On the Hodge conjecture for $q$–complete manifolds

Abstract

A complex manifold $X$ of dimension $n$ is said to be $q$–complete for some $q\in \left\{1,\dots ,n\right\}$ if it admits a smooth exhaustion function whose Levi form has at least $n-q+1$ positive eigenvalues at every point; thus, $1$–complete manifolds are Stein manifolds. Such an $X$ is necessarily noncompact and its highest-dimensional a priori nontrivial cohomology group is $H^{n+q-1}\left(X;\mathbb{Z}\right)$. In this paper we show that if $q<n$, $n+q-1$ is even, and $X$ has finite topology, then every cohomology class in $H^{n+q-1}\left(X;\mathbb{Z}\right)$ is Poincaré dual to an analytic cycle in $X$ consisting of proper holomorphic images of the ball. This holds in particular for the complement $X=ℂ{\mathbb{P}}^n\setminus A$ of any complex projective manifold $A$ defined by $q<n$ independent equations. If $X$ has infinite topology, then the same holds for elements of the group ${\mathcal{\mathscr{H}}}^{n+q-1}\left(X;\mathbb{Z}\right)=\underset{j}{lim}{H}^{n+q-1}\left(M_j;\mathbb{Z}\right)$, where ${\left\{M_j\right\}}_{j\in \mathbb{N}}$ is an exhaustion of $X$ by compact smoothly bounded domains. Finally, we provide an example of a quasi-projective manifold with a cohomology class which is analytic but not algebraic.