Geometry & Topology

On the Hodge conjecture for $q$–complete manifolds

Franc Forstnerič, Jaka Smrekar, and Alexandre Sukhov

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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.

Article information

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

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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]

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


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.

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