Positivity of the second exterior power of the tangent bundles
Positivity of the second exterior power of the tangent bundles
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DOI：
10.1016/j.aim.2021.107757
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发表时间：
2020-11
影响因子：
1.7
通讯作者：
Kiwamu Watanabe
中科院分区：
文献类型：
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作者：
Kiwamu Watanabe
设X是光滑复射影簇，满足nef <$2 T X和dim <$X≥ 3.我们证明了直到有限的代数覆盖X → X，Albanese映射X → Alb（X）是局部平凡纤维化，其纤维同构于一个光滑Fano簇F，且nef ∈ 2 T F.作为一个双积，我们看到T X是nef或X是Fano簇。此外，我们还研究了K X-负极值射线φ：X→ Y的一个收缩。特别地，我们证明了如果φ是双有理型的，则X同构于射影空间在一点的爆破。我们还证明了如果φ是纤维型的，则φ是光滑态射.作为结果，我们给出了具有nef ∈ 2 TX的簇的一个结构定理.
Let X be a smooth complex projective variety with nef⋀ 2 T X and dim X≥ 3. We prove that, up to a finite étale cover X˜→ X, the Albanese map X˜→ Alb (X˜) is a locally trivial fibration whose fibers are isomorphic to a smooth Fano variety F with nef⋀ 2 T F. As a bi-product, we see that T X is nef or X is a Fano variety. Moreover we study a contraction of a K X-negative extremal ray φ: X→ Y. In particular, we prove that X is isomorphic to the blow-up of a projective space at a point if φ is of birational type. We also prove that φ is a smooth morphism if φ is of fiber type. As a consequence, we give a structure theorem of varieties with nef⋀ 2 T X.