On the symplectic structures on moduli space of stable sheaves over a \(K\)3 or abelian surface and on Hilbert scheme of points (Q1411080)

Let \(X\) be a \(K3\) or abelian surface over the field of complex numbers. Fix a very ample curve \(C\) on \(X\). Let \(\mathcal M\) be the moduli space of pairs of the form \((F, s)\), where \(F\) is a stable sheaf over \(X\) whose Hilbert polynomial coincides with that of the direct image, by the inclusion map \(C\hookrightarrow X\), of a line bundle of degree \(d\) over \(C\), and \(s\) is a nonzero section of \(F\). Assume that \(d\) is so large that \(F\) has a nonzero section. There is a holomorphic \(2\)-form on \(\mathcal M\), namely, the pullback of the Mukai symplectic form on moduli space of stable sheaves over \(X\). On the other hand, there is a map of \(\mathcal M\) to a Hilbert scheme parametrizing \(0\)-dimensional subschemes of \(X\) that sends \((F, s)\) to the divisor, defined by \(s\), on the curve defined by the support of \(F\). The authors prove that the above \(2\)-form on \(\mathcal M\) coincides with the pullback of the symplectic form on the Hilbert scheme.