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Asymptotic geometry at infinity of quiver varieties

Panagiotis Dimakis, Frédéric Rochon

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.

Asymptotic geometry at infinity of quiver varieties

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 , 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 -cohomology and prove the Vafa-Witten conjecture. The other is a vanishing theorem in -cohomology for exact wedge -Sasakian metrics generalizing a result of Galicki and Salamon for closed -Sasakian manifolds.

Paper Structure

This paper contains 8 sections, 31 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Quiver varieties
  3. Manifolds with fibered corners
  4. Compactification of Nakajima quiver representation spaces
  5. $L^2$-cohomology of incomplete metrics
  6. Wedge $3$-Sasakian manifolds
  7. A vanishing in $L^2$-cohomology for wedge $3$-Sasakian metrics
  8. Reduced $L^2$-cohomology of quiver varieties

Key Result

Theorem 1.1

The Vafa-Witten conjecture holds for all Nakajima quiver varieties $\mathfrak M_{\zeta}$ under the assumption that $\zeta$ is properly generic in the sense of Definition pg.2 below.

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2: Nakajima1994 Theorem 2.8 and Corollary 4.2
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • ...and 72 more