Real Bialynicki-Birula flows in moduli spaces of Higgs bundles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908464950804480 |
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| author | Schaffhauser, Florent Scognamiglio, Tommaso |
| author_facet | Schaffhauser, Florent Scognamiglio, Tommaso |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18613 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Real Bialynicki-Birula flows in moduli spaces of Higgs bundles Schaffhauser, Florent Scognamiglio, Tommaso Algebraic Geometry Algebraic Topology Primary 14D20, Secondary 14P25 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. |
| title | Real Bialynicki-Birula flows in moduli spaces of Higgs bundles |
| topic | Algebraic Geometry Algebraic Topology Primary 14D20, Secondary 14P25 |
| url | https://arxiv.org/abs/2507.18613 |