Geometry & Topology
- Geom. Topol.
- Volume 20, Number 1 (2016), 353-388.
On the Hodge conjecture for $q$–complete manifolds
A complex manifold of dimension is said to be –complete for some if it admits a smooth exhaustion function whose Levi form has at least positive eigenvalues at every point; thus, –complete manifolds are Stein manifolds. Such an is necessarily noncompact and its highest-dimensional a priori nontrivial cohomology group is . In this paper we show that if , is even, and has finite topology, then every cohomology class in is Poincaré dual to an analytic cycle in consisting of proper holomorphic images of the ball. This holds in particular for the complement of any complex projective manifold defined by independent equations. If has infinite topology, then the same holds for elements of the group , where is an exhaustion of 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.
Geom. Topol., Volume 20, Number 1 (2016), 353-388.
Received: 9 April 2014
Revised: 6 April 2015
Accepted: 8 May 2015
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14C30: Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture 32F10: $q$-convexity, $q$-concavity
Secondary: 32E10: Stein spaces, Stein manifolds 32J25: Transcendental methods of algebraic geometry [See also 14C30]
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