

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
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Submission date: 02. Jan. 2002 (revised version: February 2003)
Pages: 11
published in: Archiv der Mathematik, 80 (2003) 5, p. 507-515
DOI number (of the published article): 10.1007/s00013-003-4613-4
Bibtex
MSC-Numbers: 53D30, 14J60, 14C05
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Abstract:
Fix a smooth very ample curve C on a K3 or abelian surface
X. Let 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
. On the other
hand,
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
coincides with the pullback
of the symplectic form on Hilbert scheme.