On the symplectic structures on moduli space of stable sheaves over a K3 or abelian surface and on Hilbert scheme of points

Fix a smooth very ample curve C on a K3 or abelian surface X. Let $ mathcal{M} $ denote themoduli space of pairs of the form (F, s), where F is a stable sheaf over X whose Hilbert polynomialcoincides with that of the direct image, by the inclusion map of C in X, of a line bundle of degree dover C, and s is a nonzero section of F. Assume d to be sufficiently large such that F has a nonzerosection. The pullback of the Mukai symplectic form on moduli spaces of stable sheaves over X isa holomorphic 2-form on $ mathcal{M} $. On the other hand, $ mathcal{M} $ has a map to a Hilbert scheme parametrizing0-dimensional subschemes of X that sends (F, s) to the divisor, defined by s, on the curve definedby the support of F. We prove that the above 2-form on $ mathcal{M} $ coincides with the pullback of thesymplectic form on the Hilbert scheme.