Symplectic structures on moduli spaces of sheaves via the Atiyah class

Abstract

Several situations are known when a holomorphic 2-form on a moduli space of sheaves over some base S is induced by a holomorphic 2-form on S. Moreover, the closedness of the 2-form on the base implies the closedness on the moduli space, which provides a stock of symplectic structures on moduli spaces (Mukai, Kobayashi, O'Grady, Tyurin, Huybrechts--Lehn). A parallel theory was developed for bivector fields and Poisson structures (Bottacin). However, there exist symplectic moduli spaces of sheaves over bases that have no holomorphic forms at all. A well-known example (Beauville--Donagi) is the family of lines on the cubic 4-fold Y, which can be thought of as the moduli space parameterizing the structure sheaves of lines in Y. The paper produces a general construction of closed 2-forms on the moduli spaces of sheaves, using the Atiyah class of the sheaves, and proves that this construction provides symplectic structures in 2 examples: the first one is the family of lines on Y, and the second one is the moduli space of sheaves which are supported on the hyperplane sections of Y and are cokernels of the Pfaffian representations of those hyperplane sections.