Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero

TL;DR

This paper demonstrates the failure of the integral Hodge conjecture for certain threefolds of Kodaira dimension zero, providing the first known examples of such failures and non-algebraic torsion classes in this context.

Contribution

It constructs explicit examples of smooth projective threefolds with Kodaira dimension zero where the integral Hodge conjecture does not hold, revealing new phenomena in algebraic geometry.

Findings

Failure of the integral Hodge conjecture for specific threefolds

First examples of non-algebraic torsion cohomology classes in degree 4

Identification of new obstructions in the Hodge conjecture context

Abstract

We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.