On the integral Hodge conjecture for real varieties, I

Abstract

We formulate the "real integral Hodge conjecture", a version of the integral Hodge conjecture for real varieties, and raise the question of its validity for cycles of dimension 1 on uniruled and Calabi-Yau threefolds and on rationally connected varieties. We relate it to the problem of determining the image of the Borel-Haefliger cycle class map for 1-cycles, with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of computing the torsion of the Chow group of 1-cycles of real threefolds. New results about these problems are obtained along the way.

arXiv comment67 pages; v2: minor modifications; v3: Section 1.1.3 slightly expanded, final version

References in corpus (9)

• Cohomologie non ramifiée et conjecture de Hodge entière
• Stably irrational hypersurfaces of small slopes
• Index of varieties over Henselian fields and Euler characteristic of coherent sheaves
• Algebraic cycles and topology of real Enriques surfaces
• Cycles algébriques sur les surfaces K3 réelles
• On the integral Hodge conjecture for real varieties, II
• Remarks on curve classes on rationally connected varieties
• On Hilbert's 17th problem in low degree
• Degree formula for the Euler characteristic

Cited by in corpus (10)

• The Clemens-Griffiths method over non-closed fields
• The real cycle class map
• Two coniveau filtrations
• On the subvarieties with nonsingular real loci of a real algebraic variety
• Halving spaces and lower bounds in real enumerative geometry
• Maximal real varieties from moduli constructions
• Moduli spaces and algebraic cycles in real algebraic geometry
• Steenrod operations and algebraic classes
• On the integral Hodge conjecture for real abelian threefolds
• Cohomology of algebraic varieties over non-archimedean fields