Asymptotic geometry at infinity of quiver varieties

TL;DR

This paper demonstrates that the Nakajima metric on quiver varieties exhibits quasi-asymptotic conical geometry under certain conditions, enabling the computation of its reduced L^2-cohomology and supporting the Vafa-Witten conjecture.

Contribution

It establishes the quasi-asymptotic conical nature of Nakajima metrics on quiver varieties, facilitating advanced cohomological analysis and conjecture verification.

Findings

Nakajima metric is quasi-asymptotically conical under generic parameters.

The metric has bounded geometry and maximal volume growth.

Supports computation of reduced L^2-cohomology and proves the Vafa-Witten conjecture.

Abstract

Using an approach developed by Melrose to study the geometry at infinity of the Nakajima metric on the reduced Hilbert scheme of points on $\mathbb{C}^2$, we show that the Nakajima metric on a quiver variety is quasi-asymptotically conical (QAC) whenever its defining parameters satisfy an appropriate genericity assumption. As such, it is of bounded geometry and of maximal volume growth. Being QAC is one of two main ingredients allowing us to use the work of Kottke and the second author to compute its reduced $L^2$-cohomology and prove the Vafa-Witten conjecture. The other is a vanishing theorem in $L^2$-cohomology for exact wedge $3$-Sasakian metrics generalizing a result of Galicki and Salamon for closed $3$-Sasakian manifolds.