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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 author proves the following result. Let \(\textbf{P}\) be a property of noetherian local rings satisfying several conditions (regular, normal, complete intersection etc. do satisfy) and let \(\phi:A\to B\) be a local flat morphism of noetherian local rings. If the formal fibers of \(A\) are geometrically \textbf{P} and the closed fiber of \(\phi\) is geometrically \textbf{P}, then all the fibers of \(\phi\) are geometrically \textbf{P}. This is the positive answer to the well-known Grothendieck Localization Problem. The result was known for some properties, like \textbf{P}=regular (André), \textbf{P}=complete intersection (Marot) etc. But the present proof, not only solves the problem for all the properties \textbf{P}, it moreover gives a uniform proof for all of them. The proof uses Gabber's weak local uniformization theorem, instead of Hironaka's resolution of singularities used in several other approaches. This enables the author to obtain positive results about the Localization problem for other properties, like weak normality, seminormality, F-rationality, Cohen-Macaulay and F-injective. As an application it is shown that for all the properties considered, the local lifting problem of Grothendieck has a positive answer: if \(A\) is a semilocal noetherian Nagata ring that is \(I\)-adically complete with respect to an ideal \(I\) and \(A/I\) has geometrically \textbf{P} formal fibers, then \(A\) has geometrically \textbf{P} formal fibers. formal fibers; localization problem; weak normality; seminormality Local structure of morphisms in algebraic geometry: étale, flat, etc., Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Deformations of singularities, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure A uniform treatment of Grothendieck's localization problem | 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 thesis we have tried to figure out some algebraic aspects of noncommutative tori, aiming at generalizing them to arbitrary noncommutative spaces. In the second section all relevant definitions, some examples and motivations have been provided.
In the third section we look at the example of noncommutative tori and see how they can be related to similar objects called noncommutative elliptic curves. We extract a suitably well-behaved subcategory of the category of holomorphic bundles over noncommutative tori. This category turns out to admit a Tannakian structure with \(Z+\theta Z\) as the fundamental group. The key to this construction is an equivariant version of the classical Riemann--Hilbert correspondence. The aim was to construct homotopy theoretic invariants of noncommutative tori, e.g., fundamental groups and we make a proposal to that end.
The last two sections constitute an attempt to rewrite some parts of noncommutative algebraic geometry in the framework of DG categories. We provide a description of the category of noncommutative spaces and their associated noncommutative motives. We had some arithmetic applications in mind, namely, introducing and studying motivic zeta functions of noncommutative tori. We propose a universal motivic measure on the category of noncommutative spaces. In it lies a subcategory consisting of noncommutative Calabi--Yau spaces containing elliptic curves and noncommutative tori. In this setting we introduce a motivic zeta function of noncommutative tori; more generally that of noncommutative Calabi--Yau spaces. Our work should be put in perspective with the Real Multiplication programme of Manin. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Noncommutative algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Derived categories, triangulated categories, Calabi-Yau manifolds (algebro-geometric aspects), Elliptic curves Algebraic aspects of noncommutative tori. The Riemann--Hilbert correspondence | 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 a BGG-type category of infinite-dimensional representations of
\(\mathcal H[W]\), a semidirect product of the quantum torus with parameter \(q\), built on the root lattice of a semisimple group \(G\), and the Weyl group of \(G\). Irreducible objects of our category turn out to be parametrized by semistable \(G\)-bundles on the elliptic curve \(\mathbb C^*/q^\mathbb Z\).
We introduce a noncommutative deformation of the algebra of regular functions
on a torus. This deformation \(\mathcal H\), called quantum torus algebra, depends on a complex parameter \(q\in\mathbb C^*\). We further introduce a certain category \(\mathcal M(\mathcal H,\mathcal A)\) of representations of \(\mathcal H\) which are locally-finite with respect to a commutative subalgebra \(\mathcal A\subset\mathcal H\) whose ``size'' is one-half of that of \(\mathcal H\) (our definition is modeled on the definition of the category \(\mathcal O\) of Bernstein-Gelfand-Gelfand). We classify all simple objects of \(\mathcal M(\mathcal H,\mathcal A)\) and show that any object of \(\mathcal M(\mathcal H,\mathcal A)\) has finite length.
In \S3 we consider quantum tori arising from a pair of lattices coming from
a finite reduced root system. Let \(W\) be the Weyl group of this root system. We
classify all simple modules over the twisted group ring \(\mathcal H[W]\) which belong to \(\mathcal M(\mathcal H,\mathcal A)\) as \(\mathcal H\)-modules. In \S4 we show that the twisted group ring \(\mathcal H[W]\) is Morita equivalent to \(\mathcal HW\), the ring of \(W\)-invariants.
In \S5 we establish a bijection between the set of simple modules over the algebra \(\mathcal H[W]\) associated with a semisimple simply-connected group \(G\), and the set of pairs \((P, \alpha)\), where \(P\) is a semistable principal \(G\)-bundle on the elliptic curve \(E = \mathbb C^*/q^\mathbb Z\), and \(\alpha\) is a certain ``admissible representation'' (cf. Definition 5.4) of the finite group \(\Aut(P)/(\Aut P)^\circ\). Our bijection is constructed by combining the results of \S3 with a bijection between \(q\)-conjugacy classes in a loop group and \(G\)-bundles the elliptic curve \(E\), established earlier by some of us in [Baranovsky and Ginzburg, Int. Math. Res. Not. 1996, No. 15, 733--751 (1996; Zbl 0992.20034)]. Baranovsky, V.; Evens, S.; Ginzburg, V., Representations of quantum tori and \(G\)-bundles on elliptic curves, (The Orbit Method in Geometry and Physics (Marseille, 2000), Progr. Math., vol. 213, (2003), Birkhauser Boston Boston, MA), 29-48 Quantum groups (quantized enveloping algebras) and related deformations, Ring-theoretic aspects of quantum groups, Vector bundles on curves and their moduli Representations of quantum tori and \(G\)-bundles of elliptic 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 For a Grothendieck category having a noetherian generator, we prove that there are only finitely many minimal atoms. This is a noncommutative analogue of the fact that every noetherian scheme has only finitely many irreducible components. It is also shown that each minimal atom is represented by a compressible object. minimal atom; Grothendieck category; Noetherian generator; compressible object Abelian categories, Grothendieck categories, Noetherian rings and modules (associative rings and algebras), Module categories in associative algebras, Noncommutative algebraic geometry, Module categories and commutative rings Finiteness of the number of minimal atoms in Grothendieck categories | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this article, the author presents a quantum cohomology analogue of skew Schur polynomials. These are certain symmetric polynomials labeled by shapes that are embedded in a torus. The author shows that the Gromov-Witten invariants are the expansion coefficients of these toric Schur polynomials in the basis of the ordinary Schur polynomials. The toric Schur polynomials are defined as sums over certain cylindrical semistandard tableaux. This paper makes an important contribution to the quantum cohomology theory of the Grassmannians. quantum cohomology; Schur polynomials; Gronov-Witten invariants A. Postnikov. ''Affine approach to quantum Schubert calculus''. Duke Math. J. 128 (2005), pp. 473--509.DOI. Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Affine approach to quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main purpose of this paper is to provide a survey of different notions of algebraic geometry, which one may associate to an arbitrary noncommutative ring \(R\). In the first part, we will mainly deal with the prime spectrum of \(R\), endowed both with the Zariski topology and the stable topology. In the second part we focus on quantum groups and, in particular, on schematic algebras and show how a noncommutative site may be associated to the latter. In the last part, we concentrate on regular algebras, and present a rather complete up to date overview of their main properties. noncommutative algebraic geometry; algebras satisfying a polynomial identity; quantum groups; schematic algebras; regular algebras Noncommutative algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations, Rings with polynomial identity, Rings arising from noncommutative algebraic geometry Noncommutative algebraic geometry: From pi-algebras to 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 There are two different ways to deform a quantum curve along the flows of the KP hierarchy. We clarify the relation between the two KP orbits: In the framework of suitable connections attached to the quantum curve they are related by a local Fourier duality. As an application we give a conceptual proof of duality results in 2D quantum gravity. Luu, M., Schwarz, A.: Fourier duality of quantum curves. (Preprint) Renormalization group methods applied to problems in quantum field theory, Quantization of the gravitational field, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relationships between algebraic curves and physics Fourier duality of quantum curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author investigates quadratic relations for Gromov-Witten invariants, arising from the associativity of the quantum product (i.e., the famous WDVV equations). He concentrates to the case of rational curves (i.e. invariants associated to the moduli space of stable maps of genus zero).
Here, the relations can be described by Feynman diagrams relating trees of rational curves.
In several examples, the associativity has been used to compute all Gromov-Witten numbers recursively from some starting data. Considered as a purely algebraic problem, one may ask which invariants are needed in general as starting data such that the WDVV equations can be solved uniquely and consistently. The main result obtained by the author is the following.
Strong reconstruction theorem. Let \(X\) be a complex projective manifold, and assume that \(H^{2*}(X,\mathbb{Q})\) is generated by divisors. Then it is sufficient to know all Gromov-Witten invariants \(N(\beta,d)\) with \(\sum_{i=1}^s d_s\leq 2\) (which have to fulfil a simple initial relation)
to compute uniquely and consistently all Gromov-Witten invariants of \(X\) by means of the WDVV equations.
The proof is constructive and gives in principle an algorithm (possibly quite complicated) for the computation. It is illustrated in detail with the examples of the product of a smooth quadric threefold with itself, \( \text{Sym}^2 {\mathbb{P}}^2 \), and the Grassmannians \(\text{Gr}(2,4)\) and \(\text{Gr}(2,5)\). Actually, only the first example satisfies the conditions of the reconstruction theorem, but similar techniques work well in the remaining cases. The author also points out that there are in general non-geometric solutions to a WDVV system, coming from other initial data than the geometric ones. WDVV equation; Feynman diagrams; genus zero Gromov-Witten invariants Kresch, A.: Associativity relations in quantum cohomology, Adv. math. 142, No. 1, 151-169 (1999) Enumerative problems (combinatorial problems) in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Associativity relations 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 This rather important paper indicates a precise concrete way to perform computations in the quantum equivariant ``deformation'' of the cohomology ring of \(G(k,n)\), the complex Grassmannian variety parametrizing \(k\)-dimensional vector subspaces of \({\mathbb C}^n\). It relies on the results of another important paper, regarding the same subject, by the same author [Adv. Math. 203, 1--33 (2006; Zbl 1100.14045)]. The usual singular cohomology ring of \(G(k,n)\) is a very well known object, studied since Schubert's time, at the end of the XIX Century. First of all, it is a finite free \({\mathbb Z}\)-module generated by the so-called Schubert cycles. Furthermore, the special Schubert cycles, the Chern classes of the universal quotient bundle over \(G(k,n)\), generates it as a \({\mathbb Z}\)-algebra. Multiplying two Schubert cycles then amounts to know how to multiply a special Schubert cycle with a general one (Pieri's formula) and a way to express any Schubert cycle as an explicit polynomial expression in the special Schubert cycles (Giambelli's formula).
The obvious way to deform the cohomology of a Grassmannian is to consider the cohomology of the total space of a Grassmann bundle, parametrizing \(k\)-planes in the fibers of a rank \(n\) vector bundle, which is a deformation of the cohomology of any fiber of it. In the last few decades, however, other ways to deform the cohomology ring of \(G(k,n)\) have been studied. \textit{E. Witten} [in: Geometry, topology and physics for Raoul Bott. Lectures of a conference in honor of Raoul Bott's 70th birthday, Harvard University, Cambridge, MA, USA 1993. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Geom. Topol. 4, 357--422 (1995; Zbl 0863.53054)], introduced the small quantum deformation of the cohomology ring of the Grassmannian, whose structure constants were first determined by \textit{A. Bertram} [Adv. Math. 128, No. 2, 289--305 (1997; Zbl 0945.14031)]. Finally, \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)], studied the equivariant deformation of the cohomology of the Grassmannians via the combinatorics of puzzles.
In the beautiful paper under review the author recovers the quantum and equivariant Schubert calculus within a unified framework. Basing on the algebraic properties of the Schur factorial functions, the author realizes the equivariant quantum cohomology ring in terms of generators and relations and gives an explicit basis of polynomial representatives for the equivariant quantum Schubert classes. An alternative approach is offered by \textit{D. Laksov} [Adv. Math. 217, 1869--1888 (2008; Zbl 1136.14042)], where the author proves that the basic results of equivariant Schubert calculus, the basis theorem, Pieri's formula and Giambelli's formula can be obtained from the corresponding results of a more general and elementary framework, as in [\textit{D. Laksov}, Indiana Univ. Math. J., 56, No. 2, 825--845 (2007; Zbl 1136.14042)], by a change of basis.
The paper is organized as follows. Section 1 is the introduction, where the main results are clearly stated and motivated; Section 2 is a useful and very pleasant review of the algebra of factorial Schur functions. The quantum equivariant cohomology of Grassmannians is treated in Section 3, while the proof of the theorem about the presentation of the quantum equivariant cohomology ring is given in Section 4. Section 5 ends the paper with the discussion and the proof of Giambelli's formula in equivariant quantum cohomology. Giambelli's formulas; quantum equivariant Schubert calculus; factorial Schur functions L.C. Mihalcea, \textit{Giambelli formulae for the equivariant quantum cohomology of the Grassmannian}, \textit{Trans. AMS}\textbf{360} (2008) 2285 [math/0506335]. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Giambelli formulae for the equivariant quantum cohomology of the 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 Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of \(\mathrm{GL}_n\). We construct the action of the quantum loop algebra \({U_v({\mathbf L}\mathfrak{sl}_n)}\) in the \(K\)-theory of Laumon spaces by certain natural correspondences. Also we construct the action of the quantum toroidal algebra \(\text{Ü}_{v} {(\widehat{\mathfrak{sl}}_n)}\) in the \(K\)-theory of the affine version of Laumon spaces.
This note is a sequel to [\textit{B. Feigin} et al., Sel. Math., New Ser. 17, No. 2, 337--361 (2011; Zbl 1285.14011); Sel. Math., New Ser. 17, No. 3, 573--607 (2011; Zbl 1260.14015)]. Tsymbaliuk, A., Quantum affine Gelfand-tsetlin bases and quantum toroidal algebra via \(K\)-theory of affine laumon spaces, Selecta Math. (N.S.), 16, 173-200, (2010) Quantum groups (quantized enveloping algebras) and related deformations, Algebraic moduli problems, moduli of vector bundles, \(K\)-theory in geometry Quantum affine Gelfand-Tsetlin bases and quantum toroidal algebra via \(K\)-theory of affine Laumon spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be a field and let \(K_{0}(\text{Var}_{k})\) denote the Grothendieck group of varieties over \(k\). In [Séminaire Bourbaki 1999/2000, Astérisque 276, 267--297 (2002; Zbl 0996.14011)], \textit{E. Looijenga} showed that if \(k\) is an algebraically closed field of characteristic 0, and if \(G\) is a finite abelian group acting linearly on a finite dimensional \(k\)-vector space \(V\), then \([V/G]=\mathbb{L}^{\dim_{k}V}\), where \(\mathbb{L}\) denotes the class of the affine line over \(k\).
In this paper the authors investigate possible generalizations of this formula when \(k\) is not assumed to be algebraically closed. For example they show that if \(dim_{k}V\leq2\), then Looijenga's formula holds true. Their proof makes it clear that the main reason for the restriction to the case when \(\dim_{k}V\leq2\) is that in this case \(\mathbb{P}(V)\cong\mathbb{P}_{k}^{1}\) and hence \(\mathbb{P}(V)/G\cong\mathbb{P}_{k}^{1}\) as well. Grothendieck ring of varieties; rationality Esnault, H; Viehweg, E, On a rationality question in the Grothendieck ring of varieties, Acta Math. Vietnam., 35, 31-41, (2010) Rational points, Other nonalgebraically closed ground fields in algebraic geometry On a rationality question in the Grothendieck ring of varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the course of its history, algebraic geometry has undergone several fundamental changes with respect to its conceptual foundations, methods, and techniques. A brilliant analysis of the historical developments in algebraic geometry can be found in the first volume of \textit{J. Dieudonné}'s ``History of algebraic geometry''
[``Cours de Géometrie Algébrique'', Presses Universitaires de France, Paris 1974, Engl. translation by Judith D. Sally, Wadsworth (1985; Zbl 0629.14001)], according to which the history of algebraic geometry is characterized by about seven main epoches. The so far most recent epoch of algebraic geometry began in the 1950s, when A. Grothendieck launched his revolutionary program of refounding the entire theory by means of a completely new, much more general conceptual framework, including arbitrary algebraic schemes, sheaves and their cohomology, methods of categorical and homological algebra, relative geometry, classifying spaces, and other powerful tools. Grothendieck first sketched his new theories, which turned over a new leaf in the development of algebraic geometry, in a series of talks at the Seminaire Bourbaki between 1957 and 1962. The notes of these talks were then published in the famous volume [``Fondements de la Géométrie Algébrique'' (Paris: Secrétariat mathématique) (1962; Zbl 0239.14004)], commonly abbreviated FGA. These mimeographed notes became of central importance in the sequel, as they contained the only available outlines of Grothendieck's fundamental new constructions such as descent theory, Hilbert schemes and Quot schemes, formal algebraic geometry, Picard schemes, and other pioneering approaches toward a better understanding of classical problems in algebraic geometry.
Grothendieck's various theories sketched in FGA were of crucial significance for the tremendous progress in algebraic geometry thereafter, and most of them even became indispensable ingredients in allied fields of contemporary mathematics such as arithmetic, complex-analytic geometry, mathematical physics, and others. Certainly, much of Grothendieck's FGA is now common knowledge, or at least utilized folklore, but a good deal of it is still less well-known, and perhaps only a few experts are familiar with its full scope.
In view of the undiminished significance of Grothendieck's ideas outlined in FGA for the current and future research in algebraic geometry, the book under review aims at explaining its rich contents in full detail, thereby taking into account recent developments and improvements. Written by six experts in the field, this book contains the elaborated lectures delivered by the authors at the ``Advanced School in Basic Algebraic Gecmetry'', which was held at the International Centre for Theoretical Physics (ICTP) in Trieste, Italy, from July 7 to July 18, 2003, and which addressed advanced graduate students as well as beginning researchers in algebraic geometry.
As the authors point out, this book is not intended to replace Grothendieck's celebrated FGA, after more than forty years. Rather, it is to fill in Grothendieck's ingenious sketches with detailed proofs, on the one hand, and to present both newer ideas and more recent developments wherever appropriate, on the other hand. In accordance with Grothendieck's original FGA, the present book consists of five parts written by different authors, in which the following topics are treated: (1) theory of descent; (2) Hilbert schemes and Quot schemes; (3) local properties of Hilbert schemes of points; (4) Grothendieck's existence theorem in formal geometry; and (5) Picard schemes.
Part 1 was written by A. Vistoli (Bologna). This part comes with the headline ``Grothendieck topologies, fibered categories, and descent theory'' and is subdivided into four chapters. Differing from Grothendieck's original approach in FGA, descent theory is here presented in the language of Grothendieck topologies, which Grothendieck actually introduced somewhat later. Chapter 1 reviews some basic notions of scheme-theoretic algebraic geometry and of category theory, whereas chapter 2 continues the warm-up with a brief introduction to representable functors, Grothendieck topologies and their sheaves, and to group objects in categories. Chapter 3 discusses fibered categories, which provide the appropriate abstract set-up for explaining descent theory in its full generality. The main example, in this context is the category of quasi-coherent sheaves over the category of schemes. Chapter 4 develops descent theory for algebraic stacks, that is for those fibered categories in which it properly works. The author discusses, in full detail, the various ways of defining descent data, including those for quasi-coherent sheaves, morphisms of schemes, and quasi-coherent sheaves along torsors. Overall, the author has tried to develop descent theory in its greatest generality, beyond the scope of FGA, which is certainly necessary for deeper understanding and more advanced applications.
Part 2, comprising Chapter 5, explains Grothendieck's construction of Hilbert schemes and Quot schemes. Written by N. Nitsure, this chapter provides a modern treatment of these basic constructions using more recent basic tools such as faithtully flat descent, flattening stratifications, semi-continuity techniques, and Castelnuovo-Mumford regularity of sheaves. This chapter concludes with some concrete applications due to D. Mumford, A. Altman and S. Kleiman, A. Grothendieck himself, and others.
Part 3 is formed by chapters 6 and 7. Containing the lectures by B. Pantechi and L. Göttsche (Trieste), this chapter discusses local properties and Hilbert schemes of points. Chapter 6 introduces elementary deformation theory in algebraic geometry, involving the infinitesimal study of schemes, the infinitesimal deformation functor, a tangent-obstruction theory for such a functor, and local moduli spaces. The theory is thoroughly worked out in some special cases, and sketched in a few others. This is used in chapter 7, where the Hilbert scheme of points on a smooth quasi-projective variety is studied. The reader meets here the Hilbert-Chow morphism, stratifications of Hilbert schemes of points, Betti numbers of Hilbert schemes, and further more recent cohomological results in this direction.
Part 4, written by L. Illusie (Paris), is devoted to Grothendieck's FGA exposé, where he established a fundamental comparison theorem of GAGA-type between algebraic geometry and his newly created formal geometry, on the one hand, and outlined some possible applications to the theory of the algebraic fundamental group and to infinitesimal deformation theory, on the other hand. Shortly afterwards, \textit{A. Grothendieck} gave a detailed account of this particular topic in EGA III [Publ. Math., Inst. Hautes Étud. Sci. 17, 137--223 (1963; Zbl 0122.16102)] and in SGA1 [Lect. Notes Math. 224 (1971; Zbl 0234.14002)]. Here, in part 4 of the book under review, the author revisits this topic in great detail, and in a more introductory manner for non-experts, ending with a discussion of J.-P. Serre's celebrated examples of varieties in positive characteristic that do not lift to characteristic zero.
Being one of the leading experts in this realm, the author has taken the opportunity to give various improvements and updatings of Grothendieck's original approach, mainly by using the toolkit of derived categories and, especially, perfect complexes, and he has enhanced the entire discussion by several more recent applications to lifting problems in algebraic or formal geometry, respectively. Part 4 comprises chapter 8 of the book, whereas the final part 5 is identical with chapter 9. This concluding part, written by S. Kleiman (Boston), is perhaps the show-piece of the entire collection. In a masterly manner, beginning with a highly enlightening introduction of 15 pages, the author develops in great detail most of Grothendieck's theory of the Picard scheme, together with its further developments later on.
After the extensive historical introduction, which is also of independent cultural interest, the author describes the four common relative Picard functors in their comparison, and proves then Grothendieck's existence theorem for the Picard scheme. This is followed by the study of both the connected component of the identity and the torsion component of the identity of the Picard scheme, including the related deep finiteness theorems, and the entire treatise closes with two appendices. Appendix A provides detailed answers to all the exercises scattered in this chapter, and appendix B contains an elementary treatment of basic intersection theory, as far as it is used freely in some proofs.
All together, this book must be seen as a highly valuable addition to Grothendieck's fundamental classic FGA, and as a superb contribution to the propagation of his pioneering work just as well. It is fair to say that, for the first time, the wealth of Grothendieck's FGA has been made accessible to the entire community of algebraic geometers, including non-specialists, young researchers, and seasoned graduate students. The authors have endeavoured to elaborate Grothendieck's ingenious, epoch-making outlines in the greatest possible clarity and detailedness, with complete proofs given throughout, and with various improvements, simplifications, updatings, and topical hints wherever appropriate. Due to this rewarding undertaking, FGA has come down to earth that is much closer to the community of all algebraic geometers, and therefore the book under review ought to be in the library of anyone using modern algebraic geometry in his research. textbook; algebraic geometry; Grothendieck topologies; fibred categories; descent theory; Hilbert schemes; Picard schemes; formal geometry B. Fantechi, L. Göttsche, L. Illusie, S.L. Kleiman, N. Nitsure and A. Vistoli, Fundamental Algebraic Geometry. Grothendieck's FGA Explained, Math. Surveys Monogr., \textbf{123}, Amer. Math. Soc., Providence, RI, 2005. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Divisors, linear systems, invertible sheaves, Formal methods and deformations in algebraic geometry, Picard schemes, higher Jacobians, Fibered categories Fundamental algebraic geometry: Grothendieck's FGA explained | 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 well-written paper on zeta-functions, Grothendieck groups, motivic measures and the Witt ring. For any commutative ring \(A\) with identity denote the big Witt ring with addition and with the multiplication \(\ast\) by \( W(A)\). For any natural \(n\) there are a Frobenius ring homomorphsm \(F_n:W(A)\to W(A)\) and an additive Verschiebung ring homomorphism \(V_n:W(A) \to W(A)\). Let \(k\) be a field. By \(GK_k\) the author of the paper under review denotes the Grothendieck ring \(K_0(\mathrm{Var}_k)\) of schemes of finite type over \(k\).
The author proves the following. Theorem 2.1: Let \(X\) and \(Y\) be schemes of finite type over \(\mathbb F_q\).
(i) The zeta function of the product \(X\times Y\) is the Witt product of the zeta functions of \(X\) and \(Y\). In particular, \(Z(X^n,t)=Z(X,t)\ast\cdots\ast Z(X,t)\).
(ii) The map \(\kappa:GK_{\mathbb F_q}\to W(\mathbb Z)\) is a ring homomorphism. Hence \(X\mapsto Z(X,t)\) is a motivic measure.
(iii) If \(X\to B\) is a (Zariski locally trivial) fiber bundle with fibre \(F\), namely, there is a covering of \(B\) by Zariski opens \(U\) with \(X\times_BU\) isomorphic to \(U\times_{\mathrm{Spec }\mathbb F_q}F\), then \(Z(X, t)=Z(B,t)\ast Z(F,t)\).
(iv) For any \(m\in\mathbb N\), let \(X_m\) be the variety over \(\mathbb F_{q^m}\) obtained by base change along \(b:\mathbb F_q\to\mathbb F_{q^m}\). One has \(Z(X_m/\mathbb F_{q^m},t)=F_mZ(X/\mathbb F_q,t)\).
(v) One has a commutative diagram of ring homomorphisms with upper \(b:GK_{\mathbb F_q}\to GK_{\mathbb F_q^m}\) and low \(F_m:W(\mathbb Z)\to W(\mathbb Z)\) horizontal arrows, and with left \( \kappa:GK_{\mathbb F_q}\to W(\mathbb Z)\) and right \(\kappa:GK_{\mathbb F_q^m}\to W(\mathbb Z)\) vertical arrows.
Analogical results were established by \textit{N. Naumann} [Trans. Am. Math. Soc. 359, No. 4, 1653--1683 (2007; Zbl 1115.14004)] in connection with the Grothendieck ring of varieties.
Let \(A\) be an abelian variety over \(\mathbb F_q\) and \(A'\) be an abelian variety over \(\mathbb F_q^m\).
Theorem 2.6: Let notations be as above.
(a) Let \(A'\) be an abelian variety over \(\mathbb F_q^m\). Let \(P_1(A',t)=\prod_j(1-\alpha_jt)\) and \(P_1(A,t)=\prod_r(1-\beta_rt)\). One has \(P_1(A,t) =V_mP_1(A',t)=P_1(A',t^m)=\prod_j(1-\alpha_jt^m)\). The set \(\{\beta_1^m,\cdots\}\) coincides with the set \(\{\alpha_1,\cdots\}\).
(b) For any smooth projective variety \(X'\), one has \( P_1(X,t)=V_mP_1( X',t) \).
(c) Let \(X'\) be a smooth proper geometrically connected variety over \(\mathbb F_q^m\). For each integer \(0<i\leq 2\dim X'\), the polynomial \(P_i(X,t)\) is divisible by \(V_mP_i( X',t)\). In general, \(Z(X,t)\neq V_mZ(X',t) \), \(Z(X\times_{\mathbb F_q}\mathbb F_q^m,t)=Z((X')^m,t)=Z(X,t)\ast\cdots\ast Z(X,t)\).
The relation between \(a_r=\sharp X'(\mathbb F_q^r)\) and \(b_r = \sharp X(\mathbb F_q^{mr})\) can be described explicitly (using \(d=gcd(m,r)\) and \(r=sd):b_r = a^d_s\).
The paper is an excellent survey of the results that connecting zeta functions of varieties (over finite fields) with motivic measures on the category of schemes of finite type (over a field) and with the big Witt ring over \(\mathbb Z\). For the sake of completeness the reader should also refer to the paper by \textit{S. Lichtenbaum} [Fields Inst. Commun. 56, 249--255 (2009; Zbl 1245.14023)], to the book dealing with zeta elements by \textit{K. Kato} et al. [Number theory 1. Fermat's dream. Providence, RI: AMS (2000; Zbl 0953.11003)] and also to the preprint dealing with motivic zeta function with coefficients in the Grothendieck ring of varieties by \textit{M. Kapranov} [``The elliptic curve in the \(S\)-duality theory and Eisenstein series for Kac-Moody groups'', Preprint, \url{arXiv:math/0001005}]. zeta functions; symmetric products; big Witt ring; motivic measures Ramachandran, N., Zeta functions,Grothendieck groups, and thewitt ring, Bull. Sci. Math., 139, 599-627, (2015) Algebraic theory of quadratic forms; Witt groups and rings, Varieties over finite and local fields, Grothendieck groups, \(K\)-theory and commutative rings, (Equivariant) Chow groups and rings; motives, Other Dirichlet series and zeta functions Zeta functions, Grothendieck groups, and the Witt 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 It is known that Dolbeault-Dirac operators on Kähler manifolds can be expressed as \(D = \eth + \eth^* \in U(\mathfrak{g}) \otimes Cl\), where \(U(\mathfrak{g})\) is the enveloping algebra of \(\mathfrak{g}\) and \(Cl\) is an appropriate Clifford algebra. For a Dolbeault-Dirac operator \(D\), Parthasarathy formula gives an expression for \(D^2\) in terms of quadratic Casimirs, up to multiples of the identity.
Krähmer and Tucker-Simmons found that Dolbeault-Dirac operators on quantized irreducible flag manifolds can be expressed in the form of \(D = \eth +\eth^* \in U_q(\mathfrak{g})\otimes Cl_q\), where \(U_q(\mathfrak{g})\) is the quantized enveloping algebra of \(\mathfrak{g}\) and \(Cl_q\) is the quantum Clifford algebra.
In this paper, the author proves that, unlike the classical case, a Parthasarathy-type formula does not hold for quantized irreducible flag manifolds, For example, the Lagrangian Grassmannian LG(2, 4) is considered. Dolbeault-Dirac operators; quantum Clifford algebras; Parthasarathy formula Matassa, M.: Dolbeault-Dirac operators on quantum projective spaces (2015). arXiv:1507.01823 (preprint) Selfadjoint operator theory in quantum theory, including spectral analysis, Covariant wave equations in quantum theory, relativistic quantum mechanics, Clifford algebras, spinors, Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices, The Levi problem in complex spaces; generalizations, Grassmannians, Schubert varieties, flag manifolds Dolbeault-Dirac operators, quantum Clifford algebras and the Parthasarathy 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 article considers graded Clifford algebras on \(n\) generators over polynomial rings over algebraically closed fields of \(\text{char}\neq 2\) and two-parameter deformations thereof. Regularity, Auslander regularity and Cohen-Macaulay property are discussed as the projective geometric properties behind the defining (commutation) relations. Generic, graded Clifford algebras have defining relations which are determined by their zero locus. An example of such a Clifford algebra generated by four grade 1 generators is given which contains exactly twenty point modules. A fourfold class of one parameter families of such algebras is constructed, which are found to be iterated Ore extensions on four generators. These algebras are deformations of a graded Clifford algebra such that the generic member has precisely one point module and a 1-dimensional family of line modules. The paper closes with some remarks and open problems. regular algebras; quadratic algebras; projective geometry; quadrics; commutation relations; graded Clifford algebras; two-parameter deformations; defining relations; point modules; iterated Ore extensions; line modules Vancliff, M.; Van Rompay, K.; Willaert, L., Some quantum \({\mathbf P}^3\)s with finitely many points, Comm. Algebra, 26, 4, 1193-1208, (1998) Graded rings and modules (associative rings and algebras), Noncommutative algebraic geometry, Ordinary and skew polynomial rings and semigroup rings, Deformations of associative rings, Quantum groups (quantized enveloping algebras) and related deformations, Low codimension problems in algebraic geometry, Clifford algebras, spinors Some quantum \(\mathbb{P}^3\)s with finitely many points | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Grothendieck-Teichmüller group \(\widehat{GT}_{0}\) was introduced by Drinfel'd, based on ideas of Grothendieck, to study the absolute Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\). This group contains in particular the information of the action of the absolute Galois group on dessins d'enfants.
In [\textit{P. Guillot}, Springer Proc. Math. Stat. 159, 159--191 (2016; Zbl 1357.14031)] the author defined the Grothendieck-Teichmüller group \(\mathcal{GT}(G)\) of a finite group \(G\). In a similar way as above, this group encodes via a map \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathcal{GT}(G)\) the action of the absolute Galois group on the set of dessins d'enfants whose monodromy group is \(G\) and, actually, one can retrieve the classical Grothendieck-Teichmüller group \(\widehat{GT}_{0}\) as the inverse limit \(\lim_{G} \mathcal{GT}(G)\). In this context, the interesting part of the group \(\mathcal{GT}(G)\) consists of the subgroup \(\mathcal{GT}_{1}(G)\) corresponding to the non-abelian part of the Galois action.
The object of this paper is the study of the Grothendieck-Teichmüller group of the group \(G=\mathrm{PSL}(2,q)\). The main result is that the group \(\mathcal{GT}_{1}(\mathrm{PSL}(2,2^{s}))\) is trivial for all \(s\geq 1\) and the group \(\mathcal{GT}_{1}(\mathrm{PSL}(2,q))\) is isomorphic to certain product \(C_{2}^{n_{1}}\times D_{8}^{n_{2}}\), for \(q\) odd.
The proof is based on a characterization of this group as a subgroup of the permutation group on the set of conjugacy classes of triples of generators of \(G\). The author uses then the classification of triples of generators of \(\mathrm{PSL}(2,q)\) in [\textit{A. M. Macbeath}, Proc. Symp. Pure Math. 12, 14--32 (1969; Zbl 0192.35703)] to determine the subgroup \(\mathcal{GT}_{1}(G)\).
As a consequence of this result, the author also proves that the field of moduli of any dessin whose monodromy group is isomorphic to \(\mathrm{PSL}(2,q)\) has derived length 3. Grothendieck-Teichmüller group; Galois group; dessins d'enfants Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Dessins d'enfants theory, Limits, profinite groups, Automorphism groups of groups The Grothendieck-Teichmüller group of \(\mathrm{PSL}(2,q)\) | 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 classical Hilbert Nullstellensatz experiences since decades extensions and variants for the noncommutative setting. For this see e.g. [\textit{M. Brešar} and \textit{I. Klep}, in: Notions of positivity and the geometry of polynomials. Dedicated to the memory of Julius Borcea. Basel: Birkhäuser. 79--101 (2011; Zbl 1247.14062)]. In the present paper a new variant previously introduced by the authors [\textit{J. W. Helton} et. al., Adv. Math. 331; 589--626 (2018; Zbl 1420.16015)] and aptly called Singulärstellensatz or Singularitätsstellensatz is further developed; in the fourth section we also find theorems more closely linked to the classical Nullstellensatz.
Denote by \(\mathbb{C}\langle x \rangle^{\delta\times \epsilon} = \mathbb{C}\langle x_1, \dots, x_g \rangle^{\delta\times \epsilon}\) the set of polynomials \(f=\sum_w f_w w \) with matrix coefficients \(f_w\in M_{\delta, \epsilon}(\mathbb{C}), \) and \(w\) monomials (words) in noncommuting variables \(x_1,\dots, x_g.\)
If \(X=(X_1,\dots, X_g) \in M_n(\mathbb{C})^g\) is a \(g\)-uple of \(n\times n\) matrices over \(\mathbb C\), then \(w(X)\) is obtained (as usual) by substituting indeterminate \(x_i\) by matrix \(X_i,\) but the evaluation of \(f\) at \(X\) is defined via Kronecker products: \(f(X)=\sum_w f_w \otimes w(X)\in M_{\delta n, \epsilon n}(\mathbb{C}).\) If \(\delta=\epsilon\) (as predominantly will be the case) we get the algebra \(\mathbb{C}\langle x \rangle^{\delta\times \delta}.\) Then \(f(X)\in M_{n\delta }(\mathbb{C}).\) The \textit{free algebra} is the case \(\delta=1;\) in other words it is \(\mathbb C \langle x\rangle.\)
Define \(\mathcal{ Z}_n(f)=\{X\in M_n(\mathbb{C})^g:\det f(X)=0\}\) and the union \(\mathcal{ Z}(f)=\bigcup_n \mathcal{ Z}_n\) which latter set is called the \textit{free singularity locus}. In Section 2 the relation between factorizability of \(f\) and that of \(f\) evaluated at \(M_n(\mathbb C)^g\) as well as a geometric description of this is studied. Using the definitions of \textit{P. M. Cohn} [Free ideal rings and localization in general rings. Cambridge: Cambridge University Press (2006; Zbl 1114.16001)], \(f\in \mathbb{C}\langle x \rangle^{\delta\times \delta}\) is \textit{full} if it cannot be factored as \(f_1f_2\) with \(f_1\in \mathbb{C}\langle x \rangle^{\delta\times \epsilon},\) \(f_2\in \mathbb{C}\langle x \rangle^{\epsilon \times \delta},\) with \(\epsilon<\delta.\) A non-invertible \(f\) is an \textit{atom} if it cannot be written as a product of non-invertibles in \( \mathbb{C}\langle x \rangle^{\delta\times \delta}.\) The first main result is
Theorem 2.9: A matrix polynomial \(f\) is an atom if and only if \(\det f|_{M_n(\mathbb{C})^g}\) is an irreducible polynomial for almost all \(n\in \mathbb{N}.\)
For a geometric description one introduces another concept found in [loc. cit.]. Two polynomials \(f_i\in \mathbb{C}\langle x \rangle^{\delta_i \times \delta_i},\) \(i=1,2\) are \textit{stably associated} if \(f_2\oplus I_{\epsilon_2}= P(f_1\oplus I_{\epsilon_1})Q\) for some invertible matrices \(P,Q\in \mathbb{C} \langle x \rangle^{(\delta_1+\epsilon_1)\times (\delta_1+\epsilon_1)}\) and suitable \(\epsilon_1,\epsilon_2.\) On the way to their Singulärstellensatz, the authors prove a theorem telling when two matrix pencils of the form \( L=A_0+\sum_{j=1}^g A_j x_j\) have equal (in German: `gleich' ) free locus:
Theorem 2.11 (Gleichstellensatz). Let \(L,M\) be indecomposable linear quadratic pencils. Then \(\mathcal{ Z}(L)=\mathcal{ Z}(M)\) if and only if the pencils have the same size and \(M=PLQ\) for some invertible matrices over \(\mathbb C\) of proper size.
Theorem 2.12 (Singulärstellensatz). Let \(f,h\) be full matrix polynomials. Then \(\mathcal{ Z}(f)\subseteq \mathcal{ Z}(h)\) iff every atomic factor of \(f\) is stably associated to a factor of \(h.\)
The authors say these results (2.9,2.12) are farreaching generalizations of what was obtained in the earlier paper [loc. cit., Theorems A, B]. There involved polynomials were supposed to be invertible at \(0.\) They achieve these results via a new technique they call \textit{point-centered ampliation}. It consists in adding commuting variables to the entries of the matrices of a \(g\)-uple \(X\in M_n(\mathbb C)^g\) and showing for example that atomicity of \(f\) carries over to its point centered ampliation. Also, an important role is played by the fact that matrix polynomials are stably associated to linear pencils. Notable is also the following fact which shows that the intersection of free loci does not behave in the same way as algebraic varieties.
Proposition 2.14. Given matrix polynomials \(f_1,\dots, f_s, h,\) if \(\bigcap_j \mathcal{ Z}(f_j) \subset \mathcal{ Z}(h)\) then there is a \(j\) such that \(\mathcal{ Z}(f_j)\subset \mathcal{ Z}(h).\)
Section 3 modifies results of Section 2 to the real setting. Define \(x^*=(x_1^*, \dots, x_g^*)\) as the formal adjoints to \(x.\) Often we consider now polynomials in \( \mathbb C\langle x,x^* \rangle^{\delta\times \delta}.\) The map \((X,Y) \stackrel{\mathcal J}{\mapsto} (Y^*,X^*)\) from \(M_n(\mathbb{C})^g\times M_n(\mathbb{C})^g\) to itself is conjugate linear and involutive so it defines a \textit{real structure}. If \( \mathcal W_n\subseteq (M_n(\mathbb{C})^g)^2\) is preserved by \(\mathcal J,\) then let \( \mathcal W_n^{\mathrm{re}}=\mathcal W_n \cap \{(X,X^*): X\in M_n(\mathbb C)^g\}\) be the set of \textit{real points} of \(\mathcal W_n.\) One defines \(\mathcal Z_n^{\mathrm{re}}(f)=\{X\in M_n(\mathbb C)^g: \det f(X,X^*) =0\}\) but also \(\mathcal Z_n(f)=\{(X,Y) \in (M_n(\mathbb C)^g)^2: \det f(X,Y) =0\}.\) Between these two sets there reigns the relation \(\mathcal Z_n(f)^{\mathrm{re}}=\{(X,X^*) \in (M_n(\mathbb C)^g)^2: X\in \mathcal Z_n^{\mathrm{re}}(f)\}\) used in some proofs. Section 3.2 proves variants of the above cited facts 2.11, 2.12, 2.14. For example:
Theorem 3.4 (Analytic Singulärstellensatz). Let \(f\in \mathbb{C}\langle x\rangle^{\delta\times \delta}\) be an atom and \( h\in \mathbb C \langle x, x^* \rangle^{\epsilon \times \epsilon}\) a full matrix. Then \(\mathcal Z_n^{\mathrm{re}}(f)\subseteq \mathcal Z_n^{\mathrm{re}}(h)\) if and only if \(f\) or \(f^*\) is stably associated to a factor of \(h.\)
In Section 3.3 further results are obtained for polynomials \(f\in \mathbb{C}\langle x,x^* \rangle^{\delta\times \delta} \) for which \(f=f^*,\) called \textit{hermitian} polynomials. The investigations here were inspired by Theorem 4.5.1 in the book by \textit{J. Bochnak} et al. [Real algebraic geometry. Transl. from the French. Rev. and updated ed. Berlin: Springer (1998; Zbl 0912.14023)]. It characterises the principal ideal generated by an irreducible polynomial \(f\) in the commutative polynomial ring \(R[x_1,\dots,x_n]\) as real if \(f\) changes sign in \(R^n,\) \(R\) a real closed field. A hermitian polynomial \(f\) is \textit{unsignatured} if there exist \(n\in \mathbb{N}\) and \(X,Y \in M_n(\mathbb{C})^g\) such that matrices \(f(X,X^*),f(Y,Y^*)\) are invertible and differ in their Sylvester signatures. Again we find versions of the mentioned facts for hermitian unsignatured polynomials or pencils. In the final Section 4 are proved the results that are nearest to the classical Nullstellensatz. One defines for an ideal \(J\) in \(\mathbb C\langle x\rangle\) the sets \(\mathcal V_n(J)=\{X\in M_n(\mathbb{C})^g: f(X)=0 \text{ for all }f\in J \}\) and \(\mathcal V(J)= \bigcup_n \mathcal V_n(J)\) as the \textit{free zero set}. Background on formally or geometrically rationally resolvable (frr and grr) ideals and the definition of the Nullstellensatz property of \(J\) is provided in Section 4.1. The source for this is [\textit{I. Klep} et al., Linear Algebra Appl. 527, 260--293 (2017; Zbl 1403.14007)]. Combining that articles' Theorem 2.5 and Proposition 2.6 and some technical lemmas the following is shown:
Theorem 4.5. Assume \(f=f^* \in \mathbb C\langle x,x^* \rangle\) is nonconstant and \(f(X_0,X_0^*) \) is positive definite for some matrix \(X_0.\) Let \(y\) be an additional indeterminate and \(h\in \mathbb C\langle x,x^*, y, y^* \rangle.\) Then \(\mathcal V^{\mathrm{re}}(f-y^*y)\subseteq \mathcal V^{\mathrm{re}}(h)\) if and only if \(h\in \mathrm{ideal}(f-yy^*).\)
This theorem is also an ingredient in a Positivstellensatz for quadratic polynomials of the form \(f=\alpha+ \vec{x}^* v+ v^* \vec{x} + \vec{x}^* H\vec{x}\) for which \(\alpha\in \mathbb R\) and \(H\) is a hermitian matrix. Namely the polynomials \(h\in \mathbb C\langle x,x^* \rangle\) are characterized for which one has \(h|_{\{f\succ 0\}} \succeq 0.\)
The paper is certainly important in its area but a certain sloppyness and paucity of explanations and examples makes it at times difficult to read. The authors are already quite long in the noncommutativity business; so they might have lost contact to the base. noncommutative matrix polynomial; factorization; singulärstellensatz; nullstellensatz; resolvable ideal; free singularity locus; free algebra Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry, Real algebra Factorization of noncommutative polynomials and Nullstellensätze for the free algebra | 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 \(X=G/P\) be a complex projective homogeneous space endowed with the action of the maximal torus \(T\). The \(T\)-equivariant cohomology of \(X\) is generated by the equivariant Schubert cycles, and the corresponding structure constants are called the equivariant Littlewood-Richardson coefficients. These can be thought of as polynomials, after a suitable choice of generators for \(H^{\star}_{T}(\text{point})\), corresponding to the negative simple roots. Graham showed that these polynomials have nonnegative coefficients [\textit{W. Graham}, Duke Math. J. 109, No.~3, 599--614 (2001; Zbl 1069.14055)]. The paper under review proves the same statement in quantum cohomology. The author extends Graham's result, showing that the structure constants of the quantum equivariant cohomology of \(X\) are also polynomials of with non-negative coefficients. Schubert calculus; Littlewood-Richardson coefficients; quantum cohomology L. C. Mihalcea, ''Positivity in equivariant quantum Schubert calculus,'' Amer. J. Math., vol. 128, iss. 3, pp. 787-803, 2006. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Positivity in equivariant quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials About ten years ago, E. Witten has proposed a variant of the usual supersymmetric nonlinear sigma model, governing maps from a Riemann surface to an arbitrary almost complex manifold. His topological sigma models describe, in particular, \((1+1)\)-dimensional gravity theory, and have become increasingly important in string theory and its related complex geometry. Especially the investigation of the topological phase space of such a sigma model has produced deep geometrical conjectures and predictions, based on physical intuition and axiomatic assumptions, which have been lacking mathematical affirmation (and rigor) for quite a few years. The concept of quantum cohomology, first and independently proposed by \textit{E. Witten} [cf. Surv. Differ. Geom., Suppl. J. Diff. Geom. 1, 243--310 (1991; Zbl 0757.53049) and \textit{C. Vafa} (cf. `Topological mirrors and quantum rings.' (1992; Zbl 0827.58073)], is one of the most recent and celebrated examples in this realm.
The present paper, as the title says, aims at giving one possible rigorous mathematical framework for the physicists' theory of quantum cohomology rings. The authors, following E. Witten's ideas and predictions, make use of symplectic geometry and pseudo-holomorphic curves to establish one of the most important sigma model invariants, the so-called ``\(k\)-point correlation function'' for rational curves in the underlying phase space, and to verify one of the key axioms in quantum cohomology, namely the associativity of the quantum multiplication in the cohomology ring of the underlying (phase) manifold.
Section 1 of the paper gives a very beautiful, thorough and enlightening introduction to the physical background, the problems to be discussed, the interrelations with (almost) complex geometry and algebraic geometry, and to the strategy of the paper. Section 2 is devoted to semipositive symplectic manifolds and their (so-called) mixed invariants with respect to \(k\)-pointed Riemann surfaces of arbitrary genus \(g\). These invariants are then, indeed, nothing else but Witten's topological sigma model invariants, at least in a special case. The following sections 3-6 provide some deep requisites (compactification and transversality theorems for moduli spaces of ``perturbative'' holomorphic curves in symplectic manifolds) for the ultimate definition of the mixed invariants, and for the proof of their composition law (associativity under multiplication in the quantum cohomology ring). The central result of the paper, namely the establishing of just this composition law, is then derived in section 7, whereas section 8 shows that the cohomology ring of a semi-positive symplectic manifold therefore can be equipped with a quantum ring structure in the sense of Witten and Vafa. Section 9 discusses relations with (and applications to) the mirror symmetry conjecture in complex algebraic geometry, and the concluding section 10 deepens the algebro-geometric significance of the composition law by applying it to some well-known problems in the enumerative geometry of special classes of algebraic varieties. In the course of these applications, the authors recover the Gromov-Witten invariants and show how these can be computed in terms of their invariants.
Altogether, this work is a very important contribution towards the understanding and the mathematical foundation of quantum cohomology theory. In spite of the complexity and depth of the methods and results, the presentation of the material is utmost lucid, clear and thorough.
In the meantime, other (and different) approaches to a rigorous foundation of quantum cohomology have been proposed, and that mainly by algebraic geometers. Amongst them, the recent papers ``Gromov-Witten classes, quantum cohomology, and enumerative geometry'' by \textit{M. Kontsevich} and \textit{Yu. Manin} [cf. Commun. Math. Phys. 164, No. 3, 525--562 (1994; Zbl 0853.14020)], ``Frobenius manifolds, quantum cohomology, and moduli spaces'' (Chapters I--III) by \textit{Yu. Manin} (MPI Bonn, 1996), and ``Notes on stable maps and quantum cohomology'' by \textit{W. Fulton} and \textit{R. Pandharipande} (Preprint, Univ. Chicago, 1996) are certainly of particular interest. two-dimensional topological quantum field theory; symplectic manifolds; moduli spaces of algebraic curves; quantum cohomology Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), 259-367. Moduli problems for differential geometric structures, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Families, moduli of curves (analytic), Manifolds of mappings, Enumerative problems (combinatorial problems) in algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects), Topological field theories in quantum mechanics, Topological properties in algebraic geometry, Families, moduli of curves (algebraic), , Topological quantum field theories (aspects of differential topology) A mathematical theory of quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article provides both a brief survey and an announcement of the main results of the authors' subsequent, comprehensive and detailed treatise under the same title [J. Differ. Geom. 42, No. 2, 259--367 (1995; Zbl 0860.58005 above)]. two-dimensional topological quantum field theory; quantum cohomology; symplectic manifolds; algebraic curves; moduli spaces Y. Ruan and G. Tian, ''A mathematical theory of quantum cohomology,'' Math. Res. Lett., vol. 1, iss. 2, pp. 269-278, 1994. Moduli problems for differential geometric structures, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Families, moduli of curves (analytic), Manifolds of mappings, Enumerative problems (combinatorial problems) in algebraic geometry, Topological field theories in quantum mechanics, Topological properties in algebraic geometry, Families, moduli of curves (algebraic), , Topological quantum field theories (aspects of differential topology) A mathematical theory of quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct a universal solution of the generalized coboundary equation in the case of quantum affine algebras, which is an extension of our previous work to \(U_q(A^{(1)}_r)\). This universal solution has a simple Gauss decomposition which is constructed using Sevostyanov's characters of twisted quantum Borel algebras. We show that in the evaluation representations it gives a vertex-face transformation between a vertex type solution and a face type solution of the quantum dynamical Yang-Baxter equation. In particular, in the evaluation representation of \(U_q(A_1^{(1)})\), it gives Baxter's well-known transformation between the 8-vertex model and the interaction-round-faces (IRF) height model.{
\copyright 2012 American Institute of Physics} Sevostyanov's characters E. Buffenoir, Ph. Roche, V. Terras, Universal Vertex-IRF Transformation for Quantum Affine Algebras. arXiv:0707.0955 Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Vertex operators; vertex operator algebras and related structures, Yang-Baxter equations Universal vertex-IRF transformation for 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 See the review of the entire collection in [Zbl 1303.01004]. Biographies, obituaries, personalia, bibliographies, History of mathematics in the 20th century, History of algebraic geometry The Grothendieck-Serre correspondence | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main result of this paper states that the profinite fundamental group of a proper smooth curve of genus \(\geq 2\) over a number field determines the isomorphism class of the curve. This was conjectured by A. Grothendieck (for any hyperbolic curve) around 1980 and called the fundamental conjecture of anabelian algebraic geometry [see \textit{A. Grothendieck}, Lond. Math. Soc. Lect. Note Ser. 242, 49-58; English translation 285-293 (1997) and ibid., 5-48; English translation 243-283 (1997)]. The author deduces the above result using a previous work of \textit{A. Tamagawa} [Compos. Math. 109, No. 2, 135--194 (1997; Zbl 0899.14007)], where the cases of affine curves of all genera over number fields and finite fields are established.
The method is to control group theoretically the subgroups of \(\pi_1\) arising from tubular neighborhoods of irreducible components appearing in stable bad reductions (of covers) of a given proper curve (at infinitely many primes), whose main factors are related to the tame fundamental groups of affine curves over finite fields (treated by Tamagawa) obtained as the component curves minus nodes in those special fibres. In the process, the author introduces/proves two new versions of Grothendieck's conjecture: ``log-admissible version'' over finite fields and ``ordinary version'' over \(p\)-adic fields. The latter version is further generalized by [\textit{S. Mochizuki}, ``The local pro-\(p\) anabelian geometry of curves'', RIMS-1097, preprint 1996]. Grothendieck conjecture; anabelian geometry; profinite fundamental group; curve over a number field; isomorphism class Mochizuki S., The profinite Grothendieck conjecture for closed hyperbolic curves over number fields, J. Math. Sci. Tokyo 3 (1996), no. 3, 571-627. Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic ground fields for curves, Coverings of curves, fundamental group, Curves over finite and local fields The profinite Grothendieck conjecture for closed hyperbolic curves over number fields | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Aimed at graduate students in mathematics or theoretical physics, the book provides an extremely ample and friendly introduction to quantum cohomology and integrable systems. The author is able to leave aside the technical aspects involved in foundations of the subject, and to directly focus on the relevant structures, making the reader immediately able to handle most of the typical techniques and concepts from quantum cohomology, D-modules, and integrable systems. Just to make an example, Gromov-Witten invariants are given only a naïve geometrical description, and most terms appearing in the formal definition, such as the notion of stable map or the virtual fundamental class are left undefined. This has a considerably high cost: the fundamental result on associativity of the quantum product, which is in a sense the cornerstone quantum cohomology is built on, cannot be proven in the text. Yet, in the reviewer's opinion, this is a price worth paying for: the reader is not overwhelmed by the technical subtleties of integration on moduli spaces, can safely imagine quantum cohomology as a sort of singular cohomology for a space of maps, and arrive in a few pages to get acquainted with more advanced objects such as the quantum D-module and Givental's \(J\)-function. This may sound very approximative and not quite correct at first sight, but the author's aim is precisely to move the focus from the geometrical description of quantum cohomology to its D-module description, so that the central objects in the book are D-modules quantizing graded commutative algebras, and quantum cohomology is just an example of a more general (and rigorously defined in the book) abstract quantum cohomology. Even more important, this by far not the unique example: integrable systems are the classical source for these particular D-modules, and abstract quantum cohomology plays the rôle of an unifying language. This is best expressed by the author's words from the preface and the introduction: ``The main purpose of this book is to explain how quantum cohomology is related to differential geometry and the theory of integrable systems. In concrete terms, the concept of D-module unifies several aspects of quantum cohomology, harmonic maps, and soliton equations like the KdV equation. It does this by providing natural conditions on families of flat connections and their `extensions', from which these equations are derived. These conditions can be strong enough to determine the equations completely, despite their disparate geometric origins. Our goal is simply to explain this unified way of thinking. \([\dots]\) The main theme of this book is that the quantum D-module, and more generally the idea of `matching' a D-module with a commutative algebra, suggests a general scheme for constructing integrable systems. Optimistically this could contribute to a more precise definition of the term `integrable system'. Even more optimistically it could lead to a characterization of the quantum D-module of a manifold and a more efficient way of handling quantum cohomology, in the same way that de Rham cohomology has become a more efficient way of handling simplicial or singular cohomology (although it has to be said that the subject is still a long way from this point). To some extent, this justifies the lack of technical foundational material in the first three chapters of the book, as they may be regarded purely as motivation for the D-module approach which begins in Chapter 4''.
The book is organized as follows. The first chapter contains a brief introduction to simplicial, singular and de Rham homologies and cohomologies. The second chapter introduces quantum cohomology, and the quantum product is defined; instead of gong into the details of the construction, an informal definition is given, followed by several simple but important examples. The third chapter presents, in a similar informal way, the quantum differential equations. The fourth chapter contains a review of the elementary theory of linear differential equations needed in the rest of the book, and introduces the language of D-modules. Next, in Chapter 5, the quantum differential equations are recasted in the form of quantum D-module. From this point on, there is no further need of the geometrical definition of quantum cohomology, and the author is concerned only with the associated D-module. Some of the key properties of the quantum D-module are described, and the \(J\)-function is introduced. In Chapter 6, abstract quantum cohomology is presented. A construction procedure, based on [\textit{M.~A.~Guest}, Topology 44, No. 2, 263--281 (2005; Zbl 1081.53077); \textit{H.~Iritani}, Math. Z. 252, No. 3, 577--622 (2006; Zbl 1121.53062)], is presented, converting a system of scalar differential equations of a rather specific type into a D-module which `resemble' a quantum cohomology D-module. This leads to the link between quantum cohomology and integrable systems, made more precise in the following chapters. In particular, in Chapter 7 some of the most famous infinite-dimensional integrable systems are reviewed, concentrating on the KdV equation and harmonic maps into symmetric spaces, where D-modules and flat connections provide a natural framework. Next, in Chapter 8, it is reviews how the Sato-Segal-Wilson infinite-dimensional Grassmannian provides a conceptual framework for solving integrable systems with spectral parameter, emphasising the D-module point of view. In the remaining two chapters, quantum cohomology is reexamined in this new light. It is shown how quantum cohomology can be seen as a solution to the integrable system given by the WDVV equations (Chapter 9), and how the infinite-dimensional Grassmannian point of view reveals the mirror symmetry phenomenon: namely, quantum cohomology of a Calabi-Yau manifold can be interpreted as a variation of Hodge structure on a mirror dual. Quantum cohomology; Gromov-Witten invariants; D-modules; integrable systems; KdV equation; harmonic maps; infinite-dimensional Grassmannian; Frobenius manifolds; WDVV equations; mirror symmetry. Martin Guest, \textit{From Quantum Cohomology to Integrable Systems }(Oxford Graduate Texts in Mathematics; vol. 15). Oxford University Press, Oxford 2008. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to differential geometry, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) From quantum cohomology to 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 Let \(X\) be a smooth projective variety. The quantum cohomology ring \(QH^*(X)\) of \(X\) is a deformation of the ordinary cohomology ring; structure constants of the quantum product are formal power series whose coefficients consist of Gromov-Witten invariants. One generally does not know a priori whether these power series are convergent or not. If \(c_1(X)>0\) or \(c_1(X)<0\), then convergence is trivial since, by degree constraints, the power series involved in the quantum product are actually polynomials; hence the problem of convergence is in the intermediate case, i.e., when there exist two curves \(C_1\) and \(C_2\) in \(X\) such that \(\langle c_1(X),[C_1]\rangle\geq 0\) and \(\langle c_1(X),[C_2]\rangle\leq 0\).
Let now \({\mathcal L}\) be a nef line bundle on \(X\). Then one can consider the twisted quantum cohomology \(QH^*_{S^1}(X,{\mathcal L})\), where subscript \(S^1\) means that one is taking \(S^1\)-equivariant cohomology with respect to the natural \(S^1\)-action on the fibers of \({\mathcal L}\). If \(Y\) is a smooth subvariety of \(X\) cut out by a section of \({\mathcal L}\), then the part of \(QH^*(Y)\) coming from \(QH^*(X)\) is recovered as the non-equivariant limit of \(QH^*_{S^1}(X,{\mathcal L})\). Since Coates-Givental's quantum Lefshetz theorem [\textit{T.~Coates; A.~Givental}, Ann. Math. (2) 165, No. 1, 15--53 (2007; Zbl 1189.14063)] relates \(QH^*_{S^1}(X,{\mathcal L})\) with \(QH^*(X)\), this means that convergence of \(QH^*(X)\) gives (partial) information on the convergence of \(QH^*(Y)\).
The author proves that, if \(QH^*(X)\) has convergent structure constants, then so does \(QH^*_{S^1}(X,{\mathcal L})\). The main tool in the proof is a ring of formal power series with certain estimates for coefficients. Next, the author gives a description of mirror symmetry for a not necessaraly nef toric variety. More precisely, the author proves in [\textit{H.~Iritani}, Topology 47, No. 4, 225--276 (2008; Zbl 1170.53071)] that the quantum D-module \(QDM^*(X)\) of a toric variety \(X\) can be reconstructed from the equivariant Floer cohomology \(FH^*_{S^1}\) by a generalized mirror transformation. Here, the author studies this reconstruction from an analytic point of view, obtaining that if \(X\) is a smooth projective toric variety, then \(QH^*(X)\) is convergent and generically semisimple. This semisemplicity result is a successful test for the Dubrovin-Bayer-Manin conjecture [\textit{A.~Bayer; Yu.~I.~Manin}, The Fano conference. 143--173 (2004; Zbl 1077.14082)].
As an even more remarkable consequence of semisemplicity and convergence results in the paper, the author is able to prove that Givental's \(R\)-conjecture [\textit{A.~B.~Givental}, Mosc. Math. J. 1, No. 4, 551--568 (2001; Zbl 1008.53072)], and hence the Virasoro conjecture, is true for any smooth projective toric variety (with Hamiltonian torus action, or, equivalently, with at least one fixed point for the torus action). Gromov-Witten invariants; quantum cohomology; toric varieties; Virasoro conjecture H. Iritani, Convergence of quantum cohomology by quantum Lefschetz , preprint,\arxivmath/0506236v3[math.DG] Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Convergence of quantum cohomology by quantum Lefschetz | 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 $\mathcal{M}_{g,n}$ be the moduli space of genus $g$ curves with $n$ marked points. The Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the geometric fundamental groups of the ``Teichmüller tower'' of $\{ \mathcal{M}_{g,n} \}$ (i.e. the collection of all the stacks and the natural maps between them). Ihara showed that this action extends to a faithful action of the profinite Grothendieck-Teichmüller group $\widehat{\mathrm{GT}}$, which contains $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. \par The authors of the paper under review prove that $\widehat{\mathrm{GT}}$ is actually equal to the group of homotopy automorphisms of the Teichmüller tower in genus $g = 0$. They use an operadic model $\mathcal{M}$ of the genus $0$ Teichmüller tower: they replace marked point by boundary circles, which allows to glue curves along boundary components and to fill boundary components. These operations yield maps between the corresponding moduli spaces, and the operad obtained this way is equivalent to the classical framed little disks operad. The authors' first main theorem states more precisely that $\widehat{\mathrm{GT}}$ is the group of homotopy automorphisms of the profinite completion of $\mathcal{M}$. The difficult part of this theorem is showing that the obvious faithful action of $\widehat{\mathrm{GT}}$ on each $\mathcal{M}(n)$ is compatible with the operad structure (which is related to [\textit{A. Hatcher} et al., J. Reine Angew. Math. 521, 1--24 (2000; Zbl 0953.20030)]) and that all homotopy automorphisms are obtained this way. \par The authors' second main theorem deals with the compactification $\overline{\mathcal{M}}_{g,n}$ of $\mathcal{M}_{g,n}$, obtained by allowing nodal singularities in the curves. These moduli spaces admit a natural operadic structure by gluing curves along marked points. The operad $\mathcal{M}$ maps to $\overline{\mathcal{M}}_{0,\bullet+1}$ (the latter being a homotopy quotient of the former under an $\mathrm{SO}(2)$-action [\textit{G. C. Drummond-Cole}, J. Topol. 7, No. 3, 641--676 (2014; Zbl 1301.55005)]). The authors prove that the action of $\widehat{\mathrm{GT}}$ on the profinite completion of $\mathcal{M}$ extends to a nontrivial action on the profinite completion of $\overline{\mathcal{M}}_{0,\bullet+1}$. Note that actions of $\widehat{\mathrm{GT}}$ are usually constructed on schemes whose associated complex analytic spaces are $K(\pi,1)$, whereas $\overline{\mathcal{M}}_{0,n+1}$ is simply connected. infinity operads; Grothendieck-Teichmüller group; absolute Galois group; moduli space of curves , Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Loop space machines and operads in algebraic topology, Abstract and axiomatic homotopy theory in algebraic topology Operads of genus zero curves and the 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 We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-infinite wedge formalism. This reveals the strong piecewise polynomiality in the sense of Goulden-Jackson-Vakil, generalising a result of Johnson, and provides a new explicit proof of the piecewise polynomiality of the mixed case. Moreover, we derive wall-crossing formulae for the mixed case. These statements specialise to any of the three types of Hurwitz numbers, and to the mixed case of any pair. Enumerative problems (combinatorial problems) in algebraic geometry, Dessins d'enfants theory, Arithmetic aspects of dessins d'enfants, Belyĭ theory, Coverings of curves, fundamental group, Low-dimensional topology of special (e.g., branched) coverings Wall-crossing formulae and strong piecewise polynomiality for mixed Grothendieck dessins d'enfant, monotone, and double simple Hurwitz numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials ``Sa puissance technique est secondaire, jamais une fin en soi. Sa capacité à changer de point de vue, à prendre de la hauteur pour entrouvrir des portes toujours plus lointaines plonge ses élèves dans une extase presque mystique. Peu à peu, le maître devient gourou.''
Avec la pugnacité d'un véritable enquêteur, Yan Pradeau tente dans ce récit à la croisée de l'autobiographie et du roman de comprendre comment on devient Alexandre Grothendieck, mathématicien de génie. Enfant déjà, celui qui est aujourd'hui considéré comme le refondateur de la géométrie algébrique se trouve sur la piste de la découverte du nombre Pi. À l'âge de 20 ans, il résout en quelques mois quatorze problèmes demeurés jusqu'ici irrésolus. De tous les mathématiciens du siècle, il est celui qui s'avère seul capable de généraliser un problème, d'apercevoir avec recul et démonter les liens possibles entre des figures mathématiques. Ce fils d'anarchistes, dont le père a péri à Auschwitz et la mère a succombé à une tuberculose contractée dans les camps, est férocement revêche à toute autorité et farouchement solitaire. En 1966, il refuse la très convoitée Médaille Fields. Un temps enseignant au Collège de France, il est à compter de 1973 professeur à l'Université de Montpellier. Mais il choisit de rompre rapidement avec le milieu scientifique pour vivre reclus en Ariège au début des années 1990, avant de tirer sa révérence en novembre 2014.
Se détachent sous la plume alerte de l'écrivain la silhouette, puissante, de l'homme, son histoire personnelle et ses idéaux, et avec elle un siècle entier. Car Yan Pradeau ne retrace pas seulement la vie du mathématicien: il nous entraîne dans le tourbillon d'une époque, dans des milieux réputés fermés, qui apparaissent soudain dans toute leur vérité sous les yeux du lecteur. Entre le récit d'une vie et la grande histoire gravée sur le papier avec la finesse du burin et la phrase qui claque sans concession tel un couperet, cet Algèbre tient de la saga familiale comme de la grande histoire, depuis les camps de la mort jusqu'à nos jours. Research exposition (monographs, survey articles) pertaining to history and biography, Biographies, obituaries, personalia, bibliographies, History of mathematics in the 20th century, History of algebraic geometry, History of commutative algebra, History of category theory Algebra -- elements of the life of Alexander Grothendieck | 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 two different one-parameter generalizations of Littlewood-Richardson coefficients, namely Hall polynomials and generalized inverse Kostka polynomials, and derive new combinatorial formulae for them. Our combinatorial expressions are closely related to puzzles, originally introduced by \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)] in their work on the equivariant cohomology of the Grassmannian. symmetric functions; Littlewood-Richardson coefficients; integrable lattice models Wheeler, M.; Zinn-Justin, P., Hall polynomials, inverse kostka polynomials and puzzles Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Classical problems, Schubert calculus, Groups acting on specific manifolds Hall polynomials, inverse Kostka polynomials and puzzles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0731.00008.]
In this paper the Schur \(S\)-polynomials and the Schur \(Q\)-polynomials are studied. These polynomials are then applied to elimination theory, and Schubert calculus for Grassmannians of isotropic subspaces. Schur polynomials; elimination theory; Schubert calculus for Grassmannians Pragacz, Piotr, Algebro-geometric applications of Schur \(S\)- and \(Q\)-polynomials.Topics in invariant theory, Paris, 1989/1990, Lecture Notes in Math. 1478, 130-191, (1991), Springer, Berlin Grassmannians, Schubert varieties, flag manifolds Algebro-geometric applications of Schur \(S\)- and \(Q\)-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 Quantum multiplications on the cohomology of symplectic manifolds were first proposed by the physicist Vafa, based on Witten's topological sigma models. \textit{Y. Ruan} and \textit{G. Tian} [J. Differ. Geom. 42, No. 2, 259-367 (1995; Zbl 0860.58005)] gave a mathematical construction of quantum multiplications on cohomology groups of positive symplectic manifolds (chapter 1). The definition uses certain symplectic invariants, called Gromov-Witten (GW-) invariants, that were previously defined by Ruan for (semi-) positive symplectic manifolds. A large class of such manifolds is provided by Fano manifolds (complex manifolds with ample anti-canonical bundle). Examples are low degree complete intersections and compact complex homogeneous spaces like Grassmann manifolds. If \(M\) is a Fano (or positive symplectic) manifold the quantum cohomology \(QH^*_{[\omega]}(M)\) is just the cohomology space \(H^*(M,\mathbb{C})\) with a (non-homogeneous) associative, graded commutative multiplication, the quantum multiplication. This multiplication depends on the choice of a (complexified) Kähler class \([\omega]\) on \(M\). Its homogeneous part (the ``weak coupling limit'' \(\lambda\cdot[\omega]\), \(\lambda \to\infty)\) is the usual cup product.
In this note we observe that quantum cohomology rings have a nice description in terms of generators and relations: If \(H^*(M,\mathbb{C}) =\mathbb{C}[X_1, \dots,X_N]/(f_1, \dots,f_k)\) is a presentation of the cohomology ring (for simplicity we assume \(\deg X_i\) even for the moment) then \(QH^*_{[\omega]} (M)=\mathbb{C}[X_1, \dots,X_N]/ (f_1^{[\omega]}, \dots, f_k^{[\omega]})\) , where \(f_1^{[\omega]}, \dots,f_k^{[\omega]}\) are just the polynomials \(f_1,\dots,f_k\) evaluated in the quantum ring associated to \([\omega]\) (Theorem 2.2). The \(f_i^{[\omega]}\) are real-analytic in \([\omega]\) and thus have a natural analytic extension to \(H^{1,1}(M)\), and the quantum cohomology rings fit together into a flat analytic family over \(H^{1,1}(M)\).
As an application of this observation we compute the quantum cohomology of the Grassmannians. The calculations for \(G(k,n)\) reduce to the single quantum product \(c_k\wedge_Qs_{n-k}\) of the top non-vanishing Chern respectively Segre class of the tautological \(k\)-bundle (see chapter 3). We see in chapter 4 that formulas of this type occur whenever the cohomology ring has a presentation as complete intersection. In particular, we prove as corollary 4.6:
Vafa-Intriligator formula: For any Kähler class \([\omega]\) there is a finite set \(C\subset\mathbb{C}^n\) and non-zero constants \(a_x\), \(x\in C\), such that, for any \(F\in\mathbb{C} [X_1,\dots, X_k],\)
\[
\langle F\rangle_g^{[\omega]} =\sum_{x\in C}(a_x)^{g-1} \cdot F(x).
\]
Here \(\langle F\rangle_g^{[\omega]}\) is the genus \(g\) GW-invariant associated to \(F\). In fact, there is a polynomial \(W^{[\omega]}\) in the \(X_i\) having \(C\) as its set of critical points with \(a_x\) being (up to sign) the determinants of the Hessians at \(x\in C\). quantum multiplications; Gromov-Witten invariants; Fano manifolds; quantum cohomology of the Grassmannians Siebert, B.; Tian, G., \textit{on quantum cohomology rings of Fano manifolds and a formula of Vafa and intriligator}, Asian J. Math., 1, 679-695, (1997) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Fano varieties, Enumerative problems (combinatorial problems) in algebraic geometry On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review the author studies the so-called \(\mathfrak{g}\)-endomorphism algebras, associated with irreducible representations of a connected semi-simple algebraic group over an algebraically closed field of characteristic zero. These algebras were introduced by \textit{A.~A.~Kirillov} in [Family algebras. Electron. Res. Announc. Am. Math. Soc. 6, No. 2, 7--20 (2000; Zbl 0946.16019)] under the name ``family algebras''. The main result of the present paper asserts that every commutative \(\mathfrak{g}\)-endomorphism algebra is Gorenstein. During the proof a connection between \(\mathfrak{g}\)-endomorphism algebras and Dynkin polynomials is established. The latter appear as the numerators of the Poincaré series of the \(\mathfrak{g}\)-endomorphism algebra. It is worth mentioning that the Poincaré series of \(\mathfrak{g}\)-endomorphism algebras are described explicitly in the paper. The paper is finished with a discussion of a connection between \(\mathfrak{g}\)-endomorphism algebras and equivariant cohomology. algebraic group; \(\mathfrak{g}\)-endomorphism algebra; module; weight; equivariant cohomology; Dynkin polynomial D. Panyushev, Weight multiplicity free representations,
\[
\mathfrak{g}
\]
-endomorphism algebras, and Dynkin polynomials. J. Lond. Math. Soc. 69, Part 2, 273--290 (2004) Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Group actions on varieties or schemes (quotients), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) Weight multiplicity free representations, \(\mathfrak g\)-endomorphism algebras, and Dynkin 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 paper presents an algebro-combinatorial proof of a quantum Pieri's formula in the (small) quantum cohomology ring \(QH^\ast(Fl_n,\mathbb Z)\) of the flag manifold. \(QH^\ast(Fl_n,\mathbb Z)\) is canonically isomorphic to the quotient
\[
\mathbb Z[x_1,\dots,x_n;q_1,\dots,q_{n-1}]/ \langle E_1,E_2,\dots,E_n\rangle.
\]
where the \(E_k\) are certain \(q\)-deformations of the elementary symmetric polynomials. The isomorphism is given by specifying \(x_1+x_2+\dots x_m\mapsto\sigma_{s_m}\) where \(\sigma_{s_m}\) is the Schubert class corresponding to a transposition from the permutation group, \(s_m\in S_n\), \(1\leq m\leq n-1\). The quantum Pieri formula is expressed in terms of certain \(\mathbb Z[q]\)-linear operators \(t_{ij}\), \(1\leq i<j\leq n\), acting on \(QH^\ast(Fl_n,\mathbb Z)\). Particularly the quantum Monk formula for a product of Schubert classes, with \(w\in S_n\), can be written in the form \( \sigma_{s_m}\ast\sigma_w=\sum_{a\leq m<b}t_{ab}(\sigma_w)\).
This result is generalized to products \(\sigma_{c(k,m)}\ast\sigma_w\) and \(\sigma_{r(k,m)}\ast\sigma_w\) with cyclic permutations \(c(k,m)=s_{m-k+1}s_{m-k+2}\dots s_m\) and \(r(k,m)=s_{m+k-1}s_{m+k-2}\dots s_m\). In fact, the formula is first proven in a more abstract form in the framework of a quadratic algebra such that the operators \(t_{ij}\) satisfy the defining relations of that algebra. Furthermore, several corollaries of the formula are derived and discussed. flag manifold; Monk's formula; Pieri's formula; quantum cohomology Alexander Postnikov, On a quantum version of Pieri's formula, Advances in geometry, Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 371 -- 383. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Enumerative problems (combinatorial problems) in algebraic geometry On a quantum version of Pieri'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 We show how equivariant volumes of tensor product quiver varieties of type A are given by matrix elements of vertex operators of centrally extended doubles of Yangians and how these elements satisfy the rational level-one quantum Knizhnik-Zamolodchikov equation in some cases. quiver variety; quantum Knizhnik-Zamolodchikov equation; quantum integrable system; equivariant cohomology Groups and algebras in quantum theory and relations with integrable systems, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Representations of quivers and partially ordered sets Quiver varieties and the quantum Knizhnik-Zamolodchikov equation | 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. Let \(\text{Var}_k\) denote the category of varieties over \(k\), and let \(K_{0}(\mathrm{Var}_{k})\) denote the Grothendieck group associated with \(\text{Var}_k\). The class of a variety \(X\) in \(K_{0}(\mathrm{Var}_{k})\) is denoted by \([X]\). Let \(1\) denote the identity element \([\text{Spec}(k)]\) and \(\mathbb{L}\) the class of the affine line \(\mathbb{A}_k^1\). Let \(X\) be a projective hypersurface of degree \(d\) in \(\mathbb{P}_k^n\) with \(d\leq n\).
In this paper, the author investigates the question (*):
Is it true that \(X(k)\) is nonempty if and only if \([X]\equiv 1\pmod{\mathbb{L}}\)?
The main result shows that the equivalence holds when
(i) \(X\times_kL\) is a union of \(d\) hyperplanes over some finite Galois extension \(L\) of \(k\),
(ii) \(X\) is a quadric with \(\text{char}(k)=0\).
Furthermore, it shows that when \(d=3\) and \(k\) is a \(C1\) field, if \(\text{Sing}(X)(k)\neq\varphi\) then \([X]\equiv 1 \pmod{\mathbb{L}}\), and when \(d=4\) and \(k\) is algebraically closed of characteristic 0, and \(X\) is the union of two quadric hypersurfaces, one of which is smooth, the congruence \([X]\equiv 1 \pmod{\mathbb{L}}\) holds true. However \textit{L. D. T. Nguyen} gave in [C. R., Math., Acad. Sci. Paris 350, No. 11--12, 613--615 (2012; Zbl 1252.14009)] a negative answer to the question (*) in general, one still does not know whether over an algebraically closed field of characteristic 0, the class of a hypersurfact of degree \(d\leq n\) is congruent to \(1\pmod{\mathbb{L}}\). Grothendieck ring; hypersurfaces; rational points Emel Bilgin, Classes of some hypersurfaces in the grothendieck ring of varieties, arXiv:1112.2131. Algebraic cycles, Hypersurfaces and algebraic geometry, Rational points On the classes of hypersurfaces of low degree in the Grothendieck ring of varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we give a quantum modification of the relative cup product on \(H^\ast(X,S;\mathbb{R})\) by using Gromov-Witten invariants when \(S\) is a compact codimension \(2k\) symplectic submanifold of the compact symplectic manifold \((X,\omega)\). relative cohomology; Gromov-Witten invariant; quantum relative cohomology Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), de Rham cohomology and algebraic geometry A quantum modification of relative 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 In this paper the (small) quantum cohomology of some Fano threefolds is studied. More precisely, in theorems 1 and 2 the author gives the presentation (by generators and relations) of 13 smooth Fano threefolds. Namely, those which are blow-ups of projective space or quadric with center along one or two disjoint curves (two of them were studied before). If we have the presentation of the ordinary cohomology ring by generators and relations (which is known for blow-ups of threefolds along smooth curves), then to get the similar presentation for quantum cohomology we need to add additional generators (which correspond to Picard group generators) and ``quantize'' relations. Thus, to obtain such presentation we need to ``guess'' only few invariants. To do it, the author uses the associativity relations, which are given by associativity condition for the quantum cohomology ring.
Finally, the author checks the generic semisimplicity of quantum cohomology of these varieties. This is motivated by a modified Dubrovin's conjecture. Despite this conjecture is not true (one can see it on the example of the five-dimensional cubic, which is semisimple by \textit{G. Tian} and \textit{G. Xu} [Math. Res. Lett. 4, No.4, 481--488 (1997; Zbl 0919.14024)] but has no exceptional collection of maximal length), the fact of semisimplicity of these varieties is interesting by itself. Dubrovin conjecture; blow-up; derived category; complete exceptional set G. Ciolli, ''On the Quantum Cohomology of Some Fano Threefolds and a Conjecture of Dubrovin,'' Int. J. Math. 16(8), 823--839 (2005); arXiv:math/0403300. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Fano varieties, \(3\)-folds On the quantum cohomology of some Fano threefolds and a conjecture of Dubrovin | 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 Grothendieck--Teichmüler group \(\widehat{GT}\) is a certain abstractly constructed profinite group in which the absolute Galois group \(G_{\mathbb Q}\) of the rationals is naturally embedded. It is not known whether the embeding is actually an isomorphism. In this paper, the author derives a formula which the image of \(G_{\mathbb Q}\) satisfies. It is not known if the formula holds for \(\widehat{GT}\). Elements \(\sigma\) of \(\widehat{GT}\) give rise to elements \(f_\sigma\) in the free two generator profinite group.
Let the ring \(R\) be the inverse limit of the rings \(\mathbb Z[X,Y]/(X^n-1, Y^n-1)\) as \(n \to \infty\) multiplicatively. Let \(x\) and \(y\) be the limits of \(X\) and \(Y\) in \(R\). The author finds an identity satisfied in \(GL_2(R)\) for \(f_\sigma(A,B)\) where \(A\) (\(B\)) is the upper (lower) triangular matrix with off diagonal \(1-x\) (\(1-y\)) and diagonal \(1,x\) (\(y,1\)). Specializing \(x\) and \(y\) to roots of unity in \(\mathbb Q_{ab}\) then yields various explicit matrix equations that the \(f_\sigma\) must satisfy. The author's formula which the image of \(G_{\mathbb Q}\) satisfies is also phrased in terms of the \(f_\sigma\)'s. absolute Galois group of the rationals Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Galois theory Some classical views on the parameters of the 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 chapter discusses quantum error-correcting codes constructed from algebraic curves. We give an introduction to quantum coding theory including bounds on quantum codes. We describe stabilizer codes which are the quantum analog of classical linear codes and discuss the binary and \(q\)-ary CSS construction. Then we focus on quantum codes from algebraic curves including the projective line, Hermitian curves, and hyperelliptic curves. In addition, we describe the asymptotic behaviors of quantum codes from the Garcia-Stichtenoth tower attaining the Drinfeld-Vlăduţ bound. Kim, J; Mathews, GL; Martinez, E (ed.); Munuera, C (ed.); Ruano, D (ed.), Quantum error-correcting codes from algebraic curves, 419-444, (2008), Hackensack Decoding, Quantum computation, Applications to coding theory and cryptography of arithmetic geometry 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 main theorem of this paper supplies a very uniform presentation of the quantum cohomology ring of (co)minuscule homogeneous spaces, ranging from the usual and lagrangian grassmannians, quadrics, spinor varieties up to exceptional hermitian spaces like the Freudenthal variety or the Cayley plane. The methods are based upon the combinatorics of certain quivers. Pieri's and Giambelli's type formulas for all these kind of varieties are also known but, as the the authors remark, the techniques used in this paper do not apply, yet, to deduce them in a uniform way, all at once. This paper has a sequel, concerned with \textsl{hidden symmetries}, especially for grassmannians [Int. Math. Res. Not. 2007, No. 22, Article ID rnm107 (2007; Zbl 1142.14033)].
In spite of dealing with highly non trivial mathematics, the paper is written in a quite friendly way: for instance, the section that follows the introduction explains the basic terminology regarding minuscule and cominuscule homogeneous spaces. To help the potential reader to know in advance what is this paper about, recall that any parabolic subgroup contains a Borel subgroup \(B\) which contains a maximal torus \(T\) of \(G\). The pair \((G,T)\) defines \textsl{weights} (which are characters of \(T\) satisfying certain non triviality conditions) and a \textsl{root system}. A fundamental weight \(\omega\) is said to be \textsl{minuscule} if and only if \(|<\omega, \alpha>|\leq 0\), for each positive root \(\alpha\), where \(<,>\) is the pairing induced by the natural duality between the characters and the co-characters of \(T\). Furthermore \(\omega\) is said to be \textsl{co-minuscule} if and only if \(<\omega, \alpha^\vee>=1\), where \(\alpha\) denotes the highest root. To each such weight a parabolic subgroup \(P_\omega\) of \(G\) can be associated, and the corresponding quotient \(G/P_\omega\) is said to be a (co)minuscule homogeneous space.
One of the key remarks of the paper is based on the observation that the Gromow-Witten invariant of degree \(d\) of \(G/P\) can be seen as classical intersection numbers on certain auxiliary \(G\)-homogeneous varieties. This is well explained with details in the third section of the paper, suggestively entitled ``From classical to quantum invariants''. Section 4 is devoted to the quantum Chevalley formula (concerning the intersection of any Schubert cycle with a codimension \(1\) Schubert cycle) and higher Poincaré duality. In this section the problem, already studied by \textit{W. Fulton} and \textit{C. Woodward} for grassmannians [J. Algebr. Geom. 13, No. 4, 641--661 (2004; Zbl 1081.14076)], of the minimum power of the quantum parameter \(q\) occurring in the product of two Schubert classes, is also analyzed. The final section is devoted to the study of the quantum cohomology of two exceptional hermitian spaces, namely the Cayley plane and the Freudenthal variety. quantum cohomology; minuscule homogeneous spaces; quivers; Schubert calculus P. E. Chaput, L. Manivel, and N. Perrin, ''Quantum cohomology of minuscule homogeneous varieties,'' ccsd-0086927, 28 Sep 2006, 1--34. Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology of minuscule homogeneous 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 In this article a unified description of the structure of the small cohomology rings for all projective homogeneous spaces \(SL_n(\mathbb C)/P\) (with \(P\) a parabolic subgroup) is given. First the results on the classical cohomology rings are recalled. Then the algebraic structure of the quantum cohomology ring is studied. Important results are the general quantum versions of the Giambelli and Pieri formulas of the classical cohomology (classical Schubert calculus). They are obtained via geometric computations of certain Gromov-Witten invariants, which are realized as intersection numbers on hyperquot schemes. quantum cohomology; Gromov-Witten invariants; Schubert calculus; small cohomology rings; homogeneous spaces; Pieri formulas Ionuţ Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), no. 3, 485 -- 524. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Homogeneous spaces and generalizations, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On quantum cohomology rings of partial flag 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 It is known that up to deformation, there are \(21\) distinct Fano threefolds which are \(\mathbb P^1\)-bundles on smooth projective surfaces. The quantum cohomologies of \(11\) of these Fano threefolds have previously been determined. In this paper, the authors compute the quantum cohomologies of the remaining \(10\) Fano threefolds. Their main idea is to study various rational curves on these Fano threefolds. Ancona, V.; Maggesi, M.: Quantum cohomology of some Fano threefolds, Adv. geom. 5, 49-70 (2005) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Topological properties in algebraic geometry On the quantum cohomology of some Fano 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 The paper deals with the grand canonical string partition functions. It is shown that \(\tau\) functions of \(q\)-deformed Painlevé equations can be computed using spectral determinants of operators from quantization of Calaby-Yau manifolds. The explicit example of the Painlevé III\(_3\) equation is considered.
In Section 2 a review of topological strings, spectral theory and \(q\)-Painlevé equations is presented. A connection with TS/ST duality is given (see also [\textit{A. Grassi} et al., Ann. Henri Poincaré 17, No. 11, 3177--3235 (2016; Zbl 1365.81094)]). In Section 3 the Fredholm determinant solution for the \(q\)-Painlevé III\(_3\) equation is presented and generalized to matrix models in Section 4. Section 5 is devoted to relations with Aharony-Bergman-Jafferis theory. Painlevé equations; supersymmetric gauge theory; topological string theory; spectral theory; Aharony-Bergman-Jafferis theory Difference equations, scaling (\(q\)-differences), Supersymmetric field theories in quantum mechanics, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Relationships between algebraic curves and integrable systems, Selfadjoint operator theory in quantum theory, including spectral analysis, Yang-Mills and other gauge theories in quantum field theory Quantum curves and \(q\)-deformed Painlevé equations | 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 addendum the authors prove a technical lemma to which Grothendieck reduces the proof of the Bertini theorems (see the previous review). No proof of that lemma is given in Grothendieck's notes. hyperplane sections; Bertini-Zariski theorems; Bertini theorems Rational and birational maps, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Divisors, linear systems, invertible sheaves, Hypersurfaces and algebraic geometry Addendum to ``A. Grothendieck's EGA V. I'' | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The complete work of the author consists of two volumes and four chapters. In the first- under review - volume two chapters are included. In the first chapter the author considers the stable homotopy category of \(S\)-schemes \({\mathbf{SH}}(S)\) as developed by V. Voevodsky, F. Morel and others. The main point of the first chapter is to show that the Grothendieck operations \(f^{*},\) \(f_{*},\) \(f_{!}\) and \(f^{!}\) can be extended to the motivic setting. In the first introductory section the author gives preliminaries concerning \(2\)-categories. This includes the notions of \(2\)-category, \(2\)-functor, adjunctions in a \(2\)-category etc.. In the second section the exchange structures of 2-functors are studied. The cross functors are described. Recall that a cross functor from a category \(\mathcal C\) to a \(2\)-category \(\mathcal D\) is given by the following data: four \(2\)-functors: \(H^{*}\), \(H_{*}\), \(H_{!}\), \(H^{!}\), four exchange structures on each couple: \((H^{*},H^{!})\), \((H_{*},H_{!})\), \((H^{*},H_{!})\), \((H_{*},H^{!})\) satisfying certain axioms. Third section is devoted to the extension of 2-functors. Suppose that a category \(\mathcal C\) and a \(2\)-category \(\mathcal D\) are given. Assume further that we have two \(2\)-functors \(H_{1}\) and \(H_{2}\) defined on two subcategories \(\mathcal C_{1}\) and \(\mathcal C_{2}.\) The sufficient condition for the existence of a \(2\)- functor \(H\) on the category \(\mathcal C\), such that \(H_{1}\) and \(H_{2}\) come from \(H,\) is established. This fact is used in section 6, where the \(2\)-functor \(H^{!}\) is constructed. There \({\mathcal C}=Sch/S\) is the category of quasi-projective \(S\)-schemes, \({\mathcal C}_{1}=(Sch/S)^{Imm}\) is the category of quasi-projective \(S\)-schemes with closed immersions as morphisms, \({\mathcal C}_2=(Sch/S)^{Liss}\) is the category of quasi-projective \(S\)-schemes with smooth \(S\)-morphisms as morphisms. In section 4 the \(2\)-functors \(H_{1}\) and \(H_{2}\) are described. In section \(5\) the stability axiom is studied and Thom equivalence is constructed. In section \(7\) some consequences of the stability axiom are considered. Among basic theorems in étale cohomology one has: the theorems cocerning: the costructibility of cohomology sheaves \(R^{i}f_{*}{\mathcal F}\) for a morphism \(f\) of finite type and \({\mathcal F}\) a constructible sheaf of \({\Lambda}\)-modules, cohomology dimension of \(Rf_{*}\), Verdier duality. Chapter \(2\) is devoted to establishing motivic analogs of these. motives; Grothendieck six operations; Verdier duality; stable motivic homotopy type of a scheme Ayoub, J., Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique: I, Astérisque, 314, (2007), x+466 pp., 2008 Research exposition (monographs, survey articles) pertaining to algebraic geometry, Algebraic cycles, Étale and other Grothendieck topologies and (co)homologies, Homotopy theory and fundamental groups in algebraic geometry, Motivic cohomology; motivic homotopy theory, Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.), Grothendieck topologies and Grothendieck topoi, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Nonabelian homotopical algebra, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) The Grothendieck six operations and the vanishing cycles formalism in the motivic world. I | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathsf{dgCAlg}^*\) and \(\mathsf{dgAlg}^*\) be the category of augmented commutative differential graded \({\mathbb Q}\)-algebras and the category of augmented differential graded \({\mathbb Q}\)-algebras, respectively. The author proves that for any \(R\) and \(S\) in \(\mathsf{dgCAlg}^*\), the map \( \Omega \text{Map}_{\mathsf{dgCAlg}^*}(R, S) \to \Omega \text{Map}_{\mathsf{dgAlg}^*}(R, S) \) induced by the forgetful functor \(\mathsf{dgCAlg}^* \to \mathsf{dgAlg}^*\) between the loop spaces of functor complexes has a retract. In particular, the induced map gives rise to an injective map \(\pi_i(\text{Map}_{\mathsf{dgCAlg}^*}(R, S)) \to \pi_i(\text{Map}_{\mathsf{dgAlg}^*}(R, S))\) for \(i > 0\). One of the ingredients of the proof is a factorization \(\mathsf{Ass} \to \mathsf{E}_\infty \to \mathsf{Com}\) of a map \(\mathsf{Ass} \to \mathsf{Com}\) from the associative operad to the commutative operad as a cofibration followed by a trivial fibration. Quillen adjunctions between categories of algebras, which are induced by the maps between operads mentioned above, play a crucial role in the proof. DGA; CDGA; mapping space; rational homotopy theory; derived algebraic geometry Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Nonabelian homotopical algebra, Rational homotopy theory, Differential graded algebras and applications (associative algebraic aspects) Comparing commutative and associative unbounded differential graded algebras over \(\mathbb{Q}\) from a homotopical point of view | 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 \(\Sigma\) be a finite complete nonsingular fan for \(N = \mathbb{Z}^ d\). Then the toric variety \(\mathbb{P}_ \Sigma\) over \(\mathbb{C}\) corresponding to \(\Sigma\) is a \(d\)-dimensional compact nonsingular algebraic variety. Let \(n\) be the cardinality of the set of one-dimensional cones in \(\Sigma\) and denote by \(G(\Sigma) = \{v_ 1, \dots, v_ n\}\) the set of primitive lattice vectors \(v_ j \in N\) such that \(\mathbb{R}_{\geq 0} v_ 1, \dots, \mathbb{R}_{\geq 0} v_ n\) are the one-dimensional cones in \(\Sigma\), where \(\mathbb{R}_{\geq 0}\) is the set of nonnegative real numbers.
\textit{J. Jurkiewich} and \textit{V. I. Danilov} determined the ordinary cohomology ring \(H^ \bullet (\mathbb{P}_ \Sigma, \mathbb{Z})\) as the quotient \(H^ \bullet (\mathbb{P}_ \Sigma, \mathbb{Z}) = \mathbb{Z} [z_ 1, \dots, z_ n]/ \bigl( P (\Sigma)_ \mathbb{Z} + \text{SR} (\Sigma)_ \mathbb{Z} \bigr)\) of the polynomial ring \(\mathbb{Z} [z_ 1, \dots, z_ n]\), where \(P(\Sigma)_ \mathbb{Z}\) is the ideal generated by the linear forms \(\sum_{j = 1}^ n \langle m,v_ j \rangle z_ j\) for \(m \in M : = \Hom_ \mathbb{Z} (N, \mathbb{Z})\) with \(\langle m,m \rangle : M \times N \to \mathbb{Z}\) the canonical bilinear pairing, while \(\text{SR} (\Sigma)_ \mathbb{Z}\) is the Stanley- Reisner ideal generated by \(z_{j_ 1} z_{j_ 2} \dots z_{j_ s}\) with \(\{j_ 1,j_ 2, \dots,j_ s\}\) running through the subsets of \(\{1,2,\dots,n\}\) such that the cone \(\sum_{l = 1}^ s \mathbb{R}_{\geq 0} v_{j_ i}\) does not belong to the fan \(\Sigma\).
Let \(\text{PL} (\Sigma)_ \mathbb{R}\) be the \(\mathbb{R}\)-vector space consisting of \(\mathbb{R}\)-valued functions \(\varphi\) on \(N_ \mathbb{R} : N \otimes_ \mathbb{Z} \mathbb{R} = \bigcup_{\sigma \in \Sigma} \sigma\) which are linear on each cone \(\sigma \in \Sigma\), and denote by \(\text{PL} (\Sigma)_ \mathbb{Z}\) the \(\mathbb{Z}\)-submodule consisting of those \(\varphi\)'s satisfying \(\varphi (N) \subset \mathbb{Z}\). Obviously, \(M\) is a \(\mathbb{Z}\)-submodule of \(\text{PL} (\Sigma)_ \mathbb{Z}\). The map which sends \(\varphi \in \text{PL} (\Sigma)_ \mathbb{Z}\) to \(\sum_{j = 1}^ n \varphi (v_ j) z_ j\) is known to give rise to an exact sequence
\[
0 \to M \to \text{PL} (\Sigma)_ \mathbb{Z} \to H^ 2(\mathbb{P}_ \Sigma, \mathbb{Z}) \to 0.
\]
The cone in \(\text{PL} (\Sigma)_ \mathbb{R}\) consisting of the \(\varphi\)'s which are strictly convex with respect to the fan \(\Sigma\) (resp. which are convex) is known to be mapped onto the cone \(K^ 0(\mathbb{P}_ \Sigma) \subset H^ 2 (\mathbb{P}_ \Sigma, \mathbb{R})\) of Kähler classes (resp. the closed Kähler cone \(K(\mathbb{P}_ \Sigma) \subset H^ 2 (\mathbb{P}_ \Sigma, \mathbb{R}))\). (Beware of the confusion in the paper as to the distinction between convexity and upper convexity.)
For each \(\varphi \in \text{PL} (\Sigma)_ \mathbb{C} : = \text{PL} (\Sigma)_ \mathbb{R} \otimes_ \mathbb{R} \mathbb{C}\), the author defines the quantum cohomology ring of the toric variety \(\mathbb{P}_ \Sigma\) with respect to \(\varphi\) to be
\[
QH^ \bullet_ \varphi (\mathbb{P}_ \Sigma, \mathbb{C}) : = \mathbb{C} [z_ 1, \dots, z_ n]/ \bigl( P (\Sigma)_ \mathbb{C} + Q_ \varphi (\Sigma) \bigr),
\]
where \(P(\Sigma)_ \mathbb{C} : = P (\Sigma)_ \mathbb{Z} \otimes_ \mathbb{Z} \mathbb{C}\) and \(Q_ \varphi (\Sigma)\) is the ideal in \(\mathbb{C} [z_ 1, \dots, z_ n]\) generated by the binomials
\[
\exp \left( \sum_{j = 1}^ n a_ j \varphi (v_ j) \right) \prod_{j = 1}^ n z_ j^{a_ j} - \exp \left( \sum_{l = 1}^ n b_ l \varphi (v_ l) \right) \prod_{l = 1}^ n z_ l^{b_ l}
\]
with \(\sum_{j = 1}^ n a_ j v_ j = \sum_{l = 1}^ n b_ lv_ l\) running through all the linear relations among \(v_ 1, \dots, v_ n\) such that \(a_ j\)'s and \(b_ l\)'s are nonnegative integers.
By the very definition, the quantum cohomology ring depends not on the fan \(\Sigma\) but only on the set \(G(\Sigma)\) of the generators of one- dimensional cones in \(\Sigma\). In particular, compact nonsingular toric varieties which are isomorphic in codimension one have the same quantum cohomology ring.
Among other things, the author shows the following when \(\mathbb{P}_ \Sigma\) is a Kähler manifold, that is, \(K^ 0(\mathbb{P}_ \Sigma) \neq \emptyset\):
(1) As the positive real number \(t\) tends to \(\infty\), the limit of \(QH^ \bullet_{t \varphi} (\mathbb{P}_ \Sigma, \mathbb{C})\) exists in an appropriate sense if and only if \(\varphi\) belongs to the complexified Kähler cone \(K(\mathbb{P}_ \Sigma)_ \mathbb{C} : = K (\mathbb{P}_ \Sigma) + iH^ 2(\mathbb{P}_ \Sigma, \mathbb{R})\). In that case, the limit coincides with the ordinary cohomology ring \(H^ \bullet (\mathbb{P}_ \Sigma, \mathbb{C})\).
(2) If \(\varphi \in \text{PL} (\Sigma)_ \mathbb{R}\) is strictly convex with respect to \(\Sigma\) so that it gives rise to a Kähler class, then \(QH^ \bullet_ \varphi (\mathbb{P}_ \Sigma, \mathbb{C})\) is shown to coincide with the quantum cohomology ring of \(\mathbb{P}_ \Sigma\) considered by physicists in terms of the topological sigma model of \(\mathbb{P}_ \Sigma\), namely, the moduli space of holomorphic maps from the complex projective line \(\mathbb{C} \mathbb{P}^ 1\) to \(\mathbb{P}_ \Sigma\) [\textit{E. Witten} in Proc. Conf., Cambridge 1990, Surv. Differ. Geom., J. Differ. Geom., Suppl. 1, 243-310 (1991; Zbl 0757.53049)].
(3) When the first Chern class of \(\mathbb{P}_ \Sigma\) belongs to the Kähler cone so that \(\mathbb{P}_ \Sigma\) is a toric Fano manifold, the quantum cohomology ring \(QH^ \bullet_ \varphi (\mathbb{P}_ \Sigma, \mathbb{C})\) is shown to be canonically isomorphic to the Jacobian ring of the Laurent polynomial
\[
f_ \varphi (X_ 1, \dots, X_ d) = - 1 + \sum_{j = 1}^ n \exp \bigl( - \varphi (v_ j)\bigr) \prod_{l = 1}^ d X_ l^{ v_{jl}},
\]
with \(v_ j = (v_{j1}, \dots, v_{jd})\) in the coordinate system \(N = \mathbb{Z}^ d\). In the sense of the author's earlier work [``Dual polyhedra and the mirror symmetry for Calabi-Yau hypersurfaces in toric varieties,'' J. Algebr. Geom. 3, No. 3, 493-535 (1994)], the equation \(f_ \varphi = 0\) defines in the Fano variety dual to \(\mathbb{P}_ \Sigma\) a Calabi-Yau hypersurface which is mirror symmetric to the Calabi-Yau hypersurface in \(\mathbb{P}_ \Sigma\) defined as a general member of the anticanonical linear system. Kähler cone; quantum cohomology ring; generators of one-dimensional cones; Kähler manifold; mirror symmetry; moduli space of holomorphic maps; toric variety; cones V. V. Batyrev, ''Quantum Cohomology Rings of Toric Manifolds,'' Astérisque 218, 9--34 (1993); arXiv: alg-geom/9310004. Toric varieties, Newton polyhedra, Okounkov bodies, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Global differential geometry of Hermitian and Kählerian manifolds, Applications of global differential geometry to the sciences Quantum cohomology rings of toric 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 We investigate the (small) quantum cohomology ring of the moduli spaces \(\overline{\mathcal M_{0,n}}\) of stable \(n\)-pointed curves of genus \(0\). In particular, we determine an explicit presentation in the case \(n = 5\) and we outline a computational approach to the case \(n = 6\). moduli spaces; pointed rational curves; quantum cohomology; Gromov-Witten invariants Fontanari, C.: Quantum cohomology of moduli spaces of genus zero stable curves, (2007) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic), Relationships between algebraic curves and physics Quantum cohomology of moduli spaces of genus zero stable 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 articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to commutative algebra, Proceedings, conferences, collections, etc. pertaining to \(K\)-theory, Computational aspects and applications of commutative rings, Computational aspects in algebraic geometry, Collections of articles of miscellaneous specific interest Algorithmic algebraic combinatorics and Gröbner bases | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials ``We show that the quantum cohomology ring of the Grassmannian can be used to find the minimal degree of the solution to various interpolation problems involving matrices of rational functions. We also use computations in the quantum cohomology ring to formalize the notion of linearity in this context and distinguish between linear problems such as matrix and tangential interpolation and nonlinear problems such as pole placement.''
The paper is written for specialists in algebraic geometry, but there are points of interest - especially the quantum product of two cohomology classes denoted by a star - for a larger community. For this one the paper would be better readable if some prerequisite material only being found in several papers had been presented in detail. interpolation problems; quantum cohomology; Grassmannian; matrices of rational functions; pole placement Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Approximation by rational functions, Pole and zero placement problems, Multivariable systems, multidimensional control systems, Numerical interpolation, Interpolation in approximation theory Interpolation theory and quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathcal R\) be either the Grothendieck semiring (semigroup with multiplication) of complex quasi-projective varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class of the complex affine line. We define a power structure over these (semi)rings. This means that, for a power series \(A(t)=1+\sum_{i=1}^\infty [A_i] t^i\) with the coefficients \([A_i]\) from \(\mathcal R\) and for \([M]\in {\mathcal R}\), there is defined a series \((A(t))^{[M]}\), also with coefficients from \(\mathcal R\), so that all the usual properties of the exponential function hold. In the particular case \(A(t)=(1-t)^{-1}\), the series \((A(t))^{[M]}\) is the motivic zeta function introduced by M. Kapranov. As an application we express the generating function of the Hilbert scheme of points, 0-dimensional subschemes, on a surface as an exponential of the surface. Gusein-Zade, S. M.; Luengo, I.; Melle-Hernández, A., A power structure over the Grothendieck ring of varieties, Math. Res. Lett., 11, 1, 49-57, (2004) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Parametrization (Chow and Hilbert schemes) A power structure over the Grothendieck ring of varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A generating set for the Grothendieck groups (indexed by partitions) of equivariant modules supported in conjugacy classes of nilpotent matrices is described. More precisely, for each partition, \(p\), the generators are elements that are the Euler characteristic of the push down of certain vector bundles whose index involves compositions that determine the same partition as \(p\). The minimal degeneration of nilpotent orbits is also studied, and some conjectures are stated, one of which describes a smaller set of generators. Grothendieck groups; equivariant modules; Euler characteristic; minimal degeneration of nilpotent orbits Klimek, J.; Kraskiewicz, W.; Shimozono, M.; Weyman, J.: On the Grothendieck group of modules supported in a nilpotent orbit in the Lie algebra \(gl(n)\). J. pure appl. Algebra 153, 237-261 (2000) Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Representation theory for linear algebraic groups, Grothendieck groups, \(K\)-theory and commutative rings, Determinantal varieties, Classical real and complex (co)homology in algebraic geometry, Group actions on varieties or schemes (quotients) On the Grothendieck group of modules supported in a nilpotent orbit in the Lie algebra \(gl(n)\) | 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 This book pedagogically describes recent developments in gauge theory, in particular four-dimensional \(N = 2\) supersymmetric gauge theory, in relation to various fields in mathematics, including algebraic geometry, geometric representation theory, vertex operator algebras. The key concept is the instanton, which is a solution to the anti-self-dual Yang-Mills equation in four dimensions.
In the first part of the book, starting with the systematic description of the instanton, how to integrate out the instanton moduli space is explained together with the equivariant localization formula. It is then illustrated that this formalism is generalized to various situations, including quiver and fractional quiver gauge theory, supergroup gauge theory. The second part of the book is devoted to the algebraic geometric description of supersymmetric gauge theory, known as the Seiberg-Witten theory, together with string/\(M\)-theory point of view. Based on its relation to integrable systems, how to quantize such a geometric structure via the \(\Omega \)-deformation of gauge theory is addressed. The third part of the book focuses on the quantum algebraic structure of supersymmetric gauge theory. After introducing the free field realization of gauge theory, the underlying infinite dimensional algebraic structure is discussed with emphasis on the connection with representation theory of quiver, which leads to the notion of quiver \(W\)-algebra. It is then clarified that such a gauge theory construction of the algebra naturally gives rise to further affinization and elliptic deformation of \(W\)-algebra. Yang-Mills and other gauge theories in quantum field theory, Supersymmetric field theories in quantum mechanics, Applications of differential geometry to physics, Vertex operators; vertex operator algebras and related structures, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Representations of quivers and partially ordered sets, Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Yang-Mills and other gauge theories in mechanics of particles and systems, Research exposition (monographs, survey articles) pertaining to quantum theory Instanton counting, quantum geometry and algebra | 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 These are the notes of a course given at Luminy (2011), in the context of a summer school organized on the occasion of the publication of the new edition of SGA 3. The purpose is to present some existence theorems of the quotient of a scheme by a group action, or more generally by an equivalence relation. We start by giving the necessary preliminary knowledge in the theory of sheaves and the theory of descent. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Étale and other Grothendieck topologies and (co)homologies Grothendieck topologies, descent, quotients | 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 Kostant polynomials play a crucial role in the Schubert calculus on \(G/B\) for a semisimple Lie group \(G\) and its Borel subgroup \(B\). These polynomials which are characterized by vanishing properties on the orbits of a regular point under the action of the Weyl group have nonzero values on the corresponding certain elements of higher Bruhat order. The author succeeded in giving explicit forms of these values. It should be emphasized that his description is very minute. Kostant polynomials; Schubert calculus; semisimple Lie group S. C. Billey, ''Kostant Polynomials and the Cohomology Ring for G/B,'' Duke Math. J. 96(1), 205--224 (1999). Semisimple Lie groups and their representations, Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds Kostant polynomials and the cohomology ring for \(G/B\) | 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 first section of this note, we show that Theorem 1.8.1 of Bayer-Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold \(V\) is generically semisimple, then \(V\) has no odd cohomology and is of Hodge-Tate type. In particular, this answers a question discussed by \textit{G. Ciolli} [Int. J. Math. 16, No. 8, 823--839 (2005; Zbl 1081.14075)]. In the second section, we prove that an analytic (or formal) supermanifold \(M\) with a given supercommutative associative \(\mathcal{O}_M\)-bilinear multiplication on its tangent sheaf \( \mathcal{T}_M \) is an \(F\)-manifold in the sense of Hertling-Manin if and only if its spectral cover, as an analytic subspace of the cotangent bundle \(T_{M}^{*}\), is coisotropic of maximal dimension. This answers a question of V. Ginzburg. Finally, we discuss these results in the context of mirror symmetry and Landau-Ginzburg models for Fano varieties. C. Hertling, Y. I. Manin, and C. Teleman, An update on semisimple quantum cohomology and \(F\)-manifolds , Tr. Mat. Inst. Steklova 264 (2009), no. Mnogomernaya Algebraicheskaya Geometriya 69-76, English transl., Proc. Steklov Inst. Math. 264 (2009), no. 1, 62-69. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Deformations of complex singularities; vanishing cycles An update on semisimple quantum cohomology and \(F\)-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 paper under review investigates the small quantum cohomology ring of general flag varieties \(G/P\). The quantum product deforms the classical cup product by adding contributions from the count of degree \(d\) rational curves on \(G/P\) with prescribed incidence conditions. The authors determine the smallest power of the quantum parameter that can occur in a product of two Schubert classes. This minimal degree is described combinatorially in terms of the Bruhat ordering, and geometrically by the \(11\) equivalent conditions of theorem \(9.1\) in the paper. The classical Chevalley's formula computes the the product of two Schubert classes, one of of them being of codimension \(1\). The methods of this paper allow for a proof of the quantum version of this formula (theorem \(10.1\)). Gromov Witten invariants W. Fulton and C. Woodward, On the quantum product of Schubert classes, \textit{J. Alge-} \textit{braic Geom.}, 13(2004), No.4, 641-661. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds On the quantum product of Schubert classes | 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 a smooth complex projective surface \(S\) with polarization \(H\), one defines on the second homology \(H_ 2(S)\) for any large odd integer \(c\) a polynomial \(\delta=\delta_ c(S,H)\) of degree \(4c-3\chi({\mathcal O}_ S)\). Analogous to Donaldson's polynomial, which is defined for simply connected closed four-manifolds, it is defined by mapping the arguments into the cohomology of the moduli space \(M\) of the ``\(H\)-slope-stable'' rank two vector bundles on \(S\) with Chern classes \(c_ 1=0\), \(c_ 2=c\) and integrating the product of the image over the fundamental class of M. Crucial is the following property: If \(S\) has a basepoint-free pencil \(| C|\), where \(C\) is a smooth connected curve, and \(\Gamma\in H_ 2(S)\) is the Poincaré dual to a holomorphic two-form \(\omega\) with \((\omega)\) a smooth and irreducible curve, then \(\delta(C,\dots,C,\Gamma+\overline\Gamma,\dots,\Gamma+\overline\Gamma)>0\), with the number of \(C\)'s limited in terms of the genus of \(C\).
The case of a complete intersection \(S\) is investigated in particular. Donaldson's polynomial; complete intersection O'Grady K.G., Algebro-geometric analogues of Donaldson's polynomials, Invent. Math., 1992, 107(2), 351--395 Topological properties in algebraic geometry, Special surfaces, Classical real and complex (co)homology in algebraic geometry, Complete intersections Algebro-geometric analogues of Donaldson's 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 [For the entire collection see Zbl 0745.00052.]
This very technical article gives a detailed proof of a result by \textit{A. Grothendieck} in Sémin. Geométrie Algébrique, 1967-1969, SGA 7 I, Exposé IX, Lect. Notes Math. 288, 313-523 (1972; Zbl 0248.14006), which is used in \textit{M. Raynaud}'s paper in the same volume [cf. Courbes modulaires et courbes de Shimura, C. R. Sémin., Orsay/Fr. 1987-88, Astérisque 196-197, 9-25 (1991; Zbl 0781.14023)]. For a semistable curve over a valuation ring the monodromy pairing is calculated using the Picard-Lefschetz formula. semistable curve; monodromy pairing; Picard-Lefschetz formula L. Illusie, Réalisation \(l\)-adique de l'accouplement de monodromie d'après A. Grothendieck, Astérisque 7 (1992), 27--44. Local ground fields in algebraic geometry, (Co)homology theory in algebraic geometry \(l\)-adic realization of monodromy pairing after A. Grothendieck | 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 Peterson isomorphism relates the homology of the affine Grassmannian to the quantum cohomology of any flag variety. In the case of a partial flag, Peterson's map is only a surjection, and one needs to quotient by a suitable ideal on the affine side to map isomorphically onto the quantum cohomology. We provide a detailed exposition of this parabolic Peterson isomorphism in the case of the Grassmannian of \(m\)-planes in complex \(n\)-space, including an explicit recipe for doing quantum Schubert calculus in terms of the appropriate subset of non-commutative \(k\)-Schur functions. As an application, we recast Postnikov's affine approach to the quantum cohomology of the Grassmannian as a consequence of parabolic Peterson by showing that the affine nilTemperley-Lieb algebra arises naturally when forming the requisite quotient of the homology of the affine Grassmannian. Peterson isomorphism; quantum cohomology; non-commutative \(k\)-Schur functions; Grassmannian; affine nilTemperley-Lieb algebra; Schubert calculus Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Reflection and Coxeter groups (group-theoretic aspects), Classical problems, Schubert calculus, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Applying parabolic Peterson: affine algebras and the quantum cohomology of the 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 give closed combinatorial product formulas for Kazhdan-Lusztig polynomials and their parabolic analogue of type \(q\) in the case of Boolean elements, introduced in [\textit{M. Marietti}, J. Algebra 295, No. 1, 1--26 (2006; Zbl 1097.20035)], in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincaré polynomials of the Schubert varieties of Boolean elements. Coxeter groups; Kazhdan-Lusztig polynomials; Boolean elements; Poincaré polynomials Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects), Classical problems, Schubert calculus, Hecke algebras and their representations Kazhdan-Lusztig polynomials of Boolean elements | 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 about quantum cohomology and birational geometry.
The material is organized as follows: In Section 2, the author gives a brief introduction to quantum cohomology and the naturality problem. Sections 3 and 4 deal with Mori's minimal model program and, respectively, with the connection between the naturality problem and birational geometry. Finally, in Section 5, some open problems are nicely pointed out.
In conclusion, a very nice paper. surgery; quantum cohomology; birational geometry Ruan, Y.: Surgery, quantum cohomology and birational geometry, Northern California Symplectic Geometry Seminar. AMS Transl. Ser. 2 \textbf{196}, 183-198 (1999) Research exposition (monographs, survey articles) pertaining to differential geometry, Minimal model program (Mori theory, extremal rays), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Surgery, quantum cohomology and birational 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 consider the limiting behavior of \textit{discriminants}, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety \(X\) and linear systems on \(X\). These are connected -- we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization is given in terms of motivic zeta values. Motivated by our results, we ask whether the symmetric powers of geometrically irreducible varieties stabilize in the Grothendieck ring (in an appropriate sense). Our results extend parallel results in both arithmetic and topology. We give a number of reasons for considering these questions, and we propose a number of new conjectures, both arithmetic and topological. Grothendieck ring; stabilization; discriminant; configuration spaces; hypersurfaces; motivic zeta functions Arcs and motivic integration, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Fibrations, degenerations in algebraic geometry Discriminants in the Grothendieck ring | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, the author studies integrals over Hilbert schemes of points involving tautological bundles and certain ``geometric'' subsets of these Hilbert schemes. For a smooth projective complex variety \(X\) of dimension \(d\), the Hilbert scheme \(X^{[n]}\) parametrizes length-\(n\) \(0\)-dimensional closed subschemes of \(X\). A vector bundle on \(X\) induces a tautological bundle \(E^{[n]}\) on \(X^{[n]}\). Roughly speaking, a geometric subset of \(X^{[n]}\) is a constructible subset \(P\) such that if \(Z, Z' \in X^{[n]}\) satisfying \(Z \cong Z'\) as \(\mathbb C\)-schemes, then either \(Z, Z' \in P\) or \(Z, Z' \not \in P\).
The main theorem of the paper states that the integral over \(X^{[n]}\) involving the Chern classes of \(E^{[n]}\) and the fundamental class (respectively, the Chern-Mather class, the Chern-Schwartz-MacPherson class) of a geometric subset \(P\) can be written as a universal polynomial, depending on the type of \(P\), in the Chern numbers involving \(E\) and the tangent bundle \(T_X\) of \(X\). When \(X^{[n]}\) is smooth, the integral is also allowed to contain the Chern classes of \(T_{X^{[n]}}\).
The main idea in proving this theorem is to use Jun Li's concept of Hilbert scheme \(X^{[[\alpha]]}\) of \(\alpha\)-points [\textit{J. Li}, Geom. Topol. 10, 2117--2171 (2006; Zbl 1140.14012)]. The main theorem generalizes many known results when \(X\) is a surface.
As an application, the author obtains a generalized Göttsche's conjecture for all isolated singularity types and in all dimensions. More precisely, if \(L\) is a sufficiently ample line bundle on a smooth projective variety \(X\), then in a general subsystem \(\mathbb P^m \subset |L|\) of appropriate dimension \(m\), the number of hypersurfaces with given isolated singularity types is a polynomial in the Chern numbers involving \(T_X\) and \(L\). Another application is to obtain similar results, when \(X\) is a surface, for the locus of curves with fixed ``BPS spectrum'' in the sense of stable pairs theory.
Section~2 is devoted to the preliminaries such as the definition of the tautological bundle \(E^{[n]}\) on the Hilbert scheme \(X^{[n]}\), the construction of the Chern-Mather and Chern-Schwartz-MacPherson classes, the Hilbert scheme \(X^{[[\alpha]]}\) of \(\alpha\)-points, and the definition of geometric subsets in \(X^{[n]}\) and \(X^{[[\alpha]]}\). Section~3 contains an outline of the proof of the main theorem, while the formal proof of the main theorem is presented in Section~4. In Section~5, the author verifies a technical lemma which is used in Section~4. Section~6 deals with the generating series of the above-mentioned integrals over all the Hilbert schemes \(X^{[n]}\), \(n \geq 0\). In Section~7, The precise definition of sufficiently ample is given, and the main theorem is applied to the problem of counting geometric objects with prescribed singularities. Hilbert schemes; tautological bundles; Göttsche's conjecture; counting singular divisors; BPS spectrum J. V. Rennemo, Universal polynomials for tautological integrals on Hilbert schemes , preprint, [math.AG]. arXiv:1205.1851v1 Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Universal polynomials for tautological integrals on Hilbert 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 The authors consider the problem of quantization of smooth symplectic varieties in the algebro-geometric setting. They show that, under appropriate cohomological assumptions, the Fedosov quantization procedure goes through with minimal changes. The assumptions are satisfied, for example, for affine and for projective varieties. They also give a classification of all possible quantizations. symplectic structures; formal geometry; Harish-Chandra extensions Bezrukavnikov, R.; Kaledin, D., \textit{Fedosov quantization in the algebraic context}, Mosc. Math. J., 4, 559-592, (2004) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Applications of local differential geometry to the sciences, Deformation quantization, star products Fedosov quantization in algebraic context | 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 his article [in J. Am. Math. Soc. 11, No. 2, 229-259 (1998; Zbl 0904.20033)], \textit{F. Brenti} gave a non-recursive formula for the Kazhdan-Lusztig polynomials of a Coxeter group and proved it by combinatorial methods [cf. loc. cit., Theorem 4.1]. The goal of this note is to give a geometric interpretation of this formula for the Coxeter groups that are isomorphic to the Weyl group of a split group over a finite field. This interpretation rests on a result of the author [\textit{S. Morel}, J. Am. Math. Soc. 21, No. 1, 23-61 (2008; Zbl 1225.11073), Theorem 3.3.5] that expresses the intermediate extension of a pure perverse sheaf as a ``weight truncation'' of the usual direct image. Kazhdan-Lusztig polynomials; Coxeter groups; intersection complexes; Weyl groups Morel, S.: Note sur LES polynômes de Kazhdan-Lusztig. Math. Z. 268, 593-600 (2011) Hecke algebras and their representations, Reflection and Coxeter groups (group-theoretic aspects), Combinatorial aspects of representation theory, Modular and Shimura varieties, Étale and other Grothendieck topologies and (co)homologies Note on the 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 Grothendieck's ``vast unifying vision'' provided new working and conceptual foundations for geometry, and even led him to logical foundations. While many pictures here illustrate the geometry, Grothendieck himself favored apt words and commutative diagrams over pictures and did not think of geometry pictorially. cohomology; algebraic geometry; Grothendieck; topoi; schemes History of algebraic geometry, History of mathematics in the 20th century, Biographies, obituaries, personalia, bibliographies, Sheaves in algebraic geometry, Grothendieck topologies and Grothendieck topoi Grothendieck's unifying vision of 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 In the present paper, the objects of consideration include a regular semilocal Noetherian scheme \(W\), a reductive group scheme \(G\) over \(W\) and a principal \(G\)-bundle over \(\mathbb{P}^1_W\). The main theorem of the paper states that if the restriction of such a \(G\)-bundle to each closed fiber is trivial, then the original bundle is an inverse image of some principal \(G\)-bundle on \(W\). For the case when the scheme \(W\) is equicharacteristic, this theorem was proved in a paper by \textit{I. Panin} et al. [Compos. Math. 151, No. 3, 535--567 (2015; Zbl 1317.14102)] on the Grothendieck-Serre conjecture. That equicharacteristic case of the theorem was used in a paper by \textit{R. Fedorov} and \textit{I. Panin} [Publ. Math., Inst. Hautes Étud. Sci. 122, 169--193 (2015; Zbl 1330.14077)], and in another paper by \textit{I. Panin} [``Proof of Grothendieck-Serre conjecture on principal \(G\)-bundles over regular local rings containing a finite field'', Preprint, \url{arXiv:1406.0247}], to prove the Grothendieck-Serre conjecture itself in the equicharacteristic case. The main theorem of the present paper may be useful for proving the general case of the Grothendieck-Serre conjecture. reductive group schemes; principal bundles; Grothendieck-Serre conjecture; mixed characteristic; quotient sheaves Group schemes, Étale and other Grothendieck topologies and (co)homologies A step towards the mixed-characteristic Grothendieck-Serre 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 In [``The class of the affine line is a zero divisor in the Grothendieck ring'', Preprint, \url{arXiv:1412.6194}], \textit{L. A. Borisov} has shown that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. We improve the final formula by removing a factor. Martin, N., \textit{the class of the affine line is a zero divisor in the Grothendieck ring: an improvement}, C. R. Math. Acad. Sci. Paris, 354, 936-939, (2016) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Varieties and morphisms, Grassmannians, Schubert varieties, flag manifolds The class of the affine line is a zero divisor in the Grothendieck ring: an improvement | 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 results of this paper are a generalization of those in the authors' paper [Special functions, Proc. Hayashibara Forum, Okayama/Jap. 1990, ICM-90 Satell. Conf. Proc., 149-168 (1991)]. Let \(k_ q[G]\) be the quantum algebra of functions on a semisimple algebraic group \(G\) of rank \(\ell\) (in [loc. cit.], the considered \(G= \text{SL}(N)\)). Let \(B\) be a Borel subgroup of \(G\) and let \(P\supseteq B\) be a maximal parabolic subgroup of \(G\). Let \(k_ q[B]\) be the quantum Hopf algebra of functions on \(B\). Let \(w\) be an element of the Weyl group and let \(X(w) \subset G/B\) be the corresponding Schubert variety. The authors define the quantum algebras \(k_ q[G/P]\), \(k_ q[G/B]\), \(k_ q[X(w)]\); the first two are subcomodules of \(k_ q[G]\), the last is a quotient of \(k_ q[G/B]\). Each of these algebras has, in the classical case, a basis consisting of standard monomials---compatible with canonical \(\mathbb{Z}\) or \(\mathbb{Z}^ \ell\)-gradations. The authors prove the existence of such basis and gradations in the quantum case and give a presentation for \(k_ q[G/B]\). quantum algebra of functions; semisimple algebraic group; Schubert variety; basis; gradations V. Lakshmibai and N. Reshetikhin. ''Quantum flag and Schubert schemes''. Deformation Theory and Quantum Groups with Applications to Mathematical Physics. Contemp. Math., Vol. 134. American Mathematical Society, 1992, pp. 145--181. Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Quantum flag and Schubert 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 Much has been written about Grothendieck duality. This survey will make the point that most of this literature is now obsolete: there is a brilliant article by \textit{J. L. Verdier} [in: Algebr. Geom., Bombay Colloq. 1968, 393--408 (1969; Zbl 0202.19902)] with the right idea on how to approach the subject. Verdier's article was largely forgotten for two decades until \textit{J. Lipman} resurrected it, reworked it and developed the ideas to obtain the right statements for what had before been a complicated theory [in: \textit{J. Lipman} and \textit{M. Hashimoto}, Foundations of Grothendieck duality for diagrams of schemes. Berlin: Springer, 1--259 (2009; Zbl 1467.14052)].
For the reader's benefit, Sections 1 through 5, which present the (large) portion of the theory that can nowadays be obtained from formal nonsense about rigidly compactly generated tensor triangulated categories, are all post-Verdier. The major landmarks in developing this approach were a 1996 article by me [J. Am. Math. Soc. 9, No. 1, 205--236 (1996; Zbl 0864.14008)] which was later generalized and improved on by \textit{P. Balmer} et al. [Compos. Math. 152, No. 8, 1740--1776 (2016; Zbl 1408.18026)], and a much more recent article of mine [Contemp. Math. 749, 279--325 (2020; Zbl 1442.14062)] about improvements to the Verdier base-change theorem and the functor \(f^!\). Section 6 is where Verdier's 1968 ideas still play a pivotal role [loc. cit.], but in the cleaned-up version due to Lipman and with new, short and direct proofs. Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Derived categories, triangulated categories New progress on Grothendieck duality, explained to those familiar with category theory and with algebraic geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors introduce a new version of \(3d\) mirror symmetry for the toric quotient stack \([\mathbb{C}^{n}/K]\) where \(K=(\mathbb{C}^{*})^{k}.\) Moreover, they introduce \(K\)-theoretic \(I\)-function which restricts to the \(I\)-function for a particular toric variety under certain conditions. Finally, the authors prove that the \(I\)-functions of a mirror pair coincide, under the mirror map, which switches Kahler and equivariant parameters and exchanges \(q\) to \(q^{-1}\). 3d mirror symmetry; quantum \(K\)-theory; level structure Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects) Quantum \(K\)-theory of toric varieties, level structures, and 3D 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 We obtain some fundamental results, as the Bökstedt-Neeman Theorem and Grothendieck duality, about the derived category of modules on a finite ringed space. Then we see how these results are transferred to schemes in a simple way and generalized to other ringed spaces. finite spaces; quasi-coherent modules; Grothendieck duality; ringed spaces Derived categories, triangulated categories, Algebraic aspects of posets, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry Derived categories of finite spaces and Grothendieck duality | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0744.00034.]
Let \(N\) be the smooth Seshadri compactification of \(M(2,0)^ s_{{\mathcal O}_ X}\), i.e. of the isomorphism classes of stable rank 2 vector bundles with \(\text{det} E={\mathcal O}_ X\) defined over a smooth projective curve \(X\) of genus \(g\geq 2\). The authors use the strata obtained earlier by them in The Grothendieck Festschrift, Collect. Art. in Honor of the 60th Birthday of A. Grothendieck, Vol. I, Prog. Math. 86, 87-120 (1990; Zbl 0726.14012) to compute the number of \(\mathbb{F}_ q\)- rational points of \(N\) and deduce the Poincaré polynomial of \(N\) in a compact form. The starting point is to obtain the number of rational points of \(M(2,0)^ s_{{\mathcal O}_ X}\) by the Siegel formula. Then the rational points of the remaining strata are computed and this provides full information for the number of rational points of \(N\) over \(\mathbb{F}_ q\). Seshadri compactification moduli space; Poincaré polynomial; number of rational points Algebraic moduli problems, moduli of vector bundles, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Rational points, Enumerative problems (combinatorial problems) in algebraic geometry Poincaré polynomials of some moduli 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 author discuss how the recent results of \textit{H. Krause} [Compos. Math. 141, No. 5, 1128--1162 (2005; Zbl 1090.18006)] and \textit{P. Jørgensen} [Adv. Math. 193, No. 1, 223--232 (2005; Zbl 1068.18012)] can be used for a new approach to Grothendieck duality theory. The main theorem of Krause asserts that if \(\mathbf A\) is a locally noetherian Grothendieck category, then the homotopy category of degreewise injective chain complexes of \(\mathbf A\) is a compactly generated triangulated category and the full subcategory of compact objects triangulately equivalent to the bounded derived category of \(\mathbf A\). The theorem of Jørgensen asserts that if \(R\) is left and right coherent and satisfying some additional conditions then the homotopy category of degreewise projective chain complexes of \(R\)-modules is also compactly generated triangulated category. The author explains how to deduce the Grothendieck duality for affine schemes from these statements. Since Grothendieck duality as well as the theorem of Krause hold for non-affine schemes it is natural to look for generalization of the theorem of Jørgensen for non-affine schemes. This is highly nontrivial because of lack of projective objects in the category of quasi-coherent sheaves over non-affine schemes.
The author improves the theorem of Jørgensen in several directions. He proves that for any ring \(R\) the homotopy category of degreewise projective chain complexes of \(R\)-modules is so called \(\aleph_1\)-compactly generated and hence satisfies Brown representability, but in general need nit be compactly generated. Moreover, he shows that if \(R\) is right coherent then it is compactly generated.
The another major result of the paper asserts that the homotopy category of degreewise projective chain complexes of \(R\)-modules is equivalent to a Verdier quotient of the homotopy category of degreewise flat chain complexes of \(R\)-modules. Since flat objects exists over any schemes this result implies that for any scheme one can define a triangulated category which for affine schemes coincides to the homotopy category of degreewise projective chain complexes. This result opens new perspectives in theory of derived categories of schemes. triangulated categories; Brown representability; Grothendieck duality A. Neeman, ''The homotopy category of flat modules, and Grothendieck duality,'' Invent. Math., vol. 174, iss. 2, pp. 255-308, 2008. Derived categories, triangulated categories, Derived categories and associative algebras, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The homotopy category of flat modules, and Grothendieck duality | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors try to quantize some properties of the Hopf algebra \(H = {\mathcal O}(G)\), the coordinate ring of an affine algebraic group \(G\) over an algebraically closed field \(K\). These properties include: \(H^ 0\) (the Hopf algebra dual) \(= H' \# KG\), \(H'\) the hyperalgebra of those \(f\) in \(H^*\) vanishing on some power of the augmentation ideal \(m = \text{ker }\varepsilon\) of \(H\); \(H'\) is dense in \(H^*\); \(H'\) is the injective hull of the trivial module \(H/m = K\varepsilon\) as \(H\)-module and \(H'\) is an \(H\)-module algebra. In the quantum setting, links and cliques are stressed. The hyperalgebra is enlarged to the algebra of distributions supported by the clique of \(m\). They look at the locally finite injective hull \(E\) of \(H/m\). The best results are obtained when \(H\) has a pointed clique, i.e., every prime ideal linked to \(m\) has codimension one. This works for \(H = {\mathcal O}_ q(G)\) for \(G = M(n)\), \(\text{GL}(n)\) or \(\text{SL}(n)\). The hyperalgebra \(D\) here is the wedge- closure of the span of the group-likes of \(H^ 0\) linked to the identity, and turns out to be the dual with respect to the filter of ideals generated by products of ideals in the clique of \(m\). Also \(D = E \circ D_ 0\) as right \(H\)-modules, \(D_ 0\) the coradical of \(D\). The graded algebra \(\text{gr }E\) is a subalgebra of \(\text{gr }D\), and as such, \(\text{gr }D = \text{gr }E \# D_ 0\), the action given by conjugation. Details are given for \(H = {\mathcal O}_ q(SL(2))\), \(K\) of characteristic zero, \(q\) not a root of unity. Hopf algebra; coordinate ring; affine algebraic group; hyperalgebra; augmentation ideal; links; cliques; algebra of distributions; locally finite injective hull; pointed clique; filter of ideals; graded algebra DOI: 10.1080/00927879408825095 Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Quantum groups (quantized enveloping algebras) and related deformations, Affine algebraic groups, hyperalgebra constructions Hopf algebra duality, injective modules and 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 geometry of cosets in the subgroups \(H\) of the two-generator free group \(G=\langle a,b\rangle\) nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. In previous work, it was established that dessins stabilize point-line geometries whose incidence structure reflects the commutation of (generalized) Pauli operators. Now we find that the nonexistence of a dessin for which the commutator \((a,b)=a^{-1}b^{-1}ab\) precisely corresponds to the commutator of quantum observables \([\mathcal A,\mathcal B]=\mathcal A\mathcal B-\mathcal B\mathcal A\) on all lines of the geometry is a signature of quantum contextuality. This occurs first at index \(|G:H|=9\) in Mermin's square and at index \(10\) in Mermin's pentagram, as expected. Commuting sets of \(n\)-qubit observables with \(n>3\) are found to be contextual as well as most generalized polygons. A geometrical contextuality measure is introduced. quantum contextuality; dessins d'enfants; point/line geometries; coset space Planat, M., Geometry of contextuality from Grothendieck's coset space, \textit{Quantum Information Processing}, 14, 7, 2563-2575, (2015) Yang-Mills and other gauge theories in quantum field theory, Contextuality in quantum theory, Arithmetic aspects of dessins d'enfants, Belyĭ theory, Quantum information, communication, networks (quantum-theoretic aspects), Incidence structures embeddable into projective geometries, Dessins d'enfants theory, Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices Geometry of contextuality from Grothendieck's coset space | 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 the function field of a proper, smooth, geometrically irreducible curve over a finite field \(k\) of characteristic \(p\). Let \(K^{\text{sep}}\) be a separable closure of \(K\), and let \(\overline{k}\) be the algebraic closure of \(k\) in \(K\). Then we can consider the absolute Galois group \(G_K = \text{Gal} (K^{\text{sep}} \mid K)\) and its subgroup \(\overline{G}_K = \text{Gal} (K^{\text{sep}} \mid K \overline{k})\).
Let \(\Sigma\) be a set of prime numbers not containing the characteristic \(p\), with complement \(\Sigma'\), and let \(\mathcal{C}\) be the class of finite groups whose cardinality is divisible only by primes in \(\Sigma\). Let \(\overline{G}_K^{\Sigma}\) be the maximal pro-\(\mathcal{C}\) quotient of \(\overline{G}_K\), and let \(G_K^{(\Sigma)} = G_K / \ker (\overline{G}_K \to \overline{G}_K^{\Sigma})\) be the geometrically pro-\(\Sigma\) Galois group of \(K\).
The authors impose two conditions on \(\Sigma\). The second of these conditions, called \((\epsilon_X)\), is too technical to consider in this review, but the first can be explained succinctly: it is the demand that the \(\Sigma'\)-cyclotomic character \(G_k \to \prod_{\ell \in \Sigma' \setminus \left\{ p \right\}} \mathbb{Z}_{\ell}^{\times}\) be not injective. The authors call this the \textit{\(k\)-largeness condition}. Cofinite sets of primes \(\Sigma\) always satisfy both of the aforementioned conditions.
The main result of the authors in the following. Let \(K\) and \(L\) be two algebraic functions fields as considered above, and let \(\Sigma_X\), \(\Sigma_Y\) be sets of primes such that \(\Sigma_X\) is \(k\)-large and satisfies condition \((\epsilon_X)\). Then any isomorphism of profinite groups \(\sigma : G_K^{(\Sigma_X)} \to G_L^{(\Sigma_Y)}\) arises from an isomorphism of field extensions \(L^{\sim} \to K^{\sim}\) between the corresponding subfields of \(L^{\text{sep}}\) and \(K^{\text{sep}}\).
These results generalize those in [\textit{M. Saïdi} and \textit{A. Tamagawa}, Publ. Res. Inst. Math. Sci. 45, No. 1, 135--186 (2009; Zbl 1188.14016)], which are in turn a generalization of a seminal theorem by \textit{K. Uchida} [Ann. Math. (2) 106, 589--598 (1977; Zbl 0372.12017)]. The article has a clear writing style and takes great care to summarize its argumentation in the introduction. New tools are the aforementioned \(k\)-largeness condition and the use of pseudo-functions, or in other words elements of \(K^{\times} / k^{\times} \left\{ \Sigma' \right\}\), interpreted as classes of rational functions modulo \(\Sigma'\)-primary constants. These tools are combined with the fundamental theorem of projective geometry to obtain the main result. birational anabelian geometry; curves over finite fields; pro-\(\sigma\) Galois groups Saïdi, Mohamed; Tamagawa, Akio, A refined version of Grothendieck's birational anabelian conjecture for curves over finite fields, Adv. Math., 310, 610-662, (2017) Coverings of curves, fundamental group, Curves over finite and local fields A refined version of Grothendieck's birational anabelian conjecture for curves over finite fields | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0588.00014.]
A simplicial homotopy is defined in the context of a Grothendieck topos. The result generalizes the work of \textit{K. S. Brown} concerning simplicial sheaves on a topological space [Trans. Am. Math. Soc. 186 (1973), 419-458 (1974; Zbl 0245.55007)].
The fibrations in the category of simplicial sheaves on an arbitrary Grothendieck site are defined by local lifting property, the weak equivalences, via local homotopy group sheaves isomorphisms.
Via the homotopy category, cohomology groups are defined. Between other results, coincidence with étale cohomology is given. This paper trends towards a study of algebraic groups over an algebraically closed field and proves an isomorphism conjecture on G. Grothendieck topos; simplicial sheaves; homotopy category; cohomology; étale cohomology; algebraic groups Jardine, J.F.: Simplicial objects in a Grothendieck topos. Contemp. Math. \textbf{55}(I), 193-239 (1986) Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Grothendieck topologies and Grothendieck topoi, Étale and other Grothendieck topologies and (co)homologies, Homotopy theory and fundamental groups in algebraic geometry, Linear algebraic groups and related topics, Abstract and axiomatic homotopy theory in algebraic topology Simplicial objects in a Grothendieck topos | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathcal C\) be the category whose objects consist of \(R\)-algebras (possibly non-commutative) over a possibly non-commutative ring and whose opposite morphisms consist of direct sums of bi-flat epimorphisms \(A\to B_{i}\). A map \(A\to B\) of \(R\)-algebras is a bi-flat epimorphism if every \(A\)-algebra map \(g: B\to C\) is unique. Bi-flat epimorphisms are viewed dually as localizations. An \(R\)-algebra \(A\) in \(\mathcal C\), denoted \(Geo(A)\) as an element of \(\mathcal C\), acquires via such localizations a Grothendieck topology and the representation functor is a sheaf for this Grothendieck topology. An analogue of affine scheme is thus introduced in the non-commutative case. non-commutative algebra; bi-flat epimorphism; Grothendieck topology; free products; \(\mathcal F\)-étale; flaky homomorphism; sheaf; representation functor; localizations; affine scheme Grothendieck topologies and Grothendieck topoi, Étale and other Grothendieck topologies and (co)homologies, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Noncommutative algebraic geometry A Grothendieck topology on a subcategory of opposite category of non commutative algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce the notion of an alternate product of Frobenius manifolds and we give, after \textit{I. Ciocan-Fontanine} and the authors [Invent. Math. 171, No. 2, 301--343 (2008; Zbl 1164.14012)], an interpretation of the Frobenius manifold structure canonically attached to the quantum cohomology of \(G(r,n+1)\) in terms of alternate products. We also investigate the relationship with the alternate Thom-Sebastiani product of Laurent polynomials. B. Kim and C. Sabbah, Quantum cohomology of the Grassmannian and alternate Thom-Sebastiani , Compos. Math. 144 (2008), 221-246. Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Structure of families (Picard-Lefschetz, monodromy, etc.), Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Grassmannians, Schubert varieties, flag manifolds Quantum cohomology of the Grassmannian and alternate Thom-Sebastiani | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider the system of quantum differential equations for a partial flag variety and construct a basis of solutions in the form of multidimensional hypergeometric functions, that is, we construct a Landau-Ginzburg mirror for that partial flag variety. In our construction, the solutions are labeled by elements of the \(K\)-theory algebra of the partial flag variety.
To establish these facts we consider the equivariant quantum differential equations for a partial flag variety and introduce a compatible system of difference equations, which we call the \textit{qKZ} equations. We construct a basis of solutions of the joint system of the equivariant quantum differential equations and \textit{qKZ} difference equations in the form of multidimensional hypergeometric functions. Then the facts about the non-equivariant quantum differential equations are obtained from the facts about the equivariant quantum differential equations by a suitable limit.
Analyzing these constructions we obtain a formula for the fundamental Levelt solution of the quantum differential equations for a partial flag variety. Gromov-Witten invariants; quantum differential equations; qKZ difference equations; Yangian action; multidimensional hypergeometric functions Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Structure of families (Picard-Lefschetz, monodromy, etc.), Other hypergeometric functions and integrals in several variables Landau-Ginzburg mirror, quantum differential equations and \textit{qKZ} difference equations for a partial flag 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 The introduction discusses various motivations for the following chapters of the
thesis, and their relation to questions around mirror symmetry.
The main theorem of chapter 2 says that if the quantum cohomology of a smooth
projective variety \(V\) yields a generically semisimple product, then the same holds true
for its blow-up at any number of points (theorem 3.1.1). This is a positive test for a
conjecture by Dubrovin, which claims that quantum cohomology of \(V\) is generically
semisimple if and only if its bounded derived category \(Db(V)\) has a complete exceptional collection.
Chapter 3 generalizes Bridgeland's notion of stability condition on a triangulated
category. The generalization, a polynomial stability condtion (definition 2.1.4), allows
the central charge to have values in polynomials \(\mathbb{C}[N ]\) instead of complex numbers \(\mathbb{C}\).
We show that polynomial stability conditions have the same deformation properties as
Bridgeland's stability conditions (theorem 3.2.5). In section 4, it is shown that every
projective variety has a canonical family of polynomial stability conditions.
In chapter 4, we define and study the notion of a weighted stable map (definition
2.1.2), depending on a set of weights. We show the existence of moduli spaces of
weighted stable maps as proper Deligne-Mumford stacks of finite type (theorem 2.1.4),
and study in detail their birational behaviour under changes of weights (section 4).
We introduce a category of weighted marked graphs in section 6, and show that it
keeps track of the boundary components of the moduli spaces, and natural morphisms
between them. We introduce weighted Gromov-Witten invariants by defining virtual
fundamental classes, and prove that these satisfy all properties to be expected (sections
5 and 7). In particular, we show that Gromov-Witten invariants without gravitational
descendants do not depend on the choice of weights. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Derived categories, triangulated categories Semisimple quantum cohomology, deformations of stability conditions and the derived category | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The purpose of this paper is to revisit some of the problems in quantum cohomology with emphasis on how to compute small quantum cohomology rings from a good knowledge of the classical cohomology ring together with a minimal geometric input. After a presentation of GW-theory in section 1 and various versions of quantum cohomology rings in section 2, section 3 outlines joint work with G. Tian on the (small) quantum cohomology of the moduli space of stable 2-bundles of fixed determinant of odd degree over a genus \(g\) Riemann surface. Here, the proof is complete up to a conjecture. small quantum cohomology rings; moduli space of stable 2-bundles of fixed determinant Bernd Siebert, An update on (small) quantum cohomology, Mirror symmetry, III (Montreal, PQ, 1995) AMS/IP Stud. Adv. Math., vol. 10, Amer. Math. Soc., Providence, RI, 1999, pp. 279 -- 312. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Toric varieties, Newton polyhedra, Okounkov bodies An update on (small) 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 We interpret the orthogonality relation of Kostka polynomials arising from complex reflection groups [\textit{T. Shoji}, Invent. Math. 74, 239--267 (1983; Zbl 0525.20027); J. Algebra 245, No. 2, 650--694 (2001; Zbl 0997.20044); \textit{G. Lusztig}, Adv. Math. 61, 103--155 (1986; Zbl 0602.20036)] in terms of homological algebra. This leads us to the notion of Kostka system, which can be seen as a categorical counterpart of Kostka polynomials. Then, we show that every generalized Springer correspondence [\textit{G. Lusztig}, Invent. Math. 75, 205--272 (1984; Zbl 0547.20032)] in a good characteristic gives rise to a Kostka system. This enables us to see the top-term generation property of the (twisted) homology of generalized Springer fibers, and the transition formula of Kostka polynomials between two generalized Springer correspondences of type \(\mathsf{BC}\). The latter provides an inductive algorithm to compute Kostka polynomials by upgrading [\textit{D. Ciubotaru} and the author, Adv. Math. 226, No. 2, 1538--1590 (2011; Zbl 1207.22010)] \S3 to its graded version. In the appendices, we present purely algebraic proofs that Kostka systems exist for type \(\mathsf{A}\) and asymptotic type \(\mathsf{BC}\) cases, and therefore one can skip geometric sections \S3--5 to see the key ideas and basic examples/techniques. generalized Springer correspondences; Kostka polynomials; Lusztig-Shoji algorithm; \(\mathrm{Ext}\)-orthogonal collections; Kostka systems Kato, S., A homological study of Green's polynomial, Ann. sci. éc. norm. supér. (4), 48, 5, 1035-1074, (2015) Reflection and Coxeter groups (group-theoretic aspects), Symmetric functions and generalizations, Combinatorial aspects of representation theory, Representation theory for linear algebraic groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Hecke algebras and their representations A homological study of Green 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 Grothendieck-Teichmüller group \(\widehat{GT}\) was defined by \textit{V. G. Drinfel'd} [Leningr. Math. J. 2, No. 4, 829-860 (1991); translation from Algebra Anal. 2, No. 4, 149-181 (1990; Zbl 0718.16034)]. In the paper under review, a certain subgroup of \(\widehat{GT}\) is defined which contains the image of the absolute Galois group \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\) in \(\widehat{GT}\), and which gives an automorphism group of the profinite completions of certain surface mapping class groups with geometric compatibility conditions.
These results fit into the program of \textit{A. Grothendieck} [Lond. Math. Soc. Lect. Note Ser. 242, 5-48 (1997; Zbl 0901.14001)]\ suggesting the existence of a Teichmüller tower constructed out of the pure subgroups of the mapping class groups of marked surfaces. Grothendieck-Teichmüller group; mapping class groups; absolute Galois groups; profinite completions Lochak, P.; Nakamura, H.; Schneps, L.: On a new version of the Grothendieck--Teichmüller group, C. R. Acad. sci. Paris sér. I math. 325, No. 1, 11-16 (1997) Other groups related to topology or analysis, Homotopy theory and fundamental groups in algebraic geometry, Limits, profinite groups, Groups acting on specific manifolds, Knots and links in the 3-sphere, Automorphism groups of groups On a new version of the 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 The famous Pierce-Birkhoff conjecture states that a continuous and piecewise polynomial function is a supremum of infimums of finitely many polynomials (see [\textit{G. Birkhoff} and \textit{R. S. Pierce}, ``Lattice-ordered rings'', Anais Acad. Brasil. Ci. 28, 41--69 (1956; Zbl 0070.26602)]). The (finitely many) ``pieces'' are semialgebraic sets. The conjecture was proven 1984 for \(n=2\) by \textit{L. Mahé} [``On the Pierce-Birkhoff conjecture'', Rocky Mt. J. Math. 14, 983--985 (1984; Zbl 0578.41008)]. The Pierce-Birkhoff conjecture for \(n>3\) remains unproved and unrefuted despite great effort of many mathematicians.
In the paper under review the author considers so-called generalized polynomials (also called signomials). These are real polynomials with arbitrary real exponents. Their natural domain of definition is the positive orthant. The Pierce-Birkhoff conjecture is extended to generalized polynomials: A continuous and piecewise generalized polynomial function (on the positive orthant) is a supremum of infimums of finitely many generalized polynomials. Here the (finitely many) ``pieces'' are generalized semialgebraic sets. Following the reasoning of Mahé the Pierce-Birkhoff conjecture for generalized polynomials is shown in the case \(n=2\). two-variable Pierce-Birkhoff conjecture; generalized polynomials Real algebra, Real and complex fields, Semialgebraic sets and related spaces, Representation and superposition of functions Extension of the two-variable Pierce-Birkhoff conjecture to generalized 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 \(K\) and \(L\) be a number fields and \(G_ K\), \(G_ L\) be their absolute Galois groups. Then the canonical map \(\text{Hom}(K,L) \to \text{Out}(G_ K,G_ L)\) is a bijection (Neukirch, Ikeda, Iwasawa, Uchida). The author proves a generalization of this result for the function fields of one variable over a finitely generated field. This result was conjectured by Grothendieck in the frames of his anabelian geometry. absolute Galois groups; function fields of one variable; anabelian geometry F. Pop, ''On Grothendieck's conjecture of birational anabelian geometry,'' Ann. of Math., vol. 139, iss. 1, pp. 145-182, 1994. Algebraic functions and function fields in algebraic geometry, Galois theory, Global ground fields in algebraic geometry On Grothendieck's conjecture of birational 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 It is known that the problem of determining the conditions on conjugacy classes \(\overline A_{1,\dots,}\overline A_{s}\) in \(\text{SU}(n)\), so that these lift to elements \(A_{1,\dots,}A_s\in \text{SU}(n)\) with \(A_1A_2\dots A_s=1\), is controlled by quantum Schubert calculus of Grassmannians. \textit{C.~Teleman} and \textit{C.~Woodward} [Ann. Inst. Fourier 53, 713--748 (2003; Zbl 1041.14025)] have recently generalized this to an arbitrary simple simply connected compact group \(K\). If \(G\) is a complex simple group (whose real points are \(K\)), then the role played by the Grassmannians is replaced by the flag varieties \(G/P\) for \(P\) a maximal parabolic subgroup.
In the case of \(\text{SU}(n)\) (and similarly for \(K\)), there is a natural ``action'' of the center of \(\text{SU}(n)\) on the representation theory side, namely if \(c_1,\dots,c_s\) are central elements with \(c_1c_2\dots c_s=1\), then these act on the set of conjugacy classes \(\overline A_1,\dots,\overline A_s\) in \(\text{SU}(n)\), liftable to elements \(A_1,\dots,A_s\) with \(A_1A_2\dots A_s=1\), the action being just multiplying \(\overline A_i\) by \(c_i\). This action is well defined on the level of conjugacy classes because the \(c_i\) are central. This suggests a natural transformation property of Gromov-Witten numbers of the Grassmannians under the action of the center.
The first aim of the paper is to prove the transformation formulas geometrically and in complete generality (for any simple simply connected complex Lie group). The second is to show that these formulas determine the quantum Schubert calculus in the case of Grassmannians (Bertram's Schubert calculus). The author also gives a strengthening in the case of Grassmannians of a theorem of \textit{W.~Fulton} and \textit{C.~Woodward} [J. Algebr. Geom. 13, 641--661 (2004; Zbl 1075.14038)] on the lowest power of \(q\) appearing in a (quantum) product of Schubert classes in \(G/P\), where \(P\) is a maximal parabolic subgroup. Also many results in this paper are new proofs of older results using methods which seem both natural and elementary. Grassmann variety; parabolic subgroup P. Belkale. Transformation formulas in Quantum Cohomology, 2001. preprint. Classical problems, Schubert calculus, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Transformation formulas 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 We find presentations by generators and relations for the equivariant quantum cohomology rings of the maximal isotropic Grassmannians of types B, C and D, and we find polynomial representatives for the Schubert classes in these rings. These representatives are given in terms of the same Pfaffian formulas which appear in the theory of factorial \(P\)- and \(Q\)-Schur functions. After specializing to equivariant cohomology, we interpret the resulting presentations and Pfaffian formulas in terms of Chern classes of tautological bundles. T. Ikeda , L. C. Mihalcea and H. Naruse , 'Factorial \(P\) - and \(Q\) -Schur functions represent equivariant quantum Schubert classes', Osaka J. Math., to appear. Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Factorial \(P\)- and \(Q\)-Schur functions represent equivariant quantum Schubert classes | 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 An in-depth discussion on the analysis of the function \({E_a }({X;q,t})\) for the special case \(t=0\) is made in this paper, wherein the author shows that the specialized function `\({E_a}({X;q,0})\) stabilizes to \(\omega {P_\mu }({X;0,t})\)', where \({P_\mu }({X;0,t})\) denotes the Hall-Littlewood polynomials and \(\omega\) is the `involution on symmetric functions'. The author also `relates \({E_a}({X;q,0})\) to the (finite) type A Demazure characters \(E_a ({X; 0,0})\)'.
In his another paper [``Weak dual equivalence for polynomials'', Preprint, \url{arXiv:1702.04051}], the author has developed the theory of weak dual equivalence and introduced the `standard key tableaux to develop a theory of type A Demazure characters' which is invoked by him in this paper to give a combinatorial proof of the fact that on grouping `together the terms in the fundamental slide expansion of \({E_a}({X;q,0})\), the coefficients of \({E_a}({X;q,0})\), when expanded into Demazure characters, are polynomials in \(q\) with nonnegative integer coefficients.' This treatment here parallels the earlier treatment of `the use of dual equivalence' by the author in his work [Forum Math. Sigma 3, Article ID e12, 33 p. (2015; Zbl 1319.05135)] regarding the fundamental quasisymmetric expansion of \({H_\mu}({X;q,t})\) (the transformed Macdonald symmetric functions in type A).
The first significant result of the paper is:
Theorem 3.6. The specialized nonsymmetric Macdonald polynomial \({E_a}({X;q,0})\) is given by
\[
{E_a}({X;q,0}) = \sum_{T \in {\text{SKD}}(a)} {{q^{{\text{maj}}(T)}}{{\mathcal{F}}_{{\text{des}}(T)}}(X)},
\]
where \({{\text{SKD}}(a)}\) denotes the standard key tabloids of shape \(a\), \(\mathcal{F}_a\) denotes the fundamental slide polynomial (see [\textit{S. Assaf} and \textit{D. Searles}, Adv. Math. 306, 89--122 (2017; Zbl 1356.14039)]) defined on the finite set \(X\) of variables \(x_1, \ldots, x_n\) by the relation \({{\mathcal{F}}_a}(X) = \sum_{b \geqslant a; {\text{flat}}(b){\text{ refines flat}}(a)}{{X^b}}\) in which `\({\text{flat}}(a)\) is the composition obtained by removing zero parts from \(a\)', \({{\text{des}}(T)}\) denotes the weak descent composition of \(T\) for a standard filling \(T\) of a key diagram and \({{\text{maj}}(T)}\) represents `the sum of the legs of all cells \(c\) (of a key diagram) such that the entry in \(c\) is strictly greater than the entry immediately to its left.' Another important result is the following theorem:
Theorem 4.7. For a weak composition \(a\) such that \(\text{SKD}(a)\) has no virtual elements, the maps \(\left\{ \psi_i \right\}\) on \(\text{SKD}(a)\) give a weak dual equivalence for \((\text{SKD}(a),\text{des})\).
The Demazure character \({\kappa _a}(X)\) is given by the author in [loc. cit., arXiv:1702.04051] and in this paper he beautifully develops the relation between the functions \({E_a}({X;q,0})\) and Demazure characters in the following result:
Theorem 4.9. The specialized nonsymmetric Macdonald polynomial \({E_a}({X;q,0})\) given by \({E_a}({X;q,0}) = \sum_{T \in {\text{YKD}}(a)} {{q^{{\text{maj}}(T)}}{\kappa _{{\text{des}}(T)}}}. \) In particular, \({E_a}({X;q,0})\) is a positive graded sum of Demazure characters.
The following theorem is a landmark result of this paper:
Theorem 5.6. For a weak composition \(a\), we have
\[
\lim_{m \to \infty } {E_{{0^m} \times a}}({X;q,0}) = \omega {H_{{\text{sort}}(a)'}}({X;0,q}) = \omega {H_{{\text{sort}}(a)}}({X;q,0})
\]
By defining the nonsymmetric Kostka-Foulkes polynomial \({K_{a,b}}(q)\) by the relation \({E_b}({X;q,0}) = \sum_a {{K_{a,b}}(q){\kappa _{\text{a}}}(X)} \) the author redevelops the Theorem 5.6 in terms of Kostka-Foulkes polynomials as follows:
Corollary 5.7. Given a weak composition \(b\) with column lengths \(\mu\) such that \({{\text{SKT}}(b)}\) has no virtual Yamanouchi elements, we have
\[
{K_{\lambda ,\mu }}(t) = \sum_{{\text{sort}}({{\text{flat}}(a)}) = \lambda '} {{K_{a,b}}(t)}.
\]
The reviewer finds the paper an important and valuable contribution to the theory of nonsymmetric Macdonald polynomials and their interconnection with Demazure characters. Macdonald polynomials; Demazure characters; Kostka-Foulkes polynomials; Macdonald's symmetric functions; Hall-Littlewood symmetric functions; Jack symmetric functions; nonsymmetric Macdonald polynomials; Schubert calculus; plethystic substitution; Yamanouchi; weak descent composition; non-symmetric Kostka-Foulkes polynomial; Demazure atom Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Symmetric functions and generalizations, Classical problems, Schubert calculus Nonsymmetric Macdonald polynomials and a refinement of Kostka-Foulkes 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 In this paper we study the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle. Quantum sheaf cohomology is a (0,2) deformation of the ordinary quantum cohomology ring, realized as the OPE ring in A/2-twisted theories. Quantum sheaf cohomology has previously been computed for abelian gauged linear sigma models (GLSMs); here, we study \((0,2)\) deformations of nonabelian GLSMs, for which previous methods have been intractable. Combined with the classical result, the quantum ring structure is derived from the one-loop effective potential. We also utilize recent advances in supersymmetric localization to compute A/2 correlation functions and check the general result in examples. In this paper we focus on physics derivations and examples; in a companion paper, we will provide a mathematically rigorous derivation of the classical sheaf cohomology ring. quantum sheaf cohomolgy; Grassmannians; rings structures; mathematical physics, tangent bundle; deformations Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Grassmannians, Schubert varieties, flag manifolds, Formal methods and deformations in algebraic geometry, Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.), Sheaf cohomology in algebraic topology Quantum sheaf cohomology on 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 For semi-positive symplectic manifolds the Floer cohomology groups are naturally isomorphic to the ordinary cohomology groups. The paper outlines a new proof for this fact and shows that the isomorphism intertwines the quantum cup product in ordinary cohomology with the ``pair-of-pants'' product in Floer cohomology. The new proof uses a glueing theorem for J-holomorphic curves. The article provides motivations and sketches the proofs. Technical details are promised to appear elsewhere. symplectic cohomology; Floer homology; quantum cup product; quantum cohomology S. Piunikhin, D. Salamon, and M. Schwarz. Symplectic Floer--Donaldson theory and quantum cohomology. In C.B. Thomas, editor, \textit{Contact and symplectic geometry} , pages 171--200. Cambridge Univ. Press, 1996. General geometric structures on manifolds (almost complex, almost product structures, etc.), Algebraic topology on manifolds and differential topology, Hodge theory in global analysis, de Rham cohomology and algebraic geometry Symplectic Floer-Donaldson theory and quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In a previous paper, the authors introduced the notion of mixed Frobenius structure (MFS) as a generalization of the structure of a Frobenius manifold. Roughly speaking, the MFS is de fined by replacing a metric of the Frobenius manifold with a fi ltration on the tangent bundle equipped with metrics on its graded quotients. The purpose of the current paper is to construct a MFS on the cohomology of a smooth projective variety whose multiplication is the nonequivariant limit of the quantum product twisted by a concave vector bundle. We show that such a MFS is naturally obtained as the nonequivariant limit of the Frobenius structure in the equivariant setting. mixed Frobenius structure; quantum cohomology; local mirror symmetry Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Mixed Frobenius structure and local 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-Lusztig-polynomials \(P_{x,y}(t)\) are associated with two elements \(x,y\) of a given Coxeter group. They find interpretations in combinatorics (poset-recursion), algebra (entries of a transition matrix between two natural bases of the associated Hecke algebra), geometry (Poincaré polynomial of a stalk of the intersection cohomology sheaf on a Schubert variety in the corresponding flag variety), representation theory (the coefficients are the dimensions of certain Ext groups).
The combinatorial definition of Kazhdan-Lusztig polynomials is generalized to more general posets by Stanley, the resulting polynomial is called the Kazhdan-Lusztig-Stanley polynomial. The paper under review is an accessible, beautifully written survey of these polynomials, and their role in geometry. More precisely, consider the following two classes of KLS polynomials: the \(g\)-polynomial of a rational polytope, and the KL polynomial of a hyperplane arrangement. These have geometric interpretations as Poincaré polynomials of stalks of certain intersection cohomology sheafs of an associated geometric object. This result has been known before, but this paper gives a unified proof, which also clarifies which ingredients are needed for the proof.
Moreover, the author introduces a generalization of KLS polynomials called Z-polynomials. They take into account the poset structure as well as the opposite poset structure, or in other words, both right and left KLS polynomials. Geometric interpretations of Z-polynomials analogous to the ones mentioned above for KLS polynomials are proved. Kazhdan-Lusztig; intersection cohomology Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus The algebraic geometry of Kazhdan-Lusztig-Stanley 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 classical cohomology of the Grassmannian \(\text{Gr}(r, n)\) is generated additively by the Schubert classes; the structure constants for the multiplication are the well studied Littlewood-Richardson coefficients. Multiplication by special Schubert subvarieties, in particular by the unique divisorial class, are determined explicitly by the Pieri formula. Considering the natural action of the \(n\)-dimensional torus \(T\) on \(\text{Gr}(r,n)\), one may study the equivariant cohomology of the Grassmannian, the equivariant Littlewood-Richardson coefficients, and the equivariant Pieri formula. Going further, one may include ``quantum'' corrections to the multiplication in cohomology, given by counts of higher degree holomorphic maps to \(\text{Gr}(r,n)\) with incidence conditions with Schubert cycles. In this fashion, one arrives at the equivariant quantum cohomology of the Grassmannian.
The paper under review presents an explicit equivariant quantum Pieri rule for the quantum multiplication by the equivariant divisorial Schubert class. This could be compared to the non-equivariant case obtained by \textit{A. Bertram} [Adv. Math. 128, No.~2, 289--305 (1997; Zbl 0945.14031)]. Moreover, the author also shows the vanishing of equivariant quantum Littlewood-Richardson coefficients in a certain range, and proves a recursive formula satisfied by these coefficients. Schubert calculus; quantum cohomology; quantum Pieri formula L. Mihalcea, ''Equivariant quantum Schubert calculus,'' Adv. Math., vol. 203, iss. 1, pp. 1-33, 2006. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Equivariant quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(R\) be a regular local ring with fraction field \(F\), let \(G\) be a reductive group scheme over \(R\), and let \(E\) be a \(G\)-torsor. The famous Grothendieck-Serre conjecture predicts that if \(E\) has an \(F\)-point, then \(E\) has an \(R\)-point. Equivalently, the map \(H^1(R,G)\to H^1(F,G)\) between étale cohomology pointed sets is injective. The conjecture is known to hold in many cases, notably when \(R\) contains a field (the `equicharacteristic case'), but is open in general.
The paper establishes the conjecture in the mixed-characteristic case under the assumption that \(G\) is quasi-split and \(R\) is unramified. Here, a local ring \(R\) with residue field \(k\) of characteristic \(p\) is called unramified if \(R/pR\) is regular. It is further shown that a reductive group over an unramified regular local ring \(R\) is split if and only if \(G_F\) is split.
Using similar methods, the author also establishes a special case of a more general conjecture attributed to Colloit-Thélène and Panin: If \(G_F\) admits a parabolic subgroup, then \(G\) admits a parabolic subgroup of the same type. This is shown to hold for Borel subgroups when \(R\) contains a field. reductive group; principal homogenous space; torsor; etale cohomology; Grothendieck-Serre conjecture; group scheme Group schemes, Algebraic theory of quadratic forms; Witt groups and rings, Homogeneous spaces and generalizations, Cohomology theory for linear algebraic groups Grothendieck-Serre in the quasi-split unramified case | 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 Chevalley formula for the equivariant quantum multiplication of two Schubert classes in a homogeneous variety \(X=G/P\). An efficient algorithm to compute 3 point, genus zero equivariant GW-invariants on \(X\) is also given. equivariant quantum cohomology; homogeneous space; Chevalley formula L. C. Mihalcea, ''On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms,'' Duke Math. J., vol. 140, iss. 2, pp. 321-350, 2007. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Equivariant algebraic topology of manifolds, Algebraic combinatorics On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms | 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 Studying inequalities between subgraph- or homomorphism-densities is an important topic in graph theory. Sums of squares techniques have proven useful in dealing with such questions. Using an approach from real algebraic geometry, we strengthen a positivstellensatz for simple quantum graphs by \textit{L. Lovász} and \textit{B. Szegedy} [J. Graph Theory 70, No. 2, 214--225 (2012; Zbl 1242.05249)], and we prove several new positivstellensätze for nonnegativity of quantum multigraphs. We provide new examples and counterexamples. positivstellensatz; sums of squares; nonnegativity; graph parameter; homomorphism density Netzer, Tim; Thom, Andreas: Positivstellensätze for quantum multigraphs. (2013) Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.), Real algebraic and real-analytic geometry Positivstellensätze for quantum multigraphs | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to differential geometry, Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Collections of articles of miscellaneous specific interest, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Singularities in algebraic geometry Frobenius manifolds. Quantum cohomology and singularities. Proceedings of the workshop, Bonn, Germany, July 8--19, 2002 | 0 |
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