Plurisupported Currents on Compact K\"ahler Manifolds

TL;DR

This paper investigates plurisupported currents on compact K"ahler manifolds, extending key conjectures and inequalities in complex geometry by analyzing currents supported on pluripolar sets and their implications.

Contribution

It generalizes Witt-Nyström's proof of the BDPP conjecture to manifolds with certain plurisupported currents and extends inequalities of McKinnon and Roth to broader classes.

Findings

BDPP conjecture holds on manifolds with a plurisupported current whose class is K"ahler

Established an upper bound for pluripolar mass of envelopes of quasi-psh functions

Generalized inequalities of McKinnon and Roth to arbitrary pseudoeffective classes

Abstract

Let $X$ be a compact K\"ahler manifold. We study plurisupported currents on $X$, i.e. closed, positive $(1,1)$-currents which are supported on a pluripolar set. In particular, we are able present a technical generalization of Witt-Nystr\"om's proof of the BDPP conjecture on projective manifolds, showing that this conjecture holds on $X$ admitting at least one plurisupported current $T$ such that $[T]$ is K\"ahler. One of the steps in our proof is to show an upper-bound for the pluripolar mass of certain envelopes of quasi-psh functions when the cohomology class is shifted, a result of independent interest. Using this, we are able to generalize an inequality of McKinnon and Roth to arbitrary pseudoeffective classes on compact K\"ahler manifolds.