On projective manifolds with semi-positive holomorphic sectional curvature
On projective manifolds with semi-positive holomorphic sectional curvature
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DOI：
10.1353/ajm.2022.0015
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发表时间：
2018-11
影响因子：
1.7
通讯作者：
Shin-ichi Matsumura
中科院分区：
文献类型：
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作者：
Shin-ichi Matsumura
建立了具有半正全纯截面曲率的光滑射影簇X的结构定理。我们首先证明了$X$是有理连通的，如果$X$在某点上没有真正平坦的切向量（当全纯截面曲率为拟正时满足）。这一结果强有力地解决了丘氏关于正全纯截曲率的猜想。此外，我们证明了$X$允许一个局部平凡态射$\phi：X\to Y$，使得纤维$F$是有理连通的，并且像$Y$有一个有限的\'etale cover $A\to Y$ by一个阿贝尔簇$A$.我们还证明了X的泛复盖与具有平坦度量的复欧氏空间和具有诱导K ahler度量的有理连通纤维F的乘积Bbb {C}^m\times F是双全纯等距的.我们的结构定理是Howard-Smyth-Wu和Mok关于全纯双截曲率的结构定理的自然推广。
abstract:We establish structure theorems for a smooth projective variety $X$ with semi-positive holomorphic sectional curvature. We first prove that $X$ is rationally connected if $X$ has no truly flat tangent vectors at some point (which is satisfied when the holomorphic sectional curvature is quasi-positive). This result solves Yau's conjecture on positive holomorphic sectional curvature in a strong form. Moreover, we prove that $X$ admits a locally trivial morphism $\phi:X\to Y$ such that the fiber $F$ is rationally connected and the image $Y$ has a finite \'etale cover $A\to Y$ by an abelian variety $A$. We also show that the universal cover of $X$ is biholomorphic and isometric to the product $\Bbb{C}^m\times F$ of the complex Euclidean space $\Bbb{C}^m$ with the flat metric and the rationally connected fiber $F$ with the induced K\"ahler metric. Our structure theorem is a natural generalization of the structure theorem established by Howard-Smyth-Wu and Mok for holomorphic bisectional curvature.