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 Y$Y$. The first example is the Fano scheme of lines in Y$Y$. 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∩Y$H\cap Y$ of Y$Y$ and are cokernels of the Pfaffian representations of H∩Y$H\cap Y$.