Geodesic distance and Monge-Ampère measures on contact sets

Abstract

We prove a geodesic distance formula for quasi-psh functions with finite entropy, extending results by Chen and Darvas. We work with big and nef cohomology classes: a key result we establish is the convexity of the $K$-energy in this general setting. We then study Monge-Ampère measures on contact sets, generalizing a recent result by the first author and Trapani.

27 pages. In the second version we added Corollay 3.3