Topological properties of punctual Hilbert schemes of almost-complex fourfolds (II)

TL;DR

This paper investigates the cohomology rings and cobordism classes of punctual Hilbert schemes of almost-complex fourfolds, establishing universal constructions and confirming Ruan's crepant resolution conjecture under certain conditions.

Contribution

It demonstrates that the cohomology rings of these schemes can be universally built from the base fourfold's cohomology and Chern class, and confirms Ruan's conjecture when the first Chern class is torsion.

Findings

Cohomology rings are universally constructed from base fourfold data.

Ruan's crepant resolution conjecture holds if c_1(X) is torsion.

Cobordism class of schemes depends only on the base fourfold's cobordism class.

Abstract

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 $X$.