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 $\mathbf{C}$. Fix a continuation $\mathrm{Exp}$ of $\exp$. Faltings, in the curve case, and Heuer showed that any lifting $\tilde{X}$ of $X$ over $B_{\mathrm{dR}}^{+}/t^2$ induces an equivalence between the category of Higgs bundles on $X_{\text{eˊt}}$ and the category of $v$-vector bundles on $X_v$. Changing $\tilde{X}$ 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 $\tilde{X}$, a question that has not been systematically studied. In this paper, we address this question when $X$ is a curve of genus $g\ge 2$. 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 $\tilde{X}$. We prove the following results. (1) Every lift-independent Higgs bundle of rank $r\le g$ is nilpotent. (2) Every semistable lift-independent Higgs bundle of rank $r\le \sqrt{g}+1$ 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.

arXiv comment26 pages. Comments are welcome!