On the Hodge conjecture for q–complete manifolds

Forstnerič, Franc; Smrekar, Jaka; Sukhov, Alexandre

doi:10.2140/gt.2016.20.353

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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

DOI: 10.2140/gt.2016.20.353

Abstract

A complex manifold $X$ of dimension $n$ is said to be $q$ –complete for some $q \in {1, \dots, n}$ 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 $H^{n + q - 1} (X; ℤ)$ . In this paper we show that if $q < n$ , $n + q - 1$ is even, and $X$ has finite topology, then every cohomology class in $H^{n + q - 1} (X; ℤ)$ is Poincaré dual to an analytic cycle in $X$ consisting of proper holomorphic images of the ball. This holds in particular for the complement $X = ℂ ℙ^{n} ∖ A$ 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 $ℋ^{n + q - 1} (X; ℤ) = lim_{j} H^{n + q - 1} (M_{j}; ℤ)$ , where ${M_{j}}_{j \in ℕ}$ 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.

Keywords

Hodge conjecture, complex analytic cycle, $q$–complete manifold, Stein manifold, Poincaré–Lefschetz duality

Mathematical Subject Classification 2010

Primary: 14C30, 32F10

Secondary: 32E10, 32J25

References

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

Volume 20, issue 1 (2016)