Nonvanishing results for Kähler varieties
Andreas Höring, Vladimir Lazić, Christian Lehn
TL;DR
This work extends nonvanishing and abundance-type phenomena to the Kähler setting. It develops a robust analytic toolkit—Siu decompositions, currents with minimal singularities, and good bilinear forms—to compare nef and pseudoeffective classes and to derive nonvanishing for adjoint bundles when the numerical dimension is one and Euler characteristic is nonzero, as well as dichotomies for nef but non-big line bundles on hyperkähler manifolds. A key novelty is the unification of projectivity criteria, MMP-inspired semiampleness results, and hyperkähler geometry, including a precise description of maximal Lelong components and their geometric constraints. In dimension four, these methods yield stronger abundance-type conclusions. Overall, the paper provides new pathways toward the Abundance Conjecture in non-projective settings and clarifies the geometric impact of currents and Lelong behavior on Kähler and hyperkähler varieties.
Abstract
Nonvanishing theorems play a central role in birational geometry, since they derive geometric consequences from numerical information and constitute a crucial step towards abundance and semiampleness problems. General nonvanishing statements remain rare, especially in the Kähler setting. We present two types of nonvanishing results for compact Kähler varieties. First, on non-uniruled varieties with nonzero Euler-Poincaré characteristic, we prove nonvanishing for adjoint bundles of numerical dimension one on Kähler klt pairs, as well as nonvanishing for nef line bundles of numerical dimension one on $K$-trivial varieties. Second, on hyperkähler manifolds we study line bundles $\mathcal L$ which are nef but not big, and establish a dichotomy: either nonvanishing holds for $\mathcal L$, or any closed positive current in the cohomology class of $\mathcal L$ has maximal Lelong components with a rather restricted geometry. We obtain much stronger abundance-type results in dimension $4$.