Topological properties of Hilbert schemes of almost-complex four-manifolds II

In this article, we study the rational cohomology rings of Voisin's Hilbert schemes X-[n] associated with a symplectic compact four-manifold X. We prove that these rings can be universally constructed from H*(X; Q) and c(1) (X), and that Ruan's crepant resolution conjecture holds if c(1) (X) is a torsion class. Next, we prove that for any almost-complex compact four-manifold X, the complex cobordism class of X-[n] depends only on the complex cobordism class of X.