Lift-independence problem in the p-adic Simpson correspondence for curves
Abstract
Let X be a proper smooth rigid analytic variety over a complete algebraically closed p-adic field C. Fix a continuation Exp of exp. Faltings, in the curve case, and Heuer showed that any lifting X̃ of X over BdR+/t2 induces an equivalence between the category of Higgs bundles on Xét and the category of v-vector bundles on Xv. Changing X̃ is known to alter the correspondence nontrivially. This raises the natural question of how the isomorphism class of the Higgs bundle associated with a fixed v-vector bundle varies with X̃, a question that has not been systematically studied. In this paper, we address this question when X is a curve of genus g≥ 2. More precisely, we call a Higgs bundle lift-independent if it corresponds to the same v-bundle under the p-adic Simpson correspondence for every lifting X̃. We prove the following results. (1) Every lift-independent Higgs bundle of rank r≤ g is nilpotent. (2) Every semistable lift-independent Higgs bundle of rank r≤ √g+1 has zero Higgs field. (3) In contrast, vanishing fails even within the class singled out by Faltings' conjecture: there always exists a lift-independent semistable Higgs bundle of degree $0$ with nonzero Higgs field.
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BibTeX
@misc{pan2026liftindependence,
title = {{Lift-independence problem in the $p$-adic Simpson correspondence for curves}},
author = {Xiangyu Pan and Jiahong Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.29947},
eprint = {2605.29947},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.29947},
}
RIS
TY - UNPB
TI - Lift-independence problem in the p-adic Simpson correspondence for curves
AU - Xiangyu Pan
AU - Jiahong Yu
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2605.29947
UR - https://arxiv.org/abs/2605.29947
AB - Let X be a proper smooth rigid analytic variety over a complete algebraically closed p-adic field C. Fix a continuation Exp of exp. Faltings, in the curve case, and Heuer showed that any lifting X̃ of X over B_dR^+/t^2 induces an equivalence between the category of Higgs bundles on X_ét and the category of v-vector bundles on X_v. Changing X̃ is known to alter the correspondence nontrivially. This raises the natural question of how the isomorphism class of the Higgs bundle associated with a fixed v-vector bundle varies with X̃, a question that has not been systematically studied. In this paper, we address this question when X is a curve of genus g≥ 2. More precisely, we call a Higgs bundle lift-independent if it corresponds to the same v-bundle under the p-adic Simpson correspondence for every lifting X̃. We prove the following results. (1) Every lift-independent Higgs bundle of rank r≤ g is nilpotent. (2) Every semistable lift-independent Higgs bundle of rank r≤ √g+1 has zero Higgs field. (3) In contrast, vanishing fails even within the class singled out by Faltings' conjecture: there always exists a lift-independent semistable Higgs bundle of degree $0$ with nonzero Higgs field.
ER -