Tropical reduction and lifting of $q$-differentials on Berkovich curves

TL;DR

This paper develops a tropical reduction and lifting theory for meromorphic $q$-differentials on Berkovich curves over non-Archimedean fields, generalizing previous results from $q=1$ to arbitrary $q$ and connecting tropical and algebraic data.

Contribution

It introduces a tropical reduction datum for $q$-differentials and proves a lifting theorem, extending prior work from $q=1$ to general $q$ and relating to non-Archimedean geometry.

Findings

Established a compatibility condition for tropical reduction data.

Proved a lifting theorem for compatible tropical data to algebraic pairs.

Generalized previous results from $q=1$ to arbitrary natural number $q$.

Abstract

Given a complete real-valued field $k$ of residue characteristic zero, we study properties of a meromorphic $q$-differential form $\xi$ (a section of $\Omega_X^{\otimes q}$) on a smooth proper $k$-analytic curve $X$. In particular, we associate to $(X,\xi)$ a natural tropical reduction datum combining tropical-geometric data over the value group of $|k^\times|$ and algebro-geometric reduction data over the residue field $\widetilde{k}$. We show that this datum satisfies natural compatibility conditions, and prove a lifting theorem asserting that any compatible tropical reduction datum lifts to an actual pair $(X, \xi)$. This generalizes the result of \cite{TT20} from $q=1$ to a general natural number $q$. Furthermore, it is a non-Archimedean analog of \cite{BCGGM16}.