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 field $p$-adic field $\mathbf{C}$. Fix an continuation $\mathrm{Exp}$ of $\exp$. Faltings (in the curve case) and Heuer showed that any lifting $\tilde{X}$ of $X$ over ${\mathbf{B}}_{\mathrm{dR}}^{+}/t^2$ induces an equivalence bewteen the category of Higgs bundles on $X_{\acute{\mathrm{e}}\mathrm{t}}$ 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 $\tilde{X}$ when $X$ is a curve of genus $g⩾2$. 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 $\tilde{X}$ and then we will show that (1) There exists some $r(g)⩾\sqrt{g-1}$ such that any semistable lift-independent Hitchin-small Higgs bundle of rank $r⩽r(g)$ has zero Higgs field. (2) There always exists a semistable Higgs bundle of degree $0$ with non-zero Higgs field that is lift-independent.

29 pages. Some typos are corrected. Comments are welcome!