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\(L^{2}\)-Cohomology of quasi-fibered boundary metrics

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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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Notes

  1. A slightly different convention is used in [16].

  2. We use the convention opposite to the one of [11] to be consistent with the partial order of the boundary hypersurfaces of the \(\operatorname{QAC}\)-compactification of \(\operatorname{Hilb}^{n}_{0}(\mathbb{C}^{2})\); see (4.11).

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Acknowledgements

The authors are grateful to Rafe Mazzeo, Richard Melrose and Michael Singer for helpful discussions, as well as to an anonymous referee for useful suggestions. CK was supported by NSF Grant number DMS-1811995. This paper is also based in part on work supported by NSF under grant DMS-1440140 while CK was in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Fall 2019. FR was supported by NSERC and a Canada research chair.

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  1. New College of Florida, Sarasota, USA

    Chris Kottke

  2. Département de Mathématiques, Université du Québec à Montréal, Montréal, Canada

    Frédéric Rochon

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  1. Chris Kottke
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Kottke, C., Rochon, F. \(L^{2}\)-Cohomology of quasi-fibered boundary metrics. Invent. math. (2024). https://doi.org/10.1007/s00222-024-01253-5

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