Poisson structure on the moduli spaces of sheaves of pure dimension one on a surface

TL;DR

This paper studies the Poisson structure on moduli spaces of pure dimension one sheaves on a surface, revealing that symplectic leaves correspond to fibers over the symmetric power of the surface's singular locus.

Contribution

It proves that the symplectic leaves of the moduli space are fibers over the symmetric power of the singular locus, clarifying the geometric structure induced by the Poisson structure.

Findings

Symplectic leaves are fibers over the symmetric power of the singular locus.

The moduli space inherits a natural Poisson structure from the surface.

The structure relates the geometry of sheaves to the surface's Poisson geometry.

Abstract

Let S be a smooth complex projective surface equipped with a Poisson structure s and also a polarization H. The moduli space M_H(S,P) of stable sheaves on S having a fixed Hilbert polynomial P of degree one has a natural Poisson structure given by s, studied by Tyurin and Bottacin. We prove that the symplectic leaves of M_H(S,P) are the fibers of the natural map from it to the symmetric power of the effective divisor on S given by the singular locus of s.