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

Julien Grivaux (1, 2)

1 Department of Mathematics [Imperial College London]
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

In this article, we study the rational cohomology rings of Voisin's punctual Hilbert schemes $X^{[n]}$ associated to a symplectic compact fourfold $X$. We prove that these rings can be universally constructed from $H^*(X,\mathbb{Q})$ 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 fourfold $X$, the complex cobordism class of $X^{[n]}$ depends only on the cobordism class of $X$.

Mots clés: almost-complex four-manifold; symplectic four-manifold; Hilbert schemes of points; cohomological crepant resolution conjecture

Domaines: Géométrie algébrique [math.AG]; Géométrie symplectique [math.SG]

Julien Grivaux. Topological properties of Hilbert schemes of almost-complex four-manifolds II. Geometry and Topology, 2011, 15 (1), pp.261-330. ⟨10.2140/gt.2011.15.261⟩. ⟨hal-01301345⟩