Pluripotential geometry on semi-positive effective divisors of numerical dimension one

TL;DR

This paper explores the complex-analytic geometry of semi-positive line bundles on compact Kähler manifolds, establishing criteria for semi-positivity and semi-ampleness, and revealing a dichotomy for effective semi-positive divisors using pluripotential methods.

Contribution

It provides new characterizations of semi-positivity for divisors of numerical dimension one and introduces a pluripotential approach to analyze neighborhoods of divisors.

Findings

Semi-positivity is equivalent to semi-ampleness under a torsion-type assumption.

Characterization of semi-positivity via pseudoflat neighborhoods for nef divisors.

A dichotomy for semi-positive divisors: pull-back from a Riemann surface or Hartogs extension phenomenon.

Abstract

We study the complex-analytic geometry of semi-positive holomorphic line bundles on compact K\"ahler manifolds. In one of our main results, for a $\mathbb{Q}$-effective line bundle satisfying a natural torsion-type assumption, we show the equivalence between semi-positivity and semi-ampleness. More generally, for an effective nef divisor of numerical dimension one, we characterize the semi-positivity of the associated line bundle in terms of the existence of a certain type of pseudoflat fundamental system of neighborhoods of the support. Furthermore, for an effective semi-positive divisor, we prove a dichotomy: either the divisor is the pull-back of a $\mathbb{Q}$-divisor by a fibration onto a Riemann surface, or the Hartogs extension phenomenon holds on the complement of its support. Our proof is based on a pluripotential method that has previously been used for studying the boundaries of pseudoconvex domains, which allows us to investigate the complex-analytic structure of neighborhoods of the support of the divisor even when the manifold is non-compact.