#### Volume 20, issue 1 (2016)

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On the Hodge conjecture for $q$–complete manifolds

### Franc Forstnerič, Jaka Smrekar and Alexandre Sukhov

Geometry & Topology 20 (2016) 353–388
##### 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 ${H}^{n+q-1}\left(X;ℤ\right)$. In this paper we show that if $q, $n+q-1$ is even, and $X$ has finite topology, then every cohomology class in ${H}^{n+q-1}\left(X;ℤ\right)$ is Poincaré dual to an analytic cycle in $X$ consisting of proper holomorphic images of the ball. This holds in particular for the complement $X=ℂ{ℙ}^{n}\setminus A$ of any complex projective manifold $A$ defined by $q independent equations. If $X$ has infinite topology, then the same holds for elements of the group ${\mathsc{ℋ}}^{n+q-1}\left(X;ℤ\right)=\underset{j}{lim}{H}^{n+q-1}\left({M}_{j};ℤ\right)$, where ${\left\{{M}_{j}\right\}}_{j\in ℕ}$ 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.

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##### Keywords
Hodge conjecture, complex analytic cycle, $q$–complete manifold, Stein manifold, Poincaré–Lefschetz duality
##### Mathematical Subject Classification 2010
Primary: 14C30, 32F10
Secondary: 32E10, 32J25
##### Publication
Received: 9 April 2014
Revised: 6 April 2015
Accepted: 8 May 2015
Published: 29 February 2016
Proposed: Simon Donaldson
Seconded: Richard Thomas, Yasha Eliashberg
##### Authors
 Franc Forstnerič Faculty of Mathematics and Physics University of Ljubljana Jadranska 19 1000 Ljubljana Slovenia Institute of Mathematics Physics and Mechanics Jadranska 19 1000 Ljubljana Slovenia http://www.fmf.uni-lj.si/~forstneric/ Jaka Smrekar Faculty of Mathematics and Physics University of Ljubljana Jadranska 19 1000 Ljubljana Slovenia Institute of Mathematics Physics and Mechanics Jadranska 19 1000 Ljubljana Slovenia http://www.fmf.uni-lj.si/~smrekar/ Alexandre Sukhov Laboratoire Paul Painleve, UFR de Mathematiques Universite Lille-1 F-59655 Villeneuve d’Ascq Cedex France