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

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, Tyurin (Math USSR Izvest 33:139–177, 1989), Bottacin (Invent Math 121:421–436, 1995). 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.