Moduli Spaces of Twisted Sheaves on a Projective Variety

Let X be a smooth projective variety over C. Let α := {αijk ∈ H(Ui ∩ Uj ∩ Uk,O X)} be a 2-cocycle representing a torsion class [α] ∈ H2(X,O X). An α-twisted sheaf E := {(Ei, φij)} is a collection of sheaves Ei on Ui and isomorphisms φij : Ei|Ui∩Uj → Ej|Ui∩Uj such that φii = idEi , φji = φ ij and φki ◦ φjk ◦ φij = αijk idEi . We assume that there is a locally free α-twisted sheaf, that is, α gives an element of the Brauer group Br(X). A twisted sheaf naturally appears if we consider a non-fine moduli space M 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 = dimX . Than 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], Căldăraru 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. Căldăraru conjectured that a similar result also holds for the derived category of twisted sheaves. Recently this conjecture was modified and proved by Huybrechts and Stellari, if ρ(X) ≥ 12 in [H-St]. As is well-known, twisted sheaves also appear if we consider a projective bundle over X . In this paper, we define a notion of the stability for a twisted sheaf and construct the moduli space of stable twisted sheaves on X . We also construct a projective compactification of the moduli space by adding the S-equivalence classes of semi-stable twisted sheaves. In particular if H(X,OX) = 0 (e.g. X is a K3 surface), then the moduli space of locally free twisted sheaves is the moduli space of projective bundles over X . Thus we compactify the moduli space of projective bundles by using twisted sheaves. The idea of the construction is as follows. We consider a twisted sheaf as a usual sheaf on the Brauer-Severi variety. Instead of using the Hilbert polynomial associated to an ample line bundle on the Brauer-Severi variety, we use the Hilbert polynomial associated to a line bundle coming from X in order to define the stability. Then the construction is a modification of Simpson’s construction of the moduli space of usual sheaves (cf. [Y3]). M. Lieblich told us that our stability condition coincides with Simpson’s stability for modules over the associated Azumaya algebra via Morita equivalence. Hence the construction also follows from Simpson’s moduli space [S, Thm. 4.7] and the valuative criterion for properness. In section 3, we consider the moduli space of twisted sheaves on a K3 surface. We generalize known results on the moduli space of usual stable sheaves to the moduli spaces of twisted stable sheaves (cf. [Mu1], [Y1]). In particular, we consider the non-emptyness, the deformation type and the weight 2 Hodge structure. Then we can consider twisted version of the Fourier-Mukai transform by using 2 dimensional moduli spaces, which is done in section 4. As an application of our results, Huybrechts and Stellari [H-St2] prove Căldăraru’s conjecture generally. Since our main example of twisted sheaves are those on K3 surfaces or abelian surfaces, we consider twisted sheaves over C. But some of the results (e.g., subsection 2.2) also hold over any field. E. Markman and D. Huybrechts communicated to the author that M. Lieblich independently constructed the moduli of twisted sheaves. In his paper [Li], Lieblich developed a general theory of twisted sheaves in terms of algebraic stack and constructed the moduli space intrinsic way. He also studied the moduli spaces of twisted sheaves on surfaces. There are also some overlap with a paper by Norbert Hoffmann and Ulrich Stuhler [Ho-St]. They also constructed the moduli space of rank 1 twisted sheaves and studied the symplectic structure of the moduli space.