Results 1  10
of
143
Triangulated categories of singularities and Dbranes in LandauGinzburg models
, 2003
"... ..."
(Show Context)
Compactifying the space of stable maps
 electronic), 2002. OLSSON AND STARR
"... Abstract. In this paper we study a notion of twisted stable map, from a curve to a tame Deligne–Mumford stack, which generalizes the wellknown notion of stable map to a projective variety. Contents ..."
Abstract

Cited by 182 (23 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we study a notion of twisted stable map, from a curve to a tame Deligne–Mumford stack, which generalizes the wellknown notion of stable map to a projective variety. Contents
Conformal blocks and generalized theta functions
 Comm. Math. Phys
, 1994
"... The aim of this paper is to construct a canonical isomorphism between two vector spaces associated to a Riemann surface X. The first of these spaces is the space of conformal blocks Bc(r) (also called the space of vacua), which plays an important role in conformal field theory. It is defined as foll ..."
Abstract

Cited by 141 (8 self)
 Add to MetaCart
The aim of this paper is to construct a canonical isomorphism between two vector spaces associated to a Riemann surface X. The first of these spaces is the space of conformal blocks Bc(r) (also called the space of vacua), which plays an important role in conformal field theory. It is defined as follows: choose a point p ∈ X, and let AX be the
Derived categories of coherent sheaves and triangulated categories of singularities
, 2005
"... ..."
(Show Context)
Hypergeometric functions and mirror symmetry in toric varieties
, 1999
"... We study aspects related to Kontsevich’s homological mirror symmetry conjecture [42] in the case of Calabi–Yau complete intersections in toric varieties. In a 1996 lecture, Kontsevich [43] indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in ..."
Abstract

Cited by 72 (4 self)
 Add to MetaCart
(Show Context)
We study aspects related to Kontsevich’s homological mirror symmetry conjecture [42] in the case of Calabi–Yau complete intersections in toric varieties. In a 1996 lecture, Kontsevich [43] indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in the homological mirror symmetry conjecture should also be identified. Our results provide an explicit geometric construction of the correspondence between the automorphisms of the two types of categories. We compare the monodromy calculations for the Picard–Fuchs system associated with the periods of a Calabi–Yau manifold M with the algebrogeometric computations of the cohomology action of Fourier– Mukai functors on the bounded derived category of coherent sheaves on the mirror Calabi–Yau manifold W. We obtain the complete dictionary between the two sides for the one complex parameter case of Calabi–Yau complete intersections in weighted projective spaces, as well as for some two parameter cases. We also find the complex of sheaves on W × W that corresponds to a loop in the moduli space of complex structures on M induced by a phase transition of W.
Dbranes, derived categories, and Grothendieck groups
, 1999
"... In this paper we describe how Grothendieck groups of coherent sheaves and locally free sheaves can be used to describe type II Dbranes, in the case that all Dbranes are wrapped on complex varieties and all connections are holomorphic. Our proposal is in the same spirit as recent discussions of Kt ..."
Abstract

Cited by 62 (12 self)
 Add to MetaCart
In this paper we describe how Grothendieck groups of coherent sheaves and locally free sheaves can be used to describe type II Dbranes, in the case that all Dbranes are wrapped on complex varieties and all connections are holomorphic. Our proposal is in the same spirit as recent discussions of Ktheory and Dbranes; within the restricted class mentioned, Grothendieck groups encode a choice of connection on each Dbrane worldvolume, in addition to information about the C ∞ bundles. We also point out that derived categories can also be used to give insight into Dbrane constructions, and analyze how a Z2 subset of the Tduality group acting on Dbranes on tori can be understood in terms of a FourierMukai transformation.
Derived Category Automorphisms from Mirror Symmetry
 DUKE MATH. J
, 2001
"... Inspired by the homological mirror symmetry conjecture of Kontsevich [24], we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth Calabi–Yau variety. ..."
Abstract

Cited by 50 (0 self)
 Add to MetaCart
(Show Context)
Inspired by the homological mirror symmetry conjecture of Kontsevich [24], we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth Calabi–Yau variety.
Duality and equivalence of module categories in noncommutative geometry II: Mukai . . .
, 2006
"... This is the second in a series of papers intended to set up a framework to study categories of modules in the context of noncommutative geometries. In [3] we introduced ..."
Abstract

Cited by 44 (6 self)
 Add to MetaCart
(Show Context)
This is the second in a series of papers intended to set up a framework to study categories of modules in the context of noncommutative geometries. In [3] we introduced
Sheaves on Artin stacks
 J. Reine Angew. Math. (Crelle’s Journal
"... Abstract. We develop a theory of quasi–coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and MoretBailly. We study basic cohomological properties of such sheaves, and prove stack–theoretic versions of Grothendieck’s Fundamental Theorem for prop ..."
Abstract

Cited by 44 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We develop a theory of quasi–coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and MoretBailly. We study basic cohomological properties of such sheaves, and prove stack–theoretic versions of Grothendieck’s Fundamental Theorem for proper morphisms, Grothendieck’s Existence Theorem, Zariski’s Connectedness Theorem, as well as finiteness Theorems for proper pushforwards of coherent and constructible sheaves. We also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to Laumon and Moret–Bailly. 1.1. In the book ([LMB]) the lisseétale topos of an algebraic stack was introduced, and a theory of quasi–coherent and constructible sheaves in this topology was developed. Unfortunately, it was since observed by Gabber and Behrend (independently) that the lisseétale topos is not functorial as asserted in (loc. cit.), and hence the development of the theory of sheaves in this book is not satisfactory “as is”. In addition, since the publication of the book ([LMB]), several new results have been obtained such as finiteness of coherent and étale cohomology ([Fa], [Ol]) and various other consequences of Chow’s Lemma ([Ol]).