Results 1  10
of
183
Braid group actions on derived categories of coherent sheaves
 DUKE MATH. J
, 2001
"... This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is ..."
Abstract

Cited by 255 (8 self)
 Add to MetaCart
(Show Context)
This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim X ≥ 2, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.
A holomorphic Casson invariant for CalabiYau 3folds, and bundles on K3 fibrations
 J. DIFFERENTIAL GEOM
, 2000
"... We briefly review the formal picture in which a CalabiYau nfold is the complex analogue of an oriented real nmanifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a CalabiYau 3fol ..."
Abstract

Cited by 199 (8 self)
 Add to MetaCart
We briefly review the formal picture in which a CalabiYau nfold is the complex analogue of an oriented real nmanifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a CalabiYau 3fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of [LT], [BF] in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in P 3, and Donaldson – and GromovWitten – like invariants of Fano 3folds. It also allows us to define the holomorphic Casson invariant of a CalabiYau 3fold X, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general K3 fibration X, enabling us to compute the invariant for some ranks and Chern classes, and equate it to GromovWitten invariants of the “Mukaidual” 3fold for others. As an example the invariant is shown to distinguish Gross’ diffeomorphic 3folds. Finally the Mukaidual 3fold is shown to be CalabiYau and its cohomology is related to that of X.
Triangulated categories of singularities and Dbranes in LandauGinzburg models
, 2003
"... ..."
(Show Context)
Derived categories of coherent sheaves and triangulated categories of singularities
, 2005
"... ..."
(Show Context)
Equivalences of derived categories and K3 surfaces
, 1996
"... Abstract. We consider derived categories of coherent sheaves on smooth projective varieties. We prove that any equivalence between them can be represented by an object on the product. Using this, we give a necessary and sufficient condition for equivalence of derived categories of two K3 surfaces. ..."
Abstract

Cited by 126 (7 self)
 Add to MetaCart
(Show Context)
Abstract. We consider derived categories of coherent sheaves on smooth projective varieties. We prove that any equivalence between them can be represented by an object on the product. Using this, we give a necessary and sufficient condition for equivalence of derived categories of two K3 surfaces.
Equivalences of triangulated categories and FourierMukai transforms
 Bull. London Math. Soc
, 1999
"... Abstract. We give a condition for an exact functor between triangulated categories to be an equivalence. Applications to FourierMukai transforms are discussed. In particular we obtain a large number of such transforms for K3 surfaces. 1. ..."
Abstract

Cited by 124 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We give a condition for an exact functor between triangulated categories to be an equivalence. Applications to FourierMukai transforms are discussed. In particular we obtain a large number of such transforms for K3 surfaces. 1.
Noncommutative curves and noncommutative surfaces
 Bulletin of the American Mathematical Society
"... Abstract. In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative grad ..."
Abstract

Cited by 91 (8 self)
 Add to MetaCart
(Show Context)
Abstract. In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing
Flops and derived categories
 Invent. Math
"... This paper contains some applications of FourierMukai techniques to problems in birational geometry. The main new idea is that flops occur naturally as moduli ..."
Abstract

Cited by 90 (3 self)
 Add to MetaCart
(Show Context)
This paper contains some applications of FourierMukai techniques to problems in birational geometry. The main new idea is that flops occur naturally as moduli
Derived Categories of Twisted Sheaves on CalabiYau Manifolds
, 2000
"... This dissertation is primarily concerned with the study of derived categories of twisted sheaves on CalabiYau manifolds. Twisted sheaves occur naturally in a variety of problems, but the most important situation where they are relevant is in the study of moduli problems of semistable sheaves on var ..."
Abstract

Cited by 86 (3 self)
 Add to MetaCart
This dissertation is primarily concerned with the study of derived categories of twisted sheaves on CalabiYau manifolds. Twisted sheaves occur naturally in a variety of problems, but the most important situation where they are relevant is in the study of moduli problems of semistable sheaves on varieties. Although universal sheaves may not exist as such, in many cases one can construct them as twisted universal sheaves. In fact, the twisting is an intrinsic property of the moduli problem under consideration. A fundamental construction due to Mukai associates to a universal sheaf a transform between the derived category of the original space and the derived category of the moduli space, which often turns out to be an equivalence. In the present work we study what happens when the universal sheaf is replaced by a twisted one. Under these circumstances we obtain a transform between the de