One-cycles on rationally connected varieties
One-cycles on rationally connected varieties
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有理连接品种的单周期
作者：
Zhiyu Tian;Hong R. Zong
摘要 我们证明了可分离有理连接簇上的每条曲线有理地等价于有理曲线的（无效）积分和。也就是说，1-周期的 Chow 群是由有理曲线生成的。应用相同的技术，我们还证明了索引至少为 2 的可分离有理连接 Fano 完全交集上的 1 循环 Chow 群是由线生成的。因此，我们对 Totaro 教授关于有理连通 3 重上的积分 Hodge 类的问题给出了肯定的答案。根据 Voisin 教授的结果，一般情况是有限域上曲面的泰特猜想的结果。
Abstract We prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. Applying the same technique, we also show that the Chow group of 1-cycles on a separably rationally connected Fano complete intersection of index at least 2 is generated by lines. As a consequence, we give a positive answer to a question of Professor Totaro about integral Hodge classes on rationally connected 3-folds. And by a result of Professor Voisin, the general case is a consequence of the Tate conjecture for surfaces over finite fields.