论文标题
带有琐碎的Chern类的歧管I类：闪光歧管和Severi的问题
Manifolds with trivial Chern classes I: Hyperelliptic Manifolds and a question by Severi
论文作者
论文摘要
我们对Severi在1951年提出的问题给出了负面答案，Abelian品种是否是唯一具有琐碎Chern类的投影歧管。 根据Yau的庆祝结果，带有琐碎的Chern类的紧凑型Kähler歧管必须是平坦的，也就是说，它们属于高纤维流形的类（通过有限组$ g $的自由操作的复杂圆环$ t $的商的$ t/g $）。 我们展示了不是Abelian品种的投影性高纤维状歧管的简单示例，其Chern类不仅在整体共同体中，而且在Chow环中也为零。 此外，我们证明了Bagnera-de系列歧管（如上所述的商$ t/g $，但在组$ g $是循环的情况下）具有拓扑的切线捆绑包。 我们的结果自然导致了将所有紧凑的Kähler歧管与拓扑琐碎的切线束分类的问题，并将所有反例与Severi的问题分类。
We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only projective manifolds with trivial Chern classes. By Yau' s celebrated result, compact Kähler manifolds with trivial Chern classes must be flat, that is, they belong to the class of Hyperelliptic Manifolds (quotients $T/G$ of a complex torus $T$ by the free action of a finite group $G$). We exhibit simple examples of projective Hyperelliptic Manifolds which are not Abelian varieties and whose Chern classes are zero not only in integral cohomology, but also in the Chow ring. We prove moreover that the Bagnera-de Franchis manifolds (quotients $T/G$ as above but where the group $ G$ is cyclic) have topologically trivial tangent bundle. Our results naturally lead to the question of classifying all compact Kähler manifolds with topologically trivial tangent bundle, and all the counterexamples to Severi's question.
