Reduction and lifting problem for differential forms on Berkovich curves

Michael Temkin; Ilya Tyomkin

doi:10.1016/j.aim.2022.108208

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arXiv · 2005.01397

Reduction and lifting problem for differential forms on Berkovich curves

Michael Temkin·Ilya Tyomkin

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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 $\widetilde{k}$. 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 M\"oller.

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Michael Temkin, Ilya Tyomkin. 2020-05-04. Reduction and lifting problem for differential forms on Berkovich curves. https://doi.org/10.1016/j.aim.2022.108208

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