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

Mukai, Shigeru

Invent. Math. 77, 101-116 (1984).

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

Keywords:

moduli space of simple sheaves; K 3 surface; symplectic structure; symplectic transformations

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