On the integral Hodge conjecture for real abelian threefolds

TL;DR

This paper proves the real integral Hodge conjecture for specific classes of real abelian threefolds, including those with connected real loci and certain product structures, and reduces the general case to the Jacobian case.

Contribution

It establishes the conjecture for new classes of real abelian threefolds and reduces the problem to the Jacobian case, advancing understanding of the conjecture's scope.

Findings

Proved the conjecture for real abelian threefolds with connected real locus.

Established the conjecture for product structures involving abelian surfaces and elliptic curves.

Showed the conjecture holds modulo torsion for all real abelian threefolds.

Abstract

We prove the real integral Hodge conjecture for several classes of real abelian threefolds. For instance, we prove the property for real abelian threefolds $A$ whose real locus $A(\mathbb R)$ is connected, and for real abelian threefolds $A$ which are a product $A = B \times E$ of an abelian surface $B$ and an elliptic curve $E$ with connected real locus $E(\mathbb R)$. Moreover, we show that every real abelian threefold satisfies the real integral Hodge conjecture modulo torsion, and reduce the general case to the Jacobian case.