# Monotonicity of non-pluripolar products and complex Monge-Amp\`ere equations with prescribed singularity

## TL;DR

This paper proves a monotonicity property for non-pluripolar products on compact Kahler manifolds and studies complex Monge-Ampere equations with prescribed singularities, establishing existence, uniqueness, and applications to Kahler-Einstein metrics.

## Contribution

It introduces the monotonicity property for non-pluripolar products and develops a variational approach to solve Monge-Ampere equations with prescribed singularities.

## Key findings

- Monotonicity property for non-pluripolar products established.
- Existence and uniqueness of solutions to Monge-Ampere equations with prescribed singularity proven.
- Log-concavity property shown for non-pluripolar products with small unbounded locus.

## Abstract

We establish the monotonicity property for the mass of non-pluripolar products on compact Kahler manifolds, and we initiate the study of complex Monge-Ampere type equations with prescribed singularity type. Using the variational method of Berman-Boucksom-Guedj-Zeriahi we prove existence and uniqueness of solutions with small unbounded locus. We give applications to Kahler-Einstein metrics with prescribed singularity, and we show that the log-concavity property holds for non-pluripolar products with small unbounded locus.