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arXiv · 2103.16655

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

Chris Kottke·Frédéric Rochon
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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.

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Chris Kottke, Frédéric Rochon. 2021-03-30. $L^2$-cohomology of quasi-fibered boundary metrics. https://arxiv.org/abs/2103.16655

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