Poisson structures on moduli spaces of sheaves over Poisson surfaces (Q1899742)

The author introduces and studies the notion of Poisson surface, i.e., a smooth algebraic surface endowed with a nonzero Poisson structure. He proves that the moduli space \({\mathcal M}\) of stable sheaves on such a surface \(S\) is smooth and the choice of a Poisson structure on \(S\) canonically determines a Poisson structure on the moduli space \({\mathcal M}\). This result generalizes previous results obtained by \textit{S. Mukai} [Invent. Math. 77, 101-116 (1984; Zbl 0565.14002)], for abelian or \(K3\) surfaces, and by \textit{A. N. Tyurin} [Math. USSR, Izv. 33, No. 1, 139-177 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 4, 813- 852 (1988; Zbl 0673.14021)]. Finally, some remarks are made on the rank of the Poisson structure on \({\mathcal M}\).