Poisson structure on the moduli spaces of sheaves of pure dimension one on a surface
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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.
KeywordsPoisson surface Torsion sheaf Poisson moduli space Symplectic leaf
Mathematics Subject Classification (2010)14J60 53D17 32J15
We thank N. Fakhruddin and O. Villamayor for useful discussions. We thank the International Center for Theoretical Sciences at Bangalore for hospitality while this work was being completed. The first author is partially supported by a J. C. Bose Fellowship. The second author is partially supported by the European Union (Marie Curie IRSES Fellowship within the 7th Framework Programme under agreement 612534 MODULI), and Ministerio de Ciencia, Innovación y Universidades (Grants MTM2016-79400-P and ICMAT Severo Ochoa project SEV-2015-0554).
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