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 ω on a smooth proper k-analytic curve X. In particular, we associate to (X,ω) a natural tropical reduction datum combining tropical data of (X,ω) and algebra-geometric reduction data over the residue field 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,ω). In particular, we obtain a short proof of the main result of [2].