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. 1.

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 algebro-geometric 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.

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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. 1

This paper contains some applications of Fourier-Mukai techniques to problems in birational geometry. The main new idea is that flops occur naturally as moduli

In this paper we describe how Grothendieck groups of coherent sheaves and locally free sheaves can be used to describe type II D-branes, in the case that all D-branes are wrapped on complex varieties and all connections are holomorphic. Our proposal is in the same spirit as recent discussions of K-theory and D-branes; within the restricted class mentioned, Grothendieck groups encode a choice of connection on each D-brane worldvolume, in addition to information about the C ∞ bundles. We also point out that derived categories can also be used to give insight into D-brane constructions, and analyze how a Z2 subset of the T-duality group acting on D-branes on tori can be understood in terms of a Fourier-Mukai transformation.

In this review we study BPS D-branes on Calabi–Yau threefolds. Such D-branes naturally divide into two sets called A-branes and B-branes which are most easily understood from topological field theory. The main aim of this paper is to provide a self-contained guide to the derived category approach to B-branes and the idea of Π-stability. We argue that this mathematical machinery is hard to avoid for a proper understanding of B-branes. A-branes and B-branes are related in a very complicated and interesting way which ties in with the “homological mirror symmetry ” conjecture of Kontsevich. We motivate and exploit this form of mirror symmetry. The examples of the quintic 3-fold, flops and orbifolds are discussed at some length. In the latter

Abstract. Given a non-singular variety with a K3 fibration π: X → S we construct dual fibrations ˆπ: Y → S by replacing each fibre Xs of π by a two-dimensional moduli space of stable sheaves on Xs. In certain cases we prove that the resulting scheme Y is a non-singular variety and construct an equivalence of derived categories of coherent sheaves Φ: D(Y) → D(X). Our methods also apply to elliptic and abelian surface fibrations. As an application we show how the equivalences Φ identify certain moduli spaces of stable bundles on elliptic threefolds with Hilbert schemes of curves. 1.

The purpose of this paper is to extend the conjecture stated in the paper [5] to the logarithmic case and prove some supporting evidences. [5] Conjecture

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