Symplectic structure of the moduli space of sheaves on an abelian or K 3 surface (Q1058557)

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.