Plurisupported Currents on Compact K\"ahler Manifolds

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.