Poisson structures on Hilbert schemes of points of a surface and integrable systems
The author proves that if \(S\) is a Poisson surface, i.e. a smooth projective surface with a Poisson structure, then the Hilbert scheme of points of \(S\) has a natural Poisson structure, induced by the one of \(S\). This generalizes previous results obtained by \textit{A. Beauville} [J. Differ. Geom. 18, 755-782 (1983; Zbl 0537.53056)] and \textit{S. Mukai} [Invent. Math. 77, 101-116 (1984; Zbl 0565.14002)] in the symplectic case, i.e. when \(S\) is an Abelian or K3 surface. Finally, the author applies his results to give some examples of integrable Hamiltonian systems naturally defined on these Hilbert schemes. In the case \(S= \mathbb{P}^2\) he obtains a large class of integrable systems, which includes the ones studied by \textit{P. Vanhaecke} [Prog. Math. 145, 187-212 (1997; Zbl 0873.58038)].
scientific article; zbMATH DE number 1239594