## Symplectic structure of the moduli space of sheaves on an abelian or K 3 surface.(English)Zbl 0565.14002

A sheaf E on a smooth projective surface S is called simple iff the natural injection $$H^ 0(S,{\mathcal O}_ S)\to End_{{\mathcal O}_ S}(E)$$ is an isomorphism. Here it is shown that the moduli space of simple sheaves on an abelian or K 3 surface S is smooth and has a natural holomorphic symplectic structure. Some examples and applications of this result are given. A general construction of elementary symplectic transformations arising from the natural isomorphism $$PT^*{\mathbb{P}}^ n=PT^*({\mathbb{P}}^ n)^*$$ is discussed.
Reviewer: A.Givental’

### MSC:

 14D20 Algebraic moduli problems, moduli of vector bundles 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 14J25 Special surfaces
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