Nonvanishing results for K\"ahler varieties

TL;DR

This paper establishes new nonvanishing theorems for compact K"ahler varieties, including conditions under which line bundles have nonzero sections, advancing understanding in birational geometry and abundance conjectures.

Contribution

It provides novel nonvanishing results for K"ahler varieties, especially for adjoint bundles and nef line bundles, with a focus on hyperk"ahler manifolds and dimension four cases.

Findings

Nonvanishing for adjoint bundles of numerical dimension one on K"ahler klt pairs.

Nonvanishing for nef line bundles of numerical dimension one on K-trivial varieties.

Dichotomy for nef but not big line bundles on hyperk"ahler manifolds.

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 K\"ahler setting. We present two types of nonvanishing results for compact K\"ahler varieties. First, on non-uniruled varieties with nonzero Euler-Poincar\'e characteristic, we prove nonvanishing for adjoint bundles of numerical dimension one on K\"ahler klt pairs, as well as nonvanishing for nef line bundles of numerical dimension one on $K$-trivial varieties. Second, on hyperk\"ahler 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 $4$.