Pluripotential geometry on semi-positive effective divisors of numerical dimension one
Takayuki Koike
TL;DR
This work addresses the complex-analytic geometry governed by semi-positive line bundles on compact Kähler manifolds, focusing on divisors of numerical dimension one. Employing a pluripotential framework inspired by Demailly–Nemirovski–Fu–Shaw, it links semi-positivity to unitary-flatness on neighborhoods and to the existence of pseudoflat neighborhood systems, yielding a dichotomy that unifies geometric and extension phenomena. Under a torsion-type assumption, the authors prove that semi-positivity is equivalent to semi-ampleness, and they establish a precise characterization for effective nef divisors with ${\rm nd}=1$ in terms of fibrations onto a Riemann surface or Hartogs extension on the divisor’s complement. The results extend to non-compact ambient spaces and illuminate connections with the abundance conjecture, providing tools and examples that clarify when a divisor defines a fibration or admits Levi-flat level-set structures. Overall, the paper advances the understanding of how plurisubharmonic data control the global complex-geometric structure around divisors of numerical dimension one is revealed through a robust pluripotential approach.
Abstract
We study the complex-analytic geometry of semi-positive holomorphic line bundles on compact Kähler 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.