Reduction and lifting problem for differential forms on Berkovich curves

Given a complete real-valued field k of residue characteristic zero, we study properties differential form ? on smooth proper k-analytic curve X. In particular, associate to (X,?) natural tropical reduction datum combining data and algebra-geometric over the k˜. We show that this satisfies compatibility condition, prove lifting theorem asserting any compatible lifts an actual pair (X,?). obtain short proof main result [2].