Nonvanishing results for K\"ahler 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 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 $4$.