Reduction and lifting problem for differential forms on Berkovich curves

Abstract

Given a complete real-valued field $k$ of residue characteristic zero, we study properties of a differential form $\omega$ on a smooth proper $k$-analytic curve $X$. In particular, we associate to $(X,\omega )$ a natural tropical reduction datum combining tropical data of $(X,\omega )$ and algebra-geometric reduction data over the residue field $\tilde{k}$. We show that this datum satisfies natural compatibility condition, and prove a lifting theorem asserting that any compatible tropical reduction datum lifts to an actual pair $(X,\omega )$. In particular, we obtain a short proof of the main result of a work [BCGGM20] by Bainbridge, Chen, Gendron, Grushevsky, and Möller.

19 pages, final version, published in Advances in Mathematics

References in corpus (4)

• Compactification of strata of abelian differentials
• The moduli space of multi-scale differentials
• Realizability of tropical canonical divisors
• Mittag-Leffler problems on Berkovich curves

Cited by in corpus (1)

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