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Moduli spaces of twisted sheaves on a projective variety. (English) Zbl 1118.14013

Mukai, Shigeru (ed.) et al., Moduli spaces and arithmetic geometry. Papers of the 13th International Research Institute of the Mathematical Society of Japan, Kyoto, Japan, September 8–15, 2004. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-38-9/hbk). Advanced Studies in Pure Mathematics 45, 1-30 (2006).
Roughly speaking, an \(\alpha\)-twisted sheaf on a smooth projective variety \(X\) consists of a collection of coherent sheaves on an open cover of \(X\) together with gluing maps on the overlaps of these open sets, such that, on the triple overlaps, the product of these gluing maps coincide with the given Čech 2-cocycle \(\alpha\) representing a torsion class \([\alpha]\in H^2(X,{\mathcal{O}}_X^\ast)\). The category of twisted sheaves and its bounded derived category was studied by A. Căldăraru in his PhD-thesis [Cornell University, 2000]. Twisted sheaves naturally appear as replacements for universal sheaves on non-fine moduli spaces of stable sheaves. Similar to universal sheaves, they can be used to define Fourier-Mukai equivalences between derived categories [A. Căldăraru, Int. Math. Res. Not. 2002, No. 20, 1027–1056 (2002; Zbl 1057.14020)].
The main result of the paper under review shows the existence of a moduli space of stable twisted sheaves on a smooth complex projective variety \(X\). The notion of stability introduced by the author depends on the choice of a locally free twisted sheaf \(G\) on \(X\). He shows, however, that the moduli space does not depend on \(G\). Moreover, it can be shown that this notion of stability is equivalent to one introduced by C. T. Simpson [Publ. Math., Inst. Hautes Étud. Sci. 79, 47–129 (1994; Zbl 0891.14005)] for coherent sheaves of left-modules over the Azumaya algebra \({\mathcal E}nd(G^\vee)\). In fact, instead of working with twisted sheaves, the author prefers to work with ordinary sheaves on a Brauer-Severi variety (a projective bundle which represents \(\alpha\)).
In the final two sections, the author generalizes to the twisted case known results on the moduli spaces of usual stable sheaves on \(K3\) surfaces. In particular, he studies the question of non-emptiness of his new moduli spaces, introduces a symplectic structure on them and a weight-two Hodge structure on their cohomology. The paper finishes with an investigation of the Fourier-Mukai equivalences induced by twisted sheaves in case the moduli space is a surface. The results of this paper constitute the key ingredient for a proof of Căldăraru’s conjecture which is presented in an appendix by D. Huybrechts and P. Stellari [in: Moduli spaces and arithmetic geometry. Papers 13th Int. Res. Inst. Math. Soc. Japan, Kyoto 2004, 31–42 (2006; Zbl 1118.14048)].
For the entire collection see [Zbl 1108.14002].

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J28 \(K3\) surfaces and Enriques surfaces
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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