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arXiv · 2507.18613

Real Bialynicki-Birula flows in moduli spaces of Higgs bundles

Florent Schaffhauser·Tommaso Scognamiglio
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Abstract

Let $X$ be a compact Riemann surface $X$ of genus $\geqslant 2$ and let $σ:X \to X$ be an anti-holomorphic involution. Using real and quaternionic systems of Hodge bundles, we study the topology of the real locus $\mathbb{R} \mathbf{M}_{\mathrm{Dol}}(r,d)$ of the moduli space of semistable Higgs bundles of rank $r$ and degree $d$ on $X$, for the induced real structure $(E,ϕ) \to (σ^*(\overline{E}),σ^*(\overlineϕ))$. We show in particular that, when $\mathrm{gcd}(r,d)=1$, the number of connected components of $\mathbb{R} \mathbf{M}_{\mathrm{Dol}}(r,d)$ coincides with that of $\mathbb{R} \mathrm{Pic}_d(X)$, which is well-known.

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Florent Schaffhauser, Tommaso Scognamiglio. 2025-07-24. Real Bialynicki-Birula flows in moduli spaces of Higgs bundles. https://arxiv.org/abs/2507.18613

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