  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.
