On integral Hodge classes on uniruled or Calabi-Yau threefolds

TL;DR

This paper proves that the integral Hodge conjecture holds for degree 4 classes on uniruled or Calabi-Yau threefolds, extending the understanding of algebraic cycles in these special classes of threefolds.

Contribution

It establishes the validity of the integral Hodge conjecture for degree 4 classes specifically on uniruled and Calabi-Yau threefolds, which were previously known to have counterexamples.

Findings

Integral Hodge conjecture holds for these threefolds.

Counterexamples exist for general threefolds, but not for the classes studied here.

Supports the conjecture's validity in special geometric contexts.

Abstract

If X is a smooth projective complex threefold, the Hodge conjecture holds for degree 4 rational Hodge classes on X. Kollar gave examples where it does not hold for integral Hodge classes of degree 4, that is integral Hodge classes need not be classes of algebraic 1-cycles with integral coefficients. We show however that the Hodge conjecture holds for integral Hodge classes of degree 4 on a uniruled or a Calabi-Yau threefold.