Abstract
We compute the monodromy of the Hitchin fibration for the moduli space of L-twisted \(SL(n,\mathbb {C})\) and \(GL(n,\mathbb {C})\)-Higgs bundles for any n, on a compact Riemann surface of genus \(g>1\). We require the line bundle L to either be the canonical bundle or satisfy \(deg(L) > 2g-2\). The monodromy group is generated by Picard–Lefschetz transformations associated to vanishing cycles of singular spectral curves. We construct such vanishing cycles explicitly and use this to show that the \(SL(n,\mathbb {C})\) monodromy group is a skew-symmetric vanishing lattice in the sense of Janssen. Using the classification of vanishing lattices over \(\mathbb {Z}\), we completely determine the structure of the monodromy groups of the \(SL(n,\mathbb {C})\) and \(GL(n,\mathbb {C})\) Hitchin fibrations. As an application we determine the image of the restriction map from the cohomology of the moduli space of Higgs bundles to the cohomology of a non-singular fibre of the Hitchin fibration.
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Acknowledgements
We would like to thank Laura Schaposnik, Tamás Hausel and Nigel Hitchin helpful discussions.
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Communicated by Vasudevan Srinivas.
The author is supported by the Australian Research Council Grant DE160100024.
