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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 Grothendieck has formulated classical Galois theory as equivalence of categories: the étale topos over a field k is equivalent to the category of topological spaces with G-action, where G is the Galois group of the separable closure of k. This can be generalized by taking for G a groupoid \(G_ 1\rightrightarrows G_ 0\) (the arrows are the domain and codomain). Moreover, for a general topos, the notion of space has to be extended.
Topological spaces are given locally by their lattices of open sets, which is here abstracted to arbitrary complete lattices with distributive property, called locales, and so the category of extended spaces is defined as the dual of that of locales. Then, for any topos, there is a groupoid \(G_ 1\rightrightarrows G_ 0\) in the category of extended spaces such that the given topos is equivalent to that of sheaves on \(G_ 0\) with \(G_ 1\)-action.
For demonstration a descent theory for locales is developed. For instance, the notion of an open mapping of extended spaces (and also of topoi) is defined, and it is shown that open surjections are effective descent morphisms for sheaves. Galois theory; étale topos; groupoid; Topological spaces; lattices with distributive property; locales; category of extended spaces; dual; sheaves; descent theory Joyal, A., Tierney, M.: An extension of the Galois theory of Grothendieck. Mem. Amer. Math. Soc. \textbf{51}(309) (1984) Topoi, Groupoids, semigroupoids, semigroups, groups (viewed as categories), Separable extensions, Galois theory, Categories of topological spaces and continuous mappings [See also 54-XX], Étale and other Grothendieck topologies and (co)homologies, Grothendieck topologies and Grothendieck topoi, Distributive lattices, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) An extension of the Galois theory of 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 paper under review aims to give an overview of the current state of knowledge about the small quantum cohomology ring of homogeneous varieties, with an emphasis on combinatorial aspects of the theory. Let \(X=G/P\) where \(G\) is a simple complex algebraic Lie group, and \(P\) a parabolic subgroup. The author first summarizes classical results about \(H^{*}(X,\mathbb{Z})\) and explains the relation of this ring to Young tableaux when \(X\) is a Grassmannian. Next, the small quantum cohomology ring \(QH^{*}(X)\) is defined. The author then proceeds to explain some results about the ring structure of \(QH^{*}(X)\). In particular, suppose that \(\sigma_{\lambda}* \sigma_{\mu}=\sum c_{\lambda \mu}^{\nu}(d)q^{d}\sigma_{\nu}\) are the structural equations of the ring in terms of Schubert cycles. Then some of the results discussed in the paper are the quantum Giambelli formula, the computation of \(c_{\lambda\mu}^{\nu}(d)\) in the case of a Grassmannian using classical intersection theory, the smallest value of \(d\) such that \(q^{d}\) appears in the product, and the determination of all \(q^{d}\) which occur with a nonzero coefficient. The paper ends with some remarks and questions. homogeneous spaces; Young tableaux W. Fulton, On the quantum cohomology of homogeneous varieties, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 729 -- 736. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds On the quantum cohomology of homogeneous 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 has long been accepted that the foundations of Grothendieck duality are complicated. This has changed recently.
By ``Grothendieck duality'' we mean what, in the old literature, used to go by the name ``coherent duality''. This isn't to be confused with what is nowadays called ``Verdier duality'', and used to pass as ``\(\ell\)-adic duality''. (The prevailing current terminology -- for duality in étale cohomology, that is ``\(\ell\)-adic duality'' -- is historically incorrect. The idea was originally due not to Verdier but to Grothendieck, see his work in SGA5 [Seminaire de géométrie algébrique du Bois-Marie 1965-66 SGA 5 dirige par A. Grothendieck avec la collaboration de I. Bucur, C. Houzel, L. Illusie, J.-P. Jouanolou et J. -P. Serre. Cohomologie \(\ell\)-adique et fonctions L. Springer, Cham (1977; Zbl 0345.00011)] on what is nowadays called the formalism of the six operations. Since this survey is about coherent duality we elaborate no further.) derived categories Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Derived categories, triangulated categories, Derived categories and commutative rings, Resolutions; derived functors (category-theoretic aspects) Grothendieck duality made simple | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Grothendieck duality theory on noetherian schemes plays a crucial role in various branches of algebraic and arithmetic geometry, ranging from the study of moduli spaces of algebraic curves up to the arithmetic theory of modular forms. About fourty years ago, \textit{A. Grothendieck} initiated this theory, the main goal of which was to produce a certain ``trace map'' in the cohomology theory of coherent sheaves generalizing the classical Serre duality for smooth schemes over a field. The foundations of what is now called Grothendieck duality theory are worked out in \textit{R. Hartshorne}'s celebrated monograph ``Residues and duality'', Lect. Notes Math. 20 (1966; Zbl 0212.26101) published more than thirty-five years ago.
The foundational framework developed in Hartshorne's book, based on residual complexes and the notion of a dualizing sheaf, makes this duality theory quite computable in terms of differential forms and residues, and this efficient computability has turned out to be extremely useful in many concrete situations in both algebraic and arithmetic geometry.
However, in Hartshorne's construction of Grothendieck duality theory there are some assumptions on compatibility conditions, together with some explications of abstract results, which are not rigorously proven and are, in fact, quite difficult to verify. The hardest compatibiliy condition in the theory, and also one of the most important, is the base change compatibility of the trace map in the case of proper Cohen-Macaulay morphisms with pure relative dimension. For example, this important special case occurs in the study of flat families of semi-stable curves and their moduli. Although there are simpler methods for obtaining duality theorems in the projective Cohen-Macaulay case, which allow to ignore the base change compatibility problem [\textit{A. Altman} and \textit{S. Kleiman}, ``Introduction to Grothendieck duality theory'', Lect. Notes Math. 146 (1970; Zbl 0215.37201)], there remains the fundamental question of whether the hard unproven compatibilities in the foundations elaborated by R. Hartshorne can really be verified.
The aim of the book under review is to give an affirmative answer to this long-standing problem, i.e., to provide rigorous proofs of those compatibility theorems, and to derive some important consequences and examples of this (finally established) abstract theory. In this vein, and as the author himself points out, the present book should therefore be viewed as a companion (and complement) to R. Hartshorne's classical monograph from 1966. Also, it is by no means a logically independent treatment of Grothendieck duality theory from the very beginning, as it actually (and often) appeals to results proven in Hartshorne's standard book.
As to the contents, the book under review consists of five chapters and two appendices.
Chapter 1, the introduction, provides a first overview, some motivation, and the definitions of most of the basic constructions in Hartshorne's approach to Grothendieck duality theory.
Chapter 2, entitled ``Basic compatibilities'', is concerned with verifying several important compatibility conditions underlying Hartshorne's approach. The basic functorial formalism needed to this end is developed and discussed in full detail.
Chapter 3 comes with the title ``Duality foundations'' and is devoted to a thorough discussion of Grothendieck's notion of a residual complex. The material covered here includes, among other topics, dualizing complexes, residual complexes, the general trace map, Grothendieck-Serre duality, dualizing sheaves, Cohen-Macaulay maps, and the general base change theory for dualizing sheaves.
Chapter 4 culminates in the proof of the general duality theorem for proper Cohen-Macaulay maps with pure relative dimension between noetherian schemes admitting a dualizing complex. This result, the proof of which is indeed rather involved and profound, completes Hartshorne's approach to Grothendieck duality theory and, relievingly, justifies it in a concluding manner.
Also, the author compares his result to the (classical) duality theorem of Verdier [in: \textit{J.-L. Verdier}, Algebr. Geom., Bombay Colloq. 1968, 393-408 (1969; Zbl 0202.19902)] in a very enlightening manner.
Chapter 5, simply entitled ``Examples'' makes the abstract derived category duality theorem (theorem 4.3.1. in the present book) somewhat concrete. This is done by recovering from the general theory (developed here) some of the most widely used consequences for duality of higher direct image sheaves and, in the second part, by deducing the classical results of M. Rosenlicht that describe the dualizing sheaf and the trace map on a reduced proper curve over an algebraically closed field in terms of the so-called regular differentials and residues.
Appendix A addresses the topic of the residue symbol utilized in R. Hartshorn's standard text on Grothendieck duality theory. After stating the main results on residues and cohomology with supports, the author provides full and detailed proofs for them.
Appendix B deals particularly with the theory of residues for, and the trace map on smooth curves. The discussion given here makes some of the corresponding results contained in Hartshorne's book more explicit and digestible, on the one hand, and throws the bridge to the related duality theory for Jacobian varieties, on the other hand.
Whenever appropriate in the course of the text, the author compares his approach to the different approach to duality by J. Lipman, which seems to be even more general and far-reaching, on the one hand, but which is even more abstract and ``categorically'' involved than the original approach by Grothendieck and Hartshorne [cf. \textit{L. Alonso}, \textit{A. Jeremias} and \textit{J. Lipman}, ``Studies in duality on noetherian formal schemes and non-noetherian ordinary schemes'', Contemp. Math. 244 (1999; Zbl 0927.00024)], on the other hand.
The book under review provides, altogether, an important and major contribution towards a better understanding of Grothendieck duality theory in its full generality. schemes; sheaves of differentials; dualizing sheaves; residues; duality theorems; base change theorems; Cohen-Macaulay maps; trace maps; algebraic curves; morphisms B. CONRAD, Grothendieck duality and base change. Lecture Notes in Math. 1750, Springer-Verlag (2000). Zbl0992.14001 MR1804902 Schemes and morphisms, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Grothendieck duality and base change | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 algebraic topology, the Steenrod squares of cohomology classes are algebraic operations uniquely defined by a series of axioms. It is noteworthy that the uniqueness does not include a canonical construction of these operations. The main theme of this long paper stems from symplectic geometry. From an algebraic topology point of view, a natural problem in symplectic geometry is how to construct a quantum version of Steenrod squares for quantum cohomology classes of a symplectic manifold. Differing from the classical case, the quantum Steenrod square is not necessarily axiomatically defined, but only a construction which was first suggested by Fukaya based on his Morse homotopy theory.
In the paper under review, the author gives a new construction of quantum Steenrod squares which differs from the one constructed by Fukaya. The primary objects of the study of this paper are the following:
(1) The author shows that the Adem and Cartan relations fail for quantum Steenrod squares and give their quantum deformations called the quantum Cartan relation (Theorem 1.2) and quantum Adem relation (Corollary 1.8), which solves an open problem proposed by \textit{K. Fukaya} [AMS/IP Stud. Adv. Math. 2, 409--440 (1997; Zbl 0891.57035)].
(2) For certain closed monotone symplectic manifolds, the author shows the explicit computations and the solution to the quantum deformation of Steenrod squares, such as toric varieties. As applications, the author considered the quantum Steenrod square for blow-ups of varieties (Section 8). Gromov-Witten theory; quantum cohomology; Steenrod squares; symplectic geometry; symplectic topology; quantum Adem relation Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Steenrod algebra A construction of the quantum Steenrod squares and their algebraic relations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0553.00003.]
The Grothendieck conjecture on linear differential equations concerns operators of the form \(L=\sum^{n}_{i=0}a_ i(x)(d/dx)^ i\) with \(a_ i(x)\in K(x)\), K an algebraic number field. It asserts that if for almost all primes \({\mathfrak p}\) of K the reduced (mod p) equation \(L_ p\) has a full set of solutions in \(\bar K_ p(x)\) \((\bar K_ p=\) residue field at \({\mathfrak p})\), then L should have a full set of algebraic solutions. A consequence of the previous conjecture is the following statement, formulated as a question by \textit{N. Katz} [Invent. Math. 18, 1-118 (1972; Zbl 0278.14004)]: (Log) Let C be a complete non singular curve defined over K and \(\omega\) be a differential on C rational over K with only simple poles and residues in \({\mathbb{Q}}\). Suppose that for almost all primes \({\mathfrak p}\) of K the reduced (mod p) differential \(\omega_ p\) is logarithmic on the reduced curve \(C_ p\). Then an integral multiple \(n\omega\), \(n\in {\mathbb{Z}}\setminus \{0\}\), of \(\omega\) is logarithmic on C.
In this beautiful paper the authors prove statement (Log); their methods seem to offer powerful tools to approach the general Grothendieck conjecture. We list now the 3 basic ingredients of the authors' proof. (1) The assumptions of the Grothendieck conjecture imply strong estimates on the denominators of the coefficients of \(y(x)^ j\), \(j=1,2,...\), if y(x) is a solution of L with algebraic initial values. - (2) The theory of abelian functions allows one to express a solution y of \(dy/y=\omega\) (in Log) in terms of meromorphic functions (of the integrals of the first kind on C) of bounded order of growth. - (3) A striking result on Padé approximations (main theorem 5.2 of the paper) shows that a function y(x) satisfying (1) and (2) must be algebraic.
The authors also give applications to the Lamé equation (with half- integral exponent difference at infinity): this equation satisfies the Grothendieck conjecture, since its essential part is the equation satisfied by a ratio of solutions and this is a first order linear d. e. on an elliptic curve. differential on non singular curve; logarithmic differential; reduction modulo p; Grothendieck conjecture; abelian functions; Padé approximations Chudnovsky, D., Chudnovsky, G.: Applications of Padé approximation to the Grothendieck conjecture on linear differential equations. In: Number Theory, Lecture Notes in Math. vol. 1135, pp. 52-100 (1985) Algebraic functions and function fields in algebraic geometry, \(p\)-adic differential equations, Padé approximation, Analytic theory of abelian varieties; abelian integrals and differentials, Local ground fields in algebraic geometry, Applications of Padé approximations to the Grothendieck conjecture on linear differential 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 We construct a certain solution to the Witten-Dijkgraf-Verlinde-Verlinde equation related to the small quantum cohomology ring of flag variety, and study the \(t\)-deformation of quantum Schubert polynomials corresponding to this solution. Kirillov, Anatol N., {\(t\)}-deformations of quantum {S}chubert polynomials, Funkcialaj Ekvacioj. Serio Internacia, 43, 1, 57-69, (2000) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds \(t\)-deformations of quantum Schubert polynomials. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra. Bernstein-Sato polynomial; \(D\)-module; singularities; multiplier ideals Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Commutative rings of differential operators and their modules, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Rings of differential operators (associative algebraic aspects), Local cohomology and commutative rings, Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Actions of groups on commutative rings; invariant theory Bernstein-Sato polynomials in commutative 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 See the review of the entire collection in [Zbl 1303.01004]. History of algebraic geometry, History of mathematics in the 20th century, Coverings in algebraic geometry On Grothendieck's work on the fundamental 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 In this chapter, we introduce Gröbner bases in a particular non-commutative ring and we show how they can be applied in a geometric context. In Sect. 2, we introduce the Weyl algebra and we present Gröbner bases in this ring. Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Commutative rings of differential operators and their modules, Rings arising from noncommutative algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Gröbner bases in \(D\)-modules: application to Bernstein-Sato 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 Starting from the quantum differential equation associated to a weighted projective space, which was given by Coates, Corti, Lee and Tseng, we construct a Frobenius manifold. We see that the Frobenius manifold coincides with the big quantum cohomology of the weighted projective space. The construction is based on Dubrovin's reconstruction theorem. quantum cohomology; orbifolds; D-modules Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) A construction of a Frobenius manifold from the quantum differential equation of a weighted projective 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 The first part of this paper is an introduction to the periods, the conjecture of \textit{M. Kontsevich} and \textit{D. Zagier} [in: Mathematics unlimited -- 2001 and beyond. Berlin: Springer. 771--808 (2001; Zbl 1039.11002)], the conjecture of Grothendieck on the relation between the transcendence degree of fields of periods and the dimension of the motivic Galois group, the extension of Grothendieck's conjecture by \textit{Y. André} to base field of nonzero transcendence degree [Une introduction aux motifs. Motifs purs, motifs mixtes, périodes. Paris: Société Mathématique de France (2004; Zbl 1060.14001)], and the connection, due to Kontsevich, between these conjectures [\textit{A. Huber} and \textit{S. Müller-Stach}, ``On the relation between Nori motives and Kontsevich periods'', preprint, \url{arXiv:1105.0865}].
The second part of this paper is devoted to the geometric version of the conjecture of Kontsevich-Zagier, of which the author has provided a proof [\textit{J. Ayoub}, Ann. Math. 181, No. 3, 905--992 (2015; \url{doi:10.4007/annals.2015.181.3.2})]. periods; cohomology; Grothendieck conjecture; conjecture of Kontsevich-Zagier; motives; relative motivic Galois group Ayoub, J., Periods and the conjectures of Grothendieck and Kontsevich-Zagier, Eur. math. soc. newsl., 91, 12-18, (March 2014) Motivic cohomology; motivic homotopy theory, de Rham cohomology and algebraic geometry, Galois theory, (Equivariant) Chow groups and rings; motives, Classical real and complex (co)homology in algebraic geometry Periods and the conjectures of Grothendieck and Kontsevich-Zagier | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \textit{V} be a finite-dimensional positively-graded vector space. Let \(b \in V \otimes V\) be a homogeneous element whose rank is \(\dim (V)\). Let \(A = T V /(b)\), the quotient of the tensor algebra \textit{TV} modulo the 2-sided ideal generated by \textit{b}. Let \(\mathsf{gr}(A)\) be the category of finitely presented graded left \textit{A}-modules and \(\mathsf{fdim}(A)\) its full subcategory of finite dimensional modules. Let \(\mathsf{qgr}(A)\) be the quotient category \(\mathsf{gr}(A) / \mathsf{fdim}(A)\). We compute the Grothendieck group \(K_0(\mathsf{qgr}(A))\). In particular, if the reciprocal of the Hilbert series of \textit{A}, which is a polynomial, is irreducible, then \(K_0(\mathsf{qgr}(A)) \cong \mathbb{Z} [\theta] \subset \mathbb{R}\) as ordered abelian groups where \textit{ {\(\theta\)}} is the smallest positive real root of that polynomial. When \(\dim_k (V) = 2\), \(\mathsf{qgr}(A)\) is equivalent to the category of coherent sheaves on the projective line, \(\mathbb{P}^1\), or a stacky \(\mathbb{P}^1\) if \textit{V} is not concentrated in degree 1. If \(\dim_k (V) \geq 3\), results of \textit{D. Piontkovski} [J. Algebra 319, No. 8, 3280--3290 (2008; Zbl 1193.16015)] and \textit{H. Minamoto} [J. Algebra 320, No. 1, 238--252 (2008; Zbl 1165.14009)] suggest that \(\mathsf{qgr}(A)\) behaves as if it is the category of ``coherent sheaves'' on a non-commutative, non-noetherian analogue of \(\mathbb{P}^1\). regular algebras; graded rings; global dimension; Grothendieck group Grothendieck groups, \(K\)-theory, etc., Graded rings and modules (associative rings and algebras), Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.), Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry The Grothendieck group of non-commutative non-Noetherian analogues of \(\mathbb{P}^1\) and regular algebras of global dimension two | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of these lectures is to show that three-point genus zero Gromov-Witten invariants on Grassmannians are equal (or related) to classical triple intersection
numbers on homogeneous spaces of the same Lie type, and to use this to understand the multiplicative structure of their (small) quantum cohomology rings.
This theme will be explained in more detail as the lectures progress. Much of this research is part of a project with Anders S. Buch and Andrew Kresch, presented
in the papers [\textit{A. S. Buch}, Compos. Math. 137, No. 2, 227--235 (2003; Zbl 1050.14053)], [\textit{A. Kresch} and the author, J. Algebr. Geom. 12, No. 4, 777--810 (2003; Zbl 1051.53070); Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083)], and [\textit{A. S. Buch}, \textit{A. Kresch} and the author, J. Am. Math. Soc. 16, No. 4, 901--915 (2003; Zbl 1063.53090)]. I will attempt to give the original references for each result as we discuss the theory. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Gromov-Witten invariants and quantum cohomology of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials As a basis in the space of symmetric functions the Schur functions enjoy a number of special properties such as Littlewood-Richardson and Young rules, Jacobi-Trudy and Cauchy identities. They also play a central role in the representation theory of the symmetric group and cohomology of complex Grassmanians. There exist \(q\)-deformed versions of the latter, namely Hecke algebras and quantum cohomology, respectively. In a previous paper with \textit{A. Lascoux} [Duke Math. J. 116, No.\,1, 103--146 (2003; Zbl 1020.05069)] the authors introduced a new basis of \(k\)-Schur functions that play a similar role in the deformed context and have similar algebraic properties. The original definition was purely combinatorial and in this paper algebraic and geometric applications are worked out.
First, it is proved that \(k\)-Littlewood-Richardson coefficients are the same as the structure constants in the representation ring of Hecke algebra \(H_\infty(q),\,q^n=1\) when \(k=n-1\). By a work of \textit{F. M. Goodman} and \textit{H. Wenzl} [Adv. Math. 82, No.\,2, 244--265 (1990; Zbl 0714.20004)] they coincide with the fusion coefficients for the Wess-Zumino-Witten CFT associated with the affine Lie algebra \(\widehat{\mathfrak{su}}(l)\). Furthermore, as shown by \textit{E. Witten} [Conf. Proc. Lect. Notes Geom. Topol. 4, 357--422 (1995; Zbl 0863.53054)] the fusion algebra of \(\widehat{\mathfrak{u}}(l)\) Wess-Zumino-Witten CFT is isomorphic to the small quantum cohomology of the Grassmanian \(\text{Gr}_{l,n}(\mathbb{C})\). Since \(\widehat{\mathfrak{u}}(l)=\widehat{\mathfrak{su}}(l)\times\widehat{\mathfrak{u}}(1)\) one expects a connection between the \(k\)-Schur functions and quantum cohomology as well and it is described in detail in the paper. In the main result the cohomology structure constants (\(3\)-point Gromov-Witten invariants of \(\text{Gr}_{l,n}(\mathbb{C})\)) are explicitly reduced to certain \(k\)-Littlewood-Richardson coefficients.
In the above applications only a subset of \(k\)-Littlewood-Richardson coefficients is involved. On the other hand, \textit{T. Lam} proved recently [J. Am. Math. Soc. 21, No.\,1, 259--281 (2008; Zbl 1149.05045)] that all \(k\)-Schur functions form the Schubert basis in the homology of the affine Grassmanian of \(GL_{k+1}(\mathbb{C})\). The authors introduce dual \(k\)-Schur functions and show that the expansion coefficients of their product are the same as the coproduct structure constants for the \(k\)-Schur functions. This implies that the dual Schurs form the Schubert basis in the integral cohomology of the affine Grassmanian and coincide with Lam's affine Schur functions. \(k\)-Schur functions; Littlewood-Richardson coefficients; Grassmanian; Schubert basis; quantum cohomology Lapointe, L.; Morse, J., Quantum cohomology and the \textit{k}-Schur basis, Trans. amer. math. soc., 360, 4, 2021-2040, (2008) Symmetric functions and generalizations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology and the \(k\)-Schur basis | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 an absolutely connected scheme over a field \(k\) of characteristic 0.
In this paper, Grothendieck's section conjecture in anabelian geometry is reformulated in terms of the Tannaka category \(FC(X)\) of finite connections that generalize Nori's finite bundles on proper schemes to the case of open smooth schemes. In this formulation, the conjugation classes of sections of \(\pi_1(X,\bar x)\to \Gamma_k\) correspond bijectively to the equivalence classes of neutral fiber functors of \(FC(X)\). In the 1-dimensional case, the section conjecture means that those neutral fiber functors are only those coming from \(k\)-rational points on \(X\) or those from ``packets'' arising from \(k\)-rational tangential points at cusps, i.e., points at infinity. For \(V\subset U\) non-empty smooth curves, the authors conjecture that each fiber functor of \(FC(U)\) extends to one on \(FC(V)\), and under this conjecture, they show that the section conjecture for hyperbolic curves over number fields can be reduced to the special case \(X=\mathbb{P}^1-\{0,1,\infty\}\). arithmetic fundamental group; finite connections; neutral fiber functors Esnault, H., Hai, Ph.H.: Packets in Grothendieck's section conjecture. Adv. Math. \textbf{218}(2), 395-416 (2008) Coverings of curves, fundamental group, Generalizations (algebraic spaces, stacks) Packets in Grothendieck's section 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 Grothendieck point residue is considered in the context of computational complex analysis. A new effective method is proposed for computing Grothendieck point residue mappings and residues. Basic ideas of our approach are the use of Grothendieck local duality and a transformation law for local cohomology classes. A new tool is devised for efficiency to solve the extended ideal membership problems in local rings. The resulting algorithms are described with an example to illustrate them. An extension of the proposed method to parametric cases is also discussed as an application. Grothendieck point residue mappings; local cohomology Residues for several complex variables, Local cohomology of analytic spaces, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Local cohomology and algebraic geometry An effective method for computing Grothendieck point residue mappings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Given a matrix \(f = (f_{ij})_{i,j=1}^\delta\) of noncommutative \(*\)-polynomials \(f_{ij} \in \mathbb{C} \langle x_1, \ldots, x_g, x_1^*, \ldots, x_g \rangle\), one can evaluate it on any tuple of square matrices \(X = (X_1, \ldots, X_g) \in M_n(\mathbb{C})^g\), resulting in another matrix
\[
f(X, X^*) \in M_{n \delta}(\mathbb{C}).
\]
Assuming \(f(0) \neq 0\), the \emph{free invertibility set} is the family of sets \(\mathcal{K}_f = (\mathcal{K}_f(n))_{n \in \mathbb{N}}\), where \(\mathcal{K}_f(n)\) is the connected component of \(0\) of \(\{X \in M_n(\mathbb{C}) \mid \det f(X,X^*) \neq 0\}\).
A free invertibility set is considered convex if each of its levels \(\mathcal{K}_f(n)\) is convex. This paper is concerned with the question of when convexity holds and how to detect it algorithmically. It was already known that convexity holds if and only if \(\mathcal{K}_f\) is a free spectrahedron [\textit{J. W. Helton} and \textit{S. McCullough}, Ann. Math. (2) 176, No. 2, 979--1013 (2012; Zbl 1260.14011)]. The main result of the present paper (Theorem 1.1) sharpens this result by showing that if convexity holds, then a representing linear matrix inequality of minimal size can be constructed in terms of the matrix pencil appearing in the Fornasini-Marchesini realization of \(f\) [\textit{E. Fornasini} and \textit{G. Marchesini}, Math. Syst. Theory 12, 59--72 (1978; Zbl 0392.93034)]. A concrete algorithm for computing such a linear matrix inequality representation is then derived from that (Section 4.2). In combination with a new Nichtsingulärstellensatz for linear matrix polynomials (Theorem 1.5), the authors also obtain an algorithm for testing whether \(\mathcal{K}_f\) is convex to begin with (Section 4.3).
These results are completed by some more concrete results and examples on the free semialgebraic sets associated to single-variable noncommutative \(*\)-polynomials. The appendix provides some extensions to the case of noncommutative rational functions. free semialgebraic set; free invertibility set; free spectrahedron; linear matrix inequality; semidefinite programming Semialgebraic sets and related spaces, Matrix pencils, Convex sets without dimension restrictions (aspects of convex geometry), Rings with involution; Lie, Jordan and other nonassociative structures, Infinite-dimensional and general division rings, Semidefinite programming Noncommutative polynomials describing convex sets | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 recent papers the authors proved the existence of quantum rings on semi-positive symplectic manifolds using analytic techniques [cf. \textit{Y. Ruan} and \textit{G. Tian}, J. Differ. Geom. 42, No. 2, 259--367 (1995; Zbl 0860.58005) and \textit{Y. Ruan}, Duke Math. J. 83, No. 2, 461--500 (1961; Zbl 0864.53032)].
``In this note, we will provide a purely algebro-geometric proof to the existence of quantum cohomology rings for a special class of manifolds already treated there, namely homogeneous manifolds. By using an algebro-geometric approach, we can prove the existence of quantum cohomology of homogeneous varieties defined over any algebraically closed field. This should be useful in enumerative geometry. We believe the approach here can be applied to a larger class of algebraic varieties, such as toric varieties.''
The most important tool in the construction of the quantum rings is an enumerative invariant for which a simple composition law is proved. quantum cohomology rings; homogeneous varieties; enumerative geometry Kollár, J.: Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Springer, Berlin (1996) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Homogeneous spaces and generalizations, Geometric quantization, Enumerative problems (combinatorial problems) in algebraic geometry, (Co)homology theory in algebraic geometry The quantum cohomology of homogeneous 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 exhibits a quantum algorithm for determining the zeta function of a genus \(g\) curve \(C\) over a finite field \(k=\mathbb{F}_q\), which is polynomial time in \(g\) and \(\log q\). The best current classical algorithm to obtain this result is only polynomial in \(g\) and in \(\log(q)\) for a fixed characteristic \(p\).
The algorithm is based on a result of \textit{J. Watrous} [Proc. 33rd annual ACM symposium on theory of computing (STOC 2001), Hersonissos, Crete, Greece, July 6--8, 2001. (2001; Zbl 1074.68500)] which gives a quantum algorithm to compute the order of a group knowing its Monte Carlo black box group presentation. It relies also on some effective elementary algebraic geometry results (mainly Riemann-Roch theorem) to produce provably random elements of the Jacobian -- which will turn out to be a generator set-- and a final trick to recover the Weil polynomial in terms of the order of the group of rational points of the Jacobian of \(C\) over \(2g\) extensions.
An interesting question raised in the article is to know if one could do less than these \(2g\) extensions. Indeed, this question is related to the recovering of a (reciprocal, even degree) polynomial \(P\) of degree \(d\) from enough of its cyclic resultants. A conjecture of Sturmfels and Zworski assures that the first \(d/2+1\) resultants should suffice for \(P\) generic. quantum computation; class groups; algebraic curves; finite fields; cyclic resultants Kedlaya K.S.: Quantum computation of zeta functions of curves. Comput. Complex. 15(1), 1--19 (2006) Computational aspects of algebraic curves, Curves over finite and local fields, Quantum computation, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Quantum computation of zeta functions of 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 Grothendieck-Witt spectra represent higher Grothendieck-Witt groups and higher Hermitian \(K\)-theory in particular. A description of the Grothendieck-Witt spectrum of a finite dimensional projective bundle \(\mathbb{P}(\mathcal{E})\) over a base scheme \(X\) is given in terms of the Grothendieck-Witt spectra of the base, using the dg category of strictly perfect complexes, provided that \(X\) is a scheme over \(\operatorname{Spec} \mathbb{Z} [1/2]\) and satisfies the resolution property, e.g. if \(X\) has an ample family of line bundles. Hermitian \(K\)-theory, relations with \(K\)-theory of rings, Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), \(K\)-theory of schemes The projective bundle formula for Grothendieck-Witt spectra | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 interesting paper, the Ringel-Hall algebra (and some variants of it) is studied for the category \(\text{coh}(X)\) of coherent sheaves over \(X\), where \(X\) is a smooth projective curve over a finite field \(\mathbb{F}_q\). More precisely, the complex valued functions on isomorphism classes of objects of \(\text{coh}(X)\) carry an algebra structure, which is analogous to the classical Hall algebra of a discrete valuation ring [see \textit{I. G. Macdonald}, Symmetric functions and Hall polynomials. 2nd ed. (Oxford, Clarendon Press) (1998; Zbl 0899.05068)] and to Ringel's Hall algebras for module categories over hereditary algebras. Among those functions are `automorphic forms', i.e. functions on vector bundles, and Eisenstein series. The main results of the paper give precise relations for certain generating functions which are quite similar to relations in quantum affine algebras.
An important (and motivating) example is that of the projective line \(\mathbb{P}^1\) over \(\mathbb{F}_q\). A special case of \textit{A. A. Beilinson}'s theorem [Funct. Anal. Appl. 12, 214-216 (1979); translation from Funkts. Anal. Prilozh. 12, No. 3, 68-69 (1978; Zbl 0424.14003)] asserts that \(\text{coh} (\mathbb{P}^1)\) is derived equivalent to the module category of a finite dimensional hereditary algebra (of Dynkin type \(A_1^{(1)})\), the Kronecker algebra. \textit{J. A. Green}'s theorem [Invent. Math. 120, 361-377 (1995; Zbl 0836.16021)] which extends Ringel's theorem [\textit{C. M. Ringel}, Invent. Math. (1990; Zbl 0735.16009)] gives an isomorphism between the composition algebra (a subalgebra of the Hall algebra) and the positive part \(U_q(\widehat {\mathfrak n}_+)\) of the quantum affine algebra associated with the same Dynkin diagram \(A_1^{(1)}\). Kapranov's `Ringel-Hall algebra' construction for \(\text{coh} (\mathbb{P}^1)\) produces a different `nilpotent' subalgebra of the same quantum affine algebra.
For general \(X\) as above, \(\text{coh} (X)\) still is a hereditary category and much of the technology developed by Ringel and Green for module categories of hereditary algebras still works.
The new algebras produced in this way relate quantum affine algebras and automorphic forms on \(X\) by many striking analogies. These analogies are collected in a table in section 5.3 (which also poses the open problem to fill in the remaining entries of the table by new analogies). For example, positive roots correspond to cusp eigenforms, and root space decomposition of \(U_q(\widehat{\mathfrak n}_+)\) corresponds to spectral decomposition of the algebra of automorphic forms.
To pass from the `positive part' \(U_q(\widehat{\mathfrak n}_+)\) to the full quantum affine algebra \(U_q (\widehat {\mathfrak g})\), one may apply the technique of forming the Drinfeld double. This has been done by \textit{Jie Xiao} [J. Algebra 190, 100-144 (1997; Zbl 0874.16026)]. The present paper complements this approach by adding the Heisenberg double as an intermediate step. Finally, the main result (theorem 6.7) states several identities between certain generating functions which hold true in the Drinfeld double of the Hall algebra of automorphic forms. These identities are very similar to relations valid in quantum affine algebras, in particular if one defines these algebras by Drinfeld's loop realization. curves over finite fields; Ringel-Hall algebra; coherent sheaves; quantum affine algebras; automorphic forms; Drinfeld double; Heisenberg double M. Kapranov, ''Eisenstein series and quantum affine algebras,'' J. Math. Sci. (New York), vol. 84, iss. 5, pp. 1311-1360, 1997. Relationship to Lie algebras and finite simple groups, Representations of quivers and partially ordered sets, , Quantum groups (quantized enveloping algebras) and related deformations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Eisenstein series and 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]. History of algebraic geometry, History of mathematics in the 20th century Grothendieck and scheme theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck's SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure tools because one elementary topos of sets with infinity is already this strong. Grothendieck universe Axiomatics of classical set theory and its fragments, Foundations of classical theories (including reverse mathematics), Foundations of algebraic geometry, Foundations, relations to logic and deductive systems, Topoi, Derived categories, triangulated categories The large structures of Grothendieck founded on finite-order arithmetic | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 fix a ground field \(F\) and a central simple algebra \(D\) over \(F\); we put \(r= \deg D\) and denote by \(X\) the Severi-Brauer variety \(\text{SB} (D)\) corresponding to \(D\). Moreover, we put \(X^n=\text{SB} (M_n(D))\), where \(M_n(D)\) is the \(F\)-algebra of \((n \times n)\)-matrices over \(D\). -- In the first part of this paper, we decompose the Grothendieck Chow-motif \(\widetilde {X^n}\) of the variety \(X^n\) into the direct sum \(\bigoplus^{n-1}_{i=0} \widetilde X(ir)\) of twisted motifs of \(X\); note that in the trivial case where \(D=F\) this is a well-known decomposition of the motif of a projective space. Consequently, in any ``geometric'' cohomology theory \(H\) we have \(H(X^n)= \bigoplus^{n-1}_{i=0} H(X)\), with some twisting of gradings in the graded case. A list of examples is given in corollary 1.3.2.
Thus, computation of cohomology groups of Severi-Brauer varieties reduces to the nonsplit case, i.e., to the case where \(D\) is a division algebra. However, no further reduction can be obtained on the motivic level; as shown in the second part of the paper, if \(D\) is a division algebra, then the motif \(\widetilde X\) is indecomposable as an object of the category of motifs. Chow-motif of the Severi-Brauer variety; cohomology groups Karpenko, N.A.: Grothendieck Chow motives of Severi--Brauer varieties (Russian). Algebra Anal. 7(4), 196--213 (1995) (transl. in St. Petersbg. Math. J. 7(4), 649--661 (1996)) Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes, Generalizations (algebraic spaces, stacks), Parametrization (Chow and Hilbert schemes) The Grothendieck Chow-motifs of Severi-Brauer 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 New results on equivariant cohomology of a real algebraic variety are proved; in particular, the first spectral sequence is computed. These results are applied to prove analogues of the Harnack-Thom inequality and to study relations between characteristic classes. equivariant cohomology; real algebraic variety; spectral sequence; characteristic classes V. A. Krasnov, ''On the equivariant Grothendieck cohomology of a real algebraic variety and its application,'' Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 58 (1994), no. 3, 36--52. Topology of real algebraic varieties, Spectral sequences, hypercohomology, Étale and other Grothendieck topologies and (co)homologies, Characteristic classes and numbers in differential topology On equivariant Grothendieck cohomology of a real algebraic variety, and its applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 conclusion of this paper is that the theory of 2-dimensional topological gravity has a homotopy-theoretic formulation analogous to the interpretation of quantum ordinary cohomology as a topological field theory. Witten's algebra of generalized Miller-Morita-Mumford classes for surface bundles is the coefficient ring of a cohomology theory, and topological gravity becomes a functor which assigns invariants to families of curves, just as a classical topological field theory assigns invariants to individual curves.''
The author assumes the smoothness of the Deligne-Mumford-Knudsen moduli space \(\overline{M}_{g,n}\) of stable curves of arithmetic genus \(g\) with \(n\) marked points, and certain maps between them and investigates the consequences. Let \(V\) be a simply connected smooth projective variety, and
\[
M\text{U}_\Lambda(X)= M\text{U}^*(X) \widehat{\otimes} \mathbb{Q}[H_2(V;\mathbb{Z})] [z,z^{-1}].
\]
The main point is the construction of certain maps
\[
\tau_{g,n}^V: \overline{M}_{g,n}^+\to F(M\text{U}_\Lambda,V^n),
\]
compatible with Knudsen's glueing morphisms. Indeed, these maps define a topological theory with values in the category of \(M\text{U}_\Lambda\)-module spectra, and hence a theory of topological gravity assigning invariants to flat families of stable curves using the universal property of the moduli spaces.
The author also states that the paper was motivated by a desire to understood topological gravity and quantum cohomology from the point of view of the Floer homotopy theory of Cohen-Jones-Segal. J. Morava, ''Quantum generalized cohomology'' in Operads: Proceedings of Renaissance Conferences (Hartford, Conn./Luminy, France, 1995) , Contemp. Math. 202 , Amer. Math. Soc., Providence, 1997, 407--419. Generalized (extraordinary) homology and cohomology theories in algebraic topology, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Other homology theories in algebraic topology, Families, moduli of curves (algebraic) Quantum generalized 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 \(K\) be a field and \(F,G\) two vector spaces over \(K\) of dimensions \(m,n\) respectively. Consider the affine space \(X= \Hom_K(F,G)\) of linear maps from \(F\) to \(G\). We identify \(X\) with the space \(F^* \otimes G\). The coordinate ring \(A\) of \(X\) is naturally identified with the symmetric algebra \(A=\text{Sym}(F\otimes G^*)\). Under this identification, for fixed bases \(\{f_1,\dots, f_m\}\), \(\{g_1, \dots,g_n\}\) of \(F,G\) respectively, the tensor \(f_i\otimes g^*_j\) corresponds to the \((j,i)\)-th entry function \(t_{i,j}\) on \(X\). For each \(r\) with \(0\leq r\leq\min(m,n)\) we denote by \(X_r\) the determinantal variety of maps of rank \(\leq r\): \(X_r=\{\varphi: F\to G\mid\text{rank} \varphi\leq r\}\). We denote by \(A_r\) the coordinate ring of \(X_r\).
The objective of this paper is the investigation of natural modules with support in \(X_r\). By a natural module we mean the graded \(A_r\)-module with a \(GL(F) \times GL(G)\) action compatible with the module structure. We investigate the category \({\mathcal C}_r(F,G)\) of graded \(A_r\)-modules with the rational \(GL(F) \times GL(G)\) action compatible with the module structure, and equivariant degree 0 maps. We denote by \(K_0'(A_r)\) the Grothendieck group of the category \({\mathcal C}_r(F,G)\).
The main result is a complete description of \(K_0' (A_r)\). We provide three families of modules, each of which gives the generators of \(K_0'(A)\), with no relations. The three families come from three natural desingularizations of the determinantal variety \(X_r\) as the push downs of certain vector bundles on these desingularizations. equivariant modules; Grothendieck group; desingularizations of the determinantal variety; push downs Weyman, J.: The Grothendieck group of \(GL(F){\times} GL(G)\)-equivariant modules over the coordinate ring of determinantal varieties. Colloq. math. 76, 243-263 (1998) Grothendieck groups, \(K\)-theory and commutative rings, Linkage, complete intersections and determinantal ideals, Cohen-Macaulay modules, \(K_0\) of other rings, Ext and Tor, generalizations, Künneth formula (category-theoretic aspects), Determinantal varieties, Applications of methods of algebraic \(K\)-theory in algebraic geometry The Grothendieck group of \(GL(F)\times GL(G)\)-equivariant modules over the coordinate ring of determinantal 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 Let \(R\) be a discrete valuation ring with fraction field \(K\) and let \(A_K\) be an abelian variety with Néron model \(A_R\) and component group \(\phi_{A_R}\). Let \(A^t_K\) be the dual abelian variety with Néron model \(A^t_R\) and component group \(\phi_{A^t_R}\). \textit{A. Grothendieck} [``Groupes de monodromie en géométrie algébrique'' (SGA 7I), Lect. Notes Math. 288 (1972; Zbl 0237.00013)] constructed a pairing \(\phi_{A_R}\times\phi_{A^t_R}\rightarrow\mathbb{Q}/\mathbb{Z}\) that represents the obstruction to extending the Poincaré bundle to a biextension of \(A_R\times A^t_R\) by \(\mathbb{G}_m\), which conjecturally is a perfect pairing. While this holds in a variety of situations, notably the case of semi-stable reduction, the authors' main result is that it may fail for non-perfect residue field.
The authors' main device is the following: Let \(K'/K\) be a finite Galois extension of degree \(n\) with purely inseparable residue extension. Let \(X_K=\text{Res}_{K'/K} A_{K'}\), where \(A_{K'} = A_K\times_K K'\) is the base change of \(A_K\) to \(K'\). Let \(X^t_K =\text{Res}_{K'/K} A^t_{K'}\). Assuming that Grothendieck's conjecture is true for \(A_{K'}\) the authors obtain information on that pairing for \(X_K\) and \(X^t_K\) (theorem 2.1). In particular, they show (corollary 2.2) that, if \(\text{char}(k)=p\), \(K'/K\) is unramified, \(n = p^r\) and the \(p\)-part of \(\phi_{A^t_{R'}}\) is not trivial, Grothendieck's conjecture fails for \(X_K\) and \(X^t_K\). Weil restriction; Néron model; abelian variety; Grothendieck's duality conjecture; obstruction to extending the Poincaré bundle Néron, A.: Modèles Minimaux des Variétés Abéliennes sur les Corps Locaux et Globaux. Publications Mathématiques de l'IHÉS, vol. 21, Institut des Hautes Études Scientifiques, Paris (1964) Algebraic theory of abelian varieties, Other nonalgebraically closed ground fields in algebraic geometry Weil restriction and Grothendieck's duality 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 Let \(K\) be a discretely valued field, and let \(A_K\) be an abelian variety over \(K\). We denote by \(A\) its Néron model and by \(A'\) the Néron model of the dual abelian variety \(A_K'\). Let \(\varphi_A\) and \(\varphi_{A'}\) be the groups of connected components of the special fibres of \(A\) and \(A'\). Then there is a canonical pairing \((*): \varphi_A \times \varphi_{A'} \to\mathbb{Q}/ \mathbb{Z}\), which appears as obstruction of extending the Poincaré biextension over \(A_K \times A_K'\) to \(A \times A'\). A conjecture of Grothendieck states that this pairing is perfect. If \(A\) has semistable reduction, we can use the monodromy pairing to define another pairing \((**): \varphi_A \times \varphi_{A'} \to\mathbb{Q}/ \mathbb{Z}\), which is easily seen to be perfect. Following a request of \textit{A. Grothendieck} [in Sém. Géométrie Algébr. 1967-1969, SGA 7, I, No. 9, Lect. Notes Math. 288, 313-523 (1972; Zbl 0248.14006); 11.4] we show that both pairings are compatible, which proves Grothendieck's conjecture in the semistable reduction case. We use the rigid geometry of the Raynaud extensions associated to \(A_K\) and \(A_K'\) and the theory of formal Néron models of rigid analytic groups. valued field; abelian variety; semistable reduction; rigid geometry; formal Néron models Werner, A.: On Grothendieck's pairing of component groups in the semistable reduction case, J. reine angew. Math. 486, 205-215 (1997) Arithmetic ground fields for abelian varieties, Local ground fields in algebraic geometry, Valued fields On Grothendieck's pairing of component groups in the semistable reduction 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 Let \(K_{0}(Var_k)\) denote the Grothendieck ring of \(k\)-varieties over an algebraically closed field \(k\). Larsen and Lunts asked if two \(k\)-varieties having the same class in \(K_{0}(Var_k)\) are piecewise isomorphic. Gromov asked if a birational self-map of a \(k\)-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group and that the associated graded ring of the Grothendieck ring is the monoid ring \(\mathbb{Z}[\mathfrak{B}]\) where \(\mathfrak{B}\) denotes the multiplicative monoid of birational equivalence classes of irreducible \(k\)-varieties. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Rational and birational maps On 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 Let \(X\) be an irreducible variety of dimension \(n\). In the first part of this paper, the authors extend the intersection theory for subspaces of rational functions on a variety \(X\) over \(k=\mathbb{C}\) to an arbitrary algebraically closed field \(k\).
Let \(k(X)\) be the field of rational functions and \(K(X)\) the set of all non-zero finite dimensional vector subspaces of \(k(X)\). Given \(L,M \in K(X)\), define the product \(LM\) to be the subspace spanned by all the product \(fg\), \(f\in L\) and \(g\in M.\) \(K(X)\) is a commutative semigroup with this product. Using the intersection product in the Chow ring of a product of projective spaces, the authors prove two theorems.
{Theorem 1.} Let \(L_1,\dots, L_n \in K(X)\) and \(Z\subset X\) be a closed subvariety of \(X\) containing the poles of all rational functions from the \(L_i\) and all the points \(x\) at which all functions from some subspace \(L_i\) vanish. Then there is a non-empty Zariski open subset \(U\subset L_1\times\dots\times L_n\) such that for any \((f_1,\dots, f_n)\in U\), the number of solutions
\[
\{ x\in X\backslash Z, f_1(x)=\dots=f_n(x)=0 \}
\]
is finite and is independent of the choice of \(Z\) and \((f_1,\dots, f_n)\in U\). The intersection index of the subspaces \(L_i\), denoted by \([L_1,\dots, L_n]\), is the number of solutions \(\{ x\in X\backslash Z, f_1(x)=\dots=f_n(x)=0 \}\).
{ Theorem 2}. Let \(L_1', L_1'', L_2\dots, L_n \in K(X)\) and \(L_1=L_1'L_1''\). Then
\[
[L_1,\dots, L_n]= [L_1', L_2,\dots, L_n]+ [L_1'', L_2,\dots, L_n].
\]
In the second part of the paper, the authors prove that the following two groups are isomorphic.
A projective birational model is a proper birational map \(\pi\) from a projective variety \(X_{\pi}\) to \(X\). The Riemann-Zariski space space
\[
{\mathfrak{X}}=\lim_{\overleftarrow{\pi}} X_{\pi},
\]
where the limit is taken over all the birational models of \(X\). Shokurov defined the group of Cartier \(b-\)divisors as
\[
\text{CDiv}({\mathfrak{X}})=\lim_{\overrightarrow{\pi}} \text{CDiv}(X_{\pi}),
\]
where \(\text{CDiv}(X_{\pi})\) denotes the group of Cartier divisors on the variety \(X_{\pi}\) and limit is taken with respect to the pull back maps
\[
\text{CDiv}(X_{\pi})\rightarrow \text{CDiv}(X_{\pi'})
\]
when \((X_{\pi'}, \pi')\) dominates \((X_{\pi}, \pi)\).
The authors proves that the group of Cartier \(b\)-divisors \(\text{CDiv}({\mathfrak{X}})\) is isomorphic to the Grothendieck group \(G(X)\) of \(K(X)\) and the isomorphism preserves the intersection index. intersection number; Cartier divisor; Cartier \(b\)-divisor; Grothendieck group Divisors, linear systems, invertible sheaves, Rational and birational maps, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Note on the Grothendieck group of subspaces of rational functions and Shokurov's Cartier \(b\)-divisors | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 0717.00009.]
Let \(\emptyset \neq X\) be a reduced complex analytic space with \({\mathcal S}\) a Whitney stratification on X, where \(X_ i\) denotes the union of strata of dimension \(\leq i\). The rectified homotopical depth \(rhd_{{\mathcal S}}(X)\) of X is \(\geq n\) if, for any i and any point \(x\in X_ i\setminus X_{i-1}\), there exists a fundamental system \((U_{\alpha})\) of neighbourhoods of x in X such that, for any \(\alpha\) the pair \((U_{\alpha},U_{\alpha}\setminus X_ i)\) is (n-1-i)-connected. Then \(rhd_{{\mathcal S}}(X)\) is defined as the maximum of the integers n such that \(rhd_{{\mathcal S}}(X)\geq n\). This definition is shown to be equivalent to the ones of \textit{A. Grothendieck} [see Séminaire de géométrie algébrique, SGA 2 (1962; Zbl 0159.504); Exposé XIII, p. 27, definition 2; see also Adv. Stud. Pure Math. 2 (1968; Zbl 0197.472)]. Replacing the connectedness condition by the vanishing of \(H_ k(U_{\alpha},U_{\alpha}\setminus X_ i,{\mathbb{Z}})\) for any \(k<n-i\) the authors define the rectified homological depth of X. - A. Grothendieck conjectured that the notion of rectified homotopical (resp. homological) depth gives the right level of comparison for the homotopy (resp. homology) type between X and a hyperplane section, as stated in theorems of Lefschetz type for nonsingular varieties.
The authors give positive answers to Grothendieck's conjectures. The proofs become possible because of a handy formulation of the notion of rectified homotopical depth using Whitney stratifications shown in the paper. More general there are Lefschetz type theorems for open varieties replacing the hyperplane section by a good neighbourhood of a hyperplane section as has been done by \textit{P. Deligne} [see Sémin. Bourbaki, 32e année, Vol. 1979/80, Exposé 543, Lect. Notes Math. 842, 1-10 (1981; Zbl 0478.14008)]. Lefschetz theorems; complex analytic space; Whitney stratification; rectified homotopical depth; hyperplane section H. A. Hamm and Lê Dũng Tráng, ''Rectified Homotopical Depth and Grothendieck Conjectures,'' in Grothendieck Festschrift (Birkhäuser, Boston, 1990), Vol. 2, Prog. Math. 87, pp. 311--351. Homotopy theory and fundamental groups in algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.), Complex spaces, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Rectified homotopical depth and Grothendieck conjectures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give sufficient cohomological criteria for the classes of given varieties over a field \(k\) to be algebraically independent in the Grothendieck ring of varieties over \(k\) and construct some examples. Varieties and morphisms Algebraic independence 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 Let \(k\) be a field. The Grothendieck ring \(K_0(\text{Var}_k)\) of varieties over \(k\) is the abelian group generated by the isomorphism classes \([X]\) of separated \(k\)-schemes of finite type over \(k\) subject to the relations \([X] = [Y] + [X\backslash Y]\) for \(Y \subset X\) a closed subscheme; multiplication is given by \([X_1] \cdot [X_2] = [X_1 \times X_2]\).
The author gives sufficient conditions for the classes of given varieties over \(k\) to be algebraically independent in \(K_0(\text{Var}_k)\). To this end, he constructs refined versions (so-called motivic measures) of the natural ring homomorphism from \(K_0(\text{Var}_k)\) to the ring of virtual \({\mathbb Q}_\ell\)-representations of \(\text{Gal}(\bar{k}| k)\) given by \(\ell\)-adic cohomology with compact support. The algebraic independence of virtual \(\ell\)-adic representations is then reduced to a problem about representations of (possibly non-connected) reductive groups or is approached using a lemma of Skolem.
If \(k\) is a number field the author finally shows that the classes in \(K_0(\text{Var}_k)\) of elliptic curves \(E_i\) over \(k\) are algebraically independent provided the \(E_i\) are pairwise non-isogenous and satisfy \(\text{End}_{\bar{k}}(\bar{E_i}) = {\mathbb Z}\). If \(k\) is a finite field, the author constructs an infinite sequence of proper smooth and geometrically connected curves over \(k\) whose classes in \(K_0(\text{Var}_k)\) are algebraically independent; furthermore he shows that the class of a single variety is algebraically dependent if and only if its dimension is 0; this leads to the result that \(K_0(\text{Var}_k)\) contains infinitely many zero divisors. Grothendieck ring of varieties; motivic measure; \(\ell\)-adic Galois representations; weight filtration; zeta function; non-isogenous elliptic curves Naumann N.: Algebraic independence in the Grothendieck ring of varieties. Trans. Am. Math. Soc. 359(4), 1653--1683 (2007) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Varieties and morphisms, Motivic cohomology; motivic homotopy theory, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Varieties over finite and local fields Algebraic independence 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 Let \(\pi: Gr_k(A)\to X\) be the Grassmannization of the vector bundle \(A\to X\) over a smooth base space \(X\). The push-forward map \(\pi_*\) along \(\pi\) in cohomology, or in \(K\)-theory, plays an important role in equivariant algebraic geometry. One of the many formulas that expresses \(\pi_*\) is the so-called (equivariant) localization formula.
Partition functions of integrable models are fundamental objects in statistical physics. A certain partition functions of the five-vertex model may be regarded as an element of \(K(Gr_k(A))\), the K-algebra of \(Gr_k(A)\). Another partition function may be regarded as an element of \(K(X)\). The authors show that -- in these interpretations -- \(\pi_*\) of the first one is the second one.
Halfway between the objects in geometry and those in integrable systems is a commutation relation: an identity -- due to Shigechi and Uchiyama -- involving monodromy matrix elements. While this identity is expressed in non-commutative variables (hence cannot directly be related to the commutative geometric objects), it has explicit similarity to the localization formula for \(\pi_*\).
The pushforward formula \(\pi_*\) behaves nicely for various versions of Grothedieck polynomials. The authors present such Grothendieck polynomial interpretations of the their partition functions, and derive ``skew generalizations'' of some known important identities for Grothendieck polynomials. push-forward map along Grassmann bundles; Grothendieck polynomial; five-vertex model; partition function Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.), Classical problems, Schubert calculus Integrable models and \(K\)-theoretic pushforward of Grothendieck 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 This paper appeared earlier in Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020). rational curves; stable algebraic curves; quantum cohomology of a complex projective algebraic manifold; cohomological field theory; variation of Hodge structures; mirror dual manifold; Gromov-Witten classes; operads for moduli spaces of stable curves Kontsevich (M.), Mani (Y.).â Gromov-Witten classes, quantum cohomology, and enumerative geometry. In âMirror Symmetry II,â AMS/IP Stud. Adv. Math., v. 1, AMS, Providence, RI, p. 607-653 (1997). Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Fano varieties, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Enumerative problems (combinatorial problems) in algebraic geometry, Families, moduli of curves (algebraic), Applications of manifolds of mappings to the sciences Gromov-Witten classes, quantum cohomology, and enumerative 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 mathematical aspects of topological quantum field theory have recently led to the concept of quantum cohomology of a complex projective algebraic manifold. Basically, the quantum cohomology of such a manifold \(V\) is a formal deformation of its cohomology ring \(H^* (V, \mathbb{Q})\), whose parameters are the coordinates on the space \(H^* (V, \mathbb{Q})\) itself, and this object is used to construct what is called a cohomological field theory (CohFT). The physical concept of free energy (or potential) in such a CohFT corresponds mathematically to a formal series \(\Phi^V\) whose coefficients are given by the number of parametrized rational curves in \(V\) satisfying certain incidence conditions.
Under the assumption that such a generating function (potential) \(\Phi^V\) really exists, i.e., that it can be meaningfully defined with respect to the numerical requirements concerning its coefficients, physicists have predicted its analytic behavior with such a precision that its uniqueness would be a consequence, just as the exact values of some numerical invariants of the space of parametrized pointed curves lying in \(V\). Some results on this conjectural, highly challenging subject have been obtained in the special case of a Calabi-Yau manifold \(V\), where \(\Phi^V\) would describe a variation of Hodge structures of the mirror dual manifold.
In the present paper, the authors develop a formalism of defining (axiomatically) and investigating Gromov-Witten classes. Those appear as a collection of linear maps
\[
I^V_{g,n, \beta} : H^* (V, \mathbb{Q})^{\otimes n} \to H^* (\overline M_{g,n}, \mathbb{Q})
\]
with range in the cohomology ring of the moduli space \(\overline M_{g,n}\) of stable genus-\(g\) curves with \(n\) marked points and depending on integers \(g \geq 0\), \(n \geq 3 - 2g\), and homology classes \(\beta \in H_2 (V, \mathbb{Z}) \). The postulation of a series of formal and geometric properties (or axioms) for these Gromov-Witten classes, together with the geometric intuition behind them, is an elaboration of Witten's treatment [cf. \textit{E. Witten}, J. Differ. Geom. Suppl. 1, 243-310 (1991; Zbl 0808.32023)] and allows, in the sequel, to establish their existence formally, at least for some classes of Fano varieties \(V\) and \(g = 0\). This is then used to construct an appropriate potential function \(\Phi^V\), basically with the aid of zero-codimensional Gromov-Witten classes of genus zero.
Two reconstruction theorems show that the Gromov-Witten classes can be calculated recursively, at least in certain particular situations, and that the codimension-zero classes regulate the corresponding compatibility conditions. It turns out that the potential function \(\Phi^V\), defined by the Gromov-Witten classes and the linear superspace structure of the cohomology ring \(H^*(V,\mathbb{Q})\), encodes an extremely rich geometric structure on its convergence domain in \(H^*(V,\mathbb{Q})\), including the Vafa quantum cohomology rings of the Fano variety \(V\). A special section is devoted to the discussion of concrete examples, always assuming that the relevant Gromov-Witten classes exist and the potential function can be calculated along the previously described procedure. This leads to precise statements on its coefficients, i.e., to enumerative results on the spaces of rational curves in \(V\), and -- via the reconstruction theorems -- to new (conjectural) number-theoretic identities.
The concluding part of the paper provides two possible definitions of a cohomological field theory. One of them is directly based on the axiomatics of the authors' Gromov-Witten classes, and the other one is related to the so-called operads for moduli spaces of stable curves. The formalism developed here is applied to the calculation of the cohomology of the moduli space of curves of genus zero. Using a related result of \textit{S. Keel} [Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992; Zbl 0768.14002)], the authors derive a complete system of linear relations between the homology classes of the boundary strata of these moduli spaces.
Altogether, the present paper is a highly valuable contribution towards the understanding of correlation functions of topological sigma-models from a rigorous algebro-geometric point of view. rational curves; stable algebraic curves; quantum cohomology of a complex projective algebraic manifold; cohomological field theory; variation of Hodge structures; mirror dual manifold; Gromov-Witten classes; operads for moduli spaces of stable curves M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525-562. Fano varieties, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Applications of manifolds of mappings to the sciences, Enumerative problems (combinatorial problems) in algebraic geometry, Families, moduli of curves (algebraic) Gromov-Witten classes, quantum cohomology, and enumerative 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 Suppose some complex veector bundles are given over a manifold \(M\), together with some (sufficiently generic) vector bundle maps among them. Over a point in \(M\) then one has a quiver: some vector spaces with linear maps among them. Quivers may degenerate -- e.g. dimensions of intersections of images, kernels may drop, and more general degenerations also happen. A quiver degeneracy locus is the set a points in \(M\) over which the quiver degenerates in a particular way.
It is known that the fundamental class of quiver degeneracy loci can be expressed by a universal polynomial (the quiver polynomial) in the charactersitic classes of the bundles involved, both in cohomology and in \(K\)-theory. The paper under review presents such universal polynomials in \(K\)-theory if the quiver is of Dynkin type.
The formula presented is a non-conventional generating function (named Iterated Residue generating function, pioneered by Bérczi-Szenes, Kazarian, Feher-Rimányi). Advantages of the presented formula for quiver loci include that stabilization properties are explicit, and the expansion in terms of Grothendieck polynomials is well motivated.
The paper ends with interesting comments comparing \(K\)-theory and cohomology Iterated Residue formulas, as well as remarks on Buch's positivity conjecture on the coefficients of the Grothendieck polynomial expansion. quiver polynomials; Grothendieck polynomials; iterated residues; equivariant K theory Allman, Justin, Grothendieck classes of quiver cycles as iterated residues, Michigan Math. J., 63, 4, 865-888, (2014) Grassmannians, Schubert varieties, flag manifolds, Equivariant \(K\)-theory, Symmetric functions and generalizations, Representations of quivers and partially ordered sets Grothendieck classes of quiver cycles as iterated residues | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For any triple \((i,a,\mu)\) consisting of a vertex \(i\) in a quiver \(Q\), a positive integer \(a\), and a dominant \(\operatorname{GL}_a\)-weight \(\mu \), we define a quiver current \(H^{(i,a)}_\mu\) acting on the tensor power \(\Lambda^{\!Q}\) of symmetric functions over the vertices of~\(Q\). These provide a quiver generalization of parabolic Garsia-Jing creation operators in the theory of Hall-Littlewood symmetric functions. For a triple \((i=(i_1,\ldots,i_m),a=(a_1,\ldots,a_m),(\mu(1),\ldots,\mu(m)))\) of sequences of such data, we define the quiver Hall-Littlewood function \(H^{\mathbf{i},\mathbf{a}}_{\mu(\,\cdot\,)}\) as the result of acting on \(1\in\Lambda^{\!Q}\) by the corresponding sequence of quiver currents. The quiver Kostka-Shoji polynomials are the expansion coefficients of \(H^{\mathbf{i},\mathbf{a}}_{\mu(\,\cdot\,)}\) in the tensor Schur basis. These polynomials include the Kostka-Foulkes polynomials and parabolic Kostka polynomials (Jordan quiver) and the Kostka-Shoji polynomials (cyclic quiver) as special cases. We show that the quiver Kostka-Shoji polynomials are graded multiplicities in the equivariant Euler characteristic of a vector bundle (determined by \(\mu(\,\cdot\,))\) on Lusztig's convolution diagram determined by the sequences \(\mathbf{i} \)~and~\( \mathbf{a} \). For certain compositions of currents we conjecture higher cohomology vanishing of the associated vector bundle on Lusztig's convolution diagram. For quivers with no branching, we propose an explicit positive formula for the quiver Kostka-Shoji polynomials in terms of catabolizable multitableaux. We also relate our constructions to \(K\)-theoretic Hall algebras, by realizing the quiver Kostka-Shoji polynomials as natural structure constants and showing that the quiver currents provide a symmetric function lifting of the corresponding shuffle product. In the case of a cyclic quiver, we explain how the quiver currents arise in Saito's vertex representation of the quantum toroidal algebra of type \(\mathfrak{sl}_r\). quiver Hall-Littlewood functions; Kostka-Shoji polynomials Representations of quivers and partially ordered sets, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Representation theory for linear algebraic groups Quiver Hall-Littlewood functions and Kostka-Shoji 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 \(K_0(\text{Var}_k)\) denote the Grothendieck ring of algebraic \(k\)-schemes, with addition and multiplication given by disjoint sums and products, respectively. This paper computes the subring generated by the smooth conics \(C\subset\mathbb P^2\), which are the 1-dimensional Severi--Brauer varieties. There is a technical assumption on the ground field \(k\), but number fields, functions fields of complex surfaces, and more generally \(C_2\)-fields are allowed.
To describe the result, let \(G\) be a finite subgroup inside the 2-torsion of the Brauer group \(\text{Br}(k)\). Choose a basis \(C_1,\ldots,C_n\in G\) consisting of smooth conics, where \(G\) is regarded as vector space over the field with two elements, and let \(C(G)=[C_1\times\ldots\times C_n]\) be the class in the Grothendieck ring. Then the subring of the Grothendieck ring generated by smooth conics is the free abelian group generated by elements of the form \(C(G) \cdot [\mathbb P^1]^m\), for varying \(G\) and \(m\). There is also an explicit formula for multiplication.
The computation depends on another result of the paper, which characterizes when two schemes of the form \(C_1\times\ldots\times C_n\) and \(C'_1\times\ldots\times C'_{n'}\), where all factors are smooth conics, have the same class in the Grothendieck ring. It turns out that this holds if and only if the following equivalent conditions hold: The two schemes are (1) birational, (2) stably birational, or (3) we have \(n=n'\) and the subgroups of the Brauer groups generated by the factors \(C_1,\ldots,C_n\) and \(C_1',\ldots, C'_n\) are the same.
The proofs are based on work of \textit{M. Larsen} and \textit{V. A. Lunts} [Mosc. Math. J. 3, 85--95 (2003; Zbl 1056.14015)], and a nice geometric description of the Brauer product of two smooth conics \(C_1,C_2\) as a subscheme of the Hilbert scheme of divisors on \(C_1\times C_2\) of bidegree (1,1). Grothendieck ring; conics; Severi-Brauer variety János Kollár, ``Conics in the Grothendieck ring'', Adv. Math.198 (2005) no. 1, p. 27-35 Brauer groups of schemes, Other nonalgebraically closed ground fields in algebraic geometry Conics 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 Hurwitz stacks are algebraic stacks that parametrize simple coverings. In this paper, we introduce a certain geometric version of the Grothendieck conjecture for universal curves over Hurwitz stacks. This result generalizes a similar result obtained by Hoshi and Mochizuki in the case of universal curves over moduli stacks of pointed smooth curves. After introducing these results, we give a sketch of the proof of the above version of the Grothendieck conjecture in the hyperelliptic case. anabelian geometry; Hurwitz stack; universal curve Coverings of curves, fundamental group, Families, moduli of curves (algebraic), Generalizations (algebraic spaces, stacks) Geometric version of the Grothendieck conjecture for universal curves over Hurwitz stacks: a research announcement | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Assume the existence of a Fukaya category \(\mathrm{Fuk}(X)\) of a compact symplectic manifold \(X\) with some expected properties. In this paper, we show \(\mathscr{A} \subset \mathrm{Fuk}(X)\) split generates a summand \(\mathrm{Fuk}(X)_e \subset \mathrm{Fuk}(X)\) corresponding to an idempotent \(e \in QH^{\bullet}(X)\) if the Mukai pairing of \(\mathscr{A}\) is perfect. Moreover, we show \(HH^{\bullet}(\mathscr{A}) \cong QH^{\bullet}(X) e\). As an application, we compute the quantum cohomology and the Fukaya category of a blow-up of \(\mathbb{C} P^2\) at four points with a monotone symplectic structure. compact symplectic manifold; Mukai pairing ; monotone symplectic structure; blow-up of \(\mathbb{C} P^2\) Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Topological quantum field theories (aspects of differential topology) Computation of quantum cohomology from Fukaya 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 Let \(w_0,\dots,w_n\) be strictly positive integers. The author proves the following mirror theorem: the Frobenius manifold associated to the orbifold quantum cohomology of the weighted projective space \({\mathbb P}(w_0,\dots,w_n)\) is isomorphic to the one attached to the Laurent polynomial \(f(u_0,\dots,u_n)=u_0+\cdots+u_n\) restricted to the region \(U=\{(u_0,\dots,u_n)\in {\mathbb C}^{n+1}\,| \, \prod_iu_i^{w_i}=1\}\). This generalizes Barannikov's isomorphism between the quantum cohomology of \({\mathbb P}^n({\mathbb C})\) and the Frobenius manifold associated to the Laurent polynomial \(x_1+x_2+\cdots+x_n+(x_1\cdots x_n)^{-1}\) [\textit{S.~Barannikov}, Semi-infinite Hodge structures and mirror symmetry for projective spaces, \url{arXiv:math/0010157}].
In particular, at the classical level, one has an isomorphism of Frobenius algebras
\[
\left(H^{2\bullet}_{\text{ orb}}({\mathbb P}(w_0,\dots,w_n);{\mathbb C}),\cup,\langle\cdot,\cdot\rangle\right)\simeq\left(\text{ gr}^{\mathcal N}_\bullet\left(\Omega^n(U)/df\wedge\Omega^{n-1}(U)\right),\cup,[\![g]\!](\cdot,\cdot)\right).
\]
On the left hand side, \(\cup\) denotes the orbifold cup product, and \(\langle\cdot,\cdot\rangle\) the orbifold Poincaré duality [\textit{W.~Chen, Y.~Ruan}, Commun. Math. Phys. 248, No. 1, 1--31 (2004; Zbl 1063.53091)]. On the right hand side, \({\mathcal N}\) denotes the Newton filtration on the vector space \(\Omega^n(U)/df\wedge\Omega^{n-1}(U)\); the associative product \(\cup\) and the pairing \([\![g]\!](\cdot,\cdot)\) come from the Jacobian algebra of \(f\) via the linear isomorphism with \(\Omega^n(U)/df\wedge\Omega^{n-1}(U)\) induced by a volume form on \(U\) [\textit{A.~Douai, C.~ Sabbah}, Ann. Inst. Fourier 53, No. 4, 1055--1116 (2003; Zbl 1079.32016)].
Moreover, the author proves a reconstruction theorem. Namely, that the full genus 0 Gromov-Witten potential of \({\mathbb P}(w_0,\dots,w_n)\) can be reconstructed from the 3-point invariants. Frobenius manifolds; quantum cohomology; orbifolds [18] Étienne Mann, &Orbifold quantum cohomology of weighted projective spaces&#xJ. Algebraic Geom.17 (2008) no. 1, p.~137Article | &MR~23 | &Zbl~1146. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Orbifold quantum cohomology of weighted projective spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(R=S^ G\) be a quotient singularity where \(S=\mathbb{C}[[x_ 1,\dots,x_ n]]\) is the formal power series ring in \(n\) variables over the complex numbers \(\mathbb{C}\), and \(G\) be a finite subgroup of \(GL(n,\mathbb{C})\). \(G\) acts on \(S\) by those \(\mathbb{C}\)-algebra automorphisms which are induced by the linear action on the variables. \textit{M. Auslander} and \textit{I. Reiten} [J. Pure Appl. Algebra 39, 1-51 (1986; Zbl 0576.18008)] defined an epimorphism of Grothendieck groups \(\psi: G_ 0(S[G])\to G_ 0(R)\) by sending \([M]\) to \([M^ G]\), and proved that \(\text{Ker }\psi\) is the ideal generated by \([\mathbb{C}]\) if \(G\) acts freely on the ramification locus. The author of this paper shows that this condition is also necessary. In addition, the author proves that there is a canonical isomorphism \(G_ 0(R)\simeq G_ 0(R[[t]])\), and studies the structure of \(G_ 0(R)\) for some cases with \(\dim R=3\). group action; quotient singularity; formal power series ring; epimorphism of Grothendieck groups; ramification locus E.D.N. Marcos, Grothendieck groups of quotient singularities,Trans. Amer. Math. Soc 332 (1992) 93--119 \(K_0\) of other rings, Singularities in algebraic geometry Grothendieck groups of quotient singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In Dix Exposés Cohomologies Schémas, Adv. Stud. Pure Math. 3, 67-87 (1968; Zbl 0198.258), p. 75, \textit{A. Grothendieck} suggested the problem of finding a normal algebraic surface, such that the kernel of the map \(Br(X)\to Br(K)\) is nontrivial where K is the function field of X. - T. Ford told us about a possible construction in characteristic zero. Some details still have to be worked out. Here we present an example in characteristic \(p\geq 5.\) Brauer group of algebraic surface; Brauer group of function field; characteristic p Finite ground fields in algebraic geometry, Brauer groups of schemes, Surfaces and higher-dimensional varieties On a remark of 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 Let \(w_0,w_1,\dots, w_n\) be a sequence of positive integers and let \(\mathbb P^w\) be the weighted projective space \(\mathbb P(w_0,w_1,\dots, w_n)\), i.e., the quotient \([(\mathbb{C}^{n+ 1}\setminus\{0\})/\mathbb{C}^\times]\), where \(\mathbb{C}^\times\) acts with weights \(-w_0,-w_1,\dots, -w_n\). The authors calculate the small quantum orbifold cohomology ring of the weighted projective space \(\mathbb P(w_0,w_1,\dots, w_n)\). Their approach is essentially due to \textit{A. B. Givental} [Sel. Math., New Ser. 1, No.~2, 325--345 (1995; Zbl 0920.14028) in: Proceedings of the international congress of mathematicians, ICM `94, August 3--11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 472--480 (1995; Zbl 0863.14021) and Int. Math. Res. Not. 1996, No.~13, 613--663 (1996; Zbl 0881.55006)].
They begin with a heuristic argument relating the quantum cohomology of \(\mathbb P^w\) to the \(S^1\)-equivariant Floer cohomology of the loop space \(L\mathbb P^w\), and from this they conjecture a formula for a certain generating function -- the small J-function for genus-zero Gromov-Witten invariants of \(\mathbb P^w\). The small J-function determines the small quantum orbifold cohomology of \(\mathbb P^w\). Next, the authors prove that their conjectural formula for the small J-function is correct by analyzing the relationship between two compactifications of the space of parameterized rational curves in \(\mathbb P^w\) -- a toric compactification and the space of genus-zero stable maps to \(\mathbb P^w\times\mathbb P(1, r)\) of degree \(1/r\) with respect to the second factor. These compactifications carry naturally a \(\mathbb{C}^\times\)-action, which one can think of as arising from the rotation of loops, and there is a map between them which is \(\mathbb{C}^\times\)-equivariant. The authors formula for the small J-function can be expressed in terms of integrals of \(\mathbb{C}^\times\)-equivariant cohomology classes on the toric compactification. Then, using results of \textit{A. Bertram} [Invent. Math. 142, No.~3, 487--512 (2000; Zbl 1031.14027)], and localization in equivariant cohomology, they transform these into integrals of classes on the stable map compactification. This establishes the authors formula for the small J-function, and so allows them to determine the small quantum orbifold cohomology ring of \(\mathbb P^w\). weighted projective space; quantum cohomology; rational curves; Gromov-Witten invariants Coates, T; Lee, Y-P; Corti, A; Tseng, HH, \textit{the quantum orbifold cohomology of weighted projective spaces}, Acta Math., 202, 139-193, (2009) Symplectic aspects of Floer homology and cohomology, Floer homology, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Differential geometric aspects of gerbes and differential characters, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Relationships between surfaces, higher-dimensional varieties, and physics The quantum orbifold cohomology of weighted projective spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe recent progress on \(QH^*(G/P)\) with special emphasis on its functoriality property, quantum Pieri rule and their applications. quantum cohomology; homogeneous varieties; Gromov-Witten invariable Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds An update of quantum cohomology of homogeneous varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors continue their work [from K-Theory 25, No. 4, 355-371 (2002; Zbl 1055.14003)], where they introduced the concept of a Grothendieck representation of an arbitrary category. In the paper under review, it is defined the concept of a quotient of a Grothendieck representation. A certain class of representations, called spectral representations, is introduced, and it is shown that the quotient of a spectral Grothendieck representation is spectral. As an application, it is given the construction and basic noncommutative scheme theory for the scheme Proj of a \(\mathbb{Z}\)-graded ring. It is shown that Proj can be viewed as a gluing of affine noncommutative schemes. Grothendieck representations; graded rings; spectral representations; noncommutative schemes; Grothendieck categories; noncommutative algebraic geometry Rings arising from noncommutative algebraic geometry, Noncommutative algebraic geometry, Grothendieck categories, Category-theoretic methods and results in associative algebras (except as in 16D90), Schemes and morphisms Quotient Grothendieck representations. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 a \(\mathbb{Z}^2\)-filtered algebraic structure and a ``quantum to classical'' principle on the torus-equivariant quantum cohomology of a complete flag variety of general Lie type, generalizing earlier works of Leung and the second author. We also provide various applications on equivariant quantum Schubert calculus, including an equivariant quantum Pieri rule for partial flag variety \(F \ell_{n_1, \ldots, n_k; n + 1}\) of Lie type \(A\). equivariant quantum cohomology; flag varieties; quantum to classical principle Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds On equivariant quantum Schubert calculus for \(G/P\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials From the author's abstract: We show that the small quantum product of the generalized flag manifold \(G/B\) is a product operation on \(H^*(G/B)\otimes \mathbb R[q_1,\dots, q_l]\) uniquely determined by the fact that it is a deformation of the cup product on \(H^*(G/B)\), it is commutative, associative, graded with respect to \(\deg(q_i)=4\), it satisfies a certain relation (of degree two), and the corresponding Dubrovin connection is flat. We deduce that it is again the flatness of the Dubrovin connection which characterizes essentially the solutions of the ``quantum Giambelli problem'' for \(G/B\). This result gives new proofs of the quantum Chevalley formula (proved by Peterson and Fulton-Woodward), and of Fomin, Gelfand and Postnikov's description of the quantization map for \(Fl_n\). quantum cohomology; quantum multiplication; Schubert variety; quantum Chevalley formula Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds A characterization of the quantum cohomology ring of \(G/B\) and applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum cohomology ring \(H^*_q(G(k,n), \mathbb{C})\) is introduced, \(G(k,n)\) being the Grassmannian of \(k\)-plane in \(\mathbb{C}^n\), using intersection data on the moduli space of maps from \(\mathbb{P}^1\) to \(G(k,n)\). The quantum Giambelli and quantum Pieri formulae are derived. These are similar to the classical Giambelli and Pieri formulae. Two generalizations are discussed, namely, the generalization using intersection data on the moduli space of maps from positive genus curves to the Grassmannian, and the form of the quantum cohomology ring for the full flag variety. quantum Giambelli formulae; quantum cohomology; quantum cohomology ring; moduli space of maps; quantum Pieri formulae Bertram, A.: Computing Schubert's calculus with Severi residues. (1996) Grassmannians, Schubert varieties, flag manifolds, Quantization in field theory; cohomological methods, Enumerative problems (combinatorial problems) in algebraic geometry Computing Schubert's calculus with Severi residues: An introduction to quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This work is an outline of the principal ideas of the author's Ph.D. thesis [''Tangency quantum cohomology and enumerative geometry of rational curves,'' Univ. Fed. Pernambuco, Recife, 2000] and establishes a connection between gravitational quantum cohomology and enumerative geometry of rational curves (in a projective homogeneous variety) subject to conditions of infinitesimal nature like, for example, tangency. The key concept is that of modified psi classes, which are well suited for enumerative purposes and substitute the tautological psi classes of 2D gravity. The main results are two systems of differential equations for the generating function of certain top products of such classes. One is topological recursion while the other is Witten-Dijkgraaf-Verlinde-Verlinde. In both cases, however, the background metric is not the usual Poincaré metric but a certain deformation of it, which surprisingly encodes all the combinatorics of the peculiar way modified psi classes restrict to the boundary. This machinery is applied to various enumerative problems, among which characteristic numbers in any projective homogeneous variety, characteristic numbers for curves with cusp, prescribed triple contact, or double points. Enumerative geometry, characteristic numbers, quantum cohomology, Gromov-Witten invariants. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry Tangency quantum cohomology and characteristic 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 This paper is concerned with Grothendieck's standard conjectures on algebraic cycles, introduced independently by Grothendieck and Bombieri to explain the Weil conjectures on the \(\zeta\)-function of algebraic varieties. We prove that the semisimplicity of the algebra of algebraic correspondences \({\mathcal A}^*({\mathcal X}\times {\mathcal X})\) of a projective irreducible smooth variety \({\mathcal X}\) implies the standard conjecture of Lefschetz type for \({\mathcal X}\). It was proved by U. Jannsen that the algebra \({\mathcal A}^*({\mathcal X}\times {\mathcal X})\) is semisimple when the numerical and homological equivalences of algebraic cycles on \({\mathcal X}\) are the same. Thus, with Jannsen's theorem our result asserts that the standard conjecture of Lefschetz type follows from Grothendieck's conjecture about the equality of the numerical and homological equivalences. This was known before only in the presence of the standard conjecture of Hodge type. algebraic cycles; Weil conjectures; algebraic correspondences; standard conjecture of Lefschetz type doi:10.1007/s002220050140 Algebraic cycles, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Graded rings and modules (associative rings and algebras) Graded associative algebras and Grothendieck standard conjectures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be a sub-\(p\)-adic field (i.e. a subfield of some finitely generated extension of a \(p\)-adic field), and \(X\) a smooth \(k\)-variety that admits a successive fibration by hyperbolic curves (i.e. smooth curves having a smooth compactifiction of genus \(g\) with \(n\) geometric points at infinity such that \(2g-2+n>0\)).
In the case when \(k\) is finitely generated over \(\mathbb Q\), it is part of Grothendieck's famous `anabelian conjectures' that \(k\)-isomorphisms between two \(k\)-varieties of the above type should correspond bijectively to outer isomorphisms between their étale fundamental groups preserving the natural augmentation toward the absolute Galois group of \(k\). In dimensions \(1\) and \(2\) the conjecture was proven in a famous paper by \textit{S. Mochizuki} [Invent. Math. 138, No. 2, 319--423 (1999; Zbl 0935.14019), Theorems A and D] under the more general assumption that \(k\) is sub-\(p\)-adic. It is thus reasonable to expect that Grothendieck's conjecture holds in this more general setting.
In the present paper Hoshi succeeds in proving the conjecture over sub-\(p\)-adic fields for hyperbolic fibrations of total dimension \(\leq 4\). The proof is by an elaborate inductive process starting from the dimension 1 case that uses only general considerations from algebraic geometry and group theory. Along the way (as in Mochizuki's proof of the dimension 2 case) he proves certain cases of the so-called `Hom-form' of the conjecture, i.e. the correspondence between dominating morphisms from an arbitrary \(k\)-variety to \(X\) as above (still of dimension \(\leq 4\)) and conjugacy classes of open homomorphisms between their fundamental groups, compatible with the Galois augmentation.
The truth of the general conjecture would imply the finiteness of the set of Galois-augmented outer isomorphisms between fundamental groups of successive hyperbolic fibrations as above. In the last main result of the paper, Hoshi verifies this finiteness statement in arbitrary dimension. Grothendieck anabelian conjecture; hyperbolic curve Y. Hoshi, ''The Grothendieck conjecture for hyperbolic polycurves of lower dimension,'' J. Math. Sci. Univ. Tokyo, vol. 21, iss. 2, pp. 153-219, 2014. Arithmetic ground fields for curves, Arithmetic ground fields (finite, local, global) and families or fibrations, Coverings of curves, fundamental group The Grothendieck conjecture for hyperbolic polycurves of lower dimension | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 volume is composed of two related, though independently written, monographs of nearly equal size [Zbl 1467.14052, Zbl 1467.18020]. The unifying main theme is Grothendieck duality theory, which is then developed in different contexts.
The first part, written by \textit{J. Lipman}, is titled ``Notes on Derived Functors and Grothendieck Duality''. As the author points out, this is an elaborated version of his lecture notes begun in the late 1980s, largely available from his home page since then. These notes were meant to be accessible to mid-level graduate students interested in studying the powerful framework of Grothendieck duality in a profound, systematic and coherent manner. In their present polished form, these notes are organized in four chapters, each of which is divided in several sections.
Chapter 1 reviews the foundational material concerning derived and triangulated categories to such an extent as it is necessary to understand the approach to abstract Grothendieck duality for algebraic schemes developed later on. This includes homotopy categories, derived categories, mapping cones, triangulated categories, localizing subcategories of homotopy categories, complexes with homology in a so-called plump subcategory, truncation functors, and bounded functors.
Chapter 2 provides an introduction to derived functors in the abstract context of derived and triangulated categories, whereas Chapter 3 is devoted to the corresponding rich formalism for categories of sheaves of modules over ringed spaces. The latter mainly deals with the derived functors, for unbounded complexes, of the sheaf functors \(\otimes\), Hom, \(f_*\) and \(f^*\), where \(f\) is a ringed-space map and \(f_*\), \(f^*\) denote the associated functors of direct image or inverse image, respectively. A basic objective, in this context, is the development of a categorical formalization of the relations among functorial maps involving these derived functors, thereby reflecting the according philosophy of Grothendieck, Verdier, Hartshore, and others. Chapter 4 discusses then the case when the underlying ringed spaces are schemes, which finally culminates in exhibiting some basic abstract features of Grothendieck duality for these algebro-geometric objects from a unified point of view, including gobal duality, sheafified duality, tor-independent base change, twisted inverse image pseudo-functors for separated morphisms of finte type of Noetherian schemes, and a brief discussion of dualizing complexes.
In the second part of the book, written by \textit{M. Hashimoto} and titled ``Equivariant Twisted Inverses'', the abstract theory developed in the foregoing introductory notes is extended to the context of diagrams of schemes. More precisely, let \(S\) be a scheme, \(G\) a flat \(S\)-group scheme of finite type, \(X\) and \(Y\) two Noetherian \(S\)-schemes with \(G\)-actions, and \(f: X\to Y\) a separated \(G\)-morphism of finite type. Regarding this setup, the purpose of the authors' notes is to construct an equivariant version of the twisted inverse image functor \(f^!\) and exhibit its basic properties, mainly with applications to invariant theory in mind. This program is carried out in the thirty-three sections constituting this second part of the book, where sheaves on ringed sites, their derived categories and functors, sheaves over a diagram of \(S\)-schemes, generalized direct and inverse images of such objects, simplicial objects, a suitable theory of descent, groupoids of schemes, and other advanced techniques are developed to establish an equivariant Grothendieck duality theory for diagrams of schemes. Also, equivariant dualizing complexes and canonical modules are investigated. Finally, as a concrete application of the authors' equivariant version of Grothendieck duality to invariant theory, a generalization of Watanabe's theorem on the Gorenstein property of rings of invariants [cf. \textit{K. Watanabe}, Certain invariant subrings are Gorenstein. I: Osaka J. Math. 11, 1--8 (1974; Zbl 0281.13007)] is demonstrated.
Further concrete examples of diagrams of schemes are explained in the last section of this part, mainly with a view toward local algebra and local cohomology in positive characteristics.
Altogether, both parts of the book are written in an utmost lucid, comprehensive and detailed style. The first part is largely of introductory nature, though being presented on a highly abstract and advanced level, whereas the second part must be seen as a related and generalizing pure research monograph of topical character. The interested reader will certainly appreciate the wealth of general material on abstract Grothendieck duality provided by this set of lecture notes, and the authors' expository expertise likewise. schemes and morphisms; sheaves; Grothendieck duality; diagrams of schemes; equivariant Grothendieck duality; dualizing complexes; derived categories and functors Lipman, J., \textit{notes on derived functors and Grothendieck duality}, Foundations of Grothendieck duality for diagrams of schemes, 1-259, (2009), Springer, Berlin Research exposition (monographs, survey articles) pertaining to algebraic geometry, Schemes and morphisms, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Group schemes, Actions of groups on commutative rings; invariant theory, Categorical algebra Foundations of Grothendieck duality for diagrams of 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 See the review of the entire collection in [Zbl 1303.01004]. History of algebraic geometry, History of mathematics in the 20th century, History of number theory, Étale and other Grothendieck topologies and (co)homologies Grothendieck and étale 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 define a special type of hypersurface varieties inside \(\mathbb{P}^{n-1}_k\) arising from connected planar graphs and then find their equivalence classes inside the Grothendieck ring of projective varieties. Then we find a characterization for graphs in order to define irreducible hypersurfaces in general. Grothendieck ring; banana graphs; flower graphs Applications of methods of algebraic \(K\)-theory in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Graph labelling (graceful graphs, bandwidth, etc.), Structural characterization of families of graphs, Grothendieck groups and \(K_0\) Grothendieck ring class of banana and flower graphs | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Branched covers of the complex projective line ramified over \(0,1,\) and \(\infty \) (Grothen\-dieck's dessins d'enfant) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over \(\infty\) and given numbers of preimages of \(0\) and \(1\) are considered. The generating function for the numbers of such covers is shown to satisfy a partial differential equation (PDE) that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE's called the Kadomtsev-Petviashvili (KP) hierarchy. A specification of this generating function for certain values of parameters generates the numbers of dessins of given genus and degree, thus providing a fast algorithm for computing these numbers. dessins d'enfant; branched covers; Kadomtsev-Petviashvili hierarchy P. Zograf, \textit{Enumeration of Grothendieck's dessins and KP hierarchy}, arXiv:1312.2538 [INSPIRE]. Arithmetic aspects of dessins d'enfants, Belyĭ theory, Dessins d'enfants theory Enumeration of Grothendieck's dessins and KP hierarchy | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 deals with a class of bivariant theories in the sense of \textit{W. Fulton} and \textit{R. MacPherson} [Categorical framework for the study of singular spaces, Mem. Am. Math. Soc. 243 (1981; Zbl 0467.55005)]. One of them, the {operational bivariant theory}, can always be constructed from any covariant functor. Let \(C\) be a category with a final object \(pt\), \(T_*\) a covariant theory with a distinguished element \(1\) in \(T_* (pt)\) and \(\mathbb T^{op}\) the associated operational bivariant theory, \(ev:\mathbb T^{op}(X\to pt) \to T_* (X)\) the {evaluation map}. In the above-mentioned article, Fulton and MacPherson state that the operational bivariant theory is the coarsest bivariant theory one can associate to \(T_*\): If \(\mathbb B\) is any bivariant theory on \(C\) and there are homomorphisms \(\Phi (X): B_*(X)= \mathbb B(X\to pt)\) to \(T_* (X)\), covariant for confinded morphisms, and taking \(1\in B^*(pt)\) to \(1\in T_*(pt)\), then there is a unique Grothendieck transformation \(\mathbb B \to \mathbb T^{op} \) of the bivariant theories such that the associated map \(B_*(X)\to \mathbb T^{op} (X\to pt)\) followed by the evaluation map \(ev: \mathbb T^{op}(X\to pt) \to T_*(X)\) is the given map \( \Phi (X):B_*(X)\to T_*(X)\).
From the author's introduction: ``In fact, we will see that in general there does not exist such a Grothendieck transformation \(\mathbb B \to \mathbb T^{op}\).\dots So, a reasonable problem, motivated by the above statement due to Fulton and MacPherson, is how to modify or correct it \dots''
Theorem A. Suppose that in the above situation the product \(\times\) is defined on \(B_*\) and \(T_*\) and \(\Phi :B_*\to T_*\) preserves this operation. Then there is a bivariant subgroup \(\widetilde {\mathbb B} \subset \mathbb B \) and a Grothendieck transformation \(\widetilde {\mathbb B} \to \mathbb T^{op}\) such that the associated map \(B_*(X)\to \mathbb T^{op} (X\to pt)\) followed by the evaluation map \(ev: \mathbb T^{op}(X\to pt) \to T_*(X)\) is the given map \(B_*(X)\to T_*(X)\). The theorem is motivated by Theorem B. Let \(\mathbb H^{op}\) be the operational bivariant homology theory. Then there is a bivariant theory \(\widetilde {\mathbb F}\) of constructible functions and there is a Grothendieck transformation \(\gamma : \widetilde {\mathbb F} \to \mathbb H^{op}\) such that the associated map \(F(X) = \widetilde {\mathbb F} (X\to pt)\to \mathbb H^{op} (X\to pt)\) followed by the evaluation map \(ev: \mathbb H^{op} (X\to pt) \to H_*(X)\) is the Chern-Schwartz-MacPherson class \(c_*:F(X)\to H_*(X)\). Grothendieck transformation; bivariant theories; constructible functions; Chern-Schwartz-MacPherson class S. Yokura: ''Bivariant theories of constructible functions and Grothendieck transformations'', Topology and Its Applications, Vol. 123, (2002), pp. 283--296. Other homology theories in algebraic topology, Characteristic classes and numbers in differential topology, Homology of classifying spaces and characteristic classes in algebraic topology, (Co)homology theory in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Bivariant theories of constructible functions and Grothendieck transformations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials After presenting Grothendieck abelian categories as linear sites following [the author,
J. Pure Appl. Algebra 190, No. 1--3, 197--211 (2004; Zbl 1051.18007)], we present their basic deformation theory as developed in [the author and \textit{M. van den Bergh}, Trans. Am. Math. Soc. 358, No. 12, 5441--5483 (2006; Zbl 1113.13009); the author, Commun. Algebra 33, No. 9, 3195--3223 (2005; Zbl 1099.18008)]. We apply the theory to certain categories of quasi-coherent modules over \(\mathbb{Z}\)-algebras, which can be considered as non-commutative projective schemes. The cohomological conditions we require constitute an improvement upon [\textit{O. De Deken} and the author,
J. Noncommut. Geom. 5, No. 4, 477--505 (2011; Zbl 1262.14017)]. deformation theory; quasi-coherent modules; projective schemes Lowen, W.: Grothendieck categories and their deformations with an application to schemes. Mat. contemp. 41, 27-48 (2012) Grothendieck categories, Noncommutative algebraic geometry, Module categories in associative algebras, Deformations of associative rings Grothendieck categories and their deformations with an application to 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 We show that the characteristic polynomial of a hyperplane arrangement can be recovered from the class in the Grothendieck group of varieties of the complement of the arrangement. This gives a quick proof of a theorem of \textit{P. Orlik} and \textit{L. Solomon} [Invent. Math. 56, 167--189 (1980; Zbl 0432.14016)] relating the characteristic polynomial with the ranks of the cohomology of the complement of the arrangement. We also show that the characteristic polynomial can be computed from the total Chern class of the complement of the arrangement. In the case of free arrangements, we prove that this Chern class agrees with the Chern class of the dual of a bundle of differential forms with logarithmic poles along the hyperplanes in the arrangement; this follows from the work of \textit{M. Mustaţǎ} and \textit{H. K. Schenck} [J. Algebra 241, No. 2, 699--719 (2001; Zbl 1047.14007)]. We conjecture that this relation holds for any locally quasi-homogeneous free divisor. We give an explicit relation between the characteristic polynomial of an arrangement and the Segre class of its singularity (``Jacobian'') subscheme. This gives a variant of a recent result of \textit{M. Wakefield} and \textit{M. Yoshinaga} [Math. Res. Lett. 15, No. 4, 795--799 (2008; Zbl 1158.14044)], and shows that the Segre class of the singularity subscheme of an arrangement together with the degree of the arrangement determines the ranks of the cohomology of its complement. We also discuss the positivity of the Chern classes of hyperplane arrangements: we give a combinatorial interpretation of this phenomenon, and discuss the cases of generic and free arrangements. Aluffi, P., Grothendieck classes and Chern classes of hyperplane arrangements, Int. Math. Res. Not. IMRN, 2013, 1873-1900, (2013) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Configurations and arrangements of linear subspaces, Characteristic classes and numbers in differential topology, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Grothendieck classes and Chern classes of hyperplane arrangements | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Mirror symmetry has made a lot of surprising predictions in algebraic geometry. There are still many wonderful problems to work on. From a mathematical point of view, mirror symmetry is formulated as the correspondence of the A-model and B-model geometry.
This paper first studies the mirror partner (B-model) of the small quantum orbifold cohomology of weighted projective spaces \(\mathbb{P}(w):=\mathbb{P}(w_0,w_1,\dots,w_n)\) (A-model) in the framework of differential equations. Following [\textit{H. Iritani}, Adv. Math. 222, 1016--1079 (2009; Zbl 1190.14054)], one can attach a quantum differential system to any proper smooth Deligne-Mumford stack using the quantum orbifold cohomology. By the result \textit{T. Coates} et al., [Acta. Math. 202(2), 139--193 (2009; Zbl 1213.53106)], this construction can be done explicitly in the case of weighted projective spaces and yields a quantum differential system
\[
\mathcal{Q}^A=(\mathcal{M}_A,\tilde{H}^{A,\text{sm}},\tilde{\nabla}^{A,\text{sm}},\tilde{S}^{A,\text{sm}},n)
\]
where \(\mathcal{M}_A\simeq \mathbb{C}^*\), the metric \(\tilde{S}^{A,\text{sm}}\) being constructed with the help of the orbifold Poincaré duality. This quantum differential system is called the ``small A-model quantum differential system'' by the authors. On the other hand, using the method developed in [\textit{E. Mann}, J. Alg. Geom. 17, 137--166 (2008; Zbl 1146.14029)], from the Gauss-Manin system of the function \(F: U\times \mathcal{M}_B\rightarrow\mathbb{C}\) defined by
\[
F(u_1,\dots,u_n,x)=u_1+\cdots+u_n+\frac{x}{u_1^{w_1}\cdots u_n^{w_n}}
\]
where \(U=(\mathbb{C}^*)^n\) and \(\mathcal{M}_B=\mathbb{C}^*\), the authors obtain a quantum differential system
\[
\mathcal{Q}^B=(\mathcal{M}_B,H^{B},\nabla^{B},S^{B},n)
\]
which is called the ``B-model quantum differential system''. Then the authors prove that the quantum differential systems \(\mathcal{Q}^A\) and \(\mathcal{Q}^B\) are isomorphic. Identifying these two models under this isomorphism, the authors finally obtain a quantum differential system
\[
\mathcal{S}_w=(\mathcal{M},H,\nabla,S,n)
\]
where \(\mathcal{M}=\mathbb{C}^*\)(the index \(w\) recalls the weights \(w_0,\dots,w_n\)). Then the associated Frobenius type structure \(F_w\) on \(\mathcal{M}\) is a tuple
\[
\mathbb{F}_w=(\mathcal{M},E,R_0,R_\infty,\Phi,\nabla,g).
\]
In the second part of this paper, the authors study the behavior of these structures at the origin. They construct a limit quantum differential system
\[
\overline{\mathcal{S}}_w=(\overline{H},\overline{\nabla},\overline{S},n)
\]
on \(\mathbb{P}^1\), and also a limit Frobenius type structure \(\overline{\mathbb{F}}_w\).
The last part of this paper is devoted to the construction of classical, limit and logarithmic Frobenius manifolds. quantum cohomology; Frobenius manifolds; weighted projective spaces; Brieskorn lattices; mirror symmetry Douai, A; Mann, E, The small quantum cohomology of a weighted projective space, a mirror \(D\)-module and their classical limits, Geom. Dedicata, 164, 187-226, (2013) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms The small quantum cohomology of a weighted projective space, a mirror \(D\)-module and their classical limits | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the Grothendieck conjecture for the moduli spaces of hyperbolic curves of genus one. A consequence of the main results is that the isomorphism class of a certain moduli space of hyperbolic curves of genus one over a sub-\(p\)-adic field is completely determined by the isomorphism class of the étale fundamental group of the moduli space over the absolute Galois group of the sub-\(p\)-adic field. We also prove related results in absolute anabelian geometry. anabelian geometry; Grothendieck conjecture; moduli space; hyperbolic curve; configuration space Families, moduli of curves (algebraic), Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Fundamental groups and their automorphisms (group-theoretic aspects), Coverings of curves, fundamental group The Grothendieck conjecture for the moduli spaces of hyperbolic curves of genus one | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Motivated by Givental's work on mirror symmetry for toric complete intersections, the author finds an explicit relationship between solutions to the quantum differential equation and the periods for toric orbifold mirror pairs. The author also gives a detailed study of the mirror isomorphism of variations of Hodge structure for a mirror pair of Calabi-Yau hypersurfaces and shows that the A-model and B-model periods are equal. Several interesting questions are raised in the last section. quantum cohomology; mirror symmetry; gamma class; \(K\)-theory; period; oscillatory integral; variation of Hodge structure; GKZ system; toric variety; orbifold H. Iritani, Quantum cohomology and periods. \textit{Ann. Inst. Fourier} (\textit{Grenoble}) \textbf{61} (2011), 2909-2958. MR3112512 Zbl 1300.14055 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Structure of families (Picard-Lefschetz, monodromy, etc.), Variation of Hodge structures (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Period matrices, variation of Hodge structure; degenerations, Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category Quantum cohomology and periods | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We define a Grothendieck ring for basic real semialgebraic formulas, that is, for systems of real algebraic equations and inequalities. In this ring the class of a formula takes into consideration the algebraic nature of the set of points satisfying this formula and this ring contains as a subring the usual Grothendieck ring of real algebraic formulas. We give a realization of our ring that allows us to express a class as a \(\mathbb Z[\frac{1}{2}]\)-linear combination of classes of real algebraic formulas, so this realization gives rise to a notion of virtual Poincaré polynomial for basic semialgebraic formulas. We then define zeta functions with coefficients in our ring, built on semialgebraic formulas in arc spaces. We show that they are rational and relate them to the topology of real Milnor fibers. Grothendieck ring; semialgebraic sets; motivic Milnor fiber Comte, G; Fichou, G, Grothendieck ring of semialgebraic formulas and motivic real Milnor fibres, Geom. Topol., 18, 963-996, (2014) Semialgebraic sets and related spaces, Singularities in algebraic geometry, Topology of real algebraic varieties Grothendieck ring of semialgebraic formulas and motivic real Milnor fibers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 each characteristic, there is a canonical homomorphism from the Grothendieck ring of varieties to the Grothendieck ring of sets definable in the theory of algebraically closed fields. We prove that this homomorphism is an isomorphism in characteristic zero. In positive characteristics, we exhibit specific elements in the kernel of the corresponding homomorphism of Grothendieck semirings. The comparison of these two Grothendieck rings in positive characteristics seems to be an open question, related to the difficult problem of cancellativity of the Grothendieck semigroup of varieties. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Model-theoretic algebra The Grothendieck ring of varieties and of the theory of algebraically closed 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 Hurwitz numbers count genus \(g\), degree \(d\) covers of the complex projective line with fixed branched locus and fixed ramification data. An equivalent description is given by factorisations in the symmetric group. Simple double Hurwitz numbers are a class of Hurwitz-type counts of specific interest. In recent years a related counting problem in the context of random matrix theory was introduced as so-called monotone Hurwitz numbers. These can be viewed as a desymmetrised version of the Hurwitz-problem. A combinatorial interpolation between simple and monotone double Hurwitz numbers was introduced as mixed double Hurwitz numbers and it was proved that these objects are piecewise polynomial in a certain sense. Moreover, the notion of strictly monotone Hurwitz numbers has risen in interest as it is equivalent to a certain Grothendieck dessins d'enfant count. In this paper, we introduce a combinatorial interpolation between simple, monotone and strictly monotone double Hurwitz numbers as \textit {triply interpolated Hurwitz numbers}. Our aim is twofold: using a connection between triply interpolated Hurwitz numbers and tropical covers in terms of so-called monodromy graphs, we give algorithms to compute the polynomials for triply interpolated Hurwitz numbers in all genera using Erhart theory. We further use this approach to study the wall-crossing behaviour of triply interpolated Hurwitz numbers in genus 0 in terms of related Hurwitz-type counts. All those results specialise to the extremal cases of simple, monotone and Grothendieck dessins d'enfants Hurwitz numbers. Hurwitz numbers; monodromy graphs; wall-crossing Automorphisms of curves, Coverings of curves, fundamental group, Enumerative problems (combinatorial problems) in algebraic geometry, Exact enumeration problems, generating functions A monodromy graph approach to the piecewise polynomiality of simple, monotone and Grothendieck dessins d'enfants double Hurwitz numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(P\) be a parabolic subgroup of a simply connected complex simple Lie group \(G\). The presentation of the ring structure of the quantum cohomology of the flag varieties, denoted by \(QH^*(G/P)\), have been studied by many mathematicians. The structure constants of the product in \(QH^*(G/P)\) are given by the 3-point genus 0 Gromov-Witten invariants of \(G/P\). By the Peterson-Woodward comparison formula these GW invariants of \(G/P\) can be recovered from the GW invariants of the special case of the complete flag variety \(G/B\) where \(B\) is a Borel subgroup.
The main result of the paper under review proves vanishing and relations among some of the GW invariants of \(G/B\). As an application of this result, it is proven that in the \(A_n\) cases, certain GW invariants of \(G/B\) are the classical intersection numbers, in other words these invariants satisfy the ``quantum to classical'' principle. The proof of the main result uses the functorial relations resulted from the construction of the natural filtered algebra structure on \(QH^*(G/B)\) in the earlier work of the authors of the paper under review. Leung, N. C.; Li, C.: Classical aspects of quantum cohomology of generalized flag varieties, Int. math. Res. not. IMRN 2012, No. 16, 3706-3722 (2012) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Classical aspects of quantum cohomology of generalized 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 In this paper, we develop a certain combinatorial version of the theory of Belyi cuspidalization developed by Mochizuki. Write \({\overline{\mathbb{Q}}} \subseteq \mathbb{C}\) for the subfield of algebraic numbers \(\in \mathbb{C} \). We then apply this theory of combinatorial Belyi cuspidalization to certain natural closed subgroups of the Grothendieck-Teichmüller group associated to the field of \(p\)-adic numbers [where \(p\) is a prime number] and to stably \(\times \mu\)-indivisible subfields of \({\overline{\mathbb{Q}}} \), i.e., subfields for which every finite field extension satisfies the property that every nonzero divisible element in the field extension is a root of unity. anabelian geometry; Belyi cuspidalization; Grothendieck-Teichmüller group Coverings of curves, fundamental group Combinatorial Belyi cuspidalization and arithmetic subquotients 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 In 2000, a theorem of \textit{M. Levine} and \textit{F. Morel} [C. R. Acad. Sci., Paris 332, 723--728 (2001; Zbl 0991.19001)] stated that algebraic cobordism groups are isomorphic to (multiplicative) Grothendieck groups over smooth schemes. In this paper the author extended this theorem to singular schemes. As a consequence, the author provides a new proof of the singular Riemann-Roch theorem of Baum-Fulton-MacPherson and a new type of Riemann-Roch theorem with respect to pullbacks of locally complete morphisms. algebraic cobordism; Grothendieck groups; singuler schemes DOI: 10.4310/HHA.2010.v12.n1.a8 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Riemann-Roch theorems, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) Algebraic cobordism and Grothendieck groups over singular 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 Let \(\lambda\) be a root of unity of odd order \(\ell\) and \(\mathfrak{u} = \mathfrak{u}_\lambda(\mathfrak{sl}_2)\) the small quantum group of order \(\ell^3\) (a slight variation of the usual one). The author introduces a chain of finite-dimensional complex algebras \((\mathcal D_{\lambda,N} (\mathfrak{sl}_2))_{N\in \mathbb N_0}\), with \(\mathcal D_{\lambda,N-1}(\mathfrak{sl}_2) \hookrightarrow \mathcal D_{\lambda,N} (\mathfrak{sl}_2)\) a cleft extension of \(\mathfrak{u}\)-comodule algebras. Putting all of them together in \(\mathcal D_{\lambda} (\mathfrak{sl}_2) := \lim\limits_{\to} \mathcal D_{\lambda,N} (\mathfrak{sl}_2)\) one gets a quantized version of the algebra of distributions of \(SL_2\) (in positive characteristic). The algebras \(\mathcal D_{\lambda,N} (\mathfrak{sl}_2)\) have triangular decompositions, hence highest weight modules; the classification of the simple representations of \(\mathcal D_{\lambda,N} (\mathfrak{sl}_2)\) follows in a familiar way. Every simple \(\mathcal D_{\lambda,N} (\mathfrak{sl}_2)\)-module admits a tensor product decomposition, where the first factor is a simple \(\mathfrak{u}_\lambda(\mathfrak{sl}_2)\)-module and the second factor is a simple \(\mathcal D_{\lambda,N-1}(\mathfrak{sl}_2)\)-module; this factorization is meant to be a quantum version of the celebrated Steinberg decomposition theorem. The motivation behind these constructions and results is a new approach to a character formula of simple modules over a simple algebraic group proposed in [\textit{G. Lusztig}, Represent. Theory 19, 3--8 (2015; Zbl 1316.20049)], in turn stimulated by the counterexamples to a previous conjecture presented in [\textit{G. Williamson}, J. Am. Math. Soc. 30, No. 4, 1023--1046 (2017; Zbl 1380.20015)]. pointed Hopf algebras; Frobenius-Lusztig kernels; algebras of distributions Angiono, Iván Ezequiel, A quantum version of the algebra of distributions of {\({\mathrm SL}_2\)}, Publications of the Research Institute for Mathematical Sciences, 54, 1, 141-161, (2018) Affine algebraic groups, hyperalgebra constructions, Quantum groups (quantized enveloping algebras) and related deformations, Hopf algebras and their applications, Linear algebraic groups over arbitrary fields A quantum version of the algebra of distributions of \(\mathrm{SL}_2\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 note we review ``quantum sheaf cohomology,'' a deformation of sheaf cohomology that arises in a fashion closely akin to (and sometimes generalizing) ordinary quantum cohomology. Quantum sheaf cohomology arises in the study of (0,2) mirror symmetry, which we review. We then review standard topological field theories and the A/2, B/2 models, in which quantum sheaf cohomology arises, and outline basic definitions and computations. We then discuss (2,2) and (0,2) supersymmetric Landau-Ginzburg models, and quantum sheaf cohomology in that context. (0, 2) mirror symmetry; quantum sheaf cohomology; Landau-Ginzburg model Topological field theories in quantum mechanics, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) An introduction to quantum sheaf cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(A\) and \(B\) denote local rings such that \(A=B/tB\), where \(t\) is a regular nonunit, and let \(\mathfrak b\) denote an ideal in \(B\) such that the \(A\)-ideal \(\mathfrak a=\mathfrak b/(t)\) has codimension \(\geq 2\). Let \(\mathcal F\) be a reflexive \(\mathcal O_X\)-module, where \(X=\text{Spec }A\setminus V(\mathfrak a)\). Under suitable conditions on \(A\) and \(B\) and assuming that \(\text{Ext}^2_X(\mathcal F, \mathcal F)=0\) and \(\mathcal Ext^1_X(\mathcal F, \mathcal O_X)=0\), it is shown in this article that the dual sheaf \(\mathcal F^\lor\) can be extended to a reflexive coherent \(\mathcal O_Y\)-module, where \(Y=\text{Spec } B\setminus V(\mathfrak b)\). The infinitesimal procedure that leads to this sheaf extension makes use of the injective theory of sheaves. Applications to homomorphisms of divisor class groups come about as a consequence of this result, and a strong connection with Grothendieck's theorem on parafactoriality is drawn. Phillip Griffith, Approximate liftings in local algebra and a theorem of Grothendieck, J. Pure Appl. Algebra 196 (2005), no. 2-3, 185 -- 202. Completion of commutative rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Class groups, Galois theory and commutative ring extensions Approximate liftings in local algebra and a theorem of 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 Cluster varieties are relatives of cluster algebras on which cluster modular groups act by automorphisms. Certain extensions of these groups, called saturated modular groups, are used. The program of quantization of cluster \(\mathcal{X}\)-varieties, including a construction of intertwiners, was initiated in [\textit{V. V. Fock} and \textit{A. B. Goncharov}, Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865--930 (2009; Zbl 1180.53081)]. A cluster \(\mathcal{X}\)-variety is equipped with a natural Poisson structure. One of the main results is a construction of series of \(*\)-representations of quantum cluster \(\mathcal{X}\)-varieties. In the paper under review, the authors underline that in the previously quoted paper some ingredients are missing, including a proof of a crucial relation for intertwiners. The new features of the present paper are a new construction of intertwiners. A Schwartz space \(\mathcal S_{\mathcal X}\) is introduced. Since the Langlands modular double \(*\)-algebra L\(_{\mathcal X}\) actually acts in \(\mathcal S_{\mathcal X}\), the claim that the intertwiners indeed intertwine this action makes sense. It is shown that this implies the relations for the intertwiners. In the quasiclassical limit they give functional equations for the classical dilogarithm. \textit{A. B. Goncharev} [in: Geometry and dynamics of groups and spaces. In memory of Alexander Reznikov. Partly based on the international conference on geometry and dynamics of groups and spaces in memory of Alexander Reznikov, Bonn, Germany, September 22--29, 2006. Basel: Birkhäuser. Progress in Mathematics 265, 415--428 (2008; Zbl 1139.81055)] developed the simplest example of this program, quantization of the moduli space \(\mathcal M^{\text{cyc}}_{0,5}\). One of the applications of the construction is quantum higher Teichmüller theory. Let \(\widehat{S}\) be a surface \(S\) with holes and a finite collection of marked points at the boundary, considered modulo isotopy. Let \(G\) be a split reductive group. The pair \((G, \widehat{S})\) gives rise to a moduli space \(\mathcal X_{G,\widehat{S}}\) related to the moduli space of \(G\)-local systems on \(S\) The modular group \(\Gamma_S\) of \(S\) acts on \(\mathcal X_{G,\widehat{S}}\). The moduli space \(\mathcal X_{G,\widehat{S}},\) in the case when \(G\) has connected center, has a natural cluster \(\mathcal X\)-variety structure. The authors' construction provides a family of infinite-dimensional unitary projective representations of the saturated cluster modular group \(\widehat{\Gamma}_{G,\widehat{S}}\) related to the pair \((G,\widehat{S})\). The group \(\widehat{\Gamma}_{G,\widehat{S}}\) includes, as a subquotient, the classical modular group \(\Gamma_S\) of \(S,\) but can be bigger. To prove relations for the intertwiners, the authors introduce and study a geometric object encapsulating their properties: the symplectic double of a cluster \(\mathcal X\)-variety and its noncommutative \(q\)-deformation, the quantum double. cluster varieties; Langlands modular double; quantum dilogarithm; quantization; Teichmüller theory V.V. Fock and A.B. Goncharov, \textit{The quantum dilogarithm and representations of quantum cluster varieties}, \textit{Invent. Math.}\textbf{175} (2008) 223 [math/0702397]. Arithmetic varieties and schemes; Arakelov theory; heights, Poisson manifolds; Poisson groupoids and algebroids, Quantum groups (quantized enveloping algebras) and related deformations, Deformation quantization, star products, Higher logarithm functions The quantum dilogarithm and representations of quantum cluster varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We compute explicit presentations for the small quantum cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from \(\mathbb{P}^3\) or the smooth quadric. Systematic usage of the associativity property of quantum product implies that only a very small and enumerative subset of Gromov-Witten invariants is needed.
Then, for these threefolds the Dubrovin conjecture on the semisimplicity of quantum cohomology is proven by checking the computed quantum cohomology rings and by showing that a smooth Fano threefold \(X\) with \(b_3(X)=0\) admits a complete exceptional set of the appropriate length. Details are contained in [the author. Int. J. Math. 16, No. 8, 823--839 (2005; Zbl 1081.14075)]. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Fano varieties Computing the quantum cohomology of some Fano threefolds and its semisimplicity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(J\)-holomorphic curve is a \((j,J)\)-holomorphic map \(u : \Sigma \to M\) from a Riemann surface \((\Sigma,j)\) to an almost complex manifold \((M,J)\). Following the ideas and methods of [\textit{M. Gromov}, Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)] the theory of \(J\)-holomorphic curves is one of the new techniques in searching of global results in symplectic geometry. The book is devoted mainly to establish the fundamental theorems in the subject and give a useful introduction to the methods and applications of the theory of \(J\)-holomorphic curves. The authors establish the foundational Fredholm theory and compactness results necessary in the basic constructions of the theory. The Gromov- Witten invariants are discussed, and in particular their existence and applications to quantum cohomology (complete proof of associativity of the quantum cup-product) and to the theory of symplectic manifolds which satisfy some positivity condition. The extensions of the theory of \(J\)- holomorphic curves to the Calabi-Yau manifolds and relation of this theory to the Floer homology is also discussed. In Appendix A, B there are presented some special techniques (e.g. gluing techniques for \(J\)- holomorphic spheres) and detailed proves concerning elliptic regularity. \(J\)-holomorphic curve; almost complex manifold; symplectic geometry; Fredholm theory; Gromov-Witten invariants; Calabi-Yau manifolds; Floer homology McDuff, Dusa; Salamon, Dietmar, {\(J\)}-holomorphic curves and quantum cohomology, Univ. Lecture Ser., 6, viii+207 pp., (1994) Research exposition (monographs, survey articles) pertaining to differential geometry, General geometric structures on manifolds (almost complex, almost product structures, etc.), Research exposition (monographs, survey articles) pertaining to quantum theory, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Quantization in field theory; cohomological methods, Symplectic aspects of Floer homology and cohomology, , Calabi-Yau manifolds (algebro-geometric aspects), Relationships between surfaces, higher-dimensional varieties, and physics, Applications of global analysis to structures on manifolds \(J\)-holomorphic curves and quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors prove the following analog of the result of A. Grothendieck on the cohomology of a coherent sheaf.
Theorem. Let \(U\) be a local regular scheme of geometric type over a field \(k\) and \(T\to U\) be a smooth proper morphism. Let \(F\) be a locally constant constructible torsion étale sheaf on \(T\) with torsion prime to characteristic of \(k\). Then there exists a finite complex \(L\) of locally constant constructible sheaves on \(U\) and a functor isomorphism between étale hypercohomology \(H_{\text{ét}}^q(T\times_U U',F) \cong H_{\text{ét}}^q(U',L\times_U U')\), where \(q\geq 0\) and \(U'\) denotes an \(U\)-scheme. constructible étale sheaf; smooth proper base change Panin, I.; Smirnov, A.: On a theorem of Grothendieck. Zap. nauchn. Sem. S.-peterburg. Otdel. mat. Inst. Steklov (POMI) 272, 286-293 (2000) Étale and other Grothendieck topologies and (co)homologies On a theorem of 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 Let \(G\) be a semisimple algebraic group over a field and \(B\) a Borel subgroup. Consider the \(G\)-modules which can be filtered such that the successive filtration quotients all are isomorphic to \(G\)-modules induced from one-dimensional simple \(G\)-modules. These filtrations are called good. It is then a well-known theorem that the tensor product of two such modules with good filtrations has a good filtration. The main object of this paper is to prove the `quantized' version of this theorem. For this the author needs two key ingredients. The first is Lusztig's canonical basis, and the second is a translation to the quantum setup of ``Donkin's criterion'', which gives a necessary and sufficient condition in terms of Hochschild cohomology of when a rational \(G\)-module has a good filtration. As a main tool the author proves a `quantized' analog of the so called ``Kempf's Vanishing Theorem'' which states the vanishing of the higher cohomology groups of some induced modules. The paper is concluded by proving a theorem stated by Andersen saying that quantized `tilting' modules exist. quantized tilting modules; semisimple algebraic group; simple \(G\)- modules; tensor product; good filtrations; canonical basis; Hochschild cohomology; rational \(G\)-module J.~Paradowski, Filtrations of modules over the quantum algebra, In: Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods, Univ. Park, PA, 1991, Proc. Sympos. Pure Math., \textbf{56}, Amer. Math. Soc., Providence, RI, 1994, pp. 93-108. Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations, Group actions on varieties or schemes (quotients) Filtrations of modules over the quantum 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 A Mackey functor is a family of abelian groups \(\{a(K)\}_{K\leq G}\) equipped with three types of maps: induction, restriction and conjugation, satisfying some compatibility axioms. If moreover, the abelian groups \(a(K)\) are associative algebras together with natural compatible conditions, the Mackey functor will be called a Green functor. Let \(G\) be a finite group which acts on an abelian category \(\mathcal {C}\). The main result of this paper is to show that the \(K\)-theory \(\{K_{i}(\mathcal{C}^{H})\}_{H\leq G}\) is a Mackey functor. Moreover, if \(\mathcal{C}\) is a tensor category and \(G\) acts by tensor autoequivalence, then the \(\{K_{0}(\mathcal{C}^{H})\}_{H\leq G}\) is a Green functor. As an application, the author gave a description of the Grothendieck ring of \(\mathcal{C}^{G}\) with \(\mathcal{C}\) is a graded fusion category. In addition, a new formula for the tensor product of any two simple objects of \(\mathcal{C}^{G}\) was given. Mackey functor; Green functor; Tensor category Burciu, S., On the Grothendieck rings of equivariant fusion categories, J. Math. Phys., 56, 071704, (2015) Monoidal categories (= multiplicative categories) [See also 19D23], Group actions on varieties or schemes (quotients), Equivariant \(K\)-theory, Grothendieck groups, \(K\)-theory, etc., Graded rings and modules (associative rings and algebras), Grothendieck categories On the Grothendieck rings of equivariant fusion 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 We propose that the Virasoro algebra controls quantum cohomologies of general Fano manifolds \(M\) (\(c_1(M)>0\)) and determines their partition functions at all genera. We construct Virasoro operators in the case of complex projective spaces and show that they reproduce the results of Kontsevich-Manin, Getzler etc. on the genus-0,1 instanton numbers. We also construct Virasoro operators for a wider class of Fano varieties. The central charge of the algebra is equal to \(\chi{}\)(M), the Euler characteristic of the manifold \(M\). quantum cohomology; Virasoro algebra; general Fano manifolds; instanton numbers; Euler characteristic; topological sigma model; 2D gravity coupling; Riemann surfaces; worldsheet instantons Mazorchuk, V.: Lectures on \(\mathfrak {sl}_{2}({\mathbb C})\)-modules. Imperial College Press, London (2010) Quantization in field theory; cohomological methods, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Virasoro and related algebras, Fano varieties, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Applications of deformations of analytic structures to the sciences Quantum cohomology and Virasoro 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 \(G\) be a simple and simply connected complex algebraic group, \(\text{Gr}_G\) be its affine Grassmanian, and \(P\subset G\) a parabolic subgroup. The authors prove that the quantum cohomology ring of a flag manifold \(QH^*(G/P)\) is a quotient of \(H^*(\text{Gr}_G)\) after localization, and give the quotient map explicitly in terms of Schubert classes. This result was stated without a proof by Dale Peterson in 1997.
The authors' proof also extends to the equivariant setting. Partial manifestations of this correspondence for \(G/B\), where \(B\) is the Borel subgroup, appear in the work of \textit{R. Bezrukavnikov}, \textit{M. Finkelberg} and \textit{I. Mirković} [Compos. Math. 141, No. 3, 746--768 (2005; Zbl 1065.19004)] and \textit{B. Kim} [Ann. Math. (2) 149, No. 1, 129--148 (1999; Zbl 1054.14533)]. But even for \(P=B\), the ring property of the quotient map is new. For \(G=SL_{k+1}(\mathbb{C})\), a closely related ring homomorphism was studied by \textit{L. Lapointe} and \textit{J. Morse} [J. Comb. Theory, Ser. A 112, No. 1, 44--81 (2005; Zbl 1120.05093)] in terms of \(k\)-Schur functions. The authors note that it looks promising to also compare other structures, such as mirror symmetry on \(QH^*(G/P)\) and Hopf algebra structure with nil-Hecke action on \(H^*(\text{Gr}_G)\).
For \(P=B\), the proof relies on the replationship between the quantum Bruhat graph and the Bruhat order on the elements of the affine Weyl group with a large translation component. It also utilizes algebraic properties of \(QH^*(G/B)\) including the \(T\)-equivariant quantum Chevalley formula of Peterson, proved by \textit{L. C. Mihalcea} [Duke Math. J. 140, No. 2, 321--350 (2007; Zbl 1135.14042)] (\(T\subset G\) is a maximal torus). A byproduct of the proof are expressions for affine Schubert classes in terms of generating functions over paths in the quantum Bruhat graph.
For \(P\neq B\), the authors utilize the Coxeter combinatorics of the affinization of the Weyl group of the Levy factor of \(P\). It allows them to use the comparison formula of Woodward to relate quantum Chevalley formulas of \(QH^*(G/P)\) and \(QH^*(G/B)\). A formula for quantum multiplication by a Schubert class labeled by the reflection in the highest root is then deduced. It turns out that the ring homomorphism of Lapointe and Morse differs from the one here by the strange duality of \(QH^*(G/P)\) discovered by \textit{P.-E. Chaput}, \textit{L. Manivel} and \textit{N. Perrin} [Int. Math. Res. Not. 2007, No. 22, Article ID rnm107, 29 p. (2007; Zbl 1142.14033)]. flag manifold; quantum cohomology ring; Schubert class; affine Grassmanian; quantum Bruhat graph; quantum Chevalley formula; strange duality of \(QH^*(G/P)\) Lam, Thomas; Shimozono, Mark, Quantum cohomology of \(G/P\) and homology of affine Grassmannian, Acta Math., 204, 1, 49-90, (2010) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology of \(G/P\) and homology of affine 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 The authors prove a theorem analogous to Smith's theorem but for matrices with Laurent polynomials as entries. Then they show that this result is equivalent to Grothendieck's theorem about vector bundles on the projective line. The main theorem is in Section 8.2 and Grothendieck's theorem is in Section 8.3. Smith's theorem; Laurent polynomials; Grothendieck's theorem; vector bundles Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] An elementary proof of Grothendieck's theorem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is a note from a series of lectures at Encuentro Colombiano de Computación Cuántica, Universidad de los Andes, Bogotá, Colombia, 2015. The purpose is to introduce additive quantum error correcting codes, with emphasis on the use of binary representation of Pauli matrices and modules over a translation group algebra. The topics include symplectic vector spaces, Clifford group, cleaning lemma, an error correcting criterion, entanglement spectrum, implications of the locality of stabilizer group generators, and the classification of translation-invariant one-dimensional additive codes and two-dimensional CSS codes with large code distances. In particular, we describe an algorithm to find a Clifford quantum circuit (CNOTs) to transform any two-dimensional translation-invariant CSS code on qudits of a prime dimension with code distance being the linear system size, into a tensor product of finitely
many copies of the qudit toric code and a product state. Thus, the number of embedded toric codes is the complete invariant of these CSS codes under local Clifford circuits. quantum stabilizer codes; additive codes; symplectic codes; Laurent polynomial ring; toric code; Clifford circuit Quantum coding (general), Finite-dimensional groups and algebras motivated by physics and their representations, Applications to coding theory and cryptography of arithmetic geometry, Computational aspects of field theory and polynomials, Circuits, networks Algebraic methods for quantum codes on lattices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Is the Zariski topology really so canonical in algebraic geometry and if so, why? In fact, as follows from the results in this paper, once one has decided to investigate the category of (commutative) rings and to use modules to represent them, the Zariski topology appears in a canonical way. We start with a Grothendieck representation of an arbitrary category \(\underline {\mathcal B}\) into a class of Grothendieck categories with sets of exact functors between them as homomorphism sets. Then the lattice of torsion theories (or Serre quotient categories) may be used to define a Zariski topology `on the objects' but this is still a commutative topology, i.e. intersections are commutative. In noncommutative geometry, a term covering several nonidentical theories nowadays but each of the different branches having its own relation to physics, a noncommutative topology has to be introduced [cf. \textit{F. Van Oystaeyen} and \textit{A. Verschoren}, ``Non-commutative algebraic geometry'', Lect. Notes Math. 887, Springer-Verlag, Berlin (1981; Zbl 0477.16001)], in order to give meaning to the notion of `noncommutative space' associated to algebras of functions on so-called noncommutative spaces that would otherwise remain completely `virtual'. Grothendieck category; noncommutative geometry; schemes; topology DOI: 10.1023/A:1016070804549 Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry, Category-theoretic methods and results in associative algebras (except as in 16D90) Grothendieck representations of categories and canonical noncommutative topologies | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 original description (1958) of duality for coherent sheaves on a scheme \(X\) of finite type over a field \(k\) made use of the residue complex \(K^ 0_ X\), which is a direct sum of sheaves \(D(X/Y)\), with \(Y\subseteq X\) irreducible and closed. Later, Hartshorne (1966) reformulated duality theory in terms of derived categories, replacing \(K^ 0_ X\) by suitable functors \(f^ !:{\mathcal D}_ c^ +(Y)\to{\mathcal D}^ +_ c(X)\), assigned to morphisms \(f:X\to Y\). If \(X\) has finite type over \(k\), say with structure map \(\pi\), then the residue complex \(K^ 0_ X\) is the Cousin complex \(\pi^ \Delta k\) associated to \(\pi^ !k\in{\mathcal D}_ c^ +(X)\). In Hartshorne's approach, the module structure of the residue complex is lost, however, essentially due to the fact that the summands of \(\pi^ \Delta k\) are local cohomologies of \(\pi^ !k\) and, as such, are not expressed in a concrete form.
The aim of this monograph is to present an explicit construction of the residue complex \(K^ 0_ X\), at least, when \(X\) is a reduced sheaf of finite type over a perfect field \(k\), i.e., based on a concrete realisation of this complex as an \({\mathcal O}_ X\)-module as well as a concrete description of the associated coboundary operator.
In order to realize this, the author first develops the theory of semitopological rings, needed in order to remedy the fact that the topologized rings one encounters in this set-up do not have adic topologies, hence may not be treated in the traditional way. In fact, these rings are not even topological rings: although left and right multiplication on such a ring \(A\) are continuous, the multiplication map \(A\times A\to A\) is not! A systematic study of semi-topological rings shows, however, that they behave nicely with respect to most of the traditional operations, including completions, and that they admit a differential calculus.
These techniques are then applied to topological local fields, i.e., local fields \(K\) containing a fixed perfect field \(k\) and which are semitopological \(k\)-algebras as such. For technical reasons, one also requires that there should be some parametrization \(k\simeq F((t_ n))\dots((t_ 1))\) with \(F\) discrete and rk\(_ F\Omega^ 1_{F/k}<\infty\). One of the main results here is a axiomatic treatment of the residue functor defined on the category of (reduced clusters of) topological local fields, which provides an improved version of the Parshin-Lomadze residue functor adapted to the present treatment.
In a subsequent chapter, the author introduces the Beilinson completion \({\mathcal M}_ \xi\) of a quasi coherent sheaf \({\mathcal M}\) on \(X\) with respect to a chain \(\xi\) in \(X\), i.e., a sequence of points \(\xi=(x_ 0,\dots,x_ l)\) with \(x_ i>x_{i+1}\) for all \(i\). This completion is a generalization of Beilinson's sheaves of adèles, which occur as ``local factors'' in the present construction. It appears that the completion \({\mathcal O}_{X,\xi}\) is a commutative semitopological \(k\)- algebra for every chain \(\xi\) (which is semilocal if \(\xi\) is saturated) and that \({\mathcal M}_ \xi\) is a semitopological \({\mathcal O}_{X,\xi}\)- module for every quasicoherent sheaf \({\mathcal M}\), thus yielding an exact functor \((-)_ \xi\). In fact, if \(\xi=(x,\dots,y)\) is a saturated chain of length \(n\) then \(k(x)_ \xi=k(\xi)\) is an \(n\)-dimensional reduced cluster of topological local fields, whose spectrum is determined by repeated normalization. This (and other) results yield a link between the geometry of \(X\) and topological local fields. At this point the author is reado to describe explicitly the complex \((K^ 0_ X,\delta_ x)\). In fact, starting from the Parshin residue maps
\[
\text{Res}_{\xi,\sigma}:\Omega^*_{k(x)/k}\to\Omega^{*,\text{sep} }_{k(\xi)/k} @>\text{Res}_{k(\xi)/k(y),\sigma}>>\Omega^*_{k(y)/k},
\]
where \(\xi=(x,\dots,y)\) is a saturated chain in \(X\) and \(\sigma:k(y)\to{\mathcal O}_{X,(y)}=\widehat{\mathcal O}_{X,y}\) a coefficient field for \(y\), one may deduce for compatible coefficient fields \(\sigma/\tau\) for \(\xi\) a coboundary map
\[
\sigma_{\xi,\sigma/\tau}:K(\sigma)\to K(\tau).
\]
Here, \(K(\tau)=\text{Hom}^{\text{cont}}_{k(y)}(\widehat{\mathcal O}_{X,y},\omega(y))\), where \(\omega(y)=\Omega^ d_{k(y)/k}\), with \(d=rk_{k(y)}\Omega^ 1_{k(y)/k}\). One may show (through base change) that this construction is independent of the choice of coefficient fields. The above complex is then just the sum of the local components over the \(x\in X\).
The appendix (written by \textit{P. Sastry}) describes the canonical isomorphism \(K^ 0_ X\equiv\pi^ !k\). duality for coherent sheaves; residue complex; Beilinson completion; topological local fields; Parshin residue maps Yekutieli, Amnon, An explicit construction of the Grothendieck residue complex, with an appendix by Pramathanath Sastry, Astérisque, 208, 127 pp., (1992) Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Complete rings, completion, Local cohomology and algebraic geometry An explicit construction of the Grothendieck residue complex. With an appendix by Pramathanath Sastry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Within bicovariant differential calculi framework, the BRST operator \(\Omega\) is constructed. We showed that \(\Omega \) is nil-potent \((\Omega^2=0)\). noncommutative geometry; bicovariant differential calculus; quantum groups; quantum BRST symmetry; quantum cohomology; quantum Lie algebra Noncommutative geometry methods in quantum field theory, Cohomology of Lie (super)algebras, Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory, Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) On quantum BRST 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 Grothendieck's dessins d'enfants are applied to the theory of the sixth Painlevé and Gauss hypergeometric functions, two classical special functions of isomonodromy type. It is shown that higher-order transformations and the Schwarz table for the Gauss hypergeometric function are closely related to some particular Belyi functions. Moreover, deformations of the dessins d'enfants are introduced, and it is shown that one-dimensional deformations are a useful tool for construction of algebraic sixth Painlevé functions. Kitaev, A. V.: Grothendieck's dessins d'enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations, Algebra i analiz 17, 224-275 (2005) Painlevé-type functions, Classical hypergeometric functions, \({}_2F_1\), Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, Riemann surfaces; Weierstrass points; gap sequences Grothendieck's dessins d'enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric 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 We discuss the notion of a power structure over a ring and the geometric description of the power structure over the Grothendieck ring of complex quasi-projective varieties and show some examples of applications to generating series of classes of configuration spaces (for example, nested Hilbert schemes of \textit{J. Cheah} [Math. Z. 227, No. 3, 479--504 (1998; Zbl 0890.14003); J. Algebr. Geom. 5, No. 3, 479--511 (1996; Zbl 0889.14001)]) and wreath product orbifolds. S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, ''On the Power Structure over the Grothendieck Ring of Varieties and Its Applications,'' Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 258, 58--69 (2007) [Proc. Steklov Inst. Math. 258, 53--64 (2007)]. Parametrization (Chow and Hilbert schemes), Grothendieck groups, \(K\)-theory, etc., Applications of methods of algebraic \(K\)-theory in algebraic geometry On the power structure over the Grothendieck ring of varieties and its applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 separated scheme of finite type over a field $K\subseteq \mathbb{C}$. If $X$ is smooth projective, the classical period conjecture of Grothendieck asserts that if $K=\overline{\mathbb{Q}}$, the cycle map
\[
Z^k(X)_{\mathbb{Q}}\to H^{2k}_{dR}(X)
\]
is surjective over $H^{2k}(X_{\mathrm{an}},\mathbb{Q}(k))\cap H^{2k}_{dR}(X)$. In other words, that a cohomology class is algebraic if and only if it comes from an algebraic cycle. A detailed history of the conjecture can be found in [\textit{J. Ayoub}, Eur. Math. Soc. Newsl. 91, 12--18 (2014; Zbl 1306.14006)] and [\textit{J.-B. Bost} and \textit{F. Charles}, J. Reine Angew. Math. 714, 175--208 (2016; Zbl 1337.14009)].
In the paper under review, the authors formulate an analogue period conjecture for the étale motivic cohomology, removing the assumptions on $X$.
Ayoub's period isomorphism in Voevodsky's motivic category $\mathbf{DM}^{\text{eff}}_{\text{ét}}$ [\textit{J. Ayoub}, J. Reine Angew. Math. 693, 1--149 (2014; Zbl 1299.14020)] induces, for any scheme $X$, the isomorphism
\[
\varpi^{p,q}_X : H^p(X_{\mathrm{an}},\mathbb{Z}_{\mathrm{an}}(q)) \otimes_\mathbb{Z} \mathbb{C} \to H^p_{dR}(X) \otimes_K \mathbb{C},
\]
where $\mathbb{Z}_{\mathrm{an}}(\bullet)$ is the motivic complex of the analytic category $\mathbf{DM}^{\text{eff}}_{\mathrm{an}}$, which, following Ayoub [loc. cit.], computes Betti cohomology.
The authors consider the following arithmetic invariant
\[
H^{p,q}_\varpi(X):=H^p_{dR}(X) \cap H^p(X_{\mathrm{an}},\mathbb{Z}_{\mathrm{an}}(q)) \subseteq H^p(X_{\mathrm{an}},\mathbb{Z}_{\mathrm{an}}(q))
\]
and construct a regulator map from the étale motivic cohomology
\[
r^{pq}:H^{p,q}(X):=H^p_{\text{éh}}(X,\mathbb{Z}(q))\to H^{p,q}_\varpi(X).
\]
In this context the analogue of Grothendieck's period conjecture asserts that if $K=\overline{\mathbb{Q}}$, the regulator $r^{p,q}$ is surjective.
The main result of this paper is the proof of the latter conjecture in the case $p=1$ and all $q$.
In order to attack it, the authors rivisit the definitions in terms of 1-motives and make use of the description of $H^1$ via the motivic Albanese map.
The main step becomes showing that a realizaion of 1-motives in a period category, the Betti-de Rham realization, is fully faithful.
In the appendix, some divisibility properties of motivic cohomology are proved and used to link this conjecture with the classical period conjecture of Grothendieck for $X$ smooth and projective. motives; periods; motivic and de Rham cohomology Motivic cohomology; motivic homotopy theory, de Rham cohomology and algebraic geometry, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), Group schemes Motivic periods and Grothendieck arithmetic invariants | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper studies the analogue of the Grothendieck-Ogg-Shafarevich formula for curves over local fields. This formula was originally proved for curves over algebraically closed fields [\textit{A. Grothendieck}, SGA 5, Lect. Notes Math. 589, Exposé No. X, 372-406 (1977; Zbl 0356.14005)].
The main theorem in this paper is the following.
Let \(R\) be a complete discrete valuation ring with algebraically closed residue field \(k\); let \(S=\text{Spec}R\); let \(\eta\) be the generic point of \(S\); let \(X\) be a connected normal scheme, proper and flat over \(S\), of relative dimension one, with smooth generic fiber; let \(U\) be an open dense subscheme of \(X\) contained in \(X_\eta\); let \((F_i)_{i\in I}\) be the one-dimensional irreducible components of \(F:=X\setminus U\); let \(\mathbb F\) be a finite field with \(\text{char} \mathbb F\neq\text{char} k\); and let \(\mathcal F\) be a locally constant constructible étale sheaf on \(U\) of \(\mathbb F\)-vector spaces. Assume that the residue field of any point in \(X_\eta\setminus U\) is separable over \(\eta\), and that \(\mathcal F\) has no fierce ramification (see the paper for the definition). Then the main theorem asserts that, for each \(i\in I\), there is an open dense subscheme \(F_i^\circ\) of \(F_i\) such that:
(i) \(F\) is regular along \(F^\circ:=\bigcup_{i\in I}F_i^\circ\);
(ii) if \(x\in F_i^\circ\) for some \(i\in I\), then \(\text{sw}_x^{V/U}(\mathcal F)=\text{sw}_i(\mathcal F)\); and
\[
\begin{multlined}\text{totdim}_{\mathbb F}(R\Gamma_c(U_{\bar\eta},\mathcal F)) =\\ =\text{totdim}_{\mathbb F}(R\Gamma_c(U_{\bar\eta},\mathbb F)) \text{rk}_{\mathbb F}(\mathcal F) +\sum_{i\in I}\chi_c(F_i^\circ)\text{sw}_i(\mathcal F) +\sum_{x\in F\setminus F^\circ}\text{sw}_x^{V/U}(\mathcal F).\end{multlined}\tag{iii}
\]
Here \(\chi_c\) is the Euler-Poincaré characteristic if \(F_i^\circ\) is vertical, or minus the discriminant over \(S\) if it is horizontal; \(\text{totdim}_{\mathbb F}\) is the sum of the dimension over \(\mathbb F\) and the dimension over \(\mathbb F\) of the Swan conductor; and \(R\Gamma_c\) refers to the alternating sum of the groups \(H^i_c\). See the paper for the definition of \(\text{sw}_i(\mathcal F)\).
The first part of the paper is devoted to defining the Swan conductor \(\text{sw}_x^{V/U}(\mathcal F)\) in codimension 2, and the second part gives the proof of the above theorem. A key tool in that proof is the Lefschetz fixed point formula for arithmetic surfaces proved by the author [\textit{A. Abbes}, Compos. Math. 122, 23-111 (2000; see the preceding review Zbl 0986.14014)].
The paper also formulates a conjecture that, for all closed points \(x\in X\), \(\text{sw}_x^{V/U}(\mathcal F)\) is (a) an integer, and (b) independent of the choice of \(V\). Part (a) of this conjecture extends a conjecture of Serre on the existence of Artin representations. Grothendieck-Ogg-Shafarevich formula; Swan conductor; Lefschetz fixed point formula; Artin representation A. Abbès, The Grothendieck-Ogg-Shafarevich formula for arithmetic surfaces, Journal of Algebraic Geometry, 9 (2000), 529-576. Arithmetic varieties and schemes; Arakelov theory; heights, Topological properties in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies The Grothendieck-Ogg-Shafarevich formula for arithmetic surfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 famous set of notes ``Pursuing Stacks'', \textit{A. Grothendieck} raised the question: what characterizes models for the homotopy category \(\mathsf{Hot}\) of topological spaces? For instance, you can obtain \(\mathsf{Hot}\) by inverting a class of weak equivalences in either the category of simplicial sets \(\widehat{\Delta}\) (i.e., presheaves on the category \(\Delta\) of finite non-empty ordered sets) or of small categories \({\mathcal C}at\). Here the weak equivalences on \(\widehat\Delta\) are the maps sent to weak equivalences of topological spaces upon realization, and the class of weak equivalences \(\mathcal W_\infty\) in \({\mathcal C}at\) is the class of functors sent to weak equivalences of simplicial sets via the nerve.
If \({\mathcal A}\) is a small category, considered as a subcategory in the category of presheaves \(\widehat {\mathcal A}\) through the Yoneda embedding, let \(i_{\mathcal A}\colon \widehat {\mathcal A}\to {\mathcal C}at\) be given by \(i_{\mathcal A}(F)={\mathcal A}/F\). A morphism \(F\to F'\in\widehat {\mathcal A}\) is a weak equivalence if \({\mathcal A}/F\to {\mathcal A}/F'\in {\mathcal C}at\) is in \(\mathcal W_\infty\). When \({\mathcal A}=\Delta\) this is just an elaborate redefinition of weak equivalences of simplicial sets.
If the right adjoint \(i^*_{\mathcal A}\colon {\mathcal C}at\to\widehat {\mathcal A}\) of \(i_{\mathcal A}\) preserves weak equivalences, \({\mathcal A}\) is called a weak test category, and in this case \(i^*_{\mathcal A}\) induces an equivalence from \(\mathsf{Hot}\) upon inverting the weak equivalences (in both \({\mathcal C}at\) and \(\widehat {\mathcal A}\)). Prime examples for \(\widehat {\mathcal A}\), but by far the only ones, are simplicial, multisimplicial and cubical sets.
Weak test categories are characterized by the condition that for any small category \({\mathcal C}\) with a final object, the category \(i_{\mathcal A}i^*_{\mathcal A}({\mathcal C})\) is contractible.
However, being a weak test category is not a local property, and \({\mathcal A}\) is called a test category, and \(\widehat {\mathcal A}\) an elementary modelizer, if \({\mathcal A}\) and \({\mathcal A}/a\) are weak test categories for all objects \(a\) in \({\mathcal A}\). The language is chosen so as not to conflict directly with Quillen's notion of a model category. Test categories are plentiful, for instance, any small category \({\mathcal A}\) with finite products and a ``unit interval'' is a test category. Also this notion has a nice characterization, as has the various notion of test functors (generalizing the inclusion \(\Delta\subseteq {\mathcal C}at\) and the fact that the induced nerve \({\mathcal C}at\to\widehat{\Delta}\) induces an equivalence after inverting weak equivalences).
For many applications the notion of weak equivalences does not conform with the usual class of weak equivalences of topological spaces (for instance, if one is interested in rational homotopy theory), and the theory is developed with respect to a fundamental localizer \(\mathcal W\), i.e., a class of functors for which (among other things) Quillen's theorem \(A\) holds.
Grothendieck conjectures that \(\mathcal W_\infty\) is the smallest fundamental localizer, and this conjecture is verified in the thesis of Cisinski, along with the construction of closed model structures on \(\widehat {\mathcal A}\) for all fundamental localizers of interest.
One defines the left homotopy Kan extension in this context, without presupposing the existence of a closed model structure. Finally, smooth and proper functors are defined, and characterized by means of Kan extensions as a condition on a base change, inspired by the corresponding notions from algebraic geometry.
In the present monograph the author makes Grothendieck's ideas accessible (although entertaining, reading ``Pursuing Stacks'' is not always easy) and develops them with considerable care. Pursuing stacks; test category; smooth and proper functors Maltsiniotis, G., La théorie de l'homotopie de Grothendieck, Astérisque, 301, (2005), p. vi+140 Nonabelian homotopical algebra, Homotopy theory and fundamental groups in algebraic geometry, Localization and completion in homotopy theory, Research exposition (monographs, survey articles) pertaining to category theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry Grothendieck's homotopy theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Associated to a commutative graded ring, in this note, the authors aim to associate to schematic algebras (several of which occur in the framework of quantum groups) a ``geometric'' object. Schematic algebras are characterized by the fact that they possess ``many'' Ore sets, the latter permitting the authors to introduce a suitable notion of ``noncommutative Grothendieck topology'' (based upon the free monoid over the Ore sets), as well as constructing an associated notion of ``sheaf'' over these. In general, as pointed out in the text, the structure ``sheaf'' thus associated to a schematic algebra \(R\) is not a ``sheaf of rings'', nor does the construction specialize to that of \(\text{Proj} (R)\) in the commutative case (as only covers of open sets induced by global ones are considered). However, the authors introduce a suitable notion of quasicoherent sheaf over these geometric objects and prove that the category of these is equivalent to Artin's \(\text{Proj} (R)\), thus providing an interesting step towards a full geometric understanding of the latter. noncommutative Grothendieck topology; schematic algebras; quantum groups; Ore sets; quasicoherent sheaf; geometric objects Van Oystaeyen, F., Willaert, L.: Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras. J. Pure Appl. Algebra 1(104), 109--122 (1995) Graded rings and modules (associative rings and algebras), Associative rings of functions, subdirect products, sheaves of rings, Ore rings, multiplicative sets, Ore localization, Noncommutative algebraic geometry, Torsion theories; radicals on module categories (associative algebraic aspects) Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors compute the quantum cohomology ring of the varieties \({\mathbb P}(\bigoplus_{j=1}^r{\mathcal O}_{\mathbb P^1} (a_j))\), where \(0=a_1\leq a_2 \leq{\dots}\leq a_r\), and moreover \(\sum_{j=1}^r a_j\equiv\varepsilon \bmod r\), with \(\varepsilon\in\{0,1\}\).
The main computational tool is the invariance of the quantum cohomology ring under symplectic deformations: for \(\varepsilon =0\), \(X\) is symplectic deformation equivalent to \({\mathbb P}^1\times {\mathbb P}^{r-1}\), while for \(\varepsilon=1\), \(X\) is symplectic deformation equivalent to \({\mathbb P}({\mathcal O}_{\mathbb P^1}^{\oplus (r-1)}\oplus {\mathcal O}_{\mathbb P^1}(1))\). These latter ones are both toric Fano varieties, which meet the conditions required in \textit{A.~Kresch} [Mich. Math. J. 48, 369--391 (2000; Zbl 1085.14519)].
The presentation of the quantum cohomology ring follows now from results of \textit{V.~Batyrev} [Asterisque 218, 9--34 (1993; Zbl 0806.14041)], or \textit{A. Kresch} [loc. cit.], after suitable change of coordinates. For the explicit calculations of the \(3\)-point genus zero invariants, the authors use, for \(\varepsilon=0\), the product formula proved by \textit{K.~Behrend} [J. Algebr. Geom. 8, 529--541 (1999; Zbl 0938.14032)], while, for \(\varepsilon =1\), use the formulae obtained by \textit{A.~Kresch} [loc. cit.]. small quantum cohomology; non-Fano toric varieties Laura Costa and Rosa M. Miró-Roig. The Leray quantum relation for a class of non-Fano toric varieties. Preprint. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Quantum cohomology for a class of non-Fano toric varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is the third (and the last) of the series of papers of this volume. Let \(f:X\to Y\) be a separated morphism of quasi-compact, quasi-separated schemes. The author sketches a proof of the fact that the functor \({\mathbb R}f_*:{\mathbb D}^+_{qc}(X)\to{\mathbb D}^+(Y)\) has a right adjoint \(f^!\). Moreover, if \(f\) is proper, finitely presented and flat, then duality and tor-independent base-change hold for \(f^!\). The novelty of this paper is that the noetherian assumption of the schemes in question is dropped. formal schemes; duality; base change; non-noetherian schemes; right adjoint functor Lipman, J.: Non-noetherian Grothendieck duality, Contemp. math. 244, 115-123 (1999) (Co)homology theory in algebraic geometry, Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.), Schemes and morphisms Nonnoetherian 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 paper provides a systematic treatment of adjointness situations between compactly generated triangulated categories; this clarifies the relationship between Grothendieck-Neeman duality and the so-called Wirthmuller isomorphisms. The treatment is extremely elegant and leads to peculiar situations, as for example the fact that a string of adjoints between compactly generated, tensor triangulated categories is made by either three, five of infinitely many adjoints. Even though the main motivation lies in Grothendieck duality theory, there is plenty of examples where such a calculus is useful in stable homotopy theory; it is perhaps one of the most enticing suggestions of the paper that algebraic geometry à la Grothendieck and stable homotopy theory find in the theory of ``well-behaved triangulated categories'' a natural place to be developed on the same footing. It is also remarkable how the arguments employed in the paper are completely elementary, i.e. relying only on the classical theory of triangulated categories, making no use of explicitly \(\infty\)-categorical methods. Serre functor; Grothendieck duality; adjoints; compactly generated triangulated category; dualizing object; Wirthmuller isomorphism Derived categories, triangulated categories, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Abstract and axiomatic homotopy theory in algebraic topology Grothendieck-Neeman duality and the Wirthmüller isomorphism | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a complex projective variety. Heuristically, the quantum cohomology ring \(\text{QH}(X)\) is a version of the Floer cohomology of the universal cover of the free loop space \(\mathcal L X=\text{Map} (S^1, X)\). D. Peterson proved the version of this for \(G/B\), where \(G\) is a reductive group (unpublished). The aim of this paper is to prove the version of this in the case of toric Fano varieties.
Let the toric variety \(X\) be a quotient of \(Y-D\) by torus action \(S\), where \(Y\simeq \mathbb C^N\) and \(D\) is a particular union of hyperplanes in \(Y\). Consider the scheme \(L^0 Y\) of formal arcs (see the beginning of section 3). The toric arc scheme \(\Lambda^0 X\) is defined to be the categorical quotient \(\Lambda^0 X=(L^0 Y-L^0 D)/S\). Let \(A=\Hom (\mathbb C^*,S)\simeq H_2(X, \mathbb Z)\) and \(A_+\subset A\) be ``the positive cocharacters''. Then the main theorem (Theorem 4.7) says that the localized algebra \(H^*(\Lambda^0 X,\mathbb C)\otimes_{\mathbb C[A_+]}\mathbb C[A]\) is isomorphic (as \(\mathbb C[A]\)-algebra) to \(\text{QH}(X)\).
In the toric case the localized algebra is isomorphic to the Floer cohomology of the ind-scheme \(\Lambda X\), the toric algebro-geometric model of the universal cover of the loop space \(\mathcal L X\) (see 6.1). The authors do not expect that this is true in the general case. toric variety; arc scheme; Floer cohomology; quantum cohomology Arkhipov, S. \& Kapranov, M., Toric arc schemes and quantum cohomology of toric varieties. Math. Ann., 335 (2006), 953--964. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Toric arc schemes and quantum cohomology of toric varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the ring \(\mathcal{S}\) of symmetric polynomials in \(k\) variables over an arbitrary base ring \(\mathfrak{k}\). Fix \(k\) scalars \(a_1, a_2, \ldots, a_k\) in \(\mathfrak{k}\). Let \(I\) be the ideal of \(\mathcal{S}\) generated by \(h_{n-k+1}-a_1, h_{n-k+1}-a_2,\ldots, h_{n-k+1}-a_k\), where \(h_i\) is the \(i\)-th complete homogeneous symmetric polynomial.
The quotient ring \(\mathbf{S}/I\) generalizes both the usual and the quantum cohomology of the Grassmannian.
We show that \(\mathbf{S}/I\) has a \(\mathfrak{k}\)-module basis consisting of (residue classes of) Schur polynomials fitting into a \(k \times (n-k)\)-rectangle; and that its multiplicative structure constants satisfy the same \(S_3\)-symmetry as those of the Grassmannian cohomology. We conjecture the existence of a Pieri rule (proven in two particular cases) and a positivity property generalizing that of Gromov-Witten invariants. symmetric functions; partitions; Schur functions; Gröbner bases; Grassmannian; cohomology Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Classical problems, Schubert calculus A quotient of the ring of symmetric functions generalizing 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 Grothendieck abelian categories play the role of models of possibly noncommutative schemes [\textit{M. Artin} and \textit{J. J. Zhang}, Adv. Math. 109, No. 2, 228--287 (1994; Zbl 0833.14002); \textit{J. T. Stafford} and \textit{M. Van den Bergh}, Bull. Am. Math. Soc., New Ser. 38, No. 2, 171--216 (2001; Zbl 1042.16016; \textit{M. Kontsevich} and \textit{A. L. Rosenberg}, in: The Gelfand Mathematical Seminars, 1996--1999. Dedicated to the memory of Chih-Han Sah. Boston, MA: Birkhäuser. 85--108 (2000; Zbl 1003.14001)], which is motivated by the Gabriel-Rosenberg reconstruction theorem claiming that a quasi-separated scheme can be reconstructed, up to isomorphism of schemes, solely from the abelian category of quasi-coherent sheaves on the scheme, which is a Grothendieck category [\url{https://stacks.math.columbia.edu/}]. The theorem was initially established for noetherian schemes by \textit{P. Gabriel} [Bull. Soc. Math. Fr. 90, 323--448 (1962; Zbl 0201.35602)] and generalized to quasi-separated schemes by \textit{A. Rosenberg} [\url{https://archive.mpim-bonn.mpg.de/id/eprint/1516/}]. The Gabriel-Popescu theorem [\textit{N. Popesco} and \textit{P. Gabriel}, C. R. Acad. Sci., Paris 258, 4188--4190 (1964; Zbl 0126.03304)] allows of interpreting Grothendieck categories as a linear version of Grothendieck topoi [\textit{W. Lowen}, J. Pure Appl. Algebra 190, No. 1--3, 197--211 (2004; Zbl 1051.18007)].
Different \(2\)-categories obtain, depending on the choice of morphisms.
\begin{itemize}
\item[\(\mathsf{Grt}\)] the \(2\)-category of Grothendieck categories and left adjoints as morphisms
\item[\(\mathsf{Grt}_{\flat}\)] the \(2\)-category of Grothendieck categories and left exact left adjoints as morphisms
\end{itemize}
The \(2\)-category \(\mathsf{Grt}_{\flat}\)\ is the main object of study in this paper. It is shown that \(\mathsf{Grt}_{\flat}\)\ can be endowed with a monoidal structure, where the exponentiable objects are characterized. From an algebro-geometric standpoint, this can be seen as a contribution to the understanding of exponentiable schemes or Hom-schemes when restricted to the flat case.
The synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 2] aims to \(\mathsf{Grt}_{\flat}\)\ can simulate flat algebraic geometry via a collection of examples.
\item[\S 3] shows that the monoidal structure \(\boxtimes\)\ on \(\mathsf{Grt} \)\ [\textit{W. Lowen} et al., Int. Math. Res. Not. 2018, No. 21, 6698--6736 (2018; Zbl 1408.18024)] nicely restricts to \(\mathsf{Grt}_{\flat}\), which is easy on the level of objects, but a highly non-trivial task on the level of morphisms. The problem of exponentiability is introduced. It is shown that the category of linear presheaves \(\mathrm{Mod}\left( \mathfrak{a}\right) \)\ are exponentiable (Proposition 3.15).
\item[\S 4] investigates the properties of the forgetful functor
\[
:\mathsf{Grt}_{\flat}^{\circ}\rightarrow\mathsf{Cat}_{k}
\]
showing that it is representable (Prposition 4.2).
\item[\S 5] introduces and investigates quasi-injective Grothendieck categories (\S 5.1), continuous linear categories (\S 5.2) and then connect the two concepts (\S 5.3). These are technical tools for the main theorem.
\item[\S 6] gives the main theorem:
Theorem 6.1. A Grothendieck category is exponentiable in \(\mathsf{Grt}_{\flat}\) iff it is continuous. In particular, every finitely presentable Grothendieck category is exponentiable.
\item[\S 7] is a collecction of examples and instances of the main theorem. The most relevant is the following:
Proposition 7.2. Let \(X\) be a quasi-compact quasi-separated scheme over \(k\), then \(\mathsf{Qcoh}\left( X\right) \)\ is exponentiable.
\end{itemize} Grothendieck categories; noncommutative algebraic geometry; flatness; monoidal structures; exponentiability; continuous categories; quasi-coherent sheaves Noncommutative algebraic geometry, Sheaves in algebraic geometry, Topoi, Accessible and locally presentable categories, Grothendieck topologies and Grothendieck topoi, Monoidal categories, symmetric monoidal categories, Abelian categories, Grothendieck categories Exponentiable Grothendieck categories in flat 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 In our previous work [the authors, ibid. 336, No. 2, 811--830 (2015; Zbl 1328.13031)], we introduced the partition \(q\)-series for mutation loop \(\gamma\) -- a loop in exchange quiver. In this paper, we show that for a certain class of mutation sequences, called reddening sequences, the graded version of partition \(q\)-series essentially coincides with the ordered product of quantum dilogarithm associated with each mutation; the partition \(q\)-series provides a state-sum description of combinatorial Donaldson-Thomas invariants introduced by Keller. Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Cluster algebras, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Connections of hypergeometric functions with groups and algebras, and related topics, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Forms of half-integer weight; nonholomorphic modular forms, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum dilogarithms and partition \(q\)-series | 0 |
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