Singularity of non-pluripolar cohomology classes

Abstract

We establish a relation between Lelong numbers and the full mass property of relative non-pluripolar products. We use it to show that if the restricted volume of a big cohomology class $\alpha$ in a compact K\"ahler $n$-dimensional manifold $X$ to an effective divisor $D$ is of full mass, then the Lelong numbers of the non-pluripolar class $\langle \alpha^{n-1}\rangle$ at every point in the support of $D$ is zero. In particular, we obtain that on projective manifolds, the Lelong numbers of the non-pluripolar class $\langle \alpha^{n-1}\rangle$ of a big class $\alpha$ are zero.