We prove that there exists a pencil of Enriques surfaces defined over $${\mathbb {Q}}$$ Q with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question of Murre on the universality of the Abel-Jacobi maps in codimension three.

中文翻译：

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带有非代数积分 Hodge 类的 Enriques 曲面铅笔

我们证明存在一束 Enriques 曲面定义在 $${\mathbb {Q}}$$ Q 上，具有非扭转类型的非代数积分 Hodge 类。这给出了零循环的平凡 Chow 群的第一个三重示例，在该示例上积分霍奇猜想失败。作为一个应用，我们构建了一个四元组，它对 Murre 的经典问题给出了否定答案，该问题是关于 Abel-Jacobi 映射在第三维中的普遍性。