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.

中文翻译：

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小平维零三重积分霍奇猜想的失败

我们证明 Enriques 曲面和非常一般的至少为 1 的属函数曲线的乘积不满足 1 个循环的积分 Hodge 猜想。这提供了积分霍奇猜想失败的 Kodaira 维零的平滑射影复三重的第一个例子，以及平滑射影复三重上的第 4 次非代数扭转上同调类的第一个例子。