On the Hodge conjecture for $q$-complete manifolds

Abstract

We establish the Hodge conjecture for the top dimensional cohomology group with integer coefficients of any $q$-complete complex manifold $X$ with $q<\dim X$. This holds in particular for the complement $X=\mathbb{C}{\mathbb{P}}^n\setminus A$ of any complex projective manifold defined by $q<n$ independent equations.

In this version we improved most of the main results, showing that the analytic cycles representing the top dimensional cohomology classes of a $q$-complete complex manifold can be chosen to consist of holomorphic images of the ball (instead of ellipsoids that were used in Version 1)

References in corpus (2)

• Approximation of holomorphic mappings on strongly pseudoconvex domains
• Envelopes of holomorphy and holomorphic discs