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.