Geodesic distance and Monge-Amp\`ere measures on contact sets

TL;DR

This paper extends the understanding of geodesic distances for quasi-psh functions with finite entropy, proves convexity of the K-energy in big and nef classes, and generalizes Monge-Ampère measure results on contact sets.

Contribution

It introduces a geodesic distance formula for quasi-psh functions with finite entropy and establishes convexity of the K-energy in big and nef classes, extending prior results.

Findings

Proved a geodesic distance formula for quasi-psh functions with finite entropy.

Established convexity of the K-energy in big and nef cohomology classes.

Generalized Monge-Ampère measure results 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-Amp\`ere measures on contact sets, generalizing a recent result by the first author and Trapani.