On the integral Hodge conjecture for real varieties, I

TL;DR

This paper introduces the real integral Hodge conjecture for real varieties, exploring its validity for certain classes of threefolds and rationally connected varieties, and relates it to cycle class maps and Chow group torsion.

Contribution

It formulates the real integral Hodge conjecture for real varieties and investigates its validity for specific classes, connecting it to cycle class maps and Chow group torsion.

Findings

Relation between the real integral Hodge conjecture and cycle class maps.

New results on the torsion of Chow groups of 1-cycles on real threefolds.

Insights into the existence of curves of even geometric genus on real varieties without real points.

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.