N}\) push down to Jiang-Thomas cycles and the one on \(\mathbb{P}(N)\) computes the difference.
Using similar ideas we give an expression for the difference of the quintic and \(t\)-twisted quintic GW invariants of \textit{S. Guo}, \textit{F. Janda} and \textit{Y. Ruan} [``Structure of higher genus Gromov-Witten invariants of quintic 3-folds'', Preprint, \url{arXiv:1812.11908}]. virtual cycles; projective completions Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Virtual cycles on projective completions and quantum Lefschetz formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(R\) be a commutative Noetherian ring and let \(\mathrm{G}_0(R)\) be the Grothendieck group of finitely generated \(R\)-modules. Let \(\mathrm{H}(R)\) be the quotient of \(\mathrm{G}_0(R)\) by the subgroup generated by pseudo-zero modules i.e. modules \(M\) such that \(\text{ht}(\text{ann}(M))\geq 2\). The ambient Euclidean space in which majority of the work in this paper is carried out is \(\mathrm{H}(R)_{\mathbb{R}}:=\mathrm{H}(R)\otimes_{\mathbb{Z}}\mathbb{R}\), under the assumption that it is finite dimensional. The primary objects of study are \(\mathrm{C}(R)\) and \(\mathrm{C}_r(R)\), the convex cones in \(\mathrm{H}(R)_{\mathbb{R}}\) spanned by maximal Cohen-Macaulay (MCM) modules and MCM modules of rank \(r\) respectively. Topological properties of \(\mathrm{C}_r(R)\) are used to characterize the condition that \(R\) has only finitely many rank \(r\) MCM points (defined below) in \(\mathrm{C}_r(R)\). By viewing MCM modules of rank one as elements of the divisor class group \(\mathrm{Cl}(R)\), a direct approach to a conjecture of \textit{H. Dao} and \textit{K. Kurano} [Math. Ann. 364, No. 3--4, 713--736 (2016; Zbl 1346.13020)] is also considered. \par The motivation for this work is as follows. \textit{K. Kurano} [Invent. Math. 157, No. 3, 575--619 (2004; Zbl 1070.14007)] introduced and developed the notion of Grothendieck groups modulo numerical equivalence, denoted \(\overline{\mathrm{G}_0(R)}\). \textit{C-Y. J. Chan} and \textit{K. Kurano} [Trans. Am. Math. Soc. 368, No. 2, 939--964 (2016; Zbl 1342.13016) ] introduce and explore the Cohen-Macaulay cone \(\mathrm{C}_{\mathrm{CM}}(R)\), which is defined as the convex cone spanned by MCM modules in \(\overline{\mathrm{G}_0(R)}\otimes_{\mathbb{Z}}\mathbb{R}\). When \(R\) is a Cohen-Macaulay local ring of dimension at most three, the canonical map from \(\mathrm{G}_0(R)\) to \(\overline{\mathrm{G}_0(R)}\) factors through \(\mathrm{H}(R)\). This connection motivates the study of the objects in consideration in this paper. \par Section 2 of the paper deals with establishing properties of the group \(\mathrm{H}(R)\) and the space \(\mathrm{H}(R)_{\mathbb{R}}\), while section 3 proves various results concerning the interior, closure and boundary of \(\mathrm{C}(R)\). Section 4 is motivated by the open question as to whether \(\mathrm{C}_{\mathrm{CM}}(R)\) is polyhedral. In Theorem 4.1, the author considers the polyhedrality of \(\mathrm{C}(R)\) and under certain technical assumptions constructs a ``large'' convex polyhedral subcone of \(\mathrm{C}(R)\) when a module finite extension of \(R\) has a polyhedral cone. The author shows that the result applies to Gorenstein normal local rings possessing a simple (ADE) simgularity as a finite extension. \par Sections 5 and 6 approach the aforementioned conjecture of Dao-Kurano, which says that under mild hypothesis a Cohen-Macaulay normal local ring has only finitely many rank one MCM modules up to isomorphism. In Theorem 5.4, the author gives characterizations of the existence of only finitely many rank \(r\) MCM points in \(\mathrm{C}_r(R)\) in terms of the topological properties of \(\mathrm{C}_r(R)\). Here a MCM point of rank \(r\) refers to a point represented by a rank \(r\) MCM module. Section 6 approaches the conjecture more directly by giving sufficient conditions for the finiteness of the number of MCM points on the line defined by a reflexive ideal in \(\mathrm{Cl}(R)\), when \(R\) is Gorenstein. More precisely, Theorem 6.8 says that if \(R\) is Gorenstein local of dimension at least two with an isolated singularity and \(I\) is a non-zero reflexive ideal of \(R\) satisfying certain conditions on cohomological dimension or asymptotic depth, there exist only a finite number of MCM points on \(\mathbb{Z}I\) in \(\mathrm{Cl}(R)\). The author gives several examples where the hypothesis of Theorem 6.8 is satisfied. On the other hand, Theorem 6.14 assumes \(R\) is a Gorenstein normal local ring and determines the MCM points on the line defined in \(\mathrm{Cl}(R)\) by a height one, rigid ideal such that \(R/I\) is Gorenstein. asymptotic depth; Cohen-Macaulay cone; cohomological dimension; complexity; convex cone; divisor class group; finite/countable Cohen-Macaulay representation type; Grothendieck group; intersection multiplicity; maximal Cohen-Macaulay module/point; numerical equivalence; polyhedral; strongly convex Cohen-Macaulay modules, Class groups, Grothendieck groups, \(K\)-theory and commutative rings, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Grothendieck groups, convex cones and maximal Cohen-Macaulay points | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum cohomology ring of a smooth projective variety is a very important tool in the study of the enumerative geometry of \(X\): by using the associative structure in the ring one can compute its Gromov-Witten invariants and, in some cases, calculate the number of rational curves in \(X\) which meet prescribed subvarieties. In [An. Acad. Bras. Ciênc. 73, No. 3, 319--326 (2001; Zbl 1075.14521)] and [Compos. Math. 140, No. 1, 165--178 (2004; Zbl 1050.14054)], \textit{J. Kock} defined a generalization of the quantum cohomology ring of \(X\), namely the tangency quantum cohomology ring of \(X\), together with a tangency quantum product, which encodes enumerative geometry of the rational curves of \(X\) passing through certain subvarieties of \(X\) with prescribed tangencies.
In the paper under review the authors further generalize this construction by defining a \(d\)th-order contact product that takes into account higher powers of tautological classes to be prescribed, and thus encoding higher-order contact phenomena. The main technical result in the paper is the proof of the associativity of the \(d\)th-order contact quantum product, for which the authors use formal properties of the virtual fundamental class. The authors then show that this product coincide's with Kock's tangency product for \(d=1\). The applications of the contact product to enumerative questions that the authors have in mind would extend e.g. ideas of a previous work joint with \textit{L. Ernström} in [J. Algebr. Geom. 10, No. 2, 365--396 (2001; Zbl 1014.14009)], but the present paper is confined to the construction of the ring and to proving its associativity. Deligne-Mumford stack; Moduli space; Quantum cohomology; Stable map; Virtual fundamental class Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Fine and coarse moduli spaces Tangential quantum cohomology of arbitrary order | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We find generating functions for the Poincaré polynomials of hyperquot schemes for all partial flag varieties. These generating functions give the Betti numbers of hyperquot schemes, and thus give dimension information for the cohomology ring of every hyperquot scheme. This can be viewed as a step towards understanding a presentation and the structure of the cohomology rings.
Let
\({\mathbf F}(n;{\mathbf s})\) denote the partial flag variety corresponding to flags of the form
\[
V_1\subset V_2\subset \dots\subset V_l\subset V=\mathbb{C}^n; \quad \dim V_i=s_i.
\]
It is classically known that its Poincaré polynomial \({\mathcal P}({\mathbf F}(n;{\mathbf s}))=\sum_Mb_{2M}({\mathbf F}(n;{\mathbf s}))z^M\) is equal to the following generating function for the Betti numbers of the partial flag variety:
\[
{\mathcal P}\bigl({\mathbf F}(n:{\mathbf s})\bigr)= \sum_Mb_{2M}({\mathbf F})z^M=\frac {\prod^n_{i=1}(1-z^i)}{\prod^{l+1}_{j=1}\prod^{s_j-s_{j-1}}_{i=1}(1-z^i)}
\]
with \(s_{l+1}:=n\) and \(s_0:=0\). Defining \(f_k^{i,j}:=1-t_i \dots,t_jz^k\), the main result is:
Theorem 1.
\[
\sum_{d_1,\dots,d_l}{\mathcal P}\biggl( {\mathcal H}{\mathcal Q}_{\mathbf d}\bigl({\mathbf F}(n;{\mathbf s})\bigr)\biggr) t_1^{d_1}\dots t_l^{d_l}=\;{\mathcal P} \bigl({\mathbf F}(n;{\mathbf s})\bigr)\cdot\prod_{1\leq i\leq j\leq l}\prod_{s_{i-1}<k\leq s_i} \left( \frac{1}{f^{i,j}_{s_j-k}}\right) \left(\frac{1}{f^{i,j}_{s_{j+1}-k+1}} \right).
\]
Chen, L, Poincaré polynomials of hyperquot schemes, Math Ann, 321, 235-251, (2001) Grassmannians, Schubert varieties, flag manifolds, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Parametrization (Chow and Hilbert schemes) Poincaré polynomials of hyperquot schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{F. C. Kirwan} [Cohomology of quotients in symplectic and algebraic geometry. Princeton, New Jersey: Princeton University Press (1984; Zbl 0553.14020)] studied the map from the equivariant cohomology of a Hamiltonian group action to the cohomology of the symplectic quotient. The paper under review deals with the quantum version of this situation. Let \(X\) be a smooth projectively embedded variety with a connected reductive group action such that stable locus is equal to the semistable locus.
Let \(\mathrm{QH}(X/\!/G)\) be the quantum cohomology of the GIT quotient \(X/\!/G\), and let \(\mathrm{QH}_G(X)= H_G(X)\otimes \Lambda_X^G\) denote the equivariant quantum cohomology where \( \Lambda_X^G \subset \Hom(H_2^G(X,\mathbb{Z}))\) is Novikov field. Givental equipped the latter with a product structure arising from the equivariant Gromov-Witten theory. \textit{E. Gonzalez} and and the author of the paper under review [Math. Z. 273, No. 1--2, 485--514 (2013; Zbl 1258.53092)] proved that, under some conditions, the moduli space of stable gauged maps from a smooth connected projective curve is a proper Deligne-Mumford stack equipped with a perfect obstruction theory which leads to the definition of the gauged Gromov-Witten invariants.
The first result of the paper under review is the constuction of ``Quantum Kirwan Morphism''
\[
\kappa_X^G:\mathrm{QH}_G(X)\to \mathrm{QH}(X/\!/G)
\]
by means of the virtual integration over a compactified stack of affine gauged maps. \(\kappa_X^G\) is a morphism of CohFT algebras, which can be considered as a non-linear generalization of an algebra homomorphism.
The second main result of the paper under review relates in the large area limit the graph potentials via the quantum Kirwan morphism.
More precisely, suppose that \(C\) is a smooth projective curve such that all the semistable gauged maps from \(C\) to \(X\) are stable for sufficiently large stability parameters. Then
\[
\tau_{X/\!/G}\circ\kappa^G_X=\lim_{\rho\to \infty}\tau_X^G
\]
where \(\rho\) is the stability parameter, \(\tau_{X/\!/G}\) genus zero graph potential for the GIT quotient \(X/\!/G\), and \(\tau^G_X\) is the gauged Gromov-Witten potential. The paper under review also proves the localized version of this theorem that arises as the fixed point contributions for a circle acting on the domain. This localization in the case of Gromov-Witten invariants gives rise to a solution for a version of the Picard-Fuchs quantum differential equation for \(X/\!/G\).
Some of the results have overlaps and connections with the works of Givental, Lian-Liu-Yau, Iritani, Ciocan-Fontanine-Kim-Maulik, and Coates-Corti-Iritani-Tseng (see also the references of this paper) who have taken different approaches. quantum cohomology; Kirwan morphism C. Woodward, \textit{Quantum Kirwan morphism and Gromov-Witten invariants of quotients III}, preprint. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum Kirwan morphism and Gromov-Witten invariants of quotients. III | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, using Lusztig's Frobenius maps for quantum groups at roots of unity, the authors give a representation-theoretic proof of the Frobenius-split property of Schubert varieties in the generalized flag varieties. This paper makes an important contribution to the geometry and representation theory of flag varieties. quantum groups; Frobenius Burns, D.; Levenberg, N.; Ma'u, S.; Revesz, Sz., Monge-Ampère measures for convex bodies and Bernstein-Markov type inequalities, Trans. Amer. Math. Soc., 362, 6325-6340, (2010) Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations, Cohomology theory for linear algebraic groups Algebraization of Frobenius splitting via quantum groups. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we study the structure of the quantum cohomology ring of a projective hypersurface with nonpositive first Chern class. We prove a theorem which suggests that the mirror transformation of the quantum cohomology of a projective Calabi-Yau hypersurface has a close relation with the ring of symmetric functions, or with Schur polynomials. With this result in mind, we propose a generalized mirror transformation on the quantum cohomology of a hypersurface with negative first Chern class and construct an explicit prediction formula for three-point Gromov-Witten invariants up to cubic rational curves. We also construct a projective space resolution of the moduli space of polynomial maps, which is in good correspondence with the terms that appear in the generalized mirror transformation. quantum cohomology ring; hypersurface; mirror transformation; Calabi-Yau hypersurface; Schur polynomials; Gromov-Witten invariants Jinzenji, M., On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation, Int. J. Mod. Phys., A15, 1557-1596, (2000) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Hypersurfaces and algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects) On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper generalizes Noether's theorem: If \(A\) is affine then \(A^G\) is affine also, where \(A\) is a commutative algebra and \(G\) is a finite group of automorphisms acting on \(A\). This theorem has been generalized to actions of any finite dimensional cocommutative Hopf algebra \(H\) on a commutative algebra \(A\), and the third author has shown that cocommutativity of \(H\) can be replaced by semisimplicity of \(H\). In this paper the authors follow their philosophy that states that often properties that hold for a cocommutative Hopf algebra \(H\) and a commutative \(H\)-module \(A\) should hold for appropriate generalizations: a triangular Hopf algebra \((H,R)\) and a quantum-commutative \(H\)-module \(A\) (i.e. \(A\) is commutative in the category of \(H\)-modules). Indeed, they generalize the theorem to this set-up, and also to its ``dual'': \((H,\langle \mid \rangle)\) a cotriangular Hopf algebra and \(A\) a quantum-commutative \(H\)-comodule algebra.
In order to prove the theorem the authors construct a new, non-commutative, determinant function for each of the cases mentioned above. This construction involves the action of the symmetric group that is defined by the symmetric braiding of the twist map in the category of \(H\)-modules; this gives rise to a kind of Grassmann algebra. The determinant is also computed explicitly for some examples of group gradings. affine algebras; actions of finite dimensional cocommutative Hopf algebras; Noether's theorem; finite groups of automorphisms; triangular Hopf algebras; quantum-commutative modules; non-commutative determinant functions; symmetric braidings; twist maps; categories of modules; Grassmann algebras; group gradings Cohen, M.; Westreich, S.; Zhu, S., Determinants, integrality and Noether's theorem for quantum commutative algebras, Israel J. math., 96, 185-222, (1996) Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Automorphisms and endomorphisms, Geometric invariant theory, Determinants, permanents, traces, other special matrix functions Determinants, integrality and Noether's theorem for quantum commutative algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This book is at the same time an introduction, addressed to sophisticated mathematicians, to a purely algebro-geometric theory of \(D\)-modules, and a summary of the main results, most of which due to the author, in that field. The necessary and sufficient prerequisite requested of the reader, aside from basic experience with algebraic and analytic sheaves, is a good knowledge of homological algebra, and familiarity with the formalism and basic properties of derived categories. Relying on this background, the proofs are complete, with two exceptions: Kashiwara's constructibility theorem and the theorem of faithful flatness of the ring of differential operators of infinite order over the ring of differential operators (due to Sato-Kawai-Kashiwara). The original micro-differential proofs of those results, did not in fact fit into the spirit of this book: proofs based purely on the algebro-geometric formalism of \(D\)-modules were later provided by the author, but only the first could be included, as a final note (with \textit{L. Narváez-Macarro}) in the present book.
According to Grothendieck's philosophy, \(D\)-modules are regarded in this book as general coefficients for the cohomology of smooth analytic or algebraic varieties over the complex numbers. (A Monsky-Washnitzer type theory of \(D\)-modules is, however, also in the author's mind, as well as a generalization to singular varieties.) On the model of the theory of quasi-coherent coefficients over a scheme, developed in \textit{R. Hartshorne's} book ``Residues and duality'' [Berlin etc.: Springer-Verlag (1966; Zbl 0212.26101) and of the theory of discrete coefficients contained in \(SGA 4 and 5\) (SGA \(=\) Sémin. Géom. Algébr.), the author establishes in the first chapter of this book a complete formalism for the present type of coefficients including ``les six opérations de Grothendieck'' for a smooth morphism \(f: X\to Y,\) and Grothendieck's local and global duality. In particular, the global duality results here obtained include as special cases the most general known formulations of Serre and Poincaré dualities. While the author makes a point of treating the algebraic and analytic situations in parallel, some asymmetries still remain: only the hypothesis of existence of a good global filtration allows one to prove the coherence of the direct image via a proper morphism of a coherent \(D\)-module in the analytic case (see section \(I\quad 5.4).\) In the algebraic case that hypothesis is automatically verified.
The main result of the second chapter is the interpretation of discrete constructible coefficients in terms of regular holonomic \(D\)-module coefficients. This is ``Mebkhout's equivalence'' between the derived category of complexes of \(D\)-modules with bounded regular-holonomic cohomology and the derived category of complexes of sheaves of vector spaces with bounded constructible cohomology. This equivalence, obtained via the (contravariant) solution functor, and the dual equivalence via the (covariant) de Rham functor, represent the widest known generalization of the so-called Riemann-Hilbert correspondence; they are compatible with Grothendieck's operations and are interchanged by duality (in any of the two categories). The previous statements hold in both the analytic and the algebraic case; their proofs depend strongly on Hironaka's resolution of singularities, since they rely on Deligne's work [cf. \textit{P. Deligne}, ``Équations différentielles à points singuliers réguliers'', Lect. Notes Math. 163. Berlin etc.: Springer Verlag (1970; Zbl 0244.14004)] on connections with regular singularities. All of the known comparison theorems between algebraic and complex-analytic cohomology follow here from more general statements of equivalence between several seemingly independent definitions of regularity (the author \textit{proves} that the structural sheaf is a regular \(D\)-module!), together with their behaviour under direct images. Here, relevant tools are the local algebraic cohomology of analytic \(D\)-modules, and the functor of their solutions in formal functions along a closed analytic subspace. The author has recently been able to free all of the previous results of dependence upon Hironaka's resolution of singularities (see the author's article in [Publ. Math., Inst. Hautes Étud. Sci. 69, 47--89 (1989; Zbl 0709.14015)], opening the way to an extension of the theory to schemes in positive characteristic: it is clear that, if a new systematic treatment of the theory of D-modules were to be written, this new approach by the author should be followed.
These main results are complemented by a variety of topics: the problem of Cauchy-Kowalewski in the \(D\)-module setting, the study of \(D\)-modules over a 1-dimensional complex disk, the characterization of perverse sheaves and finally the theory of vanishing cycles (in the sense \(of SGA 7)\). Considerable attention is devoted to this last topic in chapter three, where the author and \textit{C. Sabbah}, show how to recover the properties of the \(V\)-filtration of Fuchs-Malgrange-Kashiwara, from the general results on \(D\)-modules obtained in the first two chapters.
We found this book extremely interesting, rich and pleasant, and recommend its reading to any mathematician possessing the necessary background. local duality; D-modules; derived categories; global duality; Mebkhout's equivalence; Riemann-Hilbert correspondence; local algebraic cohomology of analytic D-modules; problem of Cauchy-Kowalewski; perverse sheaves; vanishing cycles Mebkhout, Z., Le formalisme des six opérations de Grothendieck pour les DX-modules cohérents, Travaux en Cours, vol. 35, (1989), Hermann: Hermann Paris, With supplementary material by the author and L. Narváez Macarro Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, de Rham cohomology and algebraic geometry, Sheaves of differential operators and their modules, \(D\)-modules, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Duality theorems for analytic spaces Differential systems. The formalism of the six Grothendieck operations for coherent \(D_X\)-modules) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A symmetric polynomial \(q(x)\) in non-commutative variables \(x=\{x_{1},\dots,x_{n}\}\) is called matrix-positive if whenever the variables \(\{x_{1},\dots,x_{n}\}\) are replaced by matrices of any size, the resulting polynomial is positive semidefinite. In [\textit{J. Helton}, Ann. Math. (2) 156, 675--694 (2002; Zbl 1033.12001)] it was proved that a symmetric non-commutative polynomial is matrix-positive if and only if it is a sum of squares. Roughly speaking, the paper under review provides a weighted sum square representation for any non-commutative polynomial \(q(x)\) that is ``strictly positive'' on a bounded ``semi-algebraic'' set \(D_{\mathcal{P}}\) defined by a collection \(\mathcal{P}\) of such polynomials.
More precisely, given a collection \(\mathcal{P}\) of symmetric polynomials in non-commutative variables \(x=\{x_{1},\dots,x_{n}\}\) and a real Hilbert space \(\mathcal{H}\), denote by \(D_{\mathcal{P}}(\mathcal{H)}\) the set of tuples \(X=(X_{1},\dots,X_{n})\) where each \(X_{j}\) is an operator on \(\mathcal{H}\) and \(p(X_{1},\dots,X_{n})\succeq0\). Then we call the domain of positivity associated to \(\mathcal{P}\) the collection \(D_{\mathcal{P}}\) of all tuples \(X\in D_{\mathcal{P}}(\mathcal{H)}\) and all Hilbert spaces \(\mathcal{H}\). The domain of positivity \(D_{\mathcal{P}}\) is bounded, if there exists a constant \(c>0\) such that for any Hilbert space \(\mathcal{H}\), any \(X=(X_{1},\dots,X_{n})\in D_{\mathcal{P}}(\mathcal{H)}\) and any \(j\in\{1,\dots,n\}\), we have \(\| X_{j}\|\leq c\). The authors prove the following ``Positivstellensatz'' result: let \(\mathcal{P}\) be a collection of symmetric polynomials with a bounded domain of positivity \(D_{\mathcal{P}}\). If a polynomial \(q\) is strictly positive on \(D_{\mathcal{P}}\), it can be represented as a weighted sum of squares:
\[
q=\sum_{j=1}^{N}s_{j}^{T}p_{j}s_{j}+\sum_{k=1}^{M}r_{k}^{T}r_{k}+\sum_{m,\ell }t_{m,\ell}^{T}(C^{2}-x_{m}^{2})t_{m,\ell}
\]
for a finite number of polynomials \(p_{j}\in P\) and polynomials \(s_{j} ,r_{k},t_{m,l}\). The proof uses a Hahn-Banach separation argument. When the domain of positivity \(D_{\mathcal{P}}\) is ``convex'' (meaning that for every \(\mathcal{H}\), the set \(D_{\mathcal{P}}(\mathcal{H)}\) is convex), the Hilbert spaces (respectively, the operators) involved above can be taken to be finite-dimensional (respectively, matrices). Versions of the Positivstellensatz are presented for three classes of matrix-valued non-commutative polynomials, since the authors anticipate potential applications.
The paper finishes with a discussion on the real Nullstellensatz, and shows by means of an example that a non-commutative version is not possible along certain lines.
The paper is written in a clear and instructive way; in particular, it contains several examples and useful explanations which make it accessible to non-specialists. non-commutative polynomial; Positivstellensatz; operator positivity; semi-algebraic set J. W. Helton and S. McCullough, \textit{A Positivstellensatz for non-commutative polynomials}, Trans. Amer. Math. Soc., 356 (2004), pp. 3721--3737. Several-variable operator theory (spectral, Fredholm, etc.), Semialgebraic sets and related spaces A positivstellensatz for non-commutative polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is an expanded version of the third author's lecture in String-Math 2015 at Sanya. It summarizes some of our works in quantum cohomology.
After reviewing the quantum Lefschetz and quantum Leray-Hirsch, we discuss their applications to the functoriality properties under special smooth flops, flips and blow-ups. Finally, for conifold transitions of Calabi-Yau 3-folds, formulations for small resolutions (blow-ups along Weil divisors) are sketched. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects), Variation of Hodge structures (algebro-geometric aspects) Quantum cohomology under birational maps and transitions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Chern-Schwartz-MacPherson classes generalize the Chern classes of the tangent bundle of a variety to the case of a singular variety. Technically, the existence of a natural transformation is proved, from the constructible function functor to the homology functor. When the variety is smooth, the image of the function 1 is the (homology) total Chern class of the tangent bundle. The present paper studies the equivariant setting of the same theorem. It is proved that the equivariant version of the Chern-Schwartz-MacPherson transformation also exists, with similar properties. The technique of the author is based on the Totaro-Edidin-Graham version of algebraic Borel construction of classifying spaces. As an application, the connection with the theory of Thom polynomials of singularities is discussed. Ohmoto's classes imitate tangent Chern classes, and Thom polynomials imitate normal Euler classes of singular varieties with rich symmetries. The expected connections between the two theories is explored in Section 4. In Section 5 Ohmoto's Chern classes are applied in the setting of symmetric products of possibly singular varieties. Using the Dey-Wohlfahrt formula, the author presents the generating function of orbifold Chern homology classes of symmetric products. Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Singularities of differentiable mappings in differential topology Chern classes and Thom polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials See the joint review of the full volume [\textit{J. Lipman} and \textit{M. Hashimoto}, Foundations of Grothendieck duality for diagrams of schemes. Berlin: Springer (2009; Zbl 1163.14001)]. Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Derived categories, triangulated categories Notes on derived functors and Grothendieck duality | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In earlier work [Proc. Lond. Math. Soc. (3) 113, No. 2, 185--212 (2016; Zbl 1375.13033)], the same three authors explained how the cluster algebra structure on the homogeneous coordinate ring of a Grassmannian, as described by \textit{J. Scott} [Proc. Lond. Math. Soc., III. Ser. 92, No. 2, 345--380 (2006; Zbl 1088.22009)], is additively categorified by the category of maximal Cohen-Macaulay modules over a certain Gorenstein order.
Among other properties reflecting the combinatorics of the Grassmannian cluster algebra, this category has `rank \(1\)' indecomposable objects in bijection with Plücker coordinates, which are Ext-orthogonal if and only if the corresponding Plücker labels are non-crossing. This leads to a bijection between the cluster-tilting objects of the category (or at least those mutation equivalent to one for which all indecomposable summands have rank \(1\)) and the seeds in the Grassmannian cluster algebra. This bijection is compatible with mutation, and the quiver of a seed can be computed as the Gabriel quiver of the endomorphism algebra of the corresponding cluster-tilting object.
The Grassmannian coordinate ring admits a natural quantisation, and \textit{J. E. Grabowski} and \textit{S. Launois} [Proc. Lond. Math. Soc. (3) 109, No. 3, 697--732 (2014; Zbl 1315.13036)] show that the cluster algebra structure can be quantised compatibly. In the present paper, the authors show that additional quantum information -- precisely, the powers of \(q\) appearing in quasi-commutation relations among quantum cluster variables -- may be computed from the same category of Cohen-Macaulay modules which categorifies the unquantised cluster structure. Thus this same category, with no modification such as passing to graded modules, categorifies the quantum Grassmannian cluster algebra.
For rank \(1\) modules, the \(\kappa\)-invariant, which the authors introduce to compute the quasi-commutation power of the corresponding Plücker coordinates, may also be expressed as an invariant MaxDiag of a pair of Young diagrams, as in work of \textit{K. Rietsch} and \textit{L. Williams} [Duke Math. J. 168, No. 18, 3437--3527 (2019; Zbl 1439.14142)].
We note that while this paper yields a categorical interpretation of the quasi-commutation data (or \(L\)-matrix) for each seed in the quantum Grassmannian cluster algebra, it does not provide a quantum cluster character, computing expressions for quantum cluster variables as quantum Laurent polynomials in a chosen initial seed. This is well-known to be a difficult problem, especially for cluster algebras without acyclic seeds, as is the case for most Grassmannians, and for now remains open. quantum Grassmannian; quantum cluster algebra; categorification Cluster algebras, Grassmannians, Schubert varieties, flag manifolds, Cohen-Macaulay modules in associative algebras, Quantum groups (quantized function algebras) and their representations Categorification and the quantum Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field \(k\). We also compute the Grothendieck group of the category of \(A\)-isotypic abelian varieties, for any simple abelian variety \(A\), assuming \(k\) has characteristic 0, and for any elliptic curve \(A\) in any characteristic. abelian varieties; Grothendieck groups; elliptic curves Abelian varieties of dimension \(> 1\), Isogeny, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grothendieck groups and \(K_0\) Grothendieck groups of categories of abelian varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials As is known [see \textit{R. Benedetti} and \textit{M. Dedò}, Compos. Math. 53, 143--151 (1984; Zbl 0547.14019)], if \(X\) is a compact nonsingular real algebraic set, and \(v\) a cohomology class in \(H^2(X; \mathbb{Z}/2)\) whose Poincaré dual homology class can be represented by an algebraic subset of \(X\), then there exists an algebraic vector bundle \(\xi\) on \(X\) such that \(w_1(\xi)= 0\) and \(w_2(\xi)= v\), where \(w_i(\xi)\) is the \(i\)th Stiefel-Whitney class of \(\xi\). The goal of the present article is to give a self-contained topological proof of the statement formulated above. Bochnak J., Kucharz W.: A topological proof of the Grothendieck formula in real algebraic geometry. Enseign. Math. 48, 237--258 (2002) Topology of real algebraic varieties A topological proof of the Grothendieck formula in real algebraic geometry. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of \({\mathbb{Z}}\) generated by the class of a point. This clarifies the extent to which the graph hypersurfaces `generate the Grothendieck ring of varieties': while it is known that graph hypersurfaces generate the Grothendieck ring over a localization of \({\mathbb{Z}[\mathbb{L}]}\) in which \({\mathbb{L}}\) becomes invertible, the span of the graph hypersurfaces in the Grothendieck ring itself is nearly killed by setting the Lefschetz motive \({\mathbb{L}}\) to zero. In particular, this shows that the graph hypersurfaces do \(not\) generate the Grothendieck ring prior to localization. The same result yields some information on the mixed Hodge structures of graph hypersurfaces, in the form of a constraint on the terms in their Deligne-Hodge polynomials. These observations are certainly not surprising for the expert reader, but are somewhat hidden in the literature. The treatment in this note is straightforward and self-contained. graph hypersurfaces; Grothendieck ring; stable birational equivalence Aluffi P., Marcolli M.: Graph hypersurfaces and a dichotomy in the Grothendieck ring. Lett. Math. Phys. 95, 223--232 (2011) (Equivariant) Chow groups and rings; motives, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Perturbative methods of renormalization applied to problems in quantum field theory, Rationality questions in algebraic geometry, Grothendieck groups, \(K\)-theory, etc. Graph hypersurfaces and a dichotomy in the Grothendieck ring | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The goal of this paper is twofold. First, to provide a self contained, detailed and rigorous mathematical introduction to some aspects of the quantum error-correcting codes and especially quantum stabilizer codes and their connection to self-orthogonal linear codes. This has been done without venturing that much, if at all, into the world of physics. While most of the results presented are not new, it is not easy to extract a precise mathematical formulation of results and to provide their rigorous proofs by reading the vast number of papers in the field, quite a few of which are written by computer scientists or physicists. It is this formulation and proofs, some of which are new, that we present here. Techniques from algebra of finite fields as well as representations of finite abelian groups have been employed. The second goal is the construction of some stabilizer codes via self-orthogonal linear codes associated to algebraic curves. linear codes; selft-orthogonal codes; Goppa codes; quantum error-correcting codes; stabilizer codes; super-elliptic curves Quantum coding (general), Geometric methods (including applications of algebraic geometry) applied to coding theory, Elliptic curves over global fields, Applications to coding theory and cryptography of arithmetic geometry, Arithmetic ground fields for curves Error correcting quantum codes and algebraic curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author proves the base change for relative dualizing sheaves: For \(X\), \(Y\) noetherian schemes and \(f:X\to Y\) a Cohen-Macaulay map of finite type, if \(f:X'\to X\) is a base change of \(f\) under \(g:Y'\to Y\) (\(Y'\) also noetherian), then there is a canonical isomorphism \(\theta ^f_g:{g'}^*\omega_f\cong \omega_{f'}\), \(\omega_f\), \(\omega_{f'}\) being the relative dualizing sheaves.
In contrast to the book of \textit{B. Conrad} [``Grothendieck duality and base change'', Lect. Notes Math. 1750 (Springer Verlag) (2000; Zbl 0992.14001)], this paper does not make use of dualizing and residual complexes, being based on the approach of Deligne and Verdier to Grothendieck duality. relative dualizing sheaf; residues doi:10.1112/S0010437X03000654 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Schemes and morphisms, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Derived categories, triangulated categories Base change and Grothendieck duality for Cohen-Macaulay maps | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review the authors study different equivalent relations in the category of smooth projective varieties over a field \(\mathbb{k}\). Firstly, two varieties \(X\) and \(Y\) are said \(D\)-equivalent when their bounded derived categories of coherent sheaves \(D(X)\) and \(D(Y)\) are isomorphic. Naively one could expect that \(D\)-equivalent projective varieties have the same class in the Grothendieck ring \(K_0(\mathrm{Var}/\mathbb{k})\). As examples 1.4 and 1.5 show, this is not the case. Prompted by this fact, the authors define a weaker relation in \(K_0(\mathrm{Var}/\mathbb{k})\): they define two projective varieties \(X\) and \(Y\) to be \(L\)-equivalent whenever
\[
([X]-[Y]).\mathbb{L}^r=0
\]
in \(K_0(\mathrm{Var}/ \mathbb{k})\) for \(r\geq 0\) and where \(\mathbb{L}=[\mathbb{P}^1]-[\mathrm{Spec}(\mathbb{k})]\) is the class of the affine line. Then the authors conjectured that for simply connected smooth projective varieties \(D\)-equivalence implies \(L\)-equivalence (Conjecture 1.6). It is important to point out that the assumption of simple connectedness is necessary as it is showed by examples of \(D\)-equivalent complex abelian varieties constructed by A. Efimov and K. Ueda that are not \(L\)-equivalent.
One of the main results of this paper is Theorem 1.9, where a new example of \(L\)-equivalence between \(D\)-equivalent varieties is discussed: let \(X\) be a \(K3\) surface given as the complete intersection of three quadrics in \(\mathbb{P}^5\) over a field \(\mathbb{k}\) of characteristict \(\neq 2\) and let us assume that the corresponding double cover \(Y\rightarrow \mathbb{P}^2\) is smooth and the corresponding Brauer class \(\alpha_Y\in Br(Y)\) of \(Y\) is zero. The authors show that if \(X\) has a \(\mathbb{k}\)-point not on a line contained in \(X\) then \(X\) and \(Y\) are D-equivalent and
\[
([X]-[Y]).\mathbb{L}=0
\]
whereas the \(L\)-equivalence is in general non-trivial (i.e., \([X]\neq[Y]\) in \(K_0(\mathrm{Var}/\mathbb{k})\)). Grothendieck ring of varieties; D-equivalence; L-equivalence Quadratic spaces; Clifford algebras, \(K3\) surfaces and Enriques surfaces, Fibrations, degenerations in algebraic geometry, Sheaves in algebraic geometry Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We propose a new point of view on quantum cohomology, motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is a \(D\)-module which ``quantizes'' a commutative algebra associated to the (uncompactified) space of rational curves. Under appropriate conditions, we show that the associated flat connection may be gauged to the flat connection underlying quantum cohomology. This method clarifies the role of the Birkhoff factorization in the ``mirror transformation'', and it gives a new algorithm (requiring construction of a Gröbner basis and solution of a system of o.d.e.) for computation of the quantum product. Quantum cohomology; \(D\)-module; Birkhoff factorization M A Guest, Quantum cohomology via \(D\)-modules, Topology 44 (2005) 263 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology via \(D\)-modules | 0 |
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