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$L^2$-cohomology of quasi-fibered boundary metrics

Chris Kottke, Frédéric Rochon

Abstract

We develop new techniques to compute the weighted $L^2$-cohomology of quasi-fibered boundary metrics (QFB-metrics). Combined with the decay of $L^2$-harmonic forms obtained in a companion paper, this allows us to compute the reduced $L^2$-cohomology for various classes of QFB-metrics. Our results applies in particular to the Nakajima metric on the Hilbert scheme of $n$ points on $\mathbb{C}^2$, for which we can show that the Vafa-Witten conjecture holds. Using the compactification of the monopole moduli space announced by Fritzsch, the first author and Singer, we can also give a proof of the Sen conjecture for the monopole moduli space of magnetic charge 3.

$L^2$-cohomology of quasi-fibered boundary metrics

Abstract

We develop new techniques to compute the weighted -cohomology of quasi-fibered boundary metrics (QFB-metrics). Combined with the decay of -harmonic forms obtained in a companion paper, this allows us to compute the reduced -cohomology for various classes of QFB-metrics. Our results applies in particular to the Nakajima metric on the Hilbert scheme of points on , for which we can show that the Vafa-Witten conjecture holds. Using the compactification of the monopole moduli space announced by Fritzsch, the first author and Singer, we can also give a proof of the Sen conjecture for the monopole moduli space of magnetic charge 3.

Paper Structure

This paper contains 6 sections, 33 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Quasi-fibered boundary metrics
  3. Weighted $L^2$-cohomology of $\operatorname{QFB}$-metrics
  4. $L^2$-cohomology of the Nakajima metric
  5. $L^2$-cohomology of quasi-asymptotically conical metrics of depth 2
  6. $L^2$-cohomology of the moduli space of monopoles

Key Result

Theorem 1

The Sen conjecture holds on $\widetilde{\mathcal{M}}^0_3$ provided the natural $L^2$-metric on $\widetilde{\mathcal{M}}^0_3$ is a $\operatorname{QFB}$-metric as announced in FKS.

Theorems & Definitions (72)

  • Theorem 1
  • Theorem 2
  • Remark
  • Theorem 3
  • Definition 2.1: AM2011ALMP2012DLR
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4: CDRKR1
  • Theorem 2.5: Theorem 17.5 in KR1
  • Remark 2.6
  • ...and 62 more