On the singularity type of full mass currents in big cohomology classes

TL;DR

This paper investigates the singularity properties of full mass currents in big cohomology classes on compact Kähler manifolds, establishing key equivalences and characterizations that advance understanding in complex geometry.

Contribution

It proves that full mass currents share singularity invariants with minimal singularity currents, characterizes additive currents, and develops weak geodesic theory in big classes.

Findings

Lelong numbers and multiplier ideal sheaves coincide for full mass and minimal singularity currents.

Inclusion relation between certain energy classes and plurisubharmonic functions is established.

Characterization of big classes with additive full mass currents is provided.

Abstract

Let $X$ be a compact K\"ahler manifold and $\{\theta\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $\theta$-plurisubharmonic functions with full mass are the same as those of the current with minimal singularities. Second, given another big and nef class $\{\eta\}$, we show the inclusion $\mathcal{E}(X,\eta) \cap {PSH}(X,\theta) \subset \mathcal{E}(X,\theta).$ Third, we characterize big classes whose full mass currents are "additive". Our techniques make use of a characterization of full mass currents in terms of the envelope of their singularity type. As an essential ingredient we also develop the theory of weak geodesics in big cohomology classes. Numerous applications of our results to complex geometry are also given.