Poisson structures on Hilbert schemes of points of a surface and integrable systems

In this paper we prove that if S is a Poisson surface, i.e., a smooth algeb raic surface with a Poisson structure, the Hilbert scheme of points of S ha s a natural Poisson structure, induced by the one of S. This generalizes pr evious results obtained by A. Beauville [B1] and S. Mukai [M2] in the sympl ectic case, i.e., when S is an abelian or K3 surface. Finally we apply our results to give some examples of integrable Hamiltonian systems naturally d efined on these Hilbert schemes. In the simple case S = P-2 We obtain by th is construction a large class of integrable systems, which includes the one s studied by P. Vanhaecke in [V1] and, more generally, in [V2].