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

Grivaux, Julien

doi:10.2140/gt.2011.15.261

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

Julien Grivaux

Geometry & Topology 15 (2011) 261–330

DOI: 10.2140/gt.2011.15.261

Abstract

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 $X$ .

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

References

Publication

Received: 29 June 2009

Revised: 13 September 2010

Accepted: 13 November 2010

Published: 14 February 2011

Proposed: Lothar Göttsche

Seconded: Richard Thomas, Jim Bryan

Authors

Julien Grivaux
Centre de Mathématiques et Informatique UMR CNRS 6632 (LATP) Université de Provence 39 rue Frédéric Joliot-Curie 13453 Cedex 13 Marseille France

Volume 15, issue 1 (2011)