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On the symplectic structures on moduli space of stable sheaves over a K3 or abelian surface and on Hilbert scheme of points.
Indranil Biswas and Avijit Mukherjee
Fix a smooth very ample curve $C$ on a $K3$ or abelian surface $X$. Let $\mathcal M$ denote 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 of $C$ in $X$, of a line bundle of degree $d$ over $C$, and $s$ is a nonzero section of $F$. Assume $d$ to be sufficiently large such that $F$ has a nonzero section. The pullback of the Mukai symplectic form on moduli spaces of stable sheaves over $X$ is a holomorphic $2$--form on $\mathcal M$. On the other hand, $\mathcal M$ has a map 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 $F$. We prove that the above $2$--form on $\mathcal M$ coincides with the pullback of the symplectic form on Hilbert scheme.