Integral and Absolute Hodge Classes

TL;DR

This paper explores the relationship between the rational Hodge conjecture and the integral Hodge conjecture, showing that the former implies a torsion-modulo version of the absolute Hodge conjecture.

Contribution

It establishes a new implication from the rational Hodge conjecture to an integral form of the absolute Hodge conjecture, despite the known failure of the integral Hodge conjecture.

Findings

Rational Hodge conjecture implies an integral (torsion-modulo) version of the absolute Hodge conjecture.

The integral Hodge conjecture fails, but a weaker integral version still holds under the rational Hodge conjecture.

Provides insights into the structure of Hodge classes and their integral properties.

Abstract

Despite the failure of the integral Hodge conjecture, we show that the rational Hodge conjecture implies an integral version (modulo torsion) of the absolute Hodge conjecture.