The integrality conjecture and the cohomology of preprojective stacks

TL;DR

This paper investigates the Borel-Moore homology of preprojective algebra stacks, establishing purity and positivity results, and applying these to cohomological Hall algebras and related geometric structures.

Contribution

It introduces BPS sheaves for stacks of preprojective algebra modules, proves their purity, and applies cohomological wall-crossing and integrality theorems to new geometric and algebraic contexts.

Findings

Proved purity of BPS cohomology for preprojective algebra stacks.

Established positivity of certain Kac polynomials.

Determined the critical cohomology of Hilbert schemes of points in three-space.

Abstract

We study the Borel-Moore homology of stacks of representations of preprojective algebras $\Pi_Q$, via the study of the DT theory of the undeformed 3-Calabi-Yau completion $\Pi_Q[x]$. Via a result on the supports of the BPS sheaves for $\Pi_Q[x]$-mod, we prove purity of the BPS cohomology for the stack of $\Pi_Q[x]$-modules, and define BPS sheaves for stacks of $\Pi_Q$-modules. These are mixed Hodge modules on the coarse moduli space of $\Pi_Q$-modules that control the Borel-Moore homology and geometric representation theory associated to these stacks. We show that the hypercohomology of these objects is pure, and thus that the Borel-Moore homology of stacks of $\Pi_Q$-modules is also pure. We transport the cohomological wall-crossing and integrality theorems from DT theory to the category of $\Pi_Q$-modules. Among these and other applications, we use our results to prove positivity of a number of "restricted" Kac polynomials, determine the critical cohomology of $\mathrm{Hilb}_n(\mathbb{A}^3)$, and the Borel-Moore homology of genus one character stacks, as well as various applications to the cohomological Hall algebras associated to Borel-Moore homology of stacks of preprojective algebras, including the PBW theorem, and torsion-freeness.