Real Bialynicki-Birula flows in moduli spaces of Higgs bundles

Abstract

Let $X$ be a compact Riemann surface $X$ of genus $⩾2$ and let $\sigma :X\to X$ 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 $(E,\varphi )\to ({\sigma }^{\ast }(\overline{E}),{\sigma }^{\ast }(\overline{\varphi }))$. We show in particular that, when $\gcd (r,d)=1$, 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.

26 pages, 2 figures