Homotopy Type of Moduli Spaces of G-Higgs Bundles and Reducibility of the Nilpotent Cone

Abstract

Let $G$ be a real reductive Lie group, and $H^{\mathbb{C}}$ the complexification of its maximal compact subgroup $H\subset G$. We consider classes of semistable $G$-Higgs bundles over a Riemann surface $X$ of genus $g\ge 2$ whose underlying $H^{\mathbb{C}}$-principal bundle is unstable. This allows us to find obstructions to a deformation retract from the moduli space of $G$-Higgs bundles over $X$ to the moduli space of $H^{\mathbb{C}}$-bundles over $X$, in contrast with the situation when $g=1$, and to show reducibility of the nilpotent cone of the moduli space of $G$-Higgs bundles, for $G$ complex.

arXiv comment14 pages

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• Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces
• The topology of moduli spaces of free group representations
• Higgs bundles, the Toledo invariant and the Cayley correspondence
• Higgs bundles for real groups and the Hitchin-Kostant-Rallis section
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Cited by in corpus (1)

• Flawed groups and the topology of character varieties