Lift-independence problem in the $P$-adic Simpson correspondence for curves
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913171953942528 |
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| author | Pan, Xiangyu Yu, Jiahong |
| author_facet | Pan, Xiangyu Yu, Jiahong |
| contents | Let $X$ be a proper smooth rigid analytic variety over a complete algebraically closed field $p$-adic field $\mathbf C$. Fix an continuation $\mathrm{Exp}$ of $\exp$. Faltings (in the curve case) and Heuer showed that any lifting $\widetilde X$ of $X$ over $\mathbf{B}_{\rm dR}^+/t^2$ induces an equivalence bewteen the category of Higgs bundles on $X_{\mathrm{\acute{e}t}}$ and the category of $v$-bundles on $X_v$. In this paper, we aim to study how the equivalence depends on the choice of such a lifting $\widetilde X$ when $X$ is a curve of genus $g\geqslant 2$. More precisely, we call a Higgs bundle lift-independent if it always corresponds to the same $v$-bundle under $p$-adic Simpson correspondence with respect to any lifting $\widetilde X$ and then we will show that
(1) There exists some $r(g)\geqslant \sqrt{g-1}$ such that any semistable lift-independent Hitchin-small Higgs bundle of rank $r\leqslant r(g)$ has zero Higgs field.
(2) There always exists a semistable Higgs bundle of degree $0$ with non-zero Higgs field that is lift-independent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29947 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lift-independence problem in the $P$-adic Simpson correspondence for curves Pan, Xiangyu Yu, Jiahong Algebraic Geometry Number Theory Let $X$ be a proper smooth rigid analytic variety over a complete algebraically closed field $p$-adic field $\mathbf C$. Fix an continuation $\mathrm{Exp}$ of $\exp$. Faltings (in the curve case) and Heuer showed that any lifting $\widetilde X$ of $X$ over $\mathbf{B}_{\rm dR}^+/t^2$ induces an equivalence bewteen the category of Higgs bundles on $X_{\mathrm{\acute{e}t}}$ and the category of $v$-bundles on $X_v$. In this paper, we aim to study how the equivalence depends on the choice of such a lifting $\widetilde X$ when $X$ is a curve of genus $g\geqslant 2$. More precisely, we call a Higgs bundle lift-independent if it always corresponds to the same $v$-bundle under $p$-adic Simpson correspondence with respect to any lifting $\widetilde X$ and then we will show that (1) There exists some $r(g)\geqslant \sqrt{g-1}$ such that any semistable lift-independent Hitchin-small Higgs bundle of rank $r\leqslant r(g)$ has zero Higgs field. (2) There always exists a semistable Higgs bundle of degree $0$ with non-zero Higgs field that is lift-independent. |
| title | Lift-independence problem in the $P$-adic Simpson correspondence for curves |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2605.29947 |