Closed differential forms on moduli spaces of sheaves

Let X be a smooth projective variety, and let M be a moduli space of stable sheaves on X. For any flat family E of coherent sheaves on X parametrized by a smooth scheme Y, and for any integer m, with 1 <= m <= dim X, we construct a closed differential form \Omega = \Omega_E on Y with values in H^m(X, O_X). By using the vector-valued differential form \Omega we then prove that the choice of a (non-zero) differential m-form \sigma on X, \sigma \in H^0(X, \Omega^m_X), determines, in a natural way, a closed differential m-form \Omega_{\sigma} on M.