Fano manifolds with nef tangent bundle and large Picard number
Fano manifolds with nef tangent bundle and large Picard number
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具有 nef 切丛和大皮卡数的 Fano 流形
DOI：
10.3792/pjaa.91.89
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Kiwamu Watanabe
中科院分区：
文献类型：
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作者：
Kiwamu Watanabe
摘要：我们研究了具有Nef切丛和大Picard数的Fano流形，关键词：Fano流形；Nef切丛；齐次流形；大Picard数。导言。经典的Gauss-Bonnet定理表明，唯一具有正曲率的紧黎曼曲面是黎曼球面。在高维情形下，Frankel猜想称具有正对分曲率的紧致Ka？hler流形是射影空间。这一猜想由Mori[12]和Siu-Yau[20]独立解决。Mori的证明是纯代数的，他得到了更一般的结果。事实上，他解决了HartShort猜想，即射影空间是唯一有充足切丛的射影流形[12]。之后，在复几何中，Mok证明了具有半正对分曲率的紧Ka？hler流形上的广义Frankel猜想[11]。作为其工作的推广，具有nef切丛的复射影流形已被许多作者研究过(例如，见[14])。根据Demailly，Peternell和Schneider[4]的结果，我们的研究可以归结为Fano流形的情形。让我们回顾一下由Campana和Peternell提出的以下猜想。猜想1.1([2])。任何具有新切丛的Fano流形都是有理齐次的，如果其维度至多为4[6]，则这一猜想成立，对于Picard数大于1的five-Fold也是如此[21]。最近，Kanemitsu[9]证明了Picard数为1的fi数的上述猜想。在本文中，我们将把[21]的一个结果推广到高维情形。我们的主要结果是定理1.2。设X是具有新切丛的范数。设m为维度，n为Picard数，i
Abstract: We study Fano manifolds with nef tangent bundle and large Picard number.Key words: Fano manifold; nef tangent bundle; homogeneous manifold; large Picardnumber.1. Introduction. The classical Gauss-Bonnet Theorem implies that the only compactRiemann surface with positive curvature is theRiemann sphere. In the higher dimensional case, theFrankel conjecture claims that a compact Ka¨hlermanifold with positive bisectional curvature is theprojective space. This conjecture was solved byMori [12] and Siu-Yau [20], independently. Mori’sproof is purely algebraic and he obtained a moregeneral result. In fact, he solved the Hartshorneconjecture, which says that the projective space isthe only projective manifold with ample tangentbundle [12]. After that, in complex geometry, Mokproved the generalized Frankel conjecture on com-pact Ka¨hler manifolds with semipositive bisectionalcurvature [11]. As a generalization of their works,complex projective manifolds with nef tangentbundle have been studied by many authors (forinstance, see [14]). By the result of Demailly,Peternell and Schneider [4], the study can bereduced to the case of Fano manifolds. Let us recallthe following conjecture posed by Campana andPeternell.Conjecture 1.1 ([2]). Any Fano manifoldwith nef tangent bundle is rational homogeneous.This conjecture holds if the dimension is atmost four [6], and this is also true for five-foldswhose Picard number greater than one [21]. Re-cently Kanemitsu [9] proved the above conjecturefor five-folds of Picard number one. In this paper,we will generalize a result of [21] to the higherdimensional case. Our main result isTheorem 1.2. Let X beaFanomanifoldwith nef tangent bundle. Let m be the dimension, nthe Picard number and i
Fano 5 重，nef 切丛且皮卡德数大于 1
DOI：
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发表时间：
2012
期刊：
影响因子：
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作者：
(K. Ueda);J. Fujioka;Y. Takahashi;T. Suzuki;S. Ishiwata;Y. Taguchi and Y. Tokura;Yoshinori Gongyo;西岡斉治;H. Yamaguchi;高橋陽太郎;西岡斉治;Yoshinori Gongyo;渡邉究
通讯作者：
渡邉究