On the hard Lefschetz theorem for pseudoeffective line bundles

Xiaojun Wu

Published 2019-11-27Version 1

In this note, we obtain a number of results related to the hard Lefschetz theorem for pseudoeffective line bundles due to Demailly, Peternell and Schneider. Our first result states that the holomorphic sections produced by the theorem are in fact closed and harmonic, when viewed as currents with respect to the singular Chern connection associated with the metric. Our proof is based on a control of the covariant derivative in the approximation process used in the construction of the section. Under a suitable analyticity assumption for the curvature current, we show as an application that the closedness of such sections induces a linear subspace structure on the tangent bundle. Finally, we produce examples of pseudoeffective line bundles showing that the multiplier ideal sheaves required in the hard Lefschetz theorem are essentially optimal.