L ² Castelnuovode Franchis, the cup product lemma, and filtered ends of Kähler manifolds

Simple approaches to the proofs of the L ² Castelnuovode Franchis theorem and the cup product lemma which give new versions are developed. For example, assume that ω ₁ and ω ₂ are two linearly independent closed holomorphic 1-forms on a bounded geometry connected complete Kähler manifold X with ω ₂ in L ². According to a version of the L ² Castelnuovode Franchis theorem obtained in this paper, if ω ₁ ∧ ω ₂ ≡ 0, then there exists a surjective proper holomorphic mapping of X onto a Riemann surface for which ω ₁ and ω ₂ are pull-backs. Previous versions required both forms to be in L ².