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

Abstract

In this article, we study the rational cohomology rings of Voisin’s Hilbert schemes $X^{\left[n\right]}$ associated with a symplectic compact four-manifold $X$. We prove that these rings can be universally constructed from $H^{\ast }\left(X,\mathbb{Q}\right)$ and $c_1\left(X\right)$, and that Ruan’s crepant resolution conjecture holds if $c_1\left(X\right)$ is a torsion class. Next, we prove that for any almost-complex compact four-manifold $X$, the complex cobordism class of $X^{\left[n\right]}$ depends only on the complex cobordism class of $X$.