Monge-Amp\`ere equations in big cohomology classes

TL;DR

This paper develops a framework for defining and analyzing Monge-Ampère equations in big cohomology classes on compact Kähler manifolds, leading to new existence results for singular Kähler-Einstein metrics.

Contribution

It introduces non-pluripolar products of positive currents and proves the existence and uniqueness of solutions with minimal singularities in big classes.

Findings

Unique representation of positive measures as non-pluripolar self-intersections

Construction of singular Kähler-Einstein volume forms with minimal singularities

Extension of Monge-Ampère equation solutions to big cohomology classes

Abstract

We define non-pluripolar products of closed positive currents on a compact Kaehler manifold. We show that a positive non-pluripolar measure can be written in a unique way as the top degree self-intersection (in the non-pluripolar sense) of a closed positive current in given big cohomology class. The solution is shown to have minimal singularities in the sense of Demailly if the measure is regular enough. These results are combined with a fixed point argument to construct singular Kaehler-Einstein volume forms with minimal singularities on varieties of general type.