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Nonfine moduli spaces of sheaves on \(K3\) surfaces. (English) Zbl 1057.14020
Let \(S\) be a \(K3\) surface and \(M\) a moduli space of sheaves on \(S\). When \(M\) is of dimension \(2\) it is again a \(K3\) surface and by an important result of S. Mukai [in: Vector bundles on algebraic varieties. Tata Inst. Fundam. Res. 11, 341–413 (1987; Zbl 0674.14023)] there exists an Hodge isometry of transcendental periods \(T_S \otimes {\mathbb Q} \to T_M \otimes \mathbb Q\) induced by an inclusion of lattices of the same rank. Moreover when \(M\) is fine, as a moduli space, then by a result of D. O. Orlov [J. Math. Sci., New York 84, 1361–1381 (1997; Zbl 0938.14019)] the derived categories of \(X\) and \(Y\) are equivalent and in particular \(T_S \cong T_M\). Summing up we have the following equivalent statements:
(1) \(M\) is a fine moduli space of sheaves on \(S\);
(2) there exists an Hodge isometry \(T_S \cong T_M\) (resp. \(\widetilde{H}(S) \cong \widetilde{H}(M)\)) between the transcendental (resp. Mukai) lattices of \(S\) and \(M\);
(3) \(D^b(S) \cong D^b(M)\).
In this paper non-fine moduli spaces are considered. It is a difficult problem to find a condition analogue to the third one given above in the non-fine case. The author gets a generalization of the implications \((1) \to (2)\) and \((1) \to (3)\) in the non-fine case replacing the universal sheaf inducing the equivalence of \((3)\) by a “twisted universal sheaf” which yields an equivalence \(D^b(M,\alpha) \cong D^b(S)\). Here \(\alpha \in \text{Br}(M)\) is an element of the Brauer group of \(M\) determined by the original moduli problem data and \(D^b(M,\alpha)\) is the derived category of \(\alpha\)-twisted sheaves on \(M\). Section 2. contains a brief review of the Brauer group and twisted sheaves. The conjecture 1.4 gives some hints in generalizing the equivalence \((2) \leftrightarrow (3)\) in the non-fine case. Recently D. Huybrechts and P. Stellari [Equivalence of twisted \(K3\) surfaces, Preprint, http://arxiv.org/abs/math.AG/0409030] introduced the notion of twisted Hodge structure and gave a reformulation of the above conjecture. Then by using a result of K. Yoshioka [Moduli spaces of twisted sheaves on a projective variety, Preprint, http://arxiv.org/abs/math.AG/0411538] they gave a proof of this reformulated conjecture [Proof of Caldararu conjecture, Preprint, http://arxiv.org/abs/math.AG/0411541].

MSC:
14D22 Fine and coarse moduli spaces
14J28 \(K3\) surfaces and Enriques surfaces
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
18E30 Derived categories, triangulated categories (MSC2010)
14D20 Algebraic moduli problems, moduli of vector bundles
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