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122
Deformation quantization for algebraic varieties
"... The paper is devoted to peculiarities of the deformation quantization in the algebrogeometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. ..."
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Cited by 62 (1 self)
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The paper is devoted to peculiarities of the deformation quantization in the algebrogeometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semiformal deformations are Noetherian and have polynomial growth. We propose constructions of semiformal quantizations of projective and affine algebraic Poisson manifolds satisfying certain natural geometric conditions. Projective symplectic manifolds (e.g. K3 surfaces and abelian varieties) do not satisfy our conditions, but projective spaces with quadratic Poisson brackets and PoissonLie groups can be semiformally quantized.
Higher topos theory
, 2006
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain com ..."
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Cited by 52 (0 self)
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Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of Gvalued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category
The elliptic curve in the Sduality theory and Eisenstein series for KacMoody groups
"... (0.1) The goal of this paper is to develop a certain mathematical framework underlying the Sduality conjecture of Vafa and Witten [VW]. Let us recall the formulation. Let S be a smooth projective surface over C and G a semisimple algebraic group. Denote by MG(S,n) the moduli space of semistable pri ..."
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(0.1) The goal of this paper is to develop a certain mathematical framework underlying the Sduality conjecture of Vafa and Witten [VW]. Let us recall the formulation. Let S be a smooth projective surface over C and G a semisimple algebraic group. Denote by MG(S,n) the moduli space of semistable principal Gbundles on
Geometry of Deligne Cohomology
, 1996
"... The aim of this paper is to give a geometric interpretation of holomorphic and smooth Deligne cohomology. Before stating the main results we recall the definition and basic properties of Deligne cohomology. Let X be a smooth complex projective variety and let Ωr X be the sheaf of germs of holomorphi ..."
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Cited by 28 (0 self)
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The aim of this paper is to give a geometric interpretation of holomorphic and smooth Deligne cohomology. Before stating the main results we recall the definition and basic properties of Deligne cohomology. Let X be a smooth complex projective variety and let Ωr X be the sheaf of germs of holomorphic rforms on X. The qth Deligne complex of X is the complex of sheaves Z(q)D:
Higher dimensional algebra V: 2groups
 Theory Appl. Categ
"... A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to tw ..."
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Cited by 27 (2 self)
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A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak ’ and ‘coherent ’ 2groups. A weak 2group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2group is a weak 2group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix: 1 → x ⊗ ¯x, ex: ¯x ⊗ x → 1 forming an adjunction. We describe 2categories of weak and coherent 2groups and an ‘improvement ’ 2functor that turns weak 2groups into coherent ones, and prove that this 2functor is a 2equivalence of 2categories. We internalize the concept of coherent 2group, which gives a quick way to define Lie 2groups. We give a tour of examples, including the ‘fundamental 2group ’ of a space and various Lie 2groups. We also explain how coherent 2groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simplyconnected compact simple Lie group G a family of 2groups G � ( � ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2groups are built using Chern–Simons theory, and are closely related to the Lie 2algebras g � ( � ∈ R) described in a companion paper. 1 1
Mirror symmetry, Langlands duality and Hitchin systems
 arXiv: math.AG/0205236 56 Hausel, T. and Sturmfels, B
"... Abstract. Among the major mathematical approaches to mirror symmetry are those of BatyrevBorisov and StromingerYauZaslow (SYZ). The first is explicit and amenable to computation but is not clearly related to the physical motivation; the second is the opposite. Furthermore, it is far from obvious ..."
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Cited by 25 (7 self)
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Abstract. Among the major mathematical approaches to mirror symmetry are those of BatyrevBorisov and StromingerYauZaslow (SYZ). The first is explicit and amenable to computation but is not clearly related to the physical motivation; the second is the opposite. Furthermore, it is far from obvious that mirror partners in one sense will also be mirror partners in the other. This paper concerns a class of examples that can be shown to satisfy the requirements of SYZ, but whose Hodge numbers are also equal. This provides significant evidence in support of SYZ. Moreover, the examples are of great interest in their own right: they are spaces of flat SLrconnections on a smooth curve. The mirror is the corresponding space for the Langlands dual group PGLr. These examples therefore throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program. When it emerged in the early 1990s, mirror symmetry was an aspect of theoretical physics, and specifically a duality between quantum field theories. Since then, many people have tried to place it on a mathematical foundation. Their labors have built up a fascinating but somewhat unruly subject. It describes some sort of relation between pairs of Calabi
Essential dimension of finite pgroups
 INVENTIONES MATH
, 2008
"... We prove that the essential dimension and pdimension of a pgroup G over a field F containing a primitive pth root of unity is equal to the least dimension of a faithful representation of G over F. ..."
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Cited by 24 (4 self)
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We prove that the essential dimension and pdimension of a pgroup G over a field F containing a primitive pth root of unity is equal to the least dimension of a faithful representation of G over F.
On the passage from local to global in number theory
 Bull. Amer. Math. Soc. (N.S
, 1993
"... Would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase “in Number Theory”? The distinction “local ” versus “global”, with various connotations, has found a home in almost every part of mathematics, local problems being ..."
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Cited by 22 (0 self)
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Would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase “in Number Theory”? The distinction “local ” versus “global”, with various connotations, has found a home in almost every part of mathematics, local problems being often a steppingstone to
Moduli of Twisted Sheaves
, 2004
"... Abstract. We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of su ..."
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Cited by 22 (7 self)
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Abstract. We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that the spaces are asympotically geometrically irreducible, normal, generically smooth, and l.c.i. over the base. We also develop general tools necessary for these results: the theory of associated points and purity of sheaves on Artin stacks, twisted Bogomolov inequalities,