L^2 Castelnuovo-de Franchis, the cup product lemma, and filtered ends of Kaehler manifolds

TL;DR

This paper introduces simplified proofs and new versions of the L^2 Castelnuovo-de Franchis theorem and cup product lemma, relating holomorphic forms and holomorphic mappings on Kaehler manifolds with bounded geometry.

Contribution

It develops more straightforward approaches and extends the L^2 Castelnuovo-de Franchis theorem to cases where only one form is in L^2, broadening its applicability.

Findings

If u and v are linearly dependent holomorphic 1-forms with v in L^2, then a proper holomorphic map to a Riemann surface exists.

Previous results required both forms to be in L^2, now only one is needed.

New proofs simplify understanding of the relationship between holomorphic forms and mappings on Kaehler manifolds.

Abstract

Simple approaches to the proofs of the L^2 Castelnuovo-de Franchis theorem and the cup product lemma which give new versions are developed. For example, suppose u and v are two linearly independent closed holomorphic 1-forms on a bounded geometry connected complete Kaehler manifold X with v in L^2. According to a version of the L^2 Castelnuovo-de Franchis theorem obtained in this paper, if u and v are pointwise linearly dependent, then there exists a surjective proper holomorphic mapping of X onto a Riemann surface for which u and v are pull-backs. Previous versions required both forms to be in L^2.