Symplectic structures on moduli spaces of sheaves via the Atiyah class

It is proven that the composition of the Yoneda coupling with the semiregularity map is a closed 2-form on moduli spaces of sheaves. Two examples are given when this 2-form is symplectic. Both of them are moduli spaces of torsion sheaves on the cubic 4-fold YY. The first example is the Fano scheme of lines in YY. Beauville and Donagi showed that it is symplectic but did not construct an explicit symplectic form on it. We prove that our construction provides a symplectic form. The other example is the moduli space of torsion sheaves which are supported on the hyperplane sections H∩YH∩Y of YY and are cokernels of the Pfaffian representations of H∩YH∩Y.