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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
3/2002

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

Abstract

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.

Received:
Jan 2, 2002
Published:
Jan 2, 2002
MSC Codes:
53D30, 14J60, 14C05

Related publications

inJournal
2003 Repository Open Access
Indranil Biswas and Avijit Mukherjee

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

In: Archiv der Mathematik, 80 (2003) 5, pp. 507-515