Derived category

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Moduli space

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Coherent sheaf

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Advanced studies in pure mathematics

IntroductionLet X be a smooth projective variety over C. Let a := { aijk E H 0 (Ui n U1 n Uk, 0~)} be a 2-cocycle representing a torsion class [a] E H 2 (X,O~).An a-twisted sheaf E := {(Ei,'PiJ)} is a collection of sheaves Ei on ui and isomorphisms 'Pij : EiiUinUj -+ EJIUinUj such that 'Pii = idei, 'PJi = 'Pi/ and 'Pki o 'PJk o 'PiJ = aijk idei.We assume that there is a locally free a-twisted sheaf, that is, a gives an element of the Brauer group Br( X).A twisted sheaf naturally appears if we consider a non-fine moduli space lif of the usual stable sheaves on X. Indeed the transition functions of the local universal families satisfy the patching condition up to the multiplication by constants and gives a twisted sheaf.If the patching condition is satisfied, i.e., the moduli space M is fine, than the universal family defines an integral functor on the bounded derived categories of coherent sheaves D(M) -+ D(X).Assume that X is a K3 surface and dimM =dim X.Then Mukai, Orlov and Bridgeland showed that the integral functor is the Fourier-Mukai functor, i.e., it is an equivalence of the categories.In his thesis [C2], Cald3xaru studied the category of twisted sheaves and its bounded derived category.In particular, he generalized Mukai, Orlov and Bridgeland's results on the Fourier-Mukai transforms to non-fine moduli spaces on a K3 surface.For the usual derived category, Orlov [Or] showed that the equivalence class is described in terms of the Hodge structure of the Mukai lattice.Caldararu conjectured that a similar result also holds for the derived

Moduli spaces of twisted sheaves on a projective variety