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The intersection form on moduli spaces of twisted \(PGL_n\)-Higgs bundles vanishes

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Abstract

Hausel and Rodriguez-Villegas conjectured that the intersection form on the moduli space of stable \(\textsf {PGL}_n\)-Higgs bundles on a curve vanishes if the degree is coprime to n. In this note we prove this conjecture. Along the way we show that moduli spaces of stable chains are irreducible for stability parameters larger than the stability condition induced form stability of Higgs bundles.

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Notes

  1. Unfortunately this Lemma is only formulated for stable chains, because it allows the case \(\alpha =\alpha _{{{\mathrm{Higgs}}}}\). However for \(\alpha >\alpha _{{{\mathrm{Higgs}}}}\) the first inequality in the proof is always strict, so that the vanishing of \(H^2\) also holds for strictly semistable chains if \(\alpha >\alpha _{{{\mathrm{Higgs}}}}\).

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Acknowledgments

I am greatly indebted to T. Hausel. Discussions with him are the reason why this article exists and it was his idea to use the Poincaré-Hopf theorem to finish the proof of Theorem 1. A large part of this work was done while visiting his group at EPFL. I also thank the referee for many very helpful comments and suggestions. A part of the work was funded through the SFB/TR 45 of the DFG.

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  1. Fachbereich Mathematik, Universität Duisburg–Essen, Universitätsstrasse 2, 45117, Essen, Germany

    Jochen Heinloth

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Heinloth, J. The intersection form on moduli spaces of twisted \(PGL_n\)-Higgs bundles vanishes. Math. Ann. 365, 1499–1526 (2016). https://doi.org/10.1007/s00208-015-1301-1

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