Singularity of non-pluripolar cohomology classes

TL;DR

This paper links Lelong numbers with the full mass property of non-pluripolar products, showing that under certain conditions, the Lelong numbers of specific classes vanish, revealing new insights into the structure of big cohomology classes.

Contribution

It establishes a relation between Lelong numbers and the full mass property of non-pluripolar products, providing conditions under which Lelong numbers vanish for big classes.

Findings

Lelong numbers of the non-pluripolar class vanish at points in the support of a divisor when the restricted volume is full.

On projective manifolds, the Lelong numbers of the non-pluripolar class of a big class are zero.

The relation between Lelong numbers and the full mass property is established for relative non-pluripolar products.

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.