Volume 15, issue 1 (2011)

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

Julien Grivaux

Geometry & Topology 15 (2011) 261–330
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 ${H}^{\ast }\left(X,ℚ\right)$ 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 $X$.

Keywords
Hilbert schemes of points, symplectic four-manifold, almost-complex four-manifold, cohomological crepant resolution conjecture
Mathematical Subject Classification 2000
Primary: 32Q60
Secondary: 14C05, 14J35