有理拓扑可缩仿射三维空间上的向量丛
Vector Bundles on Rational Topologically Contractible Affine Threefolds
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中文总结 AI 辅助
研究有理连通且维数为3的拓扑可缩光滑仿射复簇上代数向量丛是否平凡的问题，通过证明给出肯定回答，如证明科拉斯 - 拉塞尔三维空间上代数向量丛是平凡的。
中文摘要 AI 辅助
广义塞尔问题询问在拓扑可缩、光滑、仿射、复簇\(X\)上的任何代数向量丛是否平凡。本文证明，如果\(X\)的维数为3且\(X\)是有理连通的，对该问题的回答是肯定的。例如，这证明了任何科拉斯 - 拉塞尔三维空间（第一类或第二类以及某些第三类）上的每个代数向量丛都是平凡的。
英文摘要
The generalized Serre question asks whether every algebraic vector bundle on a topologically contractible smooth affine complex variety is trivial. We give an affirmative answer for rational threefolds. More generally, for a topologically contractible smooth affine complex threefold $X$, we prove that $\text{CH}^2(X)=0$ whenever $X$ admits a smooth projective compactification whose Chow group of $0$-cycles is supported on a curve. This uncovers the link between the generalized van de Ven question, Bloch's conjecture and the generalized Serre question for threefolds. We also prove that every Koras-Russell threefold is rational and therefore has only trivial algebraic vector bundles, hence answer a question of Koras and Russell.
