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 algebraic surface with a Poisson structure, the Hilbert scheme of points of S has a natural Poisson structure, induced by the one of S. This generalizes previous results obtained by A. Beauville and S. Mukai in the symplectic 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 defined on these Hilbert schemes. In the simple case S = P^2 we obtain with this construction a large class of integrable systems, which includes the ones studied by P. Vanhaecke.