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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 {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+q1(X; ). 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+q1(X; ) 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 = 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 n+q1(X; ) = limjHn+q1(Mj; ), where {Mj}j 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
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