## Geometry & Topology

### On the Hodge conjecture for $q$–complete manifolds

#### Abstract

A complex manifold $X$ of dimension $n$ is said to be $q$–complete for some $q ∈{1,…,n}$ 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 $Hn+q−1(X; ℤ)$. In this paper we show that if $q < n$, $n + q − 1$ is even, and $X$ has finite topology, then every cohomology class in $Hn+q−1(X; ℤ)$ is Poincaré dual to an analytic cycle in $X$ consisting of proper holomorphic images of the ball. This holds in particular for the complement $X = ℂℙn ∖ A$ 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 $ℋn+q−1(X; ℤ) = limjHn+q−1(Mj; ℤ)$, where ${Mj}j∈ℕ$ 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.

#### Article information

Source
Geom. Topol., Volume 20, Number 1 (2016), 353-388.

Dates
Revised: 6 April 2015
Accepted: 8 May 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858928

Digital Object Identifier
doi:10.2140/gt.2016.20.353

Mathematical Reviews number (MathSciNet)
MR3470716

Zentralblatt MATH identifier
06553431

#### Citation

Forstnerič, Franc; Smrekar, Jaka; Sukhov, Alexandre. On the Hodge conjecture for $q$–complete manifolds. Geom. Topol. 20 (2016), no. 1, 353--388. doi:10.2140/gt.2016.20.353. https://projecteuclid.org/euclid.gt/1510858928

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