# A cup product lemma for continuous plurisubharmonic functions

## TL;DR

This paper extends Gromov's cup product lemma to continuous plurisubharmonic functions and demonstrates its implications for the Bochner-Hartogs property on certain noncompact Kähler manifolds.

## Contribution

It introduces a new version of Gromov's cup product lemma involving continuous plurisubharmonic functions and applies it to complex geometry.

## Key findings

- Connected noncompact complete Kähler manifolds with one end and specific plurisubharmonic functions have the Bochner-Hartogs property.
- The first compactly supported cohomology with values in the structure sheaf vanishes under these conditions.
- The paper establishes a link between plurisubharmonic functions and topological properties of Kähler manifolds.

## Abstract

A version of Gromov's cup product lemma in which one factor is the (1,0)-part of the differential of a continuous plurisubharmonic function is obtained. As an application, it is shown that a connected noncompact complete Kaehler manifold that has exactly one end and admits a continuous plurisubharmonic function that is strictly plurisubharmonic along some germ of a 2-dimensional complex analytic set at some point has the Bochner-Hartogs property; that is, the first compactly supported cohomology with values in the structure sheaf vanishes.