L2 CASTELNUOVO–DE FRANCHIS, THE CUP PRODUCT LEMMA, AND FILTERED ENDS OF KÄHLER MANIFOLDS

Simple approaches to the proofs of the L2 Castelnuovo–de Franchis theorem and the cup product lemma which give new versions are developed. For example, assume that ω1 and ω2 are two linearly independent closed holomorphic 1-forms on a bounded geometry connected complete Kähler manifold X with ω2 in L2. According to a version of the L2 Castelnuovo–de Franchis theorem obtained in this paper, if ω1 ∧ ω2 ≡ 0, then there exists a surjective proper holomorphic mapping of X onto a Riemann surface for which ω1 and ω2 are pull-backs. Previous versions required both forms to be in L2.

On proper holomorphic mappings from domains with T-action

We describe the branch locus of a proper holomorphic mapping between two smoothly bounded pseudoconvex domains of finite type in under the assumption that the first domain admits a transversal holomorphic action of the unit circle. As an application we show that any proper holomorphic self-mapping of a smoothly bounded pseudoconvex complete circular domain of finite type in is biholomorphic.