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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 The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for \({\mathfrak{sl}_2}\). The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra \({\mathfrak{gl}_n}\). For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed.We also give a categorical version of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules. Categorification; quantum groups; Lie algebras; canonical bases; flag varieties I. Frenkel, M. Khovanov and C. Stroppel, \textit{A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products}, \textit{Selecta Math. (N.S.)}\textbf{12} (2006) 379431 [math/0511467]. Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups (quantized function algebras) and their representations, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Representations of associative Artinian rings A categorification of finite-dimensional irreducible representations of quantum \({\mathfrak{sl}_2}\) and their tensor products | 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 representation of a group \(G\) on the vector space \(V\), invariant varieties \(Y \subset V\) define equivariant cohomology classes \([Y] \in H^*_G(V)=H^*(BG)\). One key example is when \(V\) is the vector space of map germs \((C^n,0) \to (C^p,0)\) and \(Y\) is a certain collection of these germs -- called a singularity. In this context
\[
[Y]\in H^*\left(B(GL(n,C)\times GL(p,C)\right) = Z[a_1,\ldots,a_n,b_1,\ldots,b_p]
\]
is called the Thom polynomial of the singularity \(Y\). Thom polynomials govern the global behavior of singularities; namely they express cohomology classes represented by singularity submanifolds.
\textit{W. Fulton} and \textit{R. Lazarsfeld} [Positive polynomials for ample vector bundles, Ann. Math. (2) 118, 35--60 (1983; Zbl 0537.14009)] considered equivariant classes \([Y]\in H^*(BG)\) when \(Y\) is a cone, and showed certain positivity properties. The present paper applies these results to the Thom polynomial setting and obtains the following theorem. The Thom polynomial of a ``stable'' singularity, when expressed in Schur functions of the quotient variables \(c_i\) (\(\sum c_it^i = \sum b_i t^i / \sum a_i t^i\)), has nonnegative coefficients. Thom polynomial; Schur functions; numerical positivity P. Pragacz and A. Weber, ''Positivity of Schur function expansions of Thom polynomials,'' Fund. Math., vol. 195, iss. 1, pp. 85-95, 2007. Symmetric functions and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classical problems, Schubert calculus, Homology of classifying spaces and characteristic classes in algebraic topology, Singularities of differentiable mappings in differential topology Positivity of Schur function expansions of 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 Let \(G\) be a complex semisimple algebraic group with maximal torus \(T\) and Borel subgroup \(B\) (corresponding to the positive roots), and let \(X\) denote the character group of \(T\). Let \(m_{\lambda}^{\mu}\) denote the multiplicity of the weight \(\mu\) in the simple \(G\)-module of highest weight \(\lambda\). \textit{G. Lusztig} [Astérisque 101--102, 208--229 (1983; Zbl 0561.22013)] introduced certain \(q\)-analogues \(m_{\lambda}^{\mu}(q)\) of weight multiplicity (also known as Kostka-Foulkes polymomials). These were based on Kostant's partition function that counts the multiplicity of a weight in a symmetric power \(S^j({\mathfrak u})\) where \({\mathfrak u}\) is the Lie algebra of the unipotent radical of \(B\). Let \(P\) be a parabolic subgroup of \(G\), and let \(N\) be a \(P\)-stable subspace of a finite-dimensional rational \(G\)-module such that the \(T\)-weights of \(N\) lie in an open half-space of \(X\otimes_{\mathbb Z}{\mathbb Q}\). The \(T\)-weights of \(N\) form a finite multiset \(\Psi\) in \(X\). Associated to \(\Psi\), the author introduces a generalization of Lusztig's \(q\)-polynomials. Here one makes use of a generalized Kostant partition function that counts the multiplicity of a weight in a symmetric power \(S^j(N)\).
By obtaining a vanishing theorem on line bundle cohomology for \(G\times_{P} N\), the author determines some conditions under which the coefficients of the generalized \(m_{\lambda}^{\mu}(q)\) are non-negative. Next, the author focuses on the special case that \(\Psi\) is the set of short positive roots (in a non-simply laced root system). Here the author obtains a precise determination of when the coefficients are non-negative. This extends work of \textit{A. Broer} [Invent. Math. 113, 1--20 (1993; Zbl 0807.14043)].
Lastly, the author introduces the notion of ``short'' Hall-Littlewood polynomials \(P_{\lambda}(q)\) (for a dominant weight \(\lambda\)) and deduces a number of basic properties. Considering all roots to be short in the simply-laced case, the results generalize the work of \textit{R. Gupta} [J. Lond. Math. Soc., II. Ser. 36, No. 1--2, 68--76 (1987; Zbl 0649.17009)], [Bull. Am. Math. Soc., New Ser. 16, 287--291 (1987; Zbl 0648.22011)]. In particular, let \(\chi_{\lambda}\) denote the character of the simple \(G\)-module with highest weight \(\lambda\). Then, it is shown that \(\chi_{\lambda} = \sum m_{\lambda}^{\mu}(q)P_{\mu}(q)\) where the sum runs over all dominant weights \(\mu\) (and the \(m_{\lambda}^{\mu}(q)\) correspond to \(\Psi\) being the set of short positive roots). semisimple Lie algebra; weight multiplicity; \(q\)-analogue; Hall-Littlewood polynomials; Kostant partition function Panyushev, D. I., Generalised Kostka-Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles, Selecta Math. (N.S.), 16, 315-342, (2010) Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Vanishing theorems in algebraic geometry, Homogeneous spaces and generalizations, Representation theory for linear algebraic groups Generalised Kostka-Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles | 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. Let \(N^p\) and \(F^p\) denote respectively the coniveau filtration and the Hodge filtration on \(H^i(X,\mathbb{C})\). Hodge proved that (*) \(N^pH^i(X,\mathbb{C})\subset F^pH^i(X,\mathbb{C})\cap H^i(X,\mathbb{Q})\), and conjectured that the equality holds. \textit{A. Grothendieck} [Topology 8, 299--303 (1969; Zbl 0177.49002)] showed, however, that for the threefold \(X=E^3\), where \(E=\mathbb{C}/\langle 1,\tau\rangle\) with \([\mathbb{Q}(\tau),\mathbb{Q}]=3\), the dimension of the right hand side of (*) is odd. Since the left hand side of (*) must be even-dimensional as a sub-Hodge structure, his threefold provides us with a counter-example for the equality. In this paper the author investigates the structures of the vector spaces on both sides of (*) without the assumption that \(\tau\) is cubic, and clarifies the reason why the cubic case violates the validity of Hodge's original conjecture. cohomology classes; supports; generalized Hodge conjecture Transcendental methods, Hodge theory (algebro-geometric aspects), Algebraic cycles On Grothendieck's counterexample to the generalized Hodge 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 From the article (section 6): ``the main goal of this article is the proof of Theorem 3.8, which, thanks to Levine's results in [\textit{M. Levine}, in: Cycles, motives and Shimura varieties. Proceedings of the international colloquium. New Delhi: Narosa Publishing House/Published for the Tata Institute of Fundamental Research. 265--392 (2011; Zbl 1227.14014)], yields a Tannakian category of mixed Tate motives over \(\mathcal{M}_{0,n}\) whose Tannakian group is given by the spectrum \(H_{/S,\mathbb{Q},n}\). This now makes it possible to describe \(H_{/S,\mathbb{Q},n}\) by explicit algebraic cycles, hence generalizing the construction in [\textit{I. Soudères}, J. Pure Appl. Algebra 220, No. 7, 2590--2647 (2016; Zbl 1408.11067)].''' This explicit description is not undertaken in the present article.
The author shows (in Theorems 3.5 and 3.8) that the Deligne-Mumford moduli spaces of stable curves of genus \(0\) with \(n\) marked points \(\overline{\mathcal{M}}_{0,n}\) as well as the moduli spaces of smooth curves of genus \(0\) with \(n\) marked points \(\mathcal{M}_{0,n}\) have the mixed Tate property and the Beilinson-Soulé vanishing property in Spitzweck's category of mixed motives over \(\mathrm{Spec}(\mathbb{Z})\) [\textit{M. Spitzweck}, ``A commutative \(\mathbb{P}^1\)-spectrum representing motivic cohomology over Dedekind domains'', Preprint, \url{arXiv:1207.4078}]. This then implies these properties over other bases, in the triangulated categories of mixed motives introduced by Cisinski and Déglise as well as Voevodsky's over a field.
Theorem 3.8 is proved with a Gysin triangle argument from Theorem 3.5, which itself is proved by induction over \(n\) and (for \(n \geq 5\)) an induction along Keel's description of a morphism \(\overline{\mathcal{M}}_{0,n} \to \overline{\mathcal{M}}_{0,n-1} \times \overline{\mathcal{M}}_{0,4}\) as a sequence of blow-ups [\textit{S. Keel}, Trans. Am. Math. Soc. 330, No. 2, 545--574 (1992; Zbl 0768.14002)].
Remark 4.11 introduces the ``motivic short exact sequence'' \(SES_n\)
\[
1 \to K^\bullet_{/S,\mathbb{Z},n} \to G^\bullet_{/S,\mathbb{Z},n} \to G^\bullet_{/S,\mathbb{Z}} \to 1
\]
where \(G^\bullet_{/S,\mathbb{Z},n}\) is the affine derived group scheme over \(\mathbb{Z}\) whose perfect representations are equivalent to the category of mixed Tate objects \(DMT_{/S,\mathbb{Z}}(\mathcal{M}_{0,n})\) (see Theorem 4.9) and the map to \(G^\bullet_{/S,\mathbb{Z}} := G^\bullet_{/S,\mathbb{Z},3}\) is explained in Proposition 4.10. In section 4 compatibility of this short exact sequence with the geometry of the tower \(\mathcal{M}_{0,n}\) is shown and exploited for a definition of a Grothendieck-Teichmüller space over \(\mathbb{Z}\) with \(\mathbb{Z}\) coefficients in Definition 4.16, built out of homotopy automorphism spaces of the ``geometric part'' \(K^\bullet_{/S,\mathbb{Z},n}\). It is also explained how tangential base points fit into the picture.
In section 5, the integral constructions of the article are compared to the rational constructions done previously by a number of authors. Definition 5.1 introduces the motivic Grothendieck-Teichmüller group over \(S = \mathrm{Spec}(\mathbb{Z})\) with rational coefficients \(GT^{mot}_{/S}(\mathbb{Q})\) in terms of the groups \(K_{/S,\mathbb{Q},n}\).
The article closes with conjectures (while conjecture 1 appears already in section 4): Conjecture 2 describe an operadic structure of the ``geometric part'' \(K^\bullet_{/S,\mathbb{Z},n}\).
Conjecture 3 is that the geometric part is in fact a non-derived affine group scheme, the spectrum of a commutative non-cocommutative Hopf algebra.
Conjecture 4 is that the Betti and de Rham realizations induce isomorphisms of \(\mathbb{Q}\)-rational points of the motivic Grothendieck-Teichmüller group \(GT^{mot}\) with the pro-unipotent Grothendieck-Teichmüller group \(GT\). mixed Tate motives; moduli spaces of curves; Grothendieck-Teichmüller; Beilinson-Soulé vanishing property I. Soudères , 'A motivic Grothendieck--Teichmüller group', Preprint, 2015, arXiv:1502.05640. Motivic cohomology; motivic homotopy theory, Families, moduli, classification: algebraic theory, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) A motivic Grothendieck-Teichmüller 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 This is a survey paper on the quantum cohomology of isotropic Grassmannians. Two questions are investigated here. First, the author explains the quantum Pieri rule, expressing the quantum product of a general Schubert class with a special Schubert class. Secondly, he gives a presentations for the quantum cohomology in terms of generators and relations.
The survey starts with the case of the classical Grassmannians. The quantum Pieri rule was found by \textit{A. Bertram} [Adv. Math. 128, No. 2, 289--305 (1997; Zbl 0945.14031)], while \textit{B. Siebert} and \textit{G. Tian} obtained a presentation of the quantum cohomology [Asian J. Math. 1, No. 4, 679--695 (1997; Zbl 0974.14040)]. The current paper follows the approach of \textit{A. K. Buch, A. Kresch} and \textit{H. Tamvakis} [J. Am. Math. Soc. 16, No. 4, 901--915 (2003; Zbl 1063.53090)], using the idea of kernel and span for a rational map to the Grassmannian to obtain the structure constants for the quantum cohomology. The next section discusses the Lagrangian and maximal isotropic orthogonal Grassmannians, using a similar point of view. Note however that the original arguments in \textit{A. Kresch} and \textit{H. Tamvakis} [J. Algebr. Geom. 12, No. 4, 777--810 (2003; Zbl 1051.53070); Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083)] made use of Quot scheme compactifications of the moduli space of rational maps to the isotropic Grassmannian. Finally, in the last part of the survey, the author considers the non-maximal isotropic Grassmannians of type B, C, D. Harry Tamvakis, Quantum cohomology of isotropic Grassmannians, Geometric methods in algebra and number theory, Progr. Math., vol. 235, Birkhäuser Boston, Boston, MA, 2005, pp. 311 -- 338. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum cohomology of isotropic 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 present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known. algebraic geometry codes; quantum error-correction; algebraic curves; finite fields Quantum computation, Geometric methods (including applications of algebraic geometry) applied to coding theory, Algebraic coding theory; cryptography (number-theoretic aspects), Quantum coding (general), Applications to coding theory and cryptography of arithmetic geometry Quantum codes from a new construction of self-orthogonal algebraic geometry codes | 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 Associated to the quantum multiplication in the quantum cohomology of, say, a Nakajima quiver variety there is a flat connection called the quantum connection (a.k.a. Dubrovin connection or quantum differential equation). The paper under review constructs and studies the \(K\)-theoretic analog of the quantum connection. Namely, the authors describe the quantum differential equations that arise in the enumerative K-theory of ``quasimap'' counts of curves in Nakajima quiver varieties.
The centrality of this concept is illustrated by the following relations. (a) For certain choices of the quiver the varieties are Hilbert schemes of points on a surface, and the resulting quantum connection plays a key role in comparing the corresponding Gromov-Witten and Donaldson-Thomas theories. (b) The quantum connection should also be interpreted as the (K-theoretic version of the) generalized Casimir connection of the Maulik-Okounkov Yangian associated to the quiver. (c) The quantum differential equations commute with another set of differential equations, the quantum Knizhnik-Zamolodchikov equations.
The actual path the authors follow to describe the quantum difference equations is also remarkable. First they define a ``quantum dynamical Weyl group(oid)'', acting on the torus equivariant \(K\)-theory. This Weyl group naturally contains a lattice and the action of this lattice is identified with the sought after quantum differential equations. \(K\)-theory of Nakajima quiver varieties; quasimap count; quantum connection Projective and enumerative algebraic geometry, Surfaces and higher-dimensional varieties, \(K\)-theory, Quantum theory Quantum difference equation for Nakajima 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 The Hilbert scheme of points Hilb\(_n(\mathbb{C}^2)\) is the crepant resolution of the orbifold quotient \((\mathbb{C}^2)^n/S_n\) containing the configurations of \(n\) distinct points in \(\mathbb{C}^2\). Its geometry and algebraic invariants have been a subject of intense study in the last decade due to connections with the string theory. In particular, the ring structure of its torus equivariant quantum cohomology has been established (the natural action of the complex torus on \((\mathbb{C}^2)^n/S_n\) extends to the Hilbert scheme).
In the paper under review, the authors give an explicit formula for \(M_D\), the operator of multiplication by the first Chern class \(D\) of the Hilbert scheme in small quantum cohomology. The matrix elements of \(M_D\) give counts of rational curves meeting three given subvarieties of the scheme, and an associated differential equation is an integrable non-stationary deformation of the Calogero-Sutherland equation for quantum particles on a torus. The formula implies that \(D\) generates the small quantum cohomology ring, and completes establishing the four-way correspondence between the quantum cohomology of Hilb\(_n(\mathbb{C}^2)\), the equivariant orbifold cohomology of \((\mathbb{C}^2)^n/S_n\), and the Gromov-Witten and Donaldson-Thomas invariants of \(\mathbb{P}^1\times\mathbb{C}^2\). Hilbert scheme of points; small quantum cohomology; Calogero-Sutherland operator A. Okounkov and R. Pandharipande, \textit{Quantum cohomology of the Hilbert scheme of points in the plane}, math/0411210. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Quantum cohomology of the Hilbert scheme of points in the plane | 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 discussing the so-called \(d\)-dimensional anabelian conjectures. The zero-dimensional anabelian conjecture can be stated as follows: The category of all finitely generated infinite fields and field isomorphisms is anabelian. This, roughly speaking, means that the geometric and arithmetic of the objects of the category in question are encoded in their étale fundamental group. This conjecture in the global field case evolved in the seventies from the work of Neukirch, Ikeda, Iwasawa and Uchida. The next step in this direction was Pop's solution of the conjecture in the case of function fields of one variable over number fields. Finally, in 1995 the author proved this conjecture in general. Next, the following one dimensional anabelian conjecture is discussed: The category of all hyperbolic curves over finitely generated fields and scheme isomorphisms is anabelian. This conjecture has an affirmative answer in the affine case (Tamagawa) and in the characteristic zero case (Mochizuki). The main aim of this paper is to discuss the ideas and the methods of all these proofs (including historical comments). profinite groups; étale fundamental group; anabelian geometry F. Pop, Glimpses of Grothendieck's anabelian geometry, in Geometric Galois Actions I, London Mathematical Society Lecture Notes, Vol. 242, (L. Schneps and P. Lochak eds.), Cambridge University Press, 1998, Cambridge, pp 133--126. Homotopy theory and fundamental groups in algebraic geometry, Inverse Galois theory, Separable extensions, Galois theory, Galois theory Glimpses of Grothendieck's anabelian 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 After introducing the relevant deformation quantization of the standard hydrodynamic Poisson bracket of dispersionless Dubrovin-Zhang (DZ) hierarchies and a study of its propagator [\textit{B. A. Dubrovin} and \textit{Y. Zhang}, ``Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants'', Preprint, \url{arXiv:math/0108160}], the authors define the quantum double ramification (qDR) hierarchy using intersection numbers of a given a cohomological field theory (CohFT) with the double ramification cycle and the Hodge and psi classes and prove commutativity of the (quantum) flows. The authors define a quantization of the double ramification hierarchies of [Commun. Math. Phys. 336, No. 3, 1085--1107 (2015; Zbl 1329.14103); Commun. Math. Phys. 342, No. 2, 533--568 (2016; Zbl 1343.37062)], using intersection numbers of the double ramification cycle, the full Chern class of the Hodge bundle and psi-classes with a given cohomological field theory. They provide effective recursion formulae which determine the full quantum hierarchy starting from just one Hamiltonian, the one associated with the first descendant of the unit of the cohomological field theory only. They study various examples which provide, in very explicit form, new \((1+1)\)-dimensional integrable quantum field theories whose classical limits are well-known integrable hierarchies such as KdV, Intermediate Long Wave, Extended Toda, etc. Finally they prove polynomiality in the ramification multiplicities of the integral of any tautological class over the double ramification cycle. moduli space of curves; cohomological field theories; quantum integrable systems; double ramification cycle Kim, H.-C., Kim, S.: M5-branes from gauge theories on the 5-sphere. J. High Energy Phys. \textbf{2013}, 144 (2013). 10.1007/JHEP05(2013)144 Families, moduli of curves (algebraic), Relationships between algebraic curves and integrable systems, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Groups and algebras in quantum theory and relations with integrable systems Double ramification cycles and quantum integrable systems | 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 an interpretation of quantum Serre theorem of Coates and Givental as a duality of twisted quantum \(D\)-modules. This interpretation admits a non-equivariant limit, and we obtain a precise relationship among (1) the quantum \(D\)-module of \(X\) twisted by a convex vector bundle \(E\) and the Euler class, (2) the quantum \(D\)-module of the total space of the dual bundle \(E^\vee \rightarrow X\), and (3) the quantum \(D\)-module of a submanifold \(Z\subset X\) cut out by a regular section of \(E\). When \(E\) is the anticanonical line bundle \(K_X^{-1}\), we identify these twisted quantum \(D\)-modules with second structure connections with different parameters, which arise as Fourier-Laplace transforms of the quantum \(D\)-module of \(X\). In this case, we show that the duality pairing is identified with Dubrovin's second metric (intersection form). Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Quantum Serre theorem as a duality between quantum \(D\)-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 The author studies the Chevalley-Warning problem in the Grothendieck ring \(K-0(Var/k)\). It is shown that the \(\mathbf A^1\)-homotopy theory yields well-defined invariants on \(K-0(Var/k)/\mathbf L\), in particular, the Brauer group is such an invariant. The author uses this to give a concrete counter-example to the Chevalley-Warning conjecture over a \(C-1\)-field. This also gives a negative answer to the question in \textit{E. Bilgin}'s PhD thesis [``Classes of some hypersurfaces in the Grothendieck ring of varieties'', Universität Duisburg-Essen (2011)]. unramified cohomology; Grothendieck ring Applications of methods of algebraic \(K\)-theory in algebraic geometry, Other combinatorial number theory, \(K_0\) of other rings Unramified cohomology, \(\mathbb A^1\)-connectedness, and the Chevalley-Warning problem in 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 To provide possibility to design the commutative encryption algorithms on the basis of new versions of the hidden discrete logarithm problem, the term ``commutativity'' is interpreted in the extended sense. Namely, the encryption algorithm is called commutative, if the double encryption on two different keys produces the ciphertext that can be correctly decrypted using the keys in arbitrary order. The introduced commutative encryption method is characterized in using the single-use random subkeys. This feature defines probabilistic nature of the encryption process. A candidate for post-quantum commutative encryption algorithm is proposed, using the computations in the 6-dimensional finite non-commutative associative algebra with a large set of the right-sided global units. The proposed algorithm is used as the base of the post-quantum no-key protocol. commutative encryption; post-quantum cryptoscheme; no-key protocol; finite non-commutative algebra; associative algebra; homomorphism Cryptography, Computational aspects of associative rings (general theory), Applications to coding theory and cryptography of arithmetic geometry, Algebraic coding theory; cryptography (number-theoretic aspects), Endomorphism rings; matrix rings, Quantum cryptography (quantum-theoretic aspects) Post-quantum commutative encryption algorithm | 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 if \(X\) is a normal projective variety in characteristic zero, \(L\) a base-point free ample line bundle on \(X\), and \(Y\) a general member of \(| L|\), then the restriction map of divisor class groups \(\text{Cl}(X)\rightarrow \text{Cl}(Y)\) is an isomorphism provided that \(\text{dim}\,X\geq 4\). variety; linear bundle; divisor class group G. V. Ravindra and V. Srinivas \({ref.surNamesEn}, The Grothendieck-Lefschetz theorem for normal projective varieties,, \)J. Alg. Geom.\(, 15, 563, (2006)\) Picard groups, Divisors, linear systems, invertible sheaves The Grothendieck-Lefschetz theorem for normal projective 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 The authors give a new exposition and proof of a nonbinary version of the generalized binary CSS construction. Using this construction and algebraic curves they obtain various parameters (lengths, dimensions, and minimum distances) for nonbinary quantum codes. Furthermore, they apply this construction to the tower of function fields to obtain asymptotically good nonbinary quantum codes which are constructible in polynomial time (the question of asymptotically good nonbinary quantum codes has not been considered until now). algebraic geometric codes; nonbinary quantum codes; CSS construction Kim J.L., Walker J.: Nonbinary quantum error-correcting codes from algebraic curves. Discrete Math. 308, 3115--3124 (2008) Other types of codes, Geometric methods (including applications of algebraic geometry) applied to coding theory, Quantum computation, Applications to coding theory and cryptography of arithmetic geometry Nonbinary quantum error-correcting codes from 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 authors present the theory of Schur and Schubert polynomials, revisited from the point of view of generalized Thom polynomials. The Schur and Schubert polynomials are realized as first obstructions of certain fiber bundles. After presenting some results about the Thom polynomials for group actions, the authors obtain the calculation of these polynomials via the method of restriction equations. Then they obtain a new definition for the Schur and Schubert polynomials by applying the general method to compute them as Thom polynomials. They also redefine the double Schubert polynomials and the Kempf-Laksov-Schur polynomials Schur polynomials; Schubert polynomials; Thom polynomials; method of restriction equations Fehér, L.; Rimányi, R., Schur and Schubert polynomials as thom polynomials--cohomology of moduli spaces, Cent. eur. J. math., 1, 4, 418-434, (2003) Singularities of differentiable mappings in differential topology, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Enumerative problems (combinatorial problems) in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Schur and Schubert polynomials as Thom polynomials -- cohomology of 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 authors study Thom polynomials of singularities. For a (so called right-left) complex singularity \(\eta\), and a map \(f\) between compact complex manifolds, one can consider the singularity subset as the collection of those points in the source where the map has singularity \(\eta\). It is known that the closure of this set (under favorable circumstances) carries a cohomology class, which can be calculated by substituting the characteristic classes of the source and target manifold (pulled back via \(f\)) into a universal polynomial, the Thom polynomial of the singularity.
The authors study the positivity properties of Thom polynomials. More generally they study the positivity properties of \(G\)-equivariant cohomology classes represented by invariant cones in representations of \(G\), where \(G\) is a product of general linear groups. They prove that if the representation is determined by a functor which preserves ``global generatedness'', then these equivariant classes, when expressed in the basis of products of Schur functions of the Chern roots of the general linear groups, have \textit{non-negative} coefficients. The proof eventually reduces this statement to an appropriate positivity result of Lazarsfeld and Fulton.
As an application, the authors show that the Thom polynomials of right-left singularities have non-negative coefficients when expressed in the basis of products of Schur functions of the source and target Chern roots. This result extends earlier results of the authors concerning less general singularity types. Thom polynomials; Schur functions Pragacz, P., Weber, A.: Thom polynomials of invariant cones, Schur functions and positivity. In: Algebraic cycles, sheaves, shtukas, and moduli, Trends Math., pp. 117-129. Birkhäuser, Basel (2008) Classical problems, Schubert calculus, Symmetric functions and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Homology of classifying spaces and characteristic classes in algebraic topology Thom polynomials of invariant cones, Schur functions and 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 Our purpose is to describe some remarks on Schur polynomials, which play an important role in the theory of Chow ring. In {\S} 2, we describe the relationship between the Schur polynomials and the Schubert cycles of a Grassmann variety. In {\S} 3, we prove the duality of Schur polynomials using the theory of the Chow ring of Grassmann variety. - In {\S} 4, we present the theorem of W. Fulton and R. Lazarsfeld for numerically positive polynomials for ample vector bundles. - In {\S} 5 and {\S} 6, we introduce the Gysin's projection formula for flag bundles and give a formal proof of this formula. Schur polynomials; Schubert cycles; Chow ring of Grassmann variety; flag bundles Grassmannians, Schubert varieties, flag manifolds, Parametrization (Chow and Hilbert schemes) Remarks on Schur 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 We compute the center and Azumaya locus in the simplest non-abelian examples of quantized multiplicative quiver varieties at a root of unity: quantum Weyl algebras of rank \(N\), and quantum differential operators on the quantum group \(\mathrm{GL}_2\). These examples illustrate in elementary terms much more general phenomena explored further in [8]. Quantum groups (quantized enveloping algebras) and related deformations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Noncommutative algebraic geometry, Representations of quivers and partially ordered sets Quantum Weyl algebras and reflection equation algebras at a root of unity | 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 It is known that the Gromov-Witten invariants of a Fano variety define the structure of a Frobenius algebra on the even part \(W = H^{ev}(V,{\mathbb C})\) of the graded cohomology space of \(V\). This structure depends on the choice of a cohomology class \(w\in W\). It is conjectured that the inhomogeneous Euler vector field \(X(w)\) of the corresponding potential function acts semi-simply with respect to this structure. In the present paper the authors check this conjecture in the case when \(V\) is a complete intersection of dimension \(n \geq 3\) and of degree \((d_1,\ldots,d_r)\) in \({\mathbb P}^{n+r}\) with \((d_1+\ldots+d_r)-(n+r+1) < -n/2\) (except \(n = 7\) and \(d_1+\ldots+d_r = r+2\)). The computations rely on the previous work of \textit{A. Beauville} who computed the quantum cohomology of complete intersections [see Mat. Fiz. Anal. Geom. 2, No. 3-4, 384-398 (1995; Zbl 0863.14029)]. Gromov-Witten invariants; Fano variety; Euler vector field; quantum cohomology of complete intersections Tian, G.; Xu, G.: On the semi-simplicity of the quantum cohomology algebras of complete intersection, Math. res. Lett. 4, 481-488 (1997) Fano varieties, Complete intersections, Projective and enumerative algebraic geometry On the semi-simplicity of the quantum cohomology algebras of complete intersections | 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 proves that Hilbert's Nullstellensatz can be generalized to an algebra generated countably over an uncountable field, and presents some geometric forms of the generalized Hilbert Nullstellensatz. algebra generated countably over an uncountable field; generalized Hilbert Nullstellensatz Relevant commutative algebra, Noncommutative algebraic geometry, Other algebras and orders, and their zeta and \(L\)-functions Hilbert Nullstellensatz of countably generated 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 The aim of this short paper is to introduce a multi-variable analogue of the Kostka-Shoji polynomial introduced by \textit{T. Shoji} [Sci. China, Math. 61, No. 2, 353--384 (2018; Zbl 1494.05115)]. The generalisation replaces integer partitions \(\lambda,\mu\) with multipartitions with \(r\) components, for \(r>1\). The authors then show that this new version has applications related to Lusztig's iterated convolution diagram for the cyclic quiver \(\tilde A_{r-1}\).
This paper is not for the faint-hearted. It is technical right from the start, and the reader is referred to the references for most of the background. There are no examples to aid the reader's understanding. The advantage of all this is that the paper is short, so gives a very concise account for experts. Kostka-Shoji polynomials; cyclic quiver; convolution diagram; Frobenius splitting; affine flag variety; Bott-Samelson-Demazure-Hansen resolution Finkelberg, M; Ionov, A, Kostka-shoji polynomials and lusztig's convolution diagram, (2016) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Kostka-Shoji polynomials and Lusztig's convolution diagram | 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 use the Thom polynomial theory developed by \textit{L. Fehér} and \textit{R. Rimányi} [Duke Math. J. 114, No.2, 193--213 (2002; Zbl 1054.14010)] to prove the component formula for quiver varieties conjectured by \textit{A. Knutson, E. Miller} and \textit{M. Shimozono} [Four positive formulae for type A quiver polynomials, preprint, \texttt{http://arxiv.org/abs/math.AG/0308142}]. This formula expresses the cohomology class of a quiver variety as a sum of products of Schubert polynomials indexed by minimal lace diagrams, and implies that the quiver coefficients of \textit{A. Buch} and \textit{W. Fulton} [Invent. Math. 135, 665--687 (1999; Zbl 0942.14027)] are non-negative. We also apply our methods to give a new proof of the component formula from the Gröbner degeneration of quiver varieties, and to give generating moves for the KMS-factorizations that form the index set in \(K\)-theoretic versions of the component formula. Buch, Anders S.; Fehér, László M.; Rimányi, Richárd, Positivity of quiver coefficients through Thom polynomials, Adv. Math., 197, 1, 306-320, (2005) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Singularities of differentiable mappings in differential topology, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Positivity of quiver coefficients through 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 A theorem of Grothendieck asserts that over a perfect field \(k\) of cohomological dimension one, all non-abelian \(H^2\)-cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of this theorem. The generalization -- to the context of perfect fields of virtual cohomological dimension one -- takes the form of a local-global principle for the \(H^2\)-sets with respect to the orderings of the field. This principle asserts in particular that an element in \(H^2\) is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of \(k\). Our techniques provide a new proof of Grothendieck's original theorem. An application to homogeneous spaces over \(k\) is also given. non-abelian \(H^2\)-cohomology sets of algebraic groups; local-global principle; real spectrum; homogeneous spaces Flicker, Y.Z.; Scheiderer, C.; Sujatha, R., Grothendieck's theorem on non-abelian \(H^2\) and local-global principles, J. amer. math. soc., 11, 3, 731-750, (1998) Étale and other Grothendieck topologies and (co)homologies, Galois cohomology, Cohomology theory for linear algebraic groups, Forms over real fields Grothendieck's theorem on non-abelian \(H^2\) and local-global principles | 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{V. V. Batyrev} [Asterisque 218, 9--34 (1993; Zbl 0806.14041)] described the quantum cohomology ring of Fano toric varieties in terms of generators (toric divisors and formal \(q\) variable) and relations (linear relations and \(q\)-deformed monomial relations). \textit{A. Kresch} [Mich. Math. J. 48, 369--391 (2000; Zbl 1085.14519)] gave a so called quantum Giambelli formula that expresses any cohomology class in \(H^*(X,\mathbb Q)\) as a polynomial in divisor classes and formal \(q\) variables for a certain class of Fano toric varieties.
In the present paper, the authors compute Gromov-Witten invariants of \(\mathbb P^d\)-bundles, \(X=\mathbb P(\oplus_{i=1}^r{\mathcal O}_{\mathbb P^1}(a_i))\) with \(\sum_{i=1}^ra_i = 2 + kr\), and give examples of quantum products with infinitely many non-trivial quantum corrections. As a main tool, the authors use the fact that Gromov-Witten invariants are invariants of the symplectic deformation class of a symplectic manifold, and combinatorial formulas by \textit{H. Spielberg} [C. R. Acad. Sci. Paris, Sér. I Math. 329, No. 8, 699--704 (1999; Zbl 1004.14014)]. Batyrev's conjecture; symplectic deformation Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli GW-invariants and quantum products with infinitely many quantum corrections | 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 discuss how the theory of quantum cohomology may be generalized to ''gravitational quantum cohomology'' by studying topological \(\sigma\) models coupled to two-dimensional gravity. We first consider \(\sigma\) models defined on a general Fano manifold \(M\) (manifold with a positive first Chern class) and derive new recursion relations for its two-point functions. We then derive bi-Hamiltonian structures of the theories and show that they are completely integrable at least at the level of genus 0. We next consider the subspace of the phase space where only a marginal perturbation (with a parameter \(t\)) is turned on and construct Lax operators (superpotentials) L whose residue integrals reproduce correlation functions. In the case of \(M = \mathbb CP^N\) the Lax operator is given by
\[
L=Z_1+Z_2+\dots +Z_N+e^tZ_1^{-1}Z_2^{-1}\dots Z_N^{-1}
\]
and agrees with the potential of the affine Toda theory of the \(A_N\) type. We also obtain Lax operators for various Fano manifolds; Grassmannians, rational surfaces, etc. In these examples the number of variables of the Lax operators is the same as the dimension of the original manifold. Our result shows that Fano manifolds exhibit a new type of mirror phenomenon where mirror partner is a noncompact Calabi-Yau manifold of the type of an algebraic torus \(C^{*N}\) equipped with a specific superpotential. T. Eguchi, K. Hori and C.-S. Xiong. Gravitational quantum cohomology, \textit{Internat. J.} \textit{Modern Phys. A}, 12(1997), No.9, 1743-1782. Applications of deformations of analytic structures to the sciences, Enumerative problems (combinatorial problems) in algebraic geometry, Spaces of embeddings and immersions, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Gravitational 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 Kazhdan and Lusztig have introduced the so-called \(P\)-polynomials \(P_{x,y}(q)\) for each pair \((x,y)\) of elements of a Coxeter group \(W\) with \(x\prec y\), where \(\prec\) denotes the Bruhat order of \(W\). The polynomial \(P_{x,y}(q)\) is a measure for the singularity of the Schubert variety \(V_y\) at the generic point of \(V_x\) in the sense that \(V_y\) is smooth along the generic point of \(V_x\) if and only if \(P_{x,y}(q)=1\).
In general it is very hard to calculate the polynomials \(P_{x,y}(q)\). The authors give some explicit formulas for \(P_{x,y}(q)\) in the case \(W\) is the symmetric group \(S_n\) and \(y\) is a particular permutation associated to any flag variety, while \(x\) is arbitrary. Kazhdan-Lusztig polynomials; \(R\)-polynomials; Schubert cells; Coxeter group; \(P\)-polynomials; symmetric group; singularity of a Schubert variety Shapiro, B.; Shapiro, M.; Vainshtein, A., Kazhdan-Lusztig polynomials for certain varieties of incomplete flags, \textit{Discrete Math.}, 180, 345-355, (1998) Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects) Kazhdan-Lusztig polynomials for certain varieties of incomplete flags | 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 advertised in the title, this excellent paper provides a categorification of the Grothendieck-Riemann-Roch (GRR) theorem, leading to applications in geometric Langlands theory, non-commutative motives, the Gauss-Manin connection, and elliptic cohomology. I recommend reading the introduction for a detailed accounting of the main results; I will try to summarize here.
For any smooth and quasiprojecture scheme \(X\) over a field, there is a \textit{Chern character} map \(\mathrm{ch} : \iota_0 \operatorname{Perf}(X) \to \mathcal{O}(\mathcal{L}X) \cong \mathrm{HH}(\mathrm{Perf}(X))\). Here \(\mathrm{Perf}\) means that \(\infty\)-category of perfect sheaves, \(\mathcal{L}\) means the derived loop space, and \(\iota_0\) means the \(\infty\)-groupoid of objects. This is often postcomposed with the Hochschild-Kostant-Rosenberg (HKR) isomorphism \(\mathrm{HH}(\operatorname{Perf}(X)) \cong \mathrm{H}^*(X, \bigoplus_{i\geq 0}\Omega^i_X)\) to de Rham forms. The classical GRR theorem is about how these maps commute with pushing forward along a map \(f : X \to Y\): the first half of the GRR theorem says that \(\mathrm{ch}\) commutes with \(f_*\); and the second half says that the HKR isomorphism does not commute naively, but does once a correction by the Todd class is incorporated. (This factorization is explained in [\textit{N. Markarian}, J. Lond. Math. Soc., II. Ser. 79, No. 1, 129--143 (2009; Zbl 1167.14005)])
The main theorem of this paper is a categorification of the first half of the GRR theorem. (Categorifying the second half is almost trivial: there is a natural categorified HKR isomorphism, and it commutes with pushforwards without any Todd class corrections and without much work.) The statement is also generalized. Most schemes of geometric interest are examples of ``1-affine derived stacks'': stacks \(X\) functorially recoverable (as the ``spectrum'') from their symmetric monoidal stable \(\infty\)-categories \(\mathrm{QCoh}(X)\) of quasicoherent sheaves. (Examples include quasi-compact quasi-separated schemes and semi-separated Artin stacks of finite type in characteristic zero.) In particular, a dualizable sheaf of categories over \(X\) is nothing more nor less than a dualizable \(\mathrm{QCoh}(X)\)-module. With this in mind, let \(\mathcal{C}\) be any symmetric monoidal stable \(\infty\)-category, with symmetric monoidal \((\infty,2)\)-category \(\mathrm{Mod}^{\mathrm{dual}}_{\mathcal{C}}\) of dualizable modules. Define the ``loop space'' of \(\mathcal{C}\) to be \(\mathcal{L}\mathcal{C} := S^1 \otimes \mathcal{C} \cong \mathcal{C} \otimes_{\mathcal{C} \otimes \mathcal{C}} \mathcal{C}\); it carries a natural \(S^1\) action. The \textit{(equivariant) categorified Chern character} is a symmetric monoidal functor \[ \mathrm{Ch}^{S^1} : \mathrm{Mod}^{\mathrm{dual}}_{\mathcal{C}} \to (\mathcal{L}\mathcal{C})^{S^1}.\] Now suppose that \(f : \mathcal{D} \to \mathcal{C}\) is a symmetric monoidal functor (which you should think of as the restriction of quasicoherent sheaves), and that \(\mathcal{C}\) is ``rigid over \(\mathcal{D}\).'' (This is a natural generalization of rigidity, and is defined in terms of certain maps of certain modules having right adjoints; in particular, there is a \(\mathcal{D}\)-linear right adjoint \(f^R\).) The main Theorem A of the paper states that \(\mathrm{Ch}^{S^1}\) intertwines the pushforward \(f_* : \mathrm{Mod}^{\mathrm{dual}}_{\mathcal{C}} \to \mathrm{Mod}^{\mathrm{dual}}_{\mathcal{D}}\) with \(\mathcal{L}f^R : (\mathcal{L}\mathcal{C})^{S^1} \to (\mathcal{L}\mathcal{D})^{S^1}\).
There are many important applications of this theorem. When \(\mathcal{C}\) is generated by its subcategory \(\mathcal{C}^\omega\) of compact objects, the \textit{additivity} of the \(\mathrm{Ch}^{S^1}\) is the statement that it factors through the category \(\mathbb{M}\mathrm{ot}(\mathcal{C}^\omega)\) of noncommutative motives, and Theorem B concludes that in this case \(\mathrm{Ch}^{S^1}\) with the natural pushforward of motives; this parallels the K-theoretic formulation of the GRR theorem. Theorem C recognizes \(\mathcal{L}\mathrm{QCoh}(X) \cong \mathrm{QCoh}(\mathcal{L}X)\) a GRR theorem for a categorified Chern character \(\mathrm{Ch}^{S^1} : \mathrm{Mod}^{\mathrm{dual}}_{\mathrm{QCoh}(X)} \to \mathrm{QCoh}(\mathcal{L}X)^{S^1}\). (The rigidity requirement is satisfied as soon as \(f\) satisfies a mild condition called ``passability.'')
Using these results, Theorem D provides an identification between two different uncatorified Chern characters \(\mathrm{ch} : \mathcal{C} \to \mathrm{HH}(\mathcal{C}) \cong \Omega\mathcal{L}\mathcal{C}\), one defined by \textit{D. Ben-Zvi} and \textit{D. Nadler} [``Nonlinear traces''. Preprint, \url{arXiv:1305.7175}] and the other by \textit{B. Toën} and \textit{G. Vezzosi} [Abel Symp. 4, 331--354 (2009; Zbl 1177.14042)]. Theorems E and F give a GRR theorem for the ``secondary (motivic) Chern character'' \(\mathrm{ch}^{(2)} : K^{(2)}(X) \to \mathcal{O}(\mathcal{L}^2 X)^{S^1 \times S^1}\). Finally, Theorem G investigates de Rham realizations, and as such categories the classical Riemann-Roch theorem. Chern character; \(K\)-theory; categorification Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), Motivic cohomology; motivic homotopy theory, \(K\)-theory and homology; cyclic homology and cohomology The categorified Grothendieck-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 Let \(S\) be a set of labels and \(\overline{M}_{g,S}\) denote the moduli space of \(S\)--labeled curves of genus \(g\). The paper gives a combinatorial description in terms of labeled trees for Gromov--Witten classes that are Chow correspondences \(I(S,\Sigma,\beta)\in A_*(\overline{M}_{0,S}\times\overline{M}_{0,\Sigma})\), where \(S,\Sigma\) are disjoint sets of labels, and \(\beta\) runs over effective classes of boundary curves. The authors view this as a first step in an ambitious program of understanding the self-referential nature of motivic quantum cohomology and Gromov-Witten theory in general. Namely, the total motives corresponding to \(\overline{M}_{0,n}\), which are components of the cyclic modular operad, have quantum cohomology of their own, i.e. they define an algebra over the same operad. The authors also suggest contemplating analogs of \(\overline{M}_{g,n}\) with curves (strings) replaced by surfaces (membranes) leading to ''membrane quantum cohomology'' in terms of motivic actions. Chow correspondence; total motive; cyclic modular operad; Gromov-Witten invariants; non-Tannakian category Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Fine and coarse moduli spaces Towards motivic quantum cohomology of \(\bar{M}_{0,S}\) | 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 There is currently a growing interest in understanding which lattice simplices have uni-modal local \( h^*\)-polynomials (sometimes called box polynomials); specifically in light of their potential applications to unimodality questions for Ehrhart \( h^*\)-polynomials. In this note, we compute a general form for the local \( h^*\)-polynomial of a well-studied family of lattice simplices whose associated toric varieties are weighted projective spaces. We then apply this formula to prove that certain such lattice simplices, whose combinatorics are naturally encoded using common systems of numeration, all have real-rooted, and thus unimodal, local \( h^*\)-polynomials. As a consequence, we discover a new restricted Eulerian polynomial that is real-rooted, symmetric, and admits intriguing number theoretic properties. box polynomial; local \( h^*\)-polynomial; Eulerian polynomial; Ehrhart theory; lattice simplex; weighted projective space; numeral systems; simplices for numeral systems; factoradics; real-rooted; unimodal; symmetric; log-concave Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies Local \(h^*\)-polynomials of some 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 We establish a direct connection between the power of a unitary map in \(d\)-dimensions (\(d < \infty\)) to generate quantum coherence and the geometry of the set \(\mathcal{M}_d\) of maximally abelian subalgebras (of the quantum system full operator algebra). This set can be seen as a topologically non-trivial subset of the Grassmannian over linear operators. The natural distance over the Grassmannian induces a metric structure on \(\mathcal{M}_d\), which quantifies the lack of commutativity between the pairs of subalgebras. Given a maximally abelian subalgebra, one can define, on physical grounds, an associated measure of quantum coherence. We show that the average quantum coherence generated by a unitary map acting on a uniform ensemble of quantum states in the algebra (the so-called coherence generating power of the map) is proportional to the distance between a pair of maximally abelian subalgebras in \(\mathcal{M}_d\) connected by the unitary transformation itself. By embedding the Grassmannian into a projective space, one can pull-back the standard Fubini-Study metric on \(\mathcal{M}_d\) and define in this way novel geometrical measures of quantum coherence generating power. We also briefly discuss the associated differential metric structures.{
\copyright 2018 American Institute of Physics} Campos Venuti, L. ''Quantum coherence and many-body localization'' (unpublished). Quantum computation, General and philosophical questions in quantum theory, Grassmannians, Schubert varieties, flag manifolds, Contextuality in quantum theory, Applications of selfadjoint operator algebras to physics Quantum coherence generating power, maximally abelian subalgebras, and Grassmannian 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 Let \(X\) be a projective non singular variety embedded in the projective space \(\mathbb{P}^n(K)=\mathbb{P}\), with \(K\) algebraically closed, and let \(Y\) be a normal surface which is a (scheme-theoretic) complete intersection of \(X\) with some hypersurfaces
of \(P (r=\)codim\(_x (Y))\). Then the natural map of restriction (1) Pic\((X)\to\)Pic\((Y)\) yields the
following ones: (2) Pic\((X)/Z[({\mathcal O}_X (1)] \to \)Pic\((Y)/Z[{\mathcal O}_y (1)]\), (3) Pic\(^\tau(X) \to\)Pic\(^\tau(Y)\),
(4) Pic\(^0 (X)\to \)Pic\(^0 (Y)\), and (5) \(NS(X)\to NS(Y)\), where \(Z[{\mathcal O}_x (1)]\) is the subgroup of Pic\((X)\) generated by the class of \(({\mathcal O}_x (1)\) (via the embedding \(X\subset P\)), Pic\(^\tau(X)\) (resp. Pic\(^0(X)\)) is the subgroup of Pic\((X)\) consisting of all classes of invertible \({\mathcal O}_x\)-modules numerically (resp. algebraically) equivalent to zero, and NS\((X)\) is the Néron-Severi group of \(X\).
Theorem. Assume moreover that \(H^q(X,{\mathcal O}_X(m))=0\) for every \(m<0\) and every \(1\leq q <\dim (X)\). Then the maps (1), (2) and (5) are injective and have cokernels all isomorphic to the same group \(E\), which is free of finite rank if char\((K) =0\), and which is \(e\)-torsion-free of finite rank if char\((K)=p\)
and \(e\) is any positive integer prime to \(p\). Moreover, the map (4) is always an isomorphism, and if char\((K) =0\), then (3) is also an isomorphism.
Note that the cohomological hypothesis about \(X\) is always fulfilled if \(K\) is the complex field by Kodaira's vanishing theorem. On the other hand, if \(X= P\), then the above theorem is proved by \textit{P. Deligne} [SGA 7 II, Lect. Notes Math. 340, Exposé
XI, 39--61 (1973; Zbl 0265.14007)], and even in a better formulation (in this case \(E\) turns out to be torsion-free in arbitrary characteristic). The methods used are different from the methods used by Deligne, and consist of the standard
Lefschetz' theory in Grothendieck's form combined with some facts from the theory of Picard schemes. If \(K\) is the complex field, one also relates this theorem with the topological Lefschetz theorem on the singular integral cohomology. Several corollaries of this theorem are derived, one of them extending a result of \textit{L. Robbiano} [Nagoya Math. J. 61, 103--111 (1976; Zbl 0309.14043)]. Bădescu L.: A remark on the Grothendieck--Lefschetz theorem about the Picard group. Nagoya Math. J. 71, 169--179 (1978) Complete intersections, Picard groups, Structure of families (Picard-Lefschetz, monodromy, etc.) A remark on the Grothendieck-Lefschetz theorem about the Picard 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 We explain how quantum affine algebra actions can be used to systematically construct ``exotic'' t-structures. The main idea, roughly speaking, is to take advantage of the two different descriptions of quantum affine algebras, the Drinfeld-Jimbo and the Kac-Moody realizations.
Our main application is to obtain exotic t-structures on certain convolution varieties defined using the Beilinson-Drinfeld and affine Grassmannians. These varieties play an important role in the geometric Langlands program, knot homology constructions, K-theoretic geometric Satake and the coherent Satake category. As a special case we also recover the exotic t-structures of \textit{R. Bezrukavnikov} and \textit{I. Mirković} [Ann. Math. (2) 178, No. 3, 835--919 (2013; Zbl 1293.17021)] on the (Grothendieck-)Springer resolution in type A. t-structures; coherent sheaves; quantum affine algebras Derived categories of sheaves, dg categories, and related constructions in algebraic geometry Exotic t-structures and actions of quantum affine 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 The author constructs the quantum \(b\)-functions of the regular prehomogeneous vector spaces of the commutative type. He also gives their explicit forms. The main result shows that the set of factors of quantum \(b\)-functions corresponds one-to-one to the set of \(b\)-functions. quantum \(b\)-functions; regular prehomogeneous vector spaces Prehomogeneous vector spaces, Homogeneous spaces and generalizations Quantum \(b\)-functions of prehomogeneous vector spaces of commutative parabolic type | 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 \({\mathfrak g}\) be a semisimple simply connected Lie algebra, and let \(U_l\) be the associated quantum group with divided powers, where \(l\) is an even order root of unity. Let \(u_l\) be the corresponding small quantum group. The authors prove that the category of \(u_l\)-modules is naturally equivalent to the category of \(U_l\)-modules which satisfy the Hecke eigen-condition with respect to representations lifted by means of the quantum Frobenius map \(U_l\rightarrow U(\check{\mathfrak g})\), where \(\check{\mathfrak g}\) is the Langlands dual Lie algebra. This result is used to describe the regular block \(u_l\text{-mod}_0\) of the category of \(u_l\)-modules in terms of perverse sheaves on the enhanced affine flag variety satisfying the Hecke eigen-condition. quantum group; small quantum group; quantum Frobenius morphism; monoidal category; affine flag variety S. Arkhipov and D. Gaitsgory, ''Another realization of the category of modules over the small quantum group,'' Adv. Math., vol. 173, iss. 1, pp. 114-143, 2003. Quantum groups (quantized enveloping algebras) and related deformations, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act Another realization of the category of modules over the small quantum 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 Let \(W=m_1P_1+\cdots+m_sP_s\) be a scheme of fat points in \(\mathbb P^n_K\), where \(K\) is a field of characteristic zero.
The authors deal with the problem of computing the Hilbert polynomial of the modules of the Kähler differential \(k\)-forms of the coordinate ring of \(W\), \(\Omega^{k}_{R_{W/K}}\). After introducing notation and basic facts in Section 2, they show in Theorem 3.7 that the Hilbert polynomial of \(\Omega^{n+1}_{R_{W/K}}\) is \(\sum_j\binom{m_j+n-2}{n},\) i.e., it is the Hilbert polynomial of the coordinate ring of the fat points scheme \((m_1-1)P_1+\cdots+(m_s-1)P_s\).
This answers positively to a conjecture the authors stated in a previous paper, see Conjecture 5.7 in [\textit{M. Kreuzer} et al., J. Algebra 501, 255--284 (2018; Zbl 1388.13051)].
Moreover, making use of Theorem 3.7, the authors compute the Hilbert polynomial of the modules of the Kähler differential of a scheme of fat points in \(\mathbb P^2\), see Proposition 4.1. Hilbert function; fat point scheme; regularity index; Kähler differential module Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Modules of differentials, Cycles and subschemes Hilbert polynomials of Kähler differential modules for fat point 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 In this paper, we study the tangent spaces of the smooth nested Hilbert scheme \(\mathrm{Hilb}^{n,n-1}(\mathbb A^2)\) of points in the plane, and give a general formula for computing the Euler characteristic of a \(\mathbb T^2\)-equivariant locally free sheaf on \(\mathrm{Hilb}^{n,n-1}(\mathbb A^2)\). Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables \(q\) and \(t\) with non-negative integer coefficients. We call this conjecturally positive polynomial as the ``nested \(q,t\)-Catalan series'', for it has many conjectural properties similar to that of the \(q,t\)-Catalan series. Atiyah-Bott Lefschetz formula; (nested) Hilbert scheme of points; tangent spaces; diagonal coinvariants Can, M.: Nested Hilbert schemes and the nested q,t-Catalan series Parametrization (Chow and Hilbert schemes), Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Nested Hilbert schemes and the nested \(q,t\)-Catalan series | 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 paper is to prove an overview of results about classification of quantum groups which were obtained by the authors [Commun. Math. Phys. 344, No. 1, 1--24 (2016; Zbl 1360.17025); J. Math. Phys. 57, No. 5, 051707, 12 p. (2016; Zbl 1338.81238)]. Drinfeld, V.: Quantum groups. In: Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, pp. 798-820 (1987) Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Applications of Lie (super)algebras to physics, etc., Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Hopf algebras and their applications Quantum groups: from the Kulish-Reshetikhin discovery to classification | 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 an earlier paper, [J. Pure Appl. Algebra 104, No. 1, 109-122 (1995; Zbl 0854.16027)], the authors developed their notion of a ``schematic algebra'', a (non-commutative) connected graded Noetherian \(k\)-algebra having a finite collection \(\{S_i\}\) of Ore sets such that a graded left ideal \(I\) with \(I\cap S_i\neq\emptyset\) for all \(i\) must contain a power of the augmentation ideal. They proposed this concept as providing a notion of scheme in the setting of non-commutative geometry. The purpose of the present paper is to provide examples of schematic algebras -- for example, the Rees rings of almost commutative rings, the coordinate ring of quantum \(2\times 2\) matrices, quantum Weyl algebras and the three dimensional Sklyanin algebra are shown to be schematic. connected graded Noetherian algebras; Ore sets; schemes; non-commutative geometry; schematic algebras; Rees rings; quantum Weyl algebras; Sklyanin algebras Van Oystaeyen, F., Willaert, L.: Examples and quantum sections of schematic algebras. J. Pure Appl. Algebra 2(120), 195--211 (1997) Ore rings, multiplicative sets, Ore localization, Noncommutative algebraic geometry, Deformations of associative rings, Graded rings and modules (associative rings and algebras), Associative rings of functions, subdirect products, sheaves of rings, Localization and associative Noetherian rings Examples and quantum sections of schematic 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 Let\({\mathcal A}_{\mathbb C}=\{H_{1},\dots ,H_{n}\}\) be a (central) arrangement of hyperplanes in \({\mathbb C}^{d}\) and \({\mathcal M}({\mathcal A}_{\mathbb C})\) the dependence matroid of the linear forms \(\{\theta_{H_i}\in ({\mathbb C}^d)^*\mid \text{Ker} (\theta_{H_i})=H_{i}\}\). The Orlik-Solomon algebra OS\(({\mathcal M})\) of a matroid \({\mathcal M}\) is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The graded algebra OA(\({\mathcal M}({\mathcal A}_{\mathbb C}))\) is isomorphic to the cohomology algebra of the manifold \({\mathfrak M}={\mathbb C}^{d}\setminus\bigcup_{H\in{\mathcal A}_{\mathbb C}}H\). The Tutte polynomial \(T_{\mathcal M}(x,y)\) is a powerful invariant of the matroid \({\mathcal M}\). When \({\mathcal M}({\mathcal A}_{\mathbb C})\) is a rank 3 matroid and the \({\theta}_{H_i}\) are complexifications of real linear forms, we prove that OS\((\mathcal M)\) determines \(T_{\mathcal M}(x,y)\). This result partially solves a conjecture of Falk. Arrangement of hyperplanes; Matroid; Orlik-Solomon algebra; Tutte polynomial FORGE, <a, href= Combinatorial aspects of matroids and geometric lattices, de Rham cohomology and algebraic geometry, Relations with arrangements of hyperplanes A note on Tutte polynomials and Orlik--Solomon 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 We show that normalized quantum K-theoretic vertex functions for cotangent bundles of partial flag varieties are the eigenfunctions of quantum trigonometric Ruijsenaars-Schneider (tRS) Hamiltonians. Using recently observed relations between quantum Knizhnik-Zamolodchikov (qKZ) equations and tRS integrable system we derive a nontrivial identity for vertex functions with relative insertions. Special quantum systems, such as solvable systems, Selfadjoint operator theory in quantum theory, including spectral analysis, \(n\)-vertex theorems via direct methods, Grassmannians, Schubert varieties, flag manifolds, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) qKZ/tRS duality via quantum K-theoretic counts | 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 a conjecture of Oblomkov-Shende, relating the geometry of the Hilbert scheme of points of a plane curve singularity to the HOMFLY polynomial of its link. Following \textit{D.-E. Diaconescu} et al. [Commun. Number Theory Phys. 6, No. 3, 517--600 (2012; Zbl 1276.14065)] the Hilbert scheme side is reformulated in terms of the moduli space of stable pairs on a small resolution \(Y\) of a three-dimensional \(A_1\)-singularity, into which the plane curve \(C\) is embedded. By definition a \(C\)-framed stable pair is a pure one-dimensional sheaf \(\mathcal{F}\) on \(Y\) with section \(\sigma\), such that \(\mathrm{Coker}(\sigma)\) is zero-dimensional and outside the exceptional curve the pair \((\mathcal{F},\sigma)\) is isomorphic to the restriction of the surjection \(\mathcal{O}_Y \twoheadrightarrow \mathcal{O}_C\). Let \(\mathcal{P}(Y,C,r,n)\) be the moduli space of \(C\)-framed stable pairs with generic multiplicity \(r\) of the support of \(\mathcal F\) along the exceptional curve and \(n=\chi(\mathcal{\overline{F}})-\chi(\mathcal{O}_{\overline{C}})\) (computed on the projective closure). Then \(\mathbf{Z}'(Y,C;q,Q)\) is the generating function \(\sum q^nQ^r\chi_{\mathrm{top}}(\mathcal{P}(Y,C,r,n))\), normalised by dividing by \(\prod_k (1+q^kQ)^k\). There is also a colored variant \(\mathbf{Z}'(Y,C,\vec{\mu};q,Q)\). On the knot side there exists a colored variant \(\mathbf{W}(\mathcal{L},\vec{\lambda};v,s)\) of the HOMFLY polynomial. The main result of the paper is that there exist integers \(a(C,\vec{\mu})\), \(b(C,\vec{\mu})\) and a sign \((-1)^{\epsilon(C,\vec{\mu})}\) such that
\[
\mathbf{Z}'(Y,C,\vec{\mu};s^2,-v^2)=(-1)^\epsilon v^as^b\mathbf{W}(\mathcal{L},\vec{\lambda};v,s)\;.
\]
The monomial shift involved is written out in the uncolored case. The strategy of the proof is to use the behaviour under a flop to prove a blow-up formula for \(\mathbf{Z}'\). There is a corresponding, compatible formula on the link side. This allows reduction to the nodal case. plane curve singularity; Hilbert scheme; framed stable pairs; HOMFLY polynomial; wall crossing Maulik, D., Stable pairs and the HOMFLY polynomial, Invent. Math., 204, 3, 787-831, (2016) Milnor fibration; relations with knot theory, Singularities of curves, local rings, Knots and links in the 3-sphere Stable pairs and the HOMFLY polynomial | 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 an almost simple simply connected complex Lie group, and let \(G/U_-\) be its base affine space. In this paper we formulate a conjecture, which provides a new geometric interpretation of the Macdonald polynomials associated to \(G\) via perverse coherent sheaves on the scheme of formal arcs in the affinization of \(G/U_-\). We prove our conjecture for \(G=\mathrm{SL}(N)\) using the so called Laumon resolution of the space of quasi-maps (using this resolution one can reformulate the statement so that only ``usual'' (not perverse) coherent sheaves are used). In the course of the proof we also give a \(K\)-theoretic version of the main result of \textit{A. Negut} [Invent. Math. 178, No. 2, 299--331 (2009; Zbl 1185.37140)]. A. Braverman, M. Finkelberg and J. Shiraishi, \textit{Macdonald polynomials, Laumon spaces and perverse coherent sheaves}, arXiv:1206.3131. Classical groups (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, (Co)homology theory in algebraic geometry, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Loop groups and related constructions, group-theoretic treatment Macdonald polynomials, Laumon spaces and perverse coherent sheaves | 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 the author proves that a finitely generated module over a Noetherian ring defines a unique cycle class in the components with codimension zero and one of the Chow group of the ring. This is an interesting result. It generalizes a classical result over integrally closed domains and implies the isomorphism between the Chow group and the Grothendieck group \((K_0\)-group) under certain conditions. In addition, the author also discusses the difference between the map constructed in this paper and the Riemann-Roch map. filtration of a module; finitely generated module; cycle class; Chow group; Grothendieck group; Riemann-Roch map Chan, C.-Y.: Filtrations of modules, the Chow group, and the Grothendieck group. J. Algebra \textbf{219}(1), 330-344 (1999) Grothendieck groups, \(K\)-theory and commutative rings, (Equivariant) Chow groups and rings; motives, \(K_0\) of other rings Filtrations of modules, the Chow group, and the 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 We describe mirror symmetry for weak Fano toric manifolds as an equivalence of filtered \( \mathcal {D}\)-modules. We discuss in particular the logarithmic degeneration behavior at the large radius limit point and express the mirror correspondence as an isomorphism of Frobenius manifolds with logarithmic poles. The main tool is an identification of the Gauß-Manin system of the mirror Landau-Ginzburg model with a hypergeometric \( \mathcal {D}\)-module, and a detailed study of a natural filtration defined on this differential system. We obtain a solution of the Birkhoff problem for lattices defined by this filtration and show the existence of a primitive form, which yields the construction of Frobenius structures with logarithmic poles associated to the mirror Laurent polynomial. As a final application, we show the existence of a pure polarized non-commutative Hodge structure on a Zariski open subset of the complexified Kähler moduli space of the variety. T. Reichelt and C. Sevenheck, Logarithmic Frobenius manifolds, hypergeometric systems and quantum \(\mathcal{D}\)-modules, J. Algebraic Geom. 24 (2015), 201--281. Mirror symmetry (algebro-geometric aspects), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Fano varieties, Toric varieties, Newton polyhedra, Okounkov bodies, Global surface theory (convex surfaces à la A. D. Aleksandrov) Logarithmic Frobenius manifolds, hypergeometric systems and quantum \(\mathcal D\)-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 This paper places itself within the rich framework of the literature regarding quantum cohomology of homogeneous varieties produced by the authors themselves. This new piece of mathematics regards certain \textsl{Quantum Giambelli} formulas for isotropic Grassmannians. If \(V\) is a vector space equipped with a non degenerate bilinear form \(\eta\), one can consider the Grassmannian \(X:=IG(k,V)\) of isotropic subspaces of \(V\) of a fixed dimension \(k\), namely the variety of those points \([W]\in G(k,V)\) such that the restriction of \(\eta\) to \(W\) is trivial (i.e. \(\eta_{|W\times W}=0\)).
If the bilinear form is skew-symmetric, the dimension of \(V\) is even and one is then concerned with symplectic vector spaces. As well known, the cohomology ring \(H^*(X,{\mathbb{Z}})\) is generated as a \({\mathbb{Z}}\)-algebra by certain special Schubert cycles and it is also a well known fact that such cycles generate the quantum cohomology of \(X\) as well. The latter is a deformation of the usual cohomology encoding the Gromov-Witten invariants which count, roughly speaking, numbers of maps of a given degree from the projective line to \(X\). The authors find and prove quantum Giambelli's formulas expressing an arbitrary Schubert class in the small quantum cohomology ring of \(X\) as a polynomial in the special Schubert classes alluded above.
The two main theorems of the article (concerning Giambelli's formulas) are analogous to those proven for the quantum cohomology of the orthogonal and Lagrangian Grassmannians in [J. Algebr. Geom. 12, No. 4, 777--810 (2003; Zbl 1051.53070)] and [Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083)], by \textit{A. Kresch} and \textit{H. Tamvakis}. The proof are however quite different, due to the fact that for non maximal isotropic Grassmannians, the explicit recursion used in the quoted references is no longer available. The latter is replaced, however, by another kind of recursion, neatly steted and proved in Proposition 3.
The reviewed paper is for all people interested in the combinatorial aspects of cohomology theories (quantum, equivariant, quantum-equivariant) of homogeneous varieties. quantum Schubert Calculus; isotropic Grassmannians; Giambelli's Formulas Buch, AS; Kresch, A; Tamvakis, H, Quantum Giambelli formulas for isotropic Grassmannians, Math. Ann., 354, 801-812, (2012) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Quantum Giambelli formulas for isotropic 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 Let \(k\) be a field of characteristic zero, \(A=k[x_ 1,\dots,x_ n]\) the \(k\)-algebra of polynomials in \(n\) indeterminates and \(D=D_{A/k}=A[\partial_ 1,\dots,\partial_ n]\). Let \(s\) be an indeterminate over \(k\); for every non-zero \(f\in A\) denote by \({\mathbf F}[s]\) the free \(A_ f[s]\)-module generated by a symbol \({\mathbf f}^ s\); \({\mathbf F}[s]\) carries a canonical structure as \(D_ f[s]\)-module defined by \(\partial g{\mathbf f}^ s=(\partial g+sgf^{-1}){\mathbf f}^ s\) for every \(\partial\in\text{Der}_ k(A)\) and every \(g\in A_ f[s]\). \textit{I. N. Bernstein} has shown [cf. Funct. Anal. Appl. 6(1972), 273-285 (1973); translation from Funkts. Anal. Prilozh. 6, No. 4, 26-40 (1972; Zbl 0282.46038)] that there exists a polynomial \(b(s)\in k[s]\backslash\{0\}\) and a differential operator \(P(s)\in D(s)\) such that \(b(s){\mathbf f}^ s=P(s)f{\mathbf f}^ s\). This result plays an important role in the theory of \({\mathcal D}_ X\)-modules, \(X\) a variety over \(k\) or \(k=\mathbb{C}\) and \(X\) an analytic variety. In particular, it can be used to prove the finiteness of De Rham cohomology for non-singular varieties. The authors of the present paper generalize the functional equation given above for the case of Tate algebras and Dwork-Monsky-Washnitzer algebras. They consider commutative \(k\)-algebras \(A\) which are noetherian and regular, equicodimensional of dimension \(n\), such that \(A/{\mathfrak m}\) is algebraic over \(k\) for every maximal ideal \({\mathfrak m}\) of \(A\) and such that there exist \(x_ 1,\dots,x_ n\in A\) and \(\partial_ 1,\dots,\partial_ n\in\text{Der}_ k(A)\) satisfying \(\partial_ i(x_ j)=\delta_{ij}\). They prove a functional equation as above [theorem 3.1.1]; in particular, for every \(D\)-module of minimal dimension, \(M_ f\) is a \(D\)-module of finite type. In section 4 the authors apply this result to the algebraic and formal case, and to the case of Tate-algebras and algebras of Dwork-Monsky-Washnitzer. Bernstein-Sato polynomial; \({\mathcal D}\)-modules; De Rham cohomology; Tate algebras; Dwork-Monsky-Washnitzer algebras Mebkhout, Z.; Narváez-Macarro, L., La théorie du polynôme de Bernstein-Sato pour les algèbres de Tate et de Dwork-Monsky-Washnitzer, Ann. Sci. École Norm. Sup. (4), 24, 2, 227-256, (1991) 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, Local ground fields in algebraic geometry Theory of the Bernstein-Sato polynomial for Tate and Dwork-Monsky-Washnitzer 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 thesis develops further the results of an article by \textit{R. Kaufmann}, \textit{Yu. Manin} and \textit{D. Zagier} [Commun. Math. Phys. 181, No. 3, 763-787 (1996; Zbl 0890.14011)] and their applications.
It has three chapters. The first one discusses the intersection of strata classes in \(\overline M_{g,n}\) and provides the formulae for the intersection matrices in cohomology of \(\overline{M_{0,n}}\) (moduli space of pointed curves of genus 0), and for Weil-Petersson volumes. These results are used in chapter two to define the tensor product of the pointed Frobenius manifolds. In chapter 3 the formulae from chapters 1 and 2 are used to derive the explicit Künneth formula for quantum cohomology. Examples of products of two and three 3-dimensional Calabi-Yau manifolds are discussed at the end. moduli space of pointed curves of genus 0; Weil-Petersson volumes; Künneth formula for quantum cohomology; Calabi-Yau manifolds Families, moduli of curves (algebraic), (Co)homology theory in algebraic geometry, Generalized (extraordinary) homology and cohomology theories in algebraic topology, String and superstring theories; other extended objects (e.g., branes) in quantum field theory 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 | 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 Weierstrass polynomial of degree \(n\geq 1\) over a topological space X is a polynomial function \(P: X\times {\mathbb{C}}\to {\mathbb{C}}\) of the form \(P(x,z)=z^ n+\sum^{n}_{i=1}a_ i(x)z^{n-i}\), where \(a_ 1,...,a_ n: X\to {\mathbb{C}}\) are continuous, complex valued functions. If P(x,z) has no multiple roots for any \(x\in X\), we call it a ``simple Weierstrass polynomial.'' We associate to a simple Weierstrass polynomial P(x,z) of degree n, an n-fold polynomial covering map \(\pi: E\to X,\) where \(E\subset X\times {\mathbb{C}}\) is the zero set for P(x,z) and \(\pi\) is the projection onto X. We say that two simple Weierstrass polynomials over X are ``topological equivalent'' if their associated polynomial covering maps are equivalent.
The first statement is that any simple Weierstrass polynomial over X is topologically equivalent to a simple Weierstrass polynomial of the form \(P(x,z)=z^ n+\sum^{n}_{i=1}\tilde a_ i(x)z^{n-i}\) with \(\tilde a_ 1(x)=0\) for all \(x\in X\). The most elementary type of Weierstrass polynomials over X are those of the form \(P(x,z)=z^ n-q(x)\) \(n\geq 1\), where \(q: X\to {\mathbb{C}}\) is a continuous function. By the above result, any simple Weierstrass polynomial of degree 2 over X is topologically equivalent to the above form.
In this paper the author tries to simplify a Weierstrass polynomial of the above form within its topological equivalence class. The statement is that a simple Weierstrass polynomial over X of the form \(P(x,z)=z^ n- q(x)\) is topologically equivalent to its discriminant radical. Here, the discriminant radical of P(x,z) is a Weierstrass polynomial defined by the discriminant function of P(x,z).
The author also studies topological structures on the associated polynomial covering maps. Weierstrass polynomial over a topological space; simple Weierstrass polynomial; polynomial covering map; discriminant radical V.L. Hansen, Algebra and topology of Weierstrass polynomials, Expositiones Mathematicae, to appear. Covering spaces and low-dimensional topology, Coverings in algebraic geometry, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) Algebra and topology of Weierstrass 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 Consider the generalized flag manifold \(G/B\) and the corresponding affine flag manifold \(\mathcal{F}\ell_{G}\). In this paper we use curve neighborhoods for Schubert varieties in \(\mathcal{F}\ell_G\) to construct certain affine Gromov -- Witten invariants of \(\mathcal{F}\ell_G\), and to obtain a family of `affine quantum Chevalley' operators \(\Lambda_0,\ldots,\Lambda_n\) indexed by the simple roots in the affine root system of \(G\). These operators act on the cohomology ring \(H^\ast(\mathcal{F}\ell_G)\) with coefficients in \(\mathbb{Z}[q_0,\ldots,q_n]\). By analyzing commutativity and invariance properties of these operators we deduce the existence of two quantum cohomology rings, which satisfy properties conjectured earlier by Guest and Otofuji for \(G= \mathrm{SL}_n(\mathbb{C})\). The first quantum ring is a deformation of the subalgebra of \(H^\ast(\mathcal{F}\ell_G)\) generated by divisors. The second ring, denoted \(QH_{aff}^\ast (G/B)\), deforms the ordinary quantum cohomology ring \(QH^\ast(G/B)\) by adding an affine quantum parameter \(q_0\). We prove that \(QH_{aff}^\ast (G/B)\) is a Frobenius algebra, and that the new quantum product determines a flat Dubrovin connection. Further, we develop an analogue of Givental and Kim formalism for this ring and we deduce a presentation of \(QH_{aff}^\ast (G/B)\) by generators and relations. The ideal of relations is generated by the integrals of motion for the periodic Toda lattice associated to the dual of the extended Dynkin diagram of \(G\). Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) An affine quantum cohomology ring for flag manifolds and the periodic Toda lattice | 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 instanton corrections to correlators in the genus-zero topological subsector of a (0, 2) supersymmetric gauged linear sigma model with target space \(\mathbb{P}^1 \times \mathbb{P}^1\), whose left-moving fermions couple to a deformation of the tangent bundle. We then deduce the theory's chiral ring from these correlators, which reduces in the limit of zero deformation to the (2, 2) ring. Finally, we compare our results with the computations carried out by \textit{A. Adams} et al. [Adv. Theor. Math. Phys. 7, No. 5, 865-885 (2003; Zbl 1058.81064)] and \textit{S.H. Katz} and \textit{E. Sharpe} [Commun. Math. Phys. 262, No. 3, 611-644 (2006; Zbl 1109.81066)]. We find immediate agreement with the latter and an interesting puzzle in completely matching the chiral ring of the former. superstrings and heterotic strings; topological field theories Guffin, J.; Katz, S., \textit{deformed quantum cohomology and} (0, 2) \textit{mirror symmetry}, JHEP, 08, 109, (2010) String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Topological field theories in quantum mechanics, Model quantum field theories, Supersymmetric field theories in quantum mechanics, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Deformed quantum cohomology and (0,2) mirror symmetry | 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 \(S\) be a fixed Noetherian scheme and \(\mathcal S\) the category of separated, essentially finite presentation \(S\)-schemes. If \(X \rightarrow S\) is an object of \(\mathcal S\) and \(\delta_X : X \rightarrow X\times_SX\) is the diagonal embedding then the \textit{pre-Hochschild functor} is
\[
{\mathcal H}_X := \text{L}\delta_X^\ast \text{R}\delta_{X\ast} : \text{D}(X) \longrightarrow \text{D}(X)
\]
where \(\text{D}(X)\) is the derived category of the category of \({\mathcal O}_X\)-modules. If \(f : X \rightarrow Y\) is a morphism in \(\mathcal S\) then one defines a morphism \(\gamma_f : \text{L}f^\ast{\mathcal H}_Y \rightarrow {\mathcal H}_X\text{L}f^\ast\) by:
\[ \text{L}f^\ast\text{L}\delta_Y^\ast \text{R}\delta_{Y\ast} = \text{L}\delta_X^\ast\text{L}(f\times_Sf)^\ast\text{R}\delta_{Y\ast} \stackrel{{\text{L}\delta_X^\ast \theta}}{\longrightarrow}\text{L}\delta_X^\ast\text{R}\delta_{X\ast}\text{L}f^\ast
\]
where \(\theta : \text{L}(f\times_Sf)^\ast\text{R}\delta_{Y\ast} \rightarrow \text{R}\delta_{X\ast}\text{L}f^\ast\) is the \textit{base-change morphism} associated to the diagram:
\[
\begin{tikzcd} X \ar[r,"f"]\ar[d, "\delta_X" '] & Y\ar[d, "\delta_Y"] \\ X\times_SX \ar[r, "f\times_Sf" ']& Y\times_SY \end{tikzcd}
\]
The authors of the paper under review show that if \(f\) is essentially étale then \(\gamma_f({\mathcal F}^\bullet)\) is an isomorphism for every \({\mathcal F}^\bullet \in \, \text{D}_{\text{qc}}(Y)\) (this means that \({\mathcal F}^\bullet\) has quasi-coherent cohomology sheaves).
\textit{S. Nayak} [Adv. Math. 222, No. 2, 527--546 (2009; Zbl 1175.14003)] proved a factorization result allowing one to extend the compactification theorem of Nagata to morphisms of essentially finite type. As a consequence, one can unify the local and global Grothendieck duality theories and show that there exists a contravariant \(\text{D}_{\text{qc}}^+\)-valued pseudofunctor \((-)^!_+\) over \(\mathcal S\) (called the \textit{twisted inverse image}) such that \(f_+^! : \text{D}_{\text{qc}}^+(Y) \rightarrow \text{D}_{\text{qc}}^+(X)\) is a right-adjoint to \(\text{R}f_\ast\) is \(f : X \rightarrow Y\) is proper (recall that \(\text{L}f^\ast\) is a left-adjoint to \(\text{R}f_\ast\)) and such that \(f_+^! = f^\ast\) if \(f : X \rightarrow Y\) is essentially étale. According to Theorem 5.9 in Nayak's paper, if \(f : X \rightarrow Y\) is a \textit{perfect} morphism in \(\mathcal S\) (this means that \({\mathcal O}_{X,x}\) is a finite Tor-dimension \({\mathcal O}_{Y,f(x)}\)-module, \(\forall \, x \in X\)) then one can extend \(f_+^!\) to a functor \(f^! : \text{D}_{\text{qc}}(Y) \rightarrow \text{D}_{\text{qc}}(X)\) defined by the formula \(f^! := f_+^!{\mathcal O}_Y\otimes^{\text{L}}_{{\mathcal O}_X}\text{L}f^\ast\). (This extension is necessary for the purposes of the paper under review because Hochschild homology involves complexes that are bounded above, not bellow). Let
\[
\begin{tikzcd} X^\prime \ar[r, "v"] \ar[d, "g" '] & X\ar[d, "f"] \\ Y^\prime \ar[r,"u" '] & Y \end{tikzcd} \tag{\(*\)}
\]
be a \textit{Cartesian square} in \(\mathcal S\). It is a basic fact of Grothendieck duality theory that if \(f\) is proper and \(u\) is flat then the base-change morphism for the twisted inverse image \(v^\ast f_+^! \rightarrow g_+^!u^\ast\) which is the adjoint to the composite map:
\[
\text{R}g_\ast v^\ast f_+^! \overset{{\theta}^{-1}}\longrightarrow u^\ast\text{R}f_\ast f_+^! \longrightarrow u^\ast
\]
(with the first map deduced from the base-change isomorphism \(\theta : u^\ast\text{R}f_\ast \overset\sim\rightarrow \text{R}g_\ast v^\ast\) and with the second map deduced from the counit \(\text{R}f_\ast f_+^! \rightarrow \text{id}\)) is an isomorphism (when applied to objects of \(\text{D}_{\text{qc}}^+(Y)\)). In their previous paper [Asian J. Math. 15, No. 3, 451--498 (2011; Zbl 1251.14010)], the authors of the paper under review showed that this isomorphism can be extended to an isomorphism \(\text{B} : v^\ast f^! \overset\sim\rightarrow g^!u^\ast\) (if \(f\) is perfect).
In the paper under review the authors define, for any flat morphism \(f : X \rightarrow Y\) in \(\mathcal S\), a functor \(\text{c}_f : {\mathcal H}_Xf^\ast \rightarrow f^!{\mathcal H}_Y\) (called the \textit{fundamental class} of \(f\)) and verify its \textit{transitivity} with respect to compositions \(X \overset{f}\rightarrow Y \overset{g}\rightarrow Z\) of flat morphisms and its compatibility with \textit{flat base change}. In order to understand the nature of the difficulties encountered by the authors of the paper under review in proving their results we shall reproduce their definition of the fundamental class.
Let \(f : X \rightarrow S\) be a flat morphism in \(\mathcal S\). Consider the diagram
\[
\begin{tikzcd} X \ar[r, "\delta_f"] & X\times_YX \ar[d,"p_2" '] \ar[r,"p_1"]& X\ar[d,"f"]\\ {}& X \ar[r,"f" ']& Y \end{tikzcd}
\]
Let \(\text{R}\delta_{f\ast} \rightarrow p_2^!\) be the morphism deduced from the counit \(\text{R}\delta_{f\ast}\delta_{f+}^!{\mathcal O}_{X\times_YX} \rightarrow {\mathcal O}_{X\times_YX}\) and from the fact that \(p_2 \circ \delta_f = \text{id}_X\) and consider the composite morphism
\[
\begin{tikzcd} \mathrm R\delta_{f\ast}f^\ast \rar & p_2^!f^\ast \rar["\mathrm{B}^{-1}"] & p_1^\ast f^! \rlap{\,.} \end{tikzcd}
\]
Consider, next, the Cartesian square:
\[
\begin{tikzcd} X\times_YX \rar["p_1"]\dar["i" '] & X\dar["\Gamma"]\\ X\times_SX \rar["\text{id}_X\times_Sf" '] & X\times_SY \end{tikzcd}
\]
where \(\Gamma\) is the graph of \(f\). Apply \(\text{R}i_\ast\) to the above defined composite morphism and consider the composite morphism:
\[
\begin{tikzcd} \text{R}\delta_{X\ast}f^\ast = \text{R}i_\ast\text{R}\delta_{f\ast}f^\ast \rar \mathrm{R}i_\ast p_1^\ast f^! \rar["\theta^{-1}f^!"] & (\text{id}_X\times_Sf)^\ast \text{R}\Gamma_\ast f^! \end{tikzcd}
\]
where \(\theta : (\text{id}_X\times_Sf)^\ast \text{R}\Gamma_\ast \rightarrow \text{R}i_\ast p_1^\ast\) is the base-change isomorphism. Applying \(\text{L}\delta_X^\ast = \text{L}\delta_f^\ast \text{L}i^\ast\) to this composite morphism and taking into account that \(p_1 \circ \delta_f = \text{id}_X\) one gets a composite morphism:
\[
\text{a}_f : \text{L}\delta_X^\ast \text{R}\delta_{X\ast} f^\ast \rightarrow \text{L}\delta_f^\ast \text{L}i^\ast (\text{id}_X\times_Sf)^\ast \text{R}\Gamma_\ast f^! \overset\sim\longrightarrow \text{L}\delta_f^\ast p_1^\ast \text{L}\Gamma^\ast \text{R}\Gamma_\ast f^! \overset\sim\longrightarrow \text{L}\Gamma^\ast \text{R}\Gamma_\ast f^! \, .
\]
On the other hand, consider the Cartesian square:
\[
\begin{tikzcd} X \rar["f"]\dar["\Gamma" '] & Y\dar["\delta_Y"]\\ X\times_SY \rar["f\times_S\text{id}_Y" '] & Y\times_SY \end{tikzcd}
\]
Applying \(\text{L}\Gamma^\ast\) to the base-change isomorphism \(\theta : (f\times_S\text{id}_Y)^\ast \text{R}\delta_{Y\ast} \overset\sim\rightarrow \text{R}\Gamma_\ast f^\ast\) one gets an isomorphism \(\phi : f^\ast \text{L}\delta_Y^\ast \text{R}\delta_{Y\ast} \overset\sim\rightarrow \text{L}\Gamma^\ast \text{R}\Gamma_\ast f^\ast\). If \(p : X\times_SY \rightarrow X\) is the projection on the first factor then the \textit{projection formula} provides, for \({\mathcal A}^\bullet,\, {\mathcal B}^\bullet \in \text{D}_{\text{qc}}(X)\), an isomorphism:
\[
\beta : \text{L}p^\ast {\mathcal A}^\bullet \otimes_{{\mathcal O}_{X\times_SY}}^{\text{L}} \text{R}\Gamma_\ast {\mathcal B}^\bullet \overset\sim\longrightarrow \text{R}\Gamma_\ast(\text{L}\Gamma^\ast \text{L}p^\ast {\mathcal A}^\bullet \otimes_{{\mathcal O}_X}^{\text{L}}{\mathcal B}^\bullet) \overset\sim\rightarrow \text{R}\Gamma_\ast({\mathcal A}^\bullet \otimes_{{\mathcal O}_X}^{\text{L}} {\mathcal B}^\bullet)
\]
from which one deduces an isomorphism:
\[
\begin{gathered} \begin{tikzcd} \text{L}\Gamma^\ast \text{R}\Gamma_\ast({\mathcal A}^\bullet \otimes_{{\mathcal O}_X}^{\text{L}} {\mathcal B}^\bullet) \rar["\text{L}\Gamma^\ast \beta^{-1}"] & \text{L}\Gamma^\ast (\text{L}p^\ast {\mathcal A}^\bullet \otimes_{{\mathcal O}_{X\times_SY}}^{\text{L}}\text{R}\Gamma_\ast {\mathcal B}^\bullet) \overset\sim\rightarrow \end{tikzcd}\\ \text{L}\Gamma^\ast \text{L}p^\ast {\mathcal A}^\bullet \otimes_{{\mathcal O}_X}^{\text{L}}\text{L}\Gamma^\ast \text{R}\Gamma_\ast {\mathcal B}^\bullet \overset\sim\rightarrow {\mathcal A}^\bullet \otimes_{{\mathcal O}_X}^{\text{L}} \text{L}\Gamma^\ast \text{R}\Gamma_\ast {\mathcal B}^\bullet \, . \end{gathered}
\]
Let \(\text{b}_f\) be the composite isomorphism:
\[
\begin{gathered} \text{b}_f : \text{L}\Gamma^\ast \text{R}\Gamma_\ast f^! = \text{L}\Gamma^\ast \text{R}\Gamma_\ast (f_+^!{\mathcal O}_Y \otimes_{{\mathcal O}_X}^{\text{L}} f^\ast ) \overset\sim\rightarrow f_+^!{\mathcal O}_Y \otimes_{{\mathcal O}_X}^{\text{L}} \text{L}\Gamma^\ast \text{R}\Gamma_\ast f^\ast \rightarrow\\ \begin{tikzcd} {}\rar["\text{id} \otimes \phi^{-1}"] & f_+^!{\mathcal O}_Y \otimes_{{\mathcal O}_X}^{\text{L}} f^\ast \text{L}\delta_Y^\ast \text{R}\delta_{Y\ast} = f^!\text{L}\delta_Y^\ast \text{R}\delta_{Y\ast} . \end{tikzcd} \end{gathered}
\]
The \textit{fundamental class} \(\text{c}_f : {\mathcal H}_Xf^\ast \rightarrow f^! {\mathcal H}_Y\) is, by definition, \(\text{b}_f \circ \text{a}_f\). The authors of the paper under review prove, after nontrivial verifications of diagram commutativities, that:
\begin{itemize}
\item[(1)] If \(f\) is essentially étale, so that \(f^! = f^\ast\), then \(\text{c}_f\) is the inverse of the already established isomorphism \(\gamma_f : f^\ast {\mathcal H}_Y \overset\sim\rightarrow {\mathcal H}_Xf^\ast\).
\item[(2)] If \(X \overset{f}\rightarrow Y \overset{g}\rightarrow Z\) are flat morphisms in \(\mathcal S\) then \(\text{c}_{gf} = f^!\text{c}_g \circ \text{c}_fg^\ast\). This result shows that fundamental classes are \textit{orientations} for the flat maps in the \textit{bivariant Hochschild theory} constructed by the authors in their previous paper mentioned above.
\item[(3)] For any Cartesian square \((*)\) in \(\mathcal S\) with \(f,\, g,\, u,\, v\) flat morphisms, the following diagram commutes:
\[
\begin{tikzcd} v^\ast {\mathcal H}_Xf^\ast \rar["v^\ast \mathrm{c}_f"]\dar["\gamma_vf^\ast" '] & v^\ast f^! \mathcal H_Y\dar["\wr" ', "\mathrm B\mathcal H_Y"]\\ \mathcal H_{X^\prime}v^\ast f^\ast \dar[equal] & g^!u^\ast \mathcal H_Y \dar["g^!\gamma_u"]\\ \mathcal H_{X^\prime}g^\ast u^\ast \rar["\mathrm c_gu^\ast" '] & g^!{\mathcal H}_{Y^\prime}u^\ast \end{tikzcd}
\]
\item[(4)] For any separated, essentially finite type, flat \(S\)-scheme \(\xi : X \rightarrow S\) one gets, by composing the fundamental class map \(\text{c}_\xi({\mathcal O}_S) : {\mathcal H}_X({\mathcal O}_X) \rightarrow \xi^!{\mathcal O}_S\) with the natural product map \({\mathcal H}_X({\mathcal O}_X) \otimes_{{\mathcal O}_X}^{\text{L}} {\mathcal H}_X({\mathcal O}_X) \rightarrow {\mathcal H}_X({\mathcal O}_X)\), a \textit{duality map}:
\[
{\mathcal H}_X({\mathcal O}_X) \longrightarrow \text{R}{\mathcal H}om_{{\mathcal O}_X}({\mathcal H}_X({\mathcal O}_X),\, \xi^!{\mathcal O}_S)\, .
\]
If \(\xi\) is \textit{essentially smooth} then this duality map is an isomorphism.
\end{itemize} Hochschild homology; Grothendieck duality; fundamental class; bivariant theory doi:10.1016/j.aim.2014.02.017 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Schemes and morphisms, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Derived categories, triangulated categories Bivariance, Grothendieck duality and Hochschild homology. II: The fundamental class of a flat scheme-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 We study factorization algebras on configuration spaces of points on a curve, colored by elements of the root lattice. Our main result says that the factorization algebra attached to Lusztig's quantum group can be obtained as a direct image of a twisted Whittaker sheaf on the Zastava space. factorization algebras; quantum groups; Whittaker category On factorization algebras arising in the quantum geometric Langlands 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 A geometric construction of Lusztig's modified quantum algebra of symmetric type is presented by using certain localized equivariant derived categories of double framed representation varieties of quivers. modified quantum algebra; canonical basis; equivariant derived category; equivariant perverse sheaf; framed representation variety Quantum groups (quantized enveloping algebras) and related deformations, Group actions on varieties or schemes (quotients), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) A geometric realization of modified quantum 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 In this paper we study a differential equation which arises from the theory of Zolotarev polynomials. By extending a symbolic algorithm for finding rational solutions of algebraic ordinary differential equations, we construct a method for computing explicit expressions for Zolotarev polynomials. This method is an algebraic geometric one and works subject to (radical) parametrization of algebraic curves. As a main application we compute the explicit form of the proper Zolotarev polynomial of degree 5. algebraic curve; algebraic ordinary differential equation; radical parametrization; rational parametrization; Zolotarev polynomial Software, source code, etc. for problems pertaining to ordinary differential equations, Computational aspects of algebraic curves, Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations, Symbolic computation and algebraic computation An algebraic-geometric method for computing Zolotarev 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 We describe the positivity of Thom polynomials of singularities of maps, Lagrangian Thom polynomials and Legendrian Thom polynomials. We show that these positivities come from Schubert calculus. positivity; Grassmannian; Lagrangian Grassmannian; Schubert class; Schur function; \(\tilde Q\)-function; singularity class; Thom polynomial; vector bundle generated by its global sections; ample vector bundle; positive polynomial; nonnegative cycle Classical problems, Schubert calculus, Global theory of complex singularities; cohomological properties, Homology of classifying spaces and characteristic classes in algebraic topology, Singularities of differentiable mappings in differential topology, Symmetric functions and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry Positivity of Thom polynomials and 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 Let \(n\) be a positive integer and \(K[x_1,\dots,x_n]\) be a polynomial ring over an infinite field. Let \(P_n\) denote the set of partitions of \(n\). Given a tableau of shape in \(P_n\), its Specht polynomial is the product of all \(x_i-x_j\), for all \(i<j\) in a same column. If \(\mathcal{F}\) is a non-empty lower filter of \(P_n\) with respect to the dominance order, then \(I_{\mathcal{F}}\) denotes the ideal generated by the Specht polynomials of tableaux \(T\) of shape in \(\mathcal{F}\).
This article consists of a short proof of an unpublished result of \textit{M. Haiman} and \textit{A. Woo} [Garnir modules, Springer fibers, and Ellingsrud-Strømme cells on the Hilbert Scheme of points. Manuscript (2010)] on Specht ideals of filters of \(P_n\). The statement is that (i) \(I_\mathcal{F}\) is a vanishing ideal and (ii) the set of Specht polynomials of tableaux \(T\) of shape in \(\mathcal{F}\) is a universal Gröbner basis of \(I_\mathcal{F}\) (Theorem 1.1). The set of points, of which \(I_\mathcal{F}\) is the vanishing ideal of, is given in terms of \(P_n\setminus \mathcal{F}\). Given a point in \(K^n\), associate to it the partition of \(n\) given by the cardinalities of the maximal subsets of indices on which the coordinates of the point are equal. Let \(H_\mu\) denote the set of points with associated partition \(\mu\). Then \(I_\mathcal{F}\) is the vanishing ideal of the union of \(H_\mu\), as \(\mu\) varies in \(P_n\setminus \mathcal{F}\). Gröbner bases; Specht ideals Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Combinatorial aspects of commutative algebra, Representations of finite symmetric groups, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomial rings and ideals; rings of integer-valued polynomials, Configurations and arrangements of linear subspaces A note on the reducedness and Gröbner bases of Specht 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 Consider \(F(k_{\bullet}, n)\), the partial flag variety of subspaces of \(\mathbb C^n\) of fixed dimensions \(k_1<k_2<\ldots<k_r\). The paper under review studies the three-point GW invariants of \(F(k_{\bullet}, n)\). Several questions about these invariants are considered.
First, the author gives conditions on a curve class that guarantee the vanishing of the three point genus 0 GW invariants in that curve class.
Secondly, to compute the non-zero three point invariants, the author is led to the study of the evaluation morphism at the three markings
\[
\mathrm{ev}_1\times \mathrm{ev}_2\times \mathrm{ev}_3:\overline M_{0, 3}(F(k_{\bullet}, n), d_{\bullet})\to F(k_{\bullet}, n)\times F(k_{\bullet}, n)\times F(k_{\bullet}, n).
\]
This morphism is shown to be birational onto its image under certain assumptions on the curve class.
Furthermore, when the evaluation morphism is birational onto its image, it is shown that, in a large number of cases, the 3-point invariants equal certain intersections of three Schubert cycles in a different flag variety. (See also [\textit{A. S. Buch, A. Kresch, H. Tamvakis}, J. Am. Math. Soc. 16, No. 4, 901--915 (2003; Zbl 1063.53090)]) for an analysis in the case of Grassmannians). Finally, this correspondence leads to certain periodicity properties of the Schubert structure constants for the ordinary cohomology of partial flag varieties. Quantum cohomology; flag varieties; GW invariants Coskun, I.: The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants, Math. ann. 346, No. 2, 419-447 (2010) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants | 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 concerns commutativity between two operations in intersection theory, namely Riemann-Roch transformation and virtual pullback. For a morphism \(f\colon X \to Y\) between nice schemes over a base field \(k\) and with a perfect obstruction theory \(\phi\colon E \to L_f\), we know that
\[
\tau_X(f^! \alpha) = \operatorname{Td}(E^\vee) \cdot f^!\tau_Y(\alpha).
\]
Here \(\tau\) denotes Riemann-Roch transformation, \(f^!\) virtual pullbacks associated to \(\phi\) (in \(G_0\) and Chow groups \(\mathrm{CH}_*\)), and \(\alpha \in G_0(Y)\). For example, the case \(Y=\operatorname{Spec} k\) gives the formula
\[
\tau_X(\mathcal{O}_X^{\text{vir}}) =\operatorname{Td}(T_X^{\text{vir}}) \cdot [X]^{\text{vir}}.
\]
The relationship between \(\tau\) and \(f^!\) is generalized in this paper to quotient stacks. Riemann-Roch type results of Edidin-Gram and Krishna-Sreedhar for stacks are essential to the formulation and the proof of the results in this paper. In particular, for a separated quotient stack \(\mathcal{X}\) (which is then Deligne-Mumford), we should consider the isomorphism
\[
I\tau_{\mathcal{X}}\colon G_0(\mathcal{X}) \to \mathrm{CH}_*(I\mathcal{X}).
\]
in place of \(\tau_{\mathcal{X}}\). Riemann-Roch theorems; equivariant Chow groups; equivariant \(K\)-theory (Equivariant) Chow groups and rings; motives, Riemann-Roch theorems, Group actions on varieties or schemes (quotients), Equivariant \(K\)-theory Virtual equivariant Grothendieck-Riemann-Roch 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 The author considers the Pham-Brieskorn polynomial \(f=z_1^{a_1}+\cdots +z_n^{a_n}\). He gives an explicit construction of Lê's vanishing polyhedra for \(f\) which he uses to give a geometric description of the monodromy associated to \(f\). The latter allows to write the matrix that determines the induced algebraic monodromy. This provides another proof for the Brieskorn-Pham theorem, which says that the characteristic polynomial associated to the monodromy of \(f\) is given by
\(\Delta (t)=\prod (t-\omega _1\omega _2\cdots \omega _n)\), where each \(\omega _j\) ranges over all \(a_j\)-th roots of unity other than \(1\). Brieskorn-Pham theorem; Brieskorn-Pham polynomial; Milnor fibration; monodromy Singularities in algebraic geometry, Local complex singularities A geometric description of the monodromy of Brieskorn-Pham 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 The author proves that if \(\sum_{d}I_{d}Q^{d}\) where \(I_{d}(z,z^{-1})\) are cohomology valued Laurent \(z\)-series (representing a point on the graph of d\(\mathcal{F}\), the differential of genus-0 descendent potential, in symplectic loop space \(\mathcal{H}\)) and if \(\Phi_{\alpha}\) are polynomials in \(p_{1},\ldots , p_{r}\) then the family
\[
I(\tau)=\sum_{d}T_{d}Q^{d}\text{exp}\Big\{ \frac{1}{z}\sum_{\alpha}\tau_{\alpha} \Phi_{\alpha}(p_{1}-zd_{1},\ldots , p_{r}-zd_{r}) \Big \}
\]
lies on the graph of d\(\mathcal{F}\). Here \(Q^{d}\) stands for the element corresponding to \(d\) in the semigroup ring of Mori cone \(\mathcal{M}\) of the compact Kähler manifold \(X\). Moreover, for arbitrary scalar power series \(C_{\alpha}(z)=\sum_{k\geq 0}\tau_{\alpha,k}z^{k}\), the linear combination \(\sum_{\alpha}c_{\alpha}(z)z\partial_{\tau_{\alpha}}I\) of the derivatives also lies on the graph. Furthermore, in case when \(p_{1}\ldots ,p_{r}\) generate \(H^{*}(X,\mathbb{Q})\), and \(\Phi_{\alpha}\) represents a linear basis, such linear combinations comprise the whole graph. genus-0 descendent potential; symplectic loop space; dilation equation; Novikov ring; \(K\)-theory Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Riemann-Roch theorems, Chern characters, Kähler manifolds Explicit reconstruction in quantum cohomology and \(K\)-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 The Hilbert scheme \(\mathrm{Hilb}^n_{p(t)}\) parametrizes closed subschemes \(X \subset \mathbb P^n_k\) with Hilbert polynomial \(p(t)\), and hence the corresponding graded ideals in \(J \subset k[x_0, \dots, x_n]\). Algebraists study the geometry of the Hilbert scheme through special ideals, such as Borel-fixed ideals. For example, \textit{A. Reeves} and \textit{M. Stillman} showed that the Hilbert scheme is smooth at a lexicographic point [J. Algebr. Geom. 6, 235--246 (1997; Zbl 0924.14004)]. \textit{P. Lella} and \textit{M. Roggero} showed that smoothness of the Hilbert scheme at certain ideals implies that the component on which they lie is rational [Rend. Semin. Mat. Univ. Padova 126, 11--45 (2011; Zbl 1236.14006)]. Recently \textit{A. P. Staal} showed that with an appropriate probability measure, more than half of all Hilbert schemes are smooth and irreducible [Math. Z. 296, 1593--1611 (2020; Zbl 1451.14010)].
Here the authors say two saturated Borel-fixed ideals \(J,J^\prime\) defining points in \(\mathrm{Hilb}^n_{p(t)}\) are \textit{Borel-adjacent} if for \(r \gg 0\), the respective monomial bases \(F, F^\prime\) of \(J_r, J_r^\prime\) have the property that the sets \(F \setminus F^\prime\) and \(F^\prime \setminus F\) have the same linear syzygies. The authors prove that in this case there is a rational curve on \(\mathbf{Hilb}^n_{p(t)}\) passing through the corresponding points, so that \(J, J^\prime\) lie on the same component. They then form the Borel Graph of \(\mathrm{Hilb}^n_{p(t)}\) by taking the vertices to be Borel-fixed ideals with edges between the Borel-adjacent ideals. This graph is a subgraph of the \(T\)-graph introduced by \textit{K. Altmann} and \textit{B. Sturmfels} [J. Pure Appl. Algebra 201, 250--263 (2005; Zbl 1088.13012)]. Each term order induces an orientation of the Borel graph, the corresponding directed graphs are called degeneration graphs, which the authors classify by means of a polyhedral fan called the Gröbner fan associated to \(\mathbf{Hilb}^n_{p(t)}\). Constructing minimal spanning trees for some of these degeneration graphs, the authors recover the connectedness theorem of \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)]. Furthermore, they conjecture that the number of irreducible components of \(\mathrm{Hilb}^n_{p(t)}\) is at least the maximum number of vertices with no incoming edge in any degeneration graph. The paper has many helpful examples and pictures. Hilbert schemes; strongly stable ideals; Gröbner degenerations; polyhedral fans; connectedness Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Combinatorial aspects of commutative algebra The Gröbner fan of the Hilbert scheme | 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 an algebraic variety is a certain deformation of its cohomology ring. Roughly speaking, in such a deformation two subvarieties are considering intersecting if they are connected by one or more rational curves. The virtual numbers of such curves, counted by Gromov-Witten invariants, appear as coefficients in the decomposition of the quantum cohomology cup product. These ideas feature prominently in mathematical physics, especially in string theory, where Gromov-Witten invariants appear as quantum corrections to classical notions of geometry, hence the name quantum cohomology.
In this paper the authors study the equivariant quantum cohomology ring of hypertoric varieties. Hypertoric varieties are a version of toric varieties, which can be obtained as an hyperKähler quotient. Well known examples are crepant resolutions of \(A_n\) singularities. One can study such varieties using combinatorial techniques from toric geometry, and as a result their geometry is captured by certain hyperplane arrangements. The paper contains two results. The first one is an explicit presentation of the equivariant cohomology ring of such varieties, given in terms of generators and relations. This result follows from an explicit formula for the quantum multiplication by a divisor.
The second result concerns a mirror formula for the quantum connection on such varieties. The quantum connection depends on the equivariant parameters, and for fixed equivariant parameters is a deformation of an ordinary connection, which involves the quantum product. This results identifies such a quantum connection on a hypertoric variety with the Gauss-Manin connection of a certain mirror family of complex manifolds (with a local system). Such a family is defined in term of the toric data and hyperplanes of the original variety. The proof follows by identifying a certain quantum differential equation for the quantum connection, with a Picard-Fuchs equation for the periods of a specific cohomology class of the mirror family.
The results of the paper are quite explicit and can be useful for people working on hypertoric varieties. The authors have also put some effort in making the paper self-contained, and the relevant concepts of hypertoric geometry and quantum cohomology are reviewed, albeit rather concisely. quantum cohomology; hypertoric variety; symplectic resolution; mirror symmetry McBreen, M.; Shenfeld, D., Quantum cohomology of hypertoric varieties, Lett. Math. Phys., 103, 11, 1273-1291, (2013) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Quantum cohomology of hypertoric 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 construct explicit monomial bases in Milnor algebras of invertible polynomials and calculate the signature of Gorenstein quadratic form. As an application we obtain exact upper estimates for the gradient degree of invertible polynomials and show that they improve the estimates by Petrovsky numbers. quasihomogeneous polynomials; invertible polynomials; singularities Local complex singularities, Singularities in algebraic geometry, Polynomials and rational functions of several complex variables On Milnor algebras of invertible 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 J be a finite-dimensional simple Jordan algebra over an algebraically closed field of characteristic 0 or let J be simple and formally real over \({\mathbb{R}}\). Let \(\lambda\) be the reduced trace of J and r=degree of J. Then the r forms \(x\to \lambda (x^ k)\), \(x\in J\), \(1\leq k\leq r\), are algebraically independent over K and generate the algebra of Aut J- invariant polynomials, where Aut J=automorphism group of J.
The theorem is proven in the same way as the corresponding theorem for Lie algebras [see e.g. \textit{N. Bourbaki}, Groupes et algèbres de Lie (1975; Zbl 0329.17002), Chap. 8, {\S}8.3, Théorème 1]. Also, a similar theorem holds for symmetric spaces of noncompact type [see \textit{S. Helgason}, Differential geometry and symmetric spaces (1962; Zbl 0111.181), Chap. X, {\S} 6.]. Note that in the formally real case (Str(J), Aut(J)), Str(J)=structure group of J is a reductive Riemannian symmetric pair. simple Jordan algebra; formally real; reduced trace; invariant polynomials; automorphism group Associated groups, automorphisms of Jordan algebras, Group actions on varieties or schemes (quotients), Simple, semisimple Jordan algebras Invariant polynomial functions on Jordan 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 We compute the small cohomology ring of the Cayley Grassmannian, that parametrizes four-dimensional subalgebras of the complexified octonions. We show that all the Gromov-Witten invariants in the multiplication table of the Schubert classes are nonnegative and deduce Golyshev's conjecture \(\mathcal{O}\) holds true for this variety. We also check that the quantum cohomology is semisimple and that there exists, as predicted by Dubrovin's conjecture, an exceptional collection of maximal length in the derived category. quasi homogeneous varieties; quantum cohomology; Dubrovin's conjecture Homogeneous spaces and generalizations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Fano varieties The small quantum cohomology of the Cayley 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 review the Robinson-Schensted-Knuth correspondence in the light of the quantum Schur-Weyl duality. The \textit{quantum plactic algebra} is defined to be a Schur functor mapping a tower of left modules of Hecke algebras into a tower of \(U_q\mathfrak{gl}\)-modules. The functions on the quantum group carry a \(U_q\mathfrak{gl}\)-bimodule structure whose combinatorial spirit emerges in the RSK algorithm. The bimodule structure on the algebra of biletter words is used for a functorial formulation of the \textit{quantum pseudo-plactic algebra}. The latter algebra has been proposed by Daniel Krob and Jean-Yves Thibon as a higher noncommutative analogue of the quantum torus. Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Toric varieties, Newton polyhedra, Okounkov bodies Quantum plactic and pseudo-plactic 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 Let \(A\) be a noetherian commutative ring of Krull dimension \(d\) and let \(P\) be a projective \(A\)-module of rank \(r\). The well-known splitting theorem of \textit{J.-P. Serre} [Algebre Theorie Nombres, Sem. P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot 11 (1957/58), No. 23, 18 p. (1958; Zbl 0132.41202)] asserts \(P\) splits off a free factor of rank one if \(r>d.\) The authors try to give obstructions for splitting off from \(P\) a free factor of rank one. In order to do this they use higher Grothendieck-Witt groups. The construction of Grothendieck-Witt groups was introduced by \textit{M. Schlichting} [``Hermitian \(K\)-theory, derived equivalences and Karoubi's fundamental theorem'', preprint, \url{http://www.math.lsu.edu/~mschlich/research/prelim.html}] and then Balmer and Walter defined derived Grothendieck-Witt groups [\textit{C. Walter}, ``Grothendieck-Witt groups of triangulated categories'', preprint, \url{http://www.math.uiuc.edu/K-theory/0643/}].
The authors define Euler classes in the derived Grothendieck-Witt groups and show the following:
Theorem. Let \(A\) be a noetherian ring of dimension \(d\) with \(1/2 \in A.\) Let \(P\) be a projective module of rank \(d.\) If \(d=2\) or \(d=3,\) then \(e(P)=0\) in \(GW^{d}(A, \det(P)^{\vee})\) if and only if \(P\simeq Q\oplus A\) for some projective module \(Q.\) Chow-Witt groups; Grothendieck-Witt group; projective module; Euler classes J. Fasel and V. Srinivas, Chow-Witt groups and Grothendieck-Witt groups of regular schemes, Adv. Math., 221 (2009), 302--329. Projective and free modules and ideals in commutative rings, Miscellaneous applications of \(K\)-theory, (Equivariant) Chow groups and rings; motives Chow-Witt groups and Grothendieck-Witt groups of regular 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 In this article, the author computes the equivariant Grothendieck groups of the Russell-Koras threefolds. These are smooth affine contractible threefolds endowed with a hyperbolic action of \(\mathbb{C}^{*}\) such that the quotient is isomorphic to the quotient of the tangent space at the unique fixed point. Many properties of these threefolds were given by \textit{M. Koras} and \textit{P. Russell} [J. Algebr. Geom. 6, 671-695 (1997; Zbl 0882.14013)]. In particular, they are hypersurfaces in \(\mathbb{C}^{4}\) given in two different ways. It has been shown by \textit{N. Mohan Kumar} and \textit{M. P. Murthy} that all vector bundles over such a threefold are stably free [see Ann. Math., II. Ser. 116, 579-591 (1982; Zbl 0519.14009)].
In the present article, the equivariant setting with respect to the given action of \(\mathbb{C}^{*}\) is studied. The equivariant Grothendieck groups are computed. In particular, it is shown that if \(X\) is not isomorphic to \(\mathbb{C}^{3}\), then there are equivariant vector bundles over \(X\) which are not stably trivial. This is in constrast to the case of equivariant vector bundles over representation spaces of \(\mathbb{C}^{*}\), which are known to be not only stably trivial but in fact trivial. contractible threefolds; equivariant Grothendieck groups \(3\)-folds, Equivariant \(K\)-theory, Group actions on varieties or schemes (quotients), Applications of methods of algebraic \(K\)-theory in algebraic geometry The equivariant Grothendieck groups of the Russell-Koras threefolds | 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's anabelian conjecture for smooth, hyperbolic curves over finite fields states that, given any two hyperbolic curves \(U\) and \(V\) over finite fields, any isomorphism between the arithmetic fundamental groups \(\pi_1(U)\) and \(\pi_1(V)\) arises from a unique isomorphism \(U \to V\). In particular, if \(U\) is a smooth hyperbolic curve over a finite field, the fundamental group of \(U\) determines the isomorphism class of \(U\). This was proven in \textit{A. Tamagawa} [Compositio Math. 109, 135--194 (1997; Zbl 0899.14007)] in the affine case and in \textit{S. Mochizuki} [J. Math. Kyoto Univ. 47 (2007; Zbl 1143.14305)] in the proper case.
The current paper proves a stronger, ``prime-to-\(p\)'' version of this result: For any smooth curve \(X\) over a finite field \(k_X\), there is a natural inclusion \(\pi_1(\overline{X} := X \times_{\text{Spec } k_X} \text{Spec } \overline{k_X}) \hookrightarrow \pi_1(X)\). Let \(\Delta_X\) be the maximal prime-to-\(p\) quotient of \(\pi_1(\overline{X})\), and let \(S\) be the kernel of the canonical map \(\pi_1(\overline{X}) \to \Delta_X\). Write \(\Pi_X := \pi_1(X)/S\). Then, if \(U\) and \(V\) are smooth hyperbolic curves defined over finite fields, any isomorphism \(\Pi_U \to \Pi_V\) arises from a unique isomorphism \(U \to V\). In particular, \(\Pi_U\) determines the isomorphism class of \(U\). A similar birational result is also proved for absolute Galois groups of function fields, strengthening a result of \textit{K. Uchida} [Ann. Math. 106 (1977; Zbl 0372.12017)].
The outline of the proof is similar to that of \textit{A. Tamagawa} [Compositio Math. 109, 135--194 (1997; Zbl 0899.14007)]. Given \(\Pi_X\), one first addresses the ``local theory'', attempting to recover the closed points of \(X\) by recovering the conjugacy classes of decomposition groups of closed points of \(X\) as subgroups of \(\Pi_X\). In the case treated by the current paper, the points of \(X\) can only be recovered up to a finite set \(E_X\), the ``exceptional set''. Then, one uses Kummer theory to recover the group \(\mathcal{O}_{E_X}^{\times}\) of rational functions on \(X\) with support away from \(E_X\). Lastly, one recovers the additive structure on the set of rational functions \(k_X(X)\). The problem of the exceptional set \(E_X\) is dealt with by a trick allowing one to pass to an infinite extension of \(k_X\) over which \(X\) has infinitely many points, followed by a descent argument. \bibSaidi-Tamagawa1article label=Saïdi-Tamagawa1, author=Saïdi, Mohamed, author=Tamagawa, Akio, title=A prime-to-\(p\) version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields of characteristic \(p>0\), journal=Publ. Res. Inst. Math. Sci., volume=45, number=1, pages=135--186, date=2009, doi=10.2977/prims/1234361157, issn=0034-5318, review=\MR2512780, Finite ground fields in algebraic geometry, Coverings of curves, fundamental group, Arithmetic ground fields for curves, Curves over finite and local fields A prime-to-\(p\) version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields of characteristic \(p>0\) | 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 notion of the Gamma integral structure for the quantum cohomology of an algebraic variety was introduced by Iritani, Katzarkov-Kontsevich-Pantev. In this paper, we define the Gamma integral structure for an invertible polynomial of chain type. Based on the \(\Gamma \)-conjecture by Iritani, we prove that the Gamma integral structure is identified with the natural integral structure for the Berglund-Hübsch transposed polynomial by the mirror isomorphism. invertible polynomial; gamma integral structure; mirror symmetry Mirror symmetry (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Gamma integral structure for an invertible polynomial of chain type | 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 explain some remarkable connections between the two-parameter symmetric polynomials discovered in 1988 by Macdonald, and the geometry of certain algebraic varieties, notably the Hilbert scheme \(\text{Hilb}^n(\mathbb{C}^2)\) of points in the plane, and the variety \(C_n\) of pairs of commuting \(n\times n\) matrices. symmetric functions; \(n!\) conjecture; Frobenius series; diagonal harmonics; commuting variety; symmetric polynomials; algebraic varieties; Hilbert scheme M. Haiman, ''Macdonald polynomials and geometry'' in New Perspectives in Algebraic Combinatorics (Berkeley, Calif., 1996--97) , Math. Sci. Res. Inst. Publ. 38 , Cambridge Univ. Press, Cambridge, 1999, 207--254. Symmetric functions and generalizations, Research exposition (monographs, survey articles) pertaining to combinatorics, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Macdonald polynomials and 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 We introduce a ring of \(\mathbb{Z}\)-valued functions on a complete fan \(\Delta\) called Grothendieck weights to describe the ordinary operational \(K\)-theory of the associated toric variety \(X\). These functions satisfy a \(K\)-theoretic analogue of the balancing condition for Minkowski weights, which is induced by a presentation of the Grothendieck group of \(X\). We explicitly give a combinatorial presentation in low dimensions, and relate Grothendieck weights to other fan-based invariants such as piecewise exponential functions and Minkowski weights. As an application, we give an example of a projective toric surface \(X\) such that the forgetful map \(K_T^\circ(X) \to K^\circ(X)\) is not surjective. toric varieties; \(K\)-theory; equivariant \(K\)-theory; polyhedral geometry; fans; polytopes; tropical geometry Toric varieties, Newton polyhedra, Okounkov bodies, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Group actions on varieties or schemes (quotients), (Equivariant) Chow groups and rings; motives, \(K\)-theory of schemes, Riemann-Roch theorems, Applications of methods of algebraic \(K\)-theory in algebraic geometry \(K\)-theoretic balancing conditions and the Grothendieck group of a toric variety | 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 complex algebraic variety, so that each cohomology group \(H^j_c(X,{\mathbb C})\) is equipped with a weight filtration as part of its mixed Hodge structure. An algebraic action of a finite group \(\Gamma \) on \(X\) yields a \(\Gamma \)-representation on the graded pieces \(W_mH^j_c(X,{\mathbb C})/W_{m-1}H^j_c(X,{\mathbb C})\), and, summing over degrees (with sign) and weights, the weight polynomial \(W^{\Gamma}_c(X,t)\in R(\Gamma)[t]\). This polynomial enjoys an additivity property with respect to decompositions into \(\Gamma\)-invariant subsets which is the key for calculations. A variety \(X\) is called separably pure if its non-zero cohomology groups have pure Hodge structures with different weights. In this case the weight polynomial determines the individual \(\Gamma \)-modules \(H^j_c(X)\). This is the case, in particular, for a toral arrangement, i.e., for the complement of a finite union of kernels of characters of an algebraic torus. Finally, the authors comment on parallel results concerning varieties over \({\mathbb F}_q\) and their \(\ell \)-adic cohomology. mixed Hodge structure; weight filtration; algebraic action; weight polynomial; toral arrangement A. Dimca and G. I. Lehrer, Purity and equivariant weight polynomials, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 161 -- 181. Finite transformation groups, Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies Purity and equivariant weight 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 We show that all the modular invariants of the quantum double of \(\mathbb Z_p\) at every twist can be produced by subfactors, with \(p\) any prime number. String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Spinor and twistor methods applied to problems in quantum theory The twisted quantum double of \(\mathbb Z_p\) modular data and subfactors | 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 \(q\)-hypergeometric solutions of the equivariant quantum differential equations and the associated \(q\)KZ difference equations for the cotangent bundle \(T^\ast \mathcal{F}_{\boldsymbol{\lambda}}\) of a partial flag variety \(\mathcal{F}_{\boldsymbol{\lambda}}\). These \(q\)-hypergeometric solutions manifest a Landau-Ginzburg mirror symmetry for the cotangent bundle. We formulate and prove Pieri rules for quantum equivariant cohomology of the cotangent bundle. Our Gamma theorem for \(T^\ast \mathcal{F}_{\boldsymbol{\lambda}}\) says that the leading term of the asymptotics of the \(q\)-hypergeometric solutions can be written as the equivariant Gamma class of the tangent bundle of \(T^\ast \mathcal{F}_{\boldsymbol{\lambda}}\) multiplied by the exponentials of the equivariant first Chern classes of the associated vector bundles. That statement is analogous to the statement of the gamma conjecture by \textit{B. A. Dubrovin} [Quantum cohomology and isomonodromic deformation, in: Lecture at Recent Progress in the Theory of Painlevé Equations: Algebraic, Asymptotic and Topological Aspects, Strasbourg (2013)] and by \textit{S. Galkin} et al. [Duke Math. J. 165, No. 11, 2005--2077 (2016; Zbl 1350.14041)], see also the gamma theorem for \(\mathcal{F}_{\boldsymbol{\lambda}}\) in Appendix B. flag varieties; quantum differential equation; dynamical connection; \(q\)-hypergeometric solutions Exactly solvable models; Bethe ansatz, Applications of Lie algebras and superalgebras to integrable systems, Classical problems, Schubert calculus, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) \(q\)-hypergeometric solutions of quantum differential equations, quantum Pieri rules, and Gamma 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 In real algebra and geometry it is an important problem to recognize whether an element \(f\in R\), where \(R\subseteq\mathbb{R}^A\) is a subring of a ring of functions defined on some set \(A\), is nonnegative or positive on a subset \(Z\subseteq A\). Important instances are semi-algebraic subsets \(A\subseteq\mathbb{R}^n\) and basic closed sets \(Z\subseteq A\). Positivstellensätze give an algebraic answer to this question.
The ring \(R\) is equipped with an Archimedean semiring \(S\subset R\) (generalizing the notion of an Archimedean partial order) and with an \(S\)-module \(M\). The module defines the set \(Z_M= \{x\in A\mid\forall f\in M: f(x)\geq 0\}\). The elements of \(M\) are nonnegative on \(Z_M\), but frequently there are also other functions that are nonnegtive on \(Z_M\). Typically, a Positivstellensatz says that a function \(g\) satisfying certain conditions belongs to \(M\), hence is nonnegative on \(Z_M\), or can be expressed in terms of the elements of \(M\) in a way which makes it obvious that \(g\) is nonnegative on \(Z_M\). There exists a large amount of literature about Positivstellensätze. The authors adapt tools from functional analysis (i.e., states) to the study of positivity in real algebra. Using these techniques they reprove several known Positivstellensätze and discover near ones with applications in semi-algebraic geometry. preordering; Archimedean module; convex cone; state; order unit; Stellensatz; Archimedean semiring Burgdorf, S; Scheiderer, C; Schweighofer, M, Pure states, nonnegative polynomials and sums of squares, Comment. Math. Helv., 87, 113-140, (2012) Semialgebraic sets and related spaces, Ordered abelian groups, Riesz groups, ordered linear spaces, Sums of squares and representations by other particular quadratic forms, Real algebra, Ordered rings, algebras, modules, Ideals and multiplicative ideal theory in commutative rings, Polynomials, rational functions in real analysis, States of selfadjoint operator algebras, General convexity Pure states, nonnegative polynomials and sums of squares | 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 [Part I, cf. Invent. Math. 79, 499-511 (1985; Zbl 0563.14023).]
We want to develop a theory of polynomials for the parabolic setup for any Coxeter group (W,S) and the subgroup \(W_ J\) generated by any subset \(J\subseteq S\). This is the starting point of our investigation. It turns out that one does get a set \(\{P^ J_{\tau,\sigma}\}\) of polynomials in \({\mathbb{Z}}[q]\) which is indexed by a pair \(\tau\), \(\sigma\) of elements in \(W^ J\), the set of minimal coset representatives of \(W/W_ J\). These polynomials give the dimensions of the intersection cohomology modules of Schubert varieties in G/P (see Theorem 4.1) for any P. They are related to \(P_{x,y}'s\) when the subgroup corresponding to P is finite (see Propositions 3.4 and 3.5). Incidentally, Proposition 3.5 provides a method for computing \(P_{x,y}'s\) which is very efficient since the number of intermediate steps is considerably smaller than that in the original setup (for any Coxeter group (W,S)). Kazhdan Lusztig polynomials; Verma modules; semisimple algebraic group; Kac-Moody groups; Coxeter group; coset representatives; intersection cohomology modules; Schubert varieties V.V. Deodhar, \textit{On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials}, \textit{J. Algebra}\textbf{111} (1987) 483. Infinite-dimensional Lie groups and their Lie algebras: general properties, Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Semisimple Lie groups and their representations, Representation theory for linear algebraic groups, Other algebraic groups (geometric aspects) On some geometric aspects of Bruhat orderings. II: The parabolic analogue of 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 \(w_0\) denote the permutation \([n,n-1,...,2,1]\). We give a new explicit formula for the Kazhdan--Lusztig polynomials \(P_{w_0w,w_0x}\) in \(S_n\) when \(x\) indexes an irreducible component of the singular locus of the Schubert variety \(X_w\). To do this, we utilize a standard identity that relates \(P_{x,w}\) and \(P_{w_0w,w_0x}\). Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds A formula for certain inverse Kazhdan--Lusztig polynomials in \(S_{n}\) | 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 will outline computations of quantum sheaf cohomology for deformations of tangent bundles of toric varieties, for those deformations describable as deformations of toric Euler sequences. Quantum sheaf cohomology is a heterotic analogue of quantum cohomology, a quantum deformation of the classical product on sheaf cohomology groups, that computes nonperturbative corrections to analogues of \(\overline{27}^3\) couplings in heterotic string compactifications. Previous computations have relied on either physics-based gauged linear sigma model (GLSM) techniques or computation-intensive brute-force Cech cohomology techniques. This paper describes methods for greatly simplifying mathematical computations, and derives more general results than previously obtainable with GLSM techniques. We will outline recent results (rigorous proofs will appear elsewhere). A. Gadde and P. Putrov, \textit{Exact solutions of} (0\(,\) 2) \textit{Landau-Ginzburg models}, arXiv:1608.07753 [INSPIRE]. String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Model quantum field theories, Formal methods and deformations in algebraic geometry, Approximation to limiting values (summation of series, etc.), Toric varieties, Newton polyhedra, Okounkov bodies Physical aspects of quantum sheaf cohomology for deformations of tangent bundles of 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 This is a detailed proof of the formula announced previously by the authors [Bull. Am. Math. Soc., New Ser. 5, 182-184 (1981; Zbl 0495.14010)]. This is a G.R.R. formula relating characteristic classes of a coherent sheaf \({\mathcal F}\) to those of its direct images under a map of complex manifolds f:\(X\to Y\), arbitrary except for the condition that f is proper on the support of \({\mathcal F}\). Except in the case where Y is a point, previous proofs of such formulae required that X and Y be subvarieties of projective spaces. By contrast, this proof uses local geometric formulae for Čech cochains representing the characteristic classes, derives the appropriate local relations between these cochains and leads to a G.R.R. formula relating classes in Hodge cohomology rather than ordinary cohomology.
As in the authors' previous work on Hirzebruch-Riemann-Roch, the local formulae are based on the notion of a 'twisting cochain'. Innovations here include the introduction of twisting cochains for perfect complexes of sheaves, direct images of twisting cochains and the use of Grauert's coherence theorem and the relative version of Serre-Grothendieck duality. Grothendieck-Riemann-Roch formula; map of complex manifolds; Hodge cohomology; twisting cochains; Serre-Grothendieck duality N. R. O'Brian, D. Toledo, and Y. L. L. Tong, ''A Grothendieck-Riemann-Roch formula for maps of complex manifolds,'' Math. Ann., vol. 271, iss. 4, pp. 493-526, 1985. Riemann-Roch theorems, Complex manifolds, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Transcendental methods, Hodge theory (algebro-geometric aspects) A Grothendieck-Riemann-Roch formula for maps of complex manifolds | 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 Grohtendieck Quot scheme over a smooth projective curve of genus \(g\) has as closed points short exact sequences of coherent sheaves \(0\to E\to V\to F\to 0\) with \(E\) of rank \(r\) and \(V\) of rank \(r+s\). When \(g\geq2\) and \(V\) is a general stable vector bundle the Quot scheme parametrizing rank \(r\) subbundles of \(V\) of maximal degree is smooth of expected dimension \(0\). On the other hand, Quot schemes of trivial bundles are not always of expected dimension. Marion-Oprea gave an interpretation of them in terms of a TQFT defined by Witten, whose vector space is the cohomology ring of the Grassmanian \(\text{Gr}(r,\mathbb{C}^{r+s})\). The main result of this paper is that one can define a well-behaved weighted TQFT that subsumes Witten's, in particular all Quot schemes have expected dimension \(0\) (even in genus \(0\) and \(1\)).
The weighted TQFT is defined by replacing trivial bundles by general vector bundles with the degree as an extra parameter, additive under the composition of maps. The degree zero maps form a usual TQFT, which corresponds to the small quantum cohomology of the Grassmanian, and the maps of Witten's TQFT are obtained from the weighted ones by an explicit relabeling. Moreover, the weighted TQFT is explicitly computable, and recovers known formulas for the number of points in the Quot schemes, such as Holla's formula in terms of the Gromov-Witten invariants of the Grassmanian, and Marion-Oprea's in terms of Verlinde numbers. algebraic curve; Grassmanian; quot scheme; stable vector bundle; topological quantum field theory; quantum cohomology of Grassmannian; enumerative geometry; Verlinde numbers; small quantum cohomology; weighted TQFT Vector bundles on curves and their moduli, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Applications of global analysis to structures on manifolds A weighted topological quantum field theory for quot schemes on 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 paper under review proposes a construction of string cohomology spaces for Calabi-Yau hypersurfaces that arise in the Batyrev mirror symmetry construction [cf. \textit{V. Batyrev}, J. Algebr. Geom. 3, No. 3, 493--535 (1994; Zbl 0829.14023)], with spaces defined explicitly in terms of the corresponding reflexive polyhedra. The construction starts with the description of the cohomology of semiample hypersurfaces in toric varieties and then uses mirror symmetry to provide a conjectural string cohomology of Calabi-Yau hypersurfaces. This construction gives the correct (bigraded) dimension of the space, and further, a finite-dimensional family of string cohomology spaces rather than just a single-string cohomology space. That is, string cohomology space depends not only on the complex structure (the defining polynomial \(f\)), but also on some extra parameter \(\omega\). For special values of this parameter of an orbifold Calabi-Yau hypersurface, the construction gives the orbifold Dolbeault cohomology, recovering the result of \textit{W. Chen} and \textit{Y. Ruan} [Commun. Math. Phys. 248, 1--31 (2004; Zbl 1063.53091)]. However, for non-orbifold Calabi-Yau hypersurfaces, there is no natural choice for \(\omega\), and this implies the dependence of the general definition of string cohomology space on some parameter. In case of Calabi-Yau hypersurfaces, this particular parameter \(\omega\) corresponds to the defining polynomial of the mirror Calabi-Yau hypersurface. In general, this parameter should be related to the ``string complexified Kähler class'' which is yet to be defined. An attempt is made to extend the definition of string cohomology space beyond the Calabi-Yau hypersurfaces. A conjectural definition of string cohomology vector spaces is presented for stratified varieties with \({\mathbb Q}\)-Gorenstein toroidal singularities that satisfy certain restrictions on the types of singular strata. This definition would involve intersection cohomology of the closures of strata as well as perverse sheaves. This conjectural definition gives the correct bigraded dimension and also reproduces orbifold cohomology of a \({\mathbb Q}\)-Gorenstein toric variety as a special case. toric varieties; intersection cohomology; orbifold Dolbeault cohomology L. A. Borisov and A. R. Mavlyutov, String cohomology of Calabi-Yau hypersurfaces via mirror symmetry , preprint, \ Calabi-Yau manifolds (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Calabi-Yau theory (complex-analytic aspects), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) String cohomology of Calabi-Yau hypersurfaces via mirror symmetry. Appendix: \(G\)-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 We evaluate induced sign characters of \(H_{n}(q)\); at certain elements of \(H_{n}(q)\); and conjecture an interpretation for the resulting polynomials as generating functions for \(P\)-tableaux by a certain statistic. Our conjecture relates the quantum chromatic symmetric functions of \textit{J. Shareshian} and \textit{M. L. Wachs} [in: Configuration spaces. Geometry, combinatorics and topology. Pisa: Edizioni della Normale. 433--460 (2012; Zbl 1328.05194)] to \(H_{n}(q)\) characters. Hecke algebra character; unit interval order; \(P\)-tableau; chromatic symmetric function; quantum analog Symmetric functions and generalizations, Combinatorial aspects of representation theory, Hecke algebras and their representations, Toric varieties, Newton polyhedra, Okounkov bodies Hecke algebra characters and quantum chromatic symmetric functions | 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 scheme \(S\) with a good action of a finite abelian group \(G\) having enough roots of unity we show that the quotient map on the \(G\)-equivariant Grothendieck ring of varieties over \(S\) is well defined with image in the Grothendieck ring of varieties over \(S/G\) in the tame case, and in the modified Grothendieck ring in the wild case. To prove this we use a result on the class of the quotient of a vector space by a quasi-linear action in the Grothendieck ring of varieties due to Esnault and Viehweg, which we also generalize to the case of wild actions. As an application we deduce that the quotient of the motivic nearby fiber is a well defined invariant. Group actions on varieties or schemes (quotients), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Rationality questions in algebraic geometry, (Equivariant) Chow groups and rings; motives The quotient map on the equivariant 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 \textit{quantum \(\mathbb P^{n-1}\)} is a not necessarily commutative, regular \(k\)-algebra of global dimension \(n\). A quantum \(\mathbb P^{n-1}\) is called \textit{quadratic} if it is defined by quadratic relations. Artin, Tate, and Van den Bergh [\textit{M. Artin} et al., Prog. Math. 86, 33--85 (1990; Zbl 0744.14024)] classified quantum \(\mathbb P^2\)'s according to their point schemes (scheme of closed points, simple modules), and the point scheme of the most generic quadratic quantum \(\mathbb P^2\) is pictured by an elliptic curve in \(\mathbb P^2\).
In the present article, the authors study the similar task for quantum \(\mathbb P^3\)'s. The point scheme is no longer sufficient, and an additional invariant is needed. The second author together with B. Shelton introduced the concept of a line scheme, which is a certain scheme of lines in some \(\mathbb P^n\), given by a precise construction of Plücker coordinates. In the case of quantum \(\mathbb P^3\)'s, the line scheme is a collection of lines in \(\mathbb P^5\). The authors state that if a generic quantum \(\mathbb P^3\) exists, then it has a point scheme consisting of exactly 20 distinct points and a 1-dimensional line scheme. A family \(\mathcal A(\gamma)\) of quantum \(\mathbb P^3\)'s appeared in a study by the second author together with T. Cassidy on a study of generalized Clifford algebras, and in this article the authors compute the line scheme of the algebras in this family where the generic member is stated as a candidate for a generic quantum \(\mathbb P^3\).
The family in question is given as follows: Let \(\gamma\in\Bbbk^\times\). Then \(\mathcal A(\gamma)\) is the \(\Bbbk\)-algebra with generators \(x_1,\dots,x_4\) with defining relations: \(x_4x_1=ix_1x_4,\;x_3^2=x_1^2,\;x_3x_1=x_1x_3-x_2^2,\;x_3x_2=ix_2x_3,\;x_4^2=x_2^2,\;x_4x_2=x_2x_4-\gamma x_1^2.\) By construction, \(\mathcal A(\gamma)\) is a regular Noetherian domain of global dimension four with Hilbert series the same as the polynomial algebra in four variables.
The authors compute the point scheme of the algebras \(A(\lambda)\), and they compute the line schemes and identify the lines in \(\mathbb P^3\) to which the points of the line schemes correspond. (Notice that this are the points through which the lines pass, and it is also a moduli of lines; hence the term \textit{line scheme}.)
The results of the computations are the following: The line scheme of the generic member is the union of seven curves: A nonplanar elliptic curve in a \(\mathbb P^3\) (a spatial ellipic curve), four planar elliptic curves and two nonsingular conics. If \(p\) is one of the generic points of the point scheme, then there are exactly six distinct lines of the line scheme that pass through \(p\).
The result of this article suggest that the line scheme of the most generic quadratic quantum \(\mathbb P^3\) is obtainable as the union of two spatial elliptic curves and four planar elliptic curves.
This is a clean cut article, with explicit computations. The line schemes are cofactors of explicit matrices, i.e. determinantal varieties, making up the Plücker coordinates, and all the resulting polynomials are listed in the appendix. The article is very nice to read, and the results are interesting and typical algebraic geometry. quantum \(\mathbb P^n\); line scheme; quantum algebra Chandler, R. G.; Vancliff, M., The one-dimensional line scheme of a certain family of quantum \(\mathbb{P}^3\)s, J. Algebra, 81, 316-333, (2015) Noncommutative algebraic geometry, Quadratic and Koszul algebras, Rings arising from noncommutative algebraic geometry The one-dimensional line scheme of a certain family of quantum \(\mathbb{P}^3\mathrm{s}\) | 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 \(O_q(G)\) be the algebra of quantized functions on an algebraic group \(G\) and \(O_q(B)\) its quotient algebra corresponding to a Borel subgroup \(B\) of \(G\). We define the category of sheaves on the ``quantum flag variety of \(G\)'' to be the \(O_q(B)\)-equivariant \(O_q(G)\)-modules and prove that this is a proj-category. We construct a category of equivariant quantum \({\mathcal D}\)-modules on this quantized flag variety and prove the Beilinson-Bernstein's localization theorem for this category in the case when \(q\) is transcendental. quantum groups; localization; noncommutative geometry Backelin, E.; Kremnizer, K., Quantum flag varieties, equivariant quantum \(\mathcal{D}\)-modules, and localization of quantum groups, Adv. Math., 203, 408-429, (2006) Quantum groups (quantized enveloping algebras) and related deformations, Geometry of quantum groups, Noncommutative algebraic geometry Quantum flag varieties, equivariant quantum \(\mathcal D\)-modules, and localization of 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 The result of the author, \textit{A. Kresch} and \textit{H. Tamvakis} [J. Am. Math. Soc. 16, No. 4, 901--915 (2003; Zbl 1063.53090)] shows that the computation of the (3- point, genus 0) Gromov-Witten invariants of Grassmann variety can be reduced to a computation in the ordinary cohomology of certain two-step flag manifolds. The Gromov-Witten invariants, in this case, have a nice enumerative interpretation. Namely, they count rational curves that meet general translates of Schubert varieties. Unfortunately, the Gromov-Witten invariants used to define more general quantum cohomology theories do not have such an interpretation. Despite this, the authors prove that the more general Gromov-Witten invariants satisfy the key identity from the paper of Buch, Kresch, and Tamvakis.
One of the main results of the paper is that an equivariant \(K\)-theoretic Gromov-Witten invariant on a Grassmannian is equal to a quantity computed in the ordinary equivariant \(K\)-theory of a two-step flag variety. In the process of proving this the authors show that the Gromov-Witten variety of curves passing through three general points is rational and irreducible. They also describe the structure of the quantum \(K\)-theory ring of Grassmanian in terms of a Pieri rule and compute the dual Schubert basis for this ring. The authors show that the formula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces. This is proven by using a construction of \textit{P.-E. Chaput, L. Manivel} and \textit{N. Perrin} [Transform. Groups 13, No. 1, 47--89 (2008; Zbl 1147.14023)]. Gromov-Witten invariants; quantum \(K\)-tkeory; Grasssmannian Buch, A. S.; Mihalcea, L. C., Quantum \textit{K}-theory of Grassmannians, Duke Math. J., 156, 3, 501-538, (2011) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), \(K\)-theory of schemes, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Rationality questions in algebraic geometry Quantum \(K\)-theory 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 Let \({\mathfrak g}\) be a symmetrizable Kac-Moody algebra, and \(U_q ({\mathfrak g})\) the quantized enveloping algebra of \({\mathfrak g}\). Let \(\lambda\) be a dominant, integral weight, and \(V(\lambda)\) the corresponding simple \(U_q ({\mathfrak g})\)-module (\(q\) being generic). Let \(V_A (\lambda)\) be the canonical \(A\)-form of \(V(\lambda)\) where \(A= \mathbb{Z} [q,q^{-1} ]\). Let \(W\) be the Weyl group of \({\mathfrak g}\). For \(w\in W\), let \(V_{A,w}\) be the (quantum) Demazure submodule of \(V_A (\lambda)\). We construct an \(A\)-basis for \(V_A (\lambda)\) compatible with \(\{V_{a,w}\), \(w\in W\}\), consisting of \(\{De\}\), where \(e\) is the highest weight vector in \(V(\lambda)\), and \(D\) is either 1 or of the form \(F_{\beta_r}^{ (n_r)} \dots F_{\beta_1}^{ (n_1)}\), \(\beta_i\) simple, \(n_i>0\) (for some suitable \(n_i\)'s), and \(s_{\beta_r} \dots s_{\beta_1}\) is reduced. We also show that for \(w\in W\), the transition matrix from our basis for \(V_{A,w}\) to Kashiwara's global basis is upper triangular with diagonal entries equal to 1 (for a suitable indexing). We also give an explicit expression for the crystal base \(B(\lambda)\). Given \(w\in W\), and \(\alpha\) a simple root such that \(w< s_\alpha w\) (\(= \tau\), say), we exhibit a unique ``Demazure'' \(U_q (sl_2)\) structure on \(V_\tau/ V_w\). symmetrizable Kac-Moody algebra; quantized enveloping algebra; dominant, integral weight; Weyl group; Demazure submodule; crystal base V. Lakshmibai, Bases for quantum Demazure modules, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 199 -- 216. Quantum groups (quantized enveloping algebras) and related deformations, Representation theory for linear algebraic groups, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds Bases for quantum Demazure 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 The maximal minors of a \(p\times (m+p)\)-matrix of univariate polynomials of degree \(n\) with indeterminate coefficients are themselves polynomials of degree \(np\). The sub-algebra generated by their coefficients is the coordinate ring of the quantum Grassmannian, a singular compactification of the space of rational curves of degree \(np\) in the Grassmannian of \(p\)-planes in \((m+p)\)-space. These subalgebra generators are shown to form a sagbi basis. The resulting flat deformation from the quantum Grassmannian to a toric variety gives a new ``Gröbner basis style'' proof of the Ravi-Rosenthal-Wang formulas in quantum Schubert calculus [\textit{M. S. Ravi}, \textit{J. Rosenthal} and \textit{X. Wang}, Math. Ann. 311, No.1, 11-26 (1998; Zbl 0902.14036)]. The coordinate ring of the quantum Grassmannian is an algebra with straightening law, which is normal, Cohen-Macaulay, and Koszul, and the ideal of quantum Plücker relations has a quadratic Gröbner basis. This holds more generally for skew quantum Schubert varieties. These results are well-known for the classical Schubert varieties \((n=0)\). We also show that the row-consecutive \((p\times p)\)-minors of a generic matrix form a sagbi basis and we give a quadratic Gröbner basis for their algebraic relations. quantum Grassmannian; sagbi basis; quantum Plücker relations; quadratic Gröbner basis; quantum Schubert varieties Sottile, Frank; Sturmfels, Bernd, A sagbi basis for the quantum Grassmannian, J. Pure Appl. Algebra, 158, 2-3, 347-366, (2001) Computational aspects of higher-dimensional varieties, Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Rings with straightening laws, Hodge algebras, Determinantal varieties A sagbi basis for 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 Inspired by the work of \textit{L. Le Bruyn} and \textit{S. P. Smith} [Proc. Am. Math. Soc. 118, No. 3, 725--730 (1993; Zbl 0795.16029)] and the work of \textit{B. Shelton} and \textit{M. Vancliff} [Commun. Algebra 30, No. 5, 2535--2552 (2002; Zbl 1056.14002)], we analyze certain graded algebras related to the Lie algebra \(\mathfrak{sl}(2,\Bbbk)\) using geometric techniques in the spirit of Artin, Tate and Van den Bergh. In particular, we discuss the point schemes and line schemes of certain quadratic quantum \(\mathbb{P}^3\) s associated to the Lie superalgebra \(\mathfrak{sl}(1|1)\), to a quantized enveloping algebra, \(\mathcal{U}_q(\mathfrak{sl}(2, \Bbbk))\), of \(\mathfrak{sl}(2,\Bbbk)\), and to a color Lie algebra \(\mathfrak{sl}_k(2,\Bbbk)\), respectively. The geometry we consider identifies certain normal elements in the universal enveloping algebra of \(\mathfrak{sl}(1|1)\) and in \(\mathcal{U}_q(\mathfrak{sl}(2,\Bbbk))\). line scheme; point scheme; Lie algebra; superalgebra; regular algebra; Plücker coordinates Quantum groups (quantized enveloping algebras) and related deformations, Noncommutative algebraic geometry, Quadratic and Koszul algebras, Universal enveloping (super)algebras, Color Lie (super)algebras The quantum spaces of certain graded algebras related to \(\mathfrak{sl}(2, \Bbbk)\) | 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 given work the model of the quantum computer is offered which is based on the theory of holomorphic vector bundles on a compact Riemann surfaces with the connection with regular singular points. connection; holomorphic bundle; Riemann surface G. Giorgadze, ''Holomorphic quantum computing,'' Bull. Georgian Acad. Sci., 166, No. 2, 228--230 (2002). Quantum computation, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Riemann surfaces; Weierstrass points; gap sequences, Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane, Holomorphic quantum computing. | 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 use classical Schubert calculus to give a direct geometric proof of the quantum version of Monk's formula [see \textit{S. Fomin, S. Gelfand} and \textit{A. Postnikov}, J. Am. Math. Soc. 10, No. 3, 565-596 (1997; Zbl 0912.14018)]. quantum Schubert polynomials Anders Skovsted Buch, A direct proof of the quantum version of Monk's formula, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2037 -- 2042. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds A direct proof of the quantum version of Monk's 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 \(\mathbb K\) be a field of characteristic 0 and let \({\mathbb K}[x_1,\ldots,x_n]\), \(W_n({\mathbb K})\), and \(P_n({\mathbb K})={\mathbb K}[x_1,\ldots,x_n,p_1,\ldots,p_n]\) be, respectively, the polynomial algebra in \(n\) variables, the \(n\)-th Weyl algebra, and the \(n\)-th Poisson algebra equipped with the standard Poisson bracket where the only nonzero brackets are \(\{p_i,x_i\}=1\). In the paper under review, the authors consider the problem for approximation of automorphisms of these three algebras by tame automorphisms. A classical theorem in [\textit{D. J. Anick}, J. Algebra 82, 459--468 (1983; Zbl 0535.13014)] gives that every endomorphism of \({\mathbb K}[x_1,\ldots,x_n]\) with invertible Jacobian is a limit of a sequence of tame automorphisms in the formal power series topology. The first main result of the paper is a slightly modified proof of this theorem with an automorphism with Jacobian equal to 1 as a limit. The proof is adapted to the problem of approximation of the automorphisms of the Poisson algebra \(P_n({\mathbb K})\). In the paper the automorphisms of \(P_n({\mathbb K})\) are called symplectomorphisms because they can be identified with the invertible polynomial mappings \({\mathbb A}_{\mathbb K}^{2n}\to {\mathbb A}_{\mathbb K}^{2n}\) of the affine space \({\mathbb A}_{\mathbb K}^{2n}\) which preserve the symplectic form \(\sum dp_i\wedge dx_i\). The second main result is that every symplectomorphism is a limit of a sequence of tame symplectomorphisms in the formal power series topology. By a theorem in [\textit{A. Belov-Kanel} and \textit{M. Kontsevich}, Lett. Math. Phys. 74, No. 2, 181--199 (2005; Zbl 1081.16031)] and [\textit{A. Belov-Kanel} and \textit{M. Kontsevich}, Mosc. Math. J. 7, No. 2, 209--218 (2007; Zbl 1128.16014)] the groups of tame automorphisms of \(W_n({\mathbb C})\) and \(P_n({\mathbb C})\) are isomorphic. The third main result of the paper is that if \(\sigma\) is a symplectomorphism of \(P_n({\mathbb C})\), then there exists a sequence of tame automorphisms of \(W_n({\mathbb C})\) such that their images in \(\text{Aut}(P_n({\mathbb C}))\) converge to \(\sigma\). In other words, the authors prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory. Finally, let \(\varphi\) be a polynomial automorphism of \({\mathbb C}[x_1,\ldots,x_n]\) and let \({\mathcal O}_{\varphi}\) be the local ring generated by the coefficients of \(\varphi\) and with maximal ideal \(\mathfrak m\). If the sequence of tame automorphisms \(\psi_1,\psi_2,\ldots\) converges to \(\varphi\) in the formal power series topology, then the coordinates of \(\psi_k\) converge to the coordinates of \(\varphi\) in the \(\mathfrak m\)-adic topology. A similar result is established for the symplectomorphisms of \(P_n({\mathbb C})\). Jacobian conjecture; polynomial automorphisms and symplectomorphisms; tame and wild automorphisms; quantization Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Polynomials over commutative rings, Jacobian problem Lifting of polynomial symplectomorphisms and deformation quantization | 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 the polynomial structure of the topological string partition functions as solutions to the holomorphic anomaly equations. We also explain the connection between the ring of propagators defined from special Kähler geometry and the ring of almost-holomorphic modular forms defined on modular curves. Calabi-Yau threefolds; mirror symmetry Calabi-Yau theory (complex-analytic aspects), \(3\)-folds Polynomial structure of topological string partition functions | 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 text is suitable for graduate students and researchers in algebraic geometry. It begins by introducing hyperbolicity cones of homogeneous polynomials, definite determinantal representations, the notion of spectrahedra and the generalized Lax conjecture. The second section defines generalized Clifford algebras of hyperbolic homogenous polynomials. The main result is a theorem relating that \(-1\) is not a sum of squares in the generalized Clifford algebra and the spectrahedral property. This is followed by a theorem relating a positive trace functional with the spectrahedral property. Traces are in turn related to determinantal representations. Finally, several computational aspects are explained and the paper concludes with a list of open questions surrounding the generalized Lax conjecture. homogenous polynomial; hyperbolicity; generalized Clifford algebra, spectrahedra; semidefinite programming; Lax conjecture; positive trace functional; determinantal representations Netzer, T; Thom, A, Hyperbolic polynomials and generalized Clifford algebras, Disc. Comput. Geom., 51, 802-814, (2014) Clifford algebras, spinors, Solving polynomial systems; resultants, Semidefinite programming, Determinantal varieties Hyperbolic polynomials and generalized Clifford 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 The authors continue their investigation of the birational abelian conjecture for function fields of curves over finite fields started in [Publ. Res. Inst. Math. Sci. 45, No. 1, 135--186 (2009; Zbl 1188.14016)]. The aim is to recover finite separable extensions of such function fields from open homomorphisms between their absolute Galois groups (or better, from geometrically prime-to-\(p\) quotients of these). Contrary to the now well-understood case of isomorphisms of Galois groups, the main obstacle here is the current lack of a satisfactory local theory for decomposition and inertia groups. So roughly the authors show that this is the only obstacle: if a satisfactory local theory exists, then the Galois characterization of finite separable extensions of function fields is possible.
More precisely, the authors call a continuous open homomorphism \(\phi: G_1\to G_2\) between absolute Galois groups of function fields of curves over finite fields rigid if, up to replacing \(G_1\) and \(G_2\) by suitable open subgroups, \(\phi\) maps each decomposition group of a closed point in \(G_1\) to a decomposition group in \(G_1\). The first main result is that, up to composing by inner automorphisms, rigid homomorphisms correspond bijectively to finite separable extensions of function fields. If instead of the full Galois group of a function field \(K\) over a finite field \(k\) of characteristic \(p\) one considers the extension of the absolute Galois group of \(k\) by the maximal prime-to-\(p\) quotient of that of \(K{\bar k}\), it is possible to recover from rigid homomorphisms only those extensions that become of degree prime to \(p\) after base change to the algebraic closure of the base field. In order to recover finite separable extensions in this case as well, the authors impose a more involved technical condition on homomorphisms of Galois groups that they call proper and inertia-rigid.
The (technically difficult) proofs are refinements of the arguments of earlier papers, in particular those of [loc. cit.]. In the case of rigid homomorphisms such a refinement is possible but the authors are able to give a quicker proof by reducing the statement to the case of isomorphisms treated by \textit{K. Uchida} [Ann. Math. (2) 106, 589--598 (1977; Zbl 0372.12017)] for the full Galois group and in their previous paper in the prime-to-\(p\) case. birational anabelian geometry; Galois group of function field M. Saïdi and A. Tamagawa, On the Hom-form of Grothendieck's birational anabelian conjec- ture in positive characteristic, Algebra Number Theory 5 (2011), 131-184. Coverings of curves, fundamental group, Curves over finite and local fields, Finite ground fields in algebraic geometry On the Hom-form of Grothendieck's birational anabelian conjecture in positive characteristic | 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 Given two Schubert classes \(\sigma_{\lambda}\) and \(\sigma_{\mu}\) in the quantum cohomology of a Grassmannian, we construct a partition \(\nu \), depending on \(\lambda\) and \(\mu \), such that \(\sigma_{\nu}\) appears with coefficient 1 in the lowest (or highest) degree part of the quantum product \(\sigma_{\lambda}\bigstar \sigma_{\mu}\). To do this, we show that for any two partitions \(\lambda\) and \(\mu\), contained in a \(k \times (n - k)\) rectangle and such that the \(180^{\circ}\)-rotation of one does not overlap the other, there is a third partition \(\nu\), also contained in the rectangle, such that the Littlewood-Richardson number \(c_{\lambda \mu}^{\nu}\) is 1. quantum cohomology; toric tableau; Littlewood-Richardson number Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory A note on quantum products of Schubert classes in a 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 A GKZ-hypergeometric system is a holonomic \(\mathcal D\)-module determined by an integral matrix \(A\) and a parameter vector \(\beta\). For homogeneous non-resonant systems Gelfand, Kapranov and Zelevinsky [\textit{I. M. Gelfand} et al., Adv. Math. 84, No. 2, 255--271 (1990; Zbl 0741.33011)] showed that the solution complex is isomorphic to a direct image of a local system, defined on the complement of the graph of an associated family of Laurent polynomials. In this paper for resonant but not strongly-resonant parameters a tight relation is established between certain direct sums of GKZ-systems and Gauss-Manin systems of associated families of Laurent polynomials. The results carry over to the category of mixed Hodge modules. It is shown that a homogeneous GKZ-system with non strongly-resonant, integer parameter vector \(\beta\) carries a mixed Hodge module structure. For rational \(\beta\) the GKZ-system is a direct summand in a mixed Hodge module, showing that the underlying perverse sheaf has quasi-unipotent local monodromy. Gauss-Manin system; Radon transform; mixed Hodge module; hypergeometric \(\mathcal D\)-module [12] Thomas Reichelt, &Laurent polynomials, GKZ-hypergeometric systems and mixed Hodge modulesompos. Math.150 (2014) no. 6, p. 911Article | &MR 32 Variation of Hodge structures (algebro-geometric aspects), Mixed Hodge theory of singular varieties (complex-analytic aspects), Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation), Other hypergeometric functions and integrals in several variables Laurent polynomials, GKZ-hypergeometric systems and mixed Hodge modules | 0 |
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