# Finite entropy vs finite energy

### Eleonora Di Nezza

École Polytechnique, Palaiseau, France, and Sorbonne Université, Paris, France### Vincent Guedj

Université de Toulouse, France### Chinh H. Lu

Université Paris-Saclay, France

## Abstract

Probability measures with either finite Monge–Ampère energy or finite entropy have played a central role in recent developments in Kähler geometry. In this note we make a systematic study of quasi-plurisubharmonic potentials whose Monge–Ampère measures have finite entropy. We show that these potentials belong to the finite energy class $\mathcal{E}^{\frac{n}{n-1}}$, where $n$ denotes the complex dimension, and provide examples showing that this critical exponent is sharp. Our proof relies on refined Moser–Trudinger inequalities for quasi-plurisubharmonic functions.

## Cite this article

Eleonora Di Nezza, Vincent Guedj, Chinh H. Lu, Finite entropy vs finite energy. Comment. Math. Helv. 96 (2021), no. 2, pp. 389–419

DOI 10.4171/CMH/515