Reduction and lifting problem for differential forms on Berkovich curves

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.