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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
##### Keywords:
vector bundles; Brauer group; twisted sheaves
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