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

TL;DR

This paper investigates the topological features of Voisin's punctual Hilbert schemes on almost-complex fourfolds, computing Betti numbers, constructing Nakajima operators, and defining tautological bundles with canonical properties in K-theory.

Contribution

It introduces new methods to compute Betti numbers, constructs Nakajima operators, and defines tautological bundles in the context of almost-complex fourfolds, extending previous work to a broader setting.

Findings

Betti numbers of the schemes are explicitly computed

Nakajima operators are constructed for these schemes

Tautological bundles are defined and shown to be canonical in K-theory

Abstract

In this article, we study topological properties of Voisin's punctual Hilbert schemes of an almost-complex fourfold $X$. In this setting, we compute their Betti numbers and construct Nakajima operators. We also define tautological bundles associated with any complex bundle on $X$, which are shown to be canonical in $K$-theory.