Degenerate complex Monge-Amp\`ere equations over compact K\"ahler manifolds

TL;DR

This paper establishes the existence and uniqueness of solutions to a broad class of degenerate complex Monge-Ampère equations, crucial for constructing Kähler-Einstein metrics on singular spaces.

Contribution

It provides a general existence and uniqueness theorem for degenerate complex Monge-Ampère equations on compact Kähler manifolds, enabling advances in Kähler-Einstein metric theory.

Findings

Proved existence of solutions for a broad class of degenerate equations

Established uniqueness of solutions under general conditions

Facilitated construction of Kähler-Einstein metrics on singular spaces

Abstract

We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge-Amp\`ere equations. This type of equations is precisely what is needed in order to construct K\"ahler-Einstein metrics over irreducible singular K\"ahler spaces with ample or trivial canonical sheaf and singular K\"ahler-Einstein metrics over varieties of general type.