On projective space bundle with nef normalized tautological line bundle
On projective space bundle with nef normalized tautological line bundle
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带有nef归一化同义反复线丛的射影空间丛
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发表时间：
2011
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影响因子：
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通讯作者：
安武 和範
中科院分区：
文献类型：
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作者：
K. Yasutake;安武 和範
本文研究了相对反标准线丛为nef的射影空间丛的结构。作为应用，我们得到了有限代数覆盖下的阿贝尔簇的一个刻画。对于光滑投射簇π：Y → X之间的态射，Y上的相对反正则因子−Kπ由反正则因子的差−Kπ：= −KY − π（−KX）定义。J. Kollár，Y. Miyaoka和S. Mori证明了非常数广义光滑态射的相对反正则因子在任意特征上都不可能是充分的[7]，[11].如果π：Y = PX（E）→ X是向量丛在X上的投影，我们知道相对反典型因子是正规重言因子的正比例。Miyaoka研究了Y是曲线的情况，并证明了归一化重言式因子的nefness等于向量丛的半稳定性[10]。Nakayama在[13]中将其推广到任意维数。本文研究了具有nef正规重言式因子的向量丛的更显式结构。在第一节中，我们回顾了定义和一些已知的结果。在第二节中，我们讨论了半样本情形，证明了这样的丛被某个有限非分歧覆盖拉回到被某个线丛扭曲是平凡的。在第三节中，我们讨论了X是光滑簇Z沿着光滑子簇的blow-up或光滑簇Z上的投射丛的情形。在这些情况下，我们表明，与nef正规重言式因子的向量丛在X上同构的拉回K上的向量丛具有相同的属性扭曲的例外因子。在第四节中，我们研究了切丛具有nef正规化重言式因子的流形。我们证明了这类曲面同构于交换曲面的商。进一步，在这样一个因子是半充分的假设下，我们可以证明阿贝尔簇的有限代数覆盖都是满足这一性质的簇。2000年数学学科分类。小学14 J 40;中学14 J10，14 J60。
In this paper, we study the structure of projective space bundles whose relative anti-canonical line bundle is nef. As an application, we get a characterization of abelian varieties up to finite étale covering. Introduction For a morphism between smooth projective varieties π : Y → X the relative anticanonical divisor −Kπ on Y is defined by the difference of anticanonical divisors −Kπ := −KY − π(−KX). J. Kollár, Y. Miyaoka and S. Mori proved that the relative anticanonical divisor of a non-constant generically smooth morphism cannot be ample in arbitrary characteristic [7], [11]. In the case where π : Y = PX(E) → X is a projectivization of vector bundle on X, we know that the relative anti-canonical divisor is positive proportion of the normalized tautological divisor. Miyaoka studied the case where Y is a curve and showed that the nefness of the normalized tautological divisor is equal to the semistability of vector bundle [10]. Nakayama generalized this to the arbitrary dimension in [13]. In this paper we study the more explicit structure of vector bundles with nef normalized tautological divisor. In Section 1, we review the definition and some known results. In Section 2, we treat semiample cases and show that a pullback of such a bundle by some finite unramified covering is trivial up to twist by some line bundle. In Section 3, we treat the case where X is a blow-up of a smooth variety Z along smooth subvariety or a projective bundle over a smooth variety Z. In these cases we show that the vector bundle with nef normalized tautological divisor on X is isomorphic to the pullback of vector bundle on K having the same property up to twist by the exceptional divisor. In Section 4 we study manifolds whose tangent bundle have a nef normalized tautological divisor. We prove such surfaces are isomorphic to a quotient of abelian surface by some finite étale morphism. Moreover under the assumption that such a divisor is semiample, we can show that finite étale covering of abelian varieties are all varieties satisfying this property. 2000 Mathematics Subject Classification. Primary 14J40; Secondary 14J10, 14J60.
影响因子：
0.9
作者：
P. Wilson
通讯作者：
P. Wilson
高维代数几何，2018年3月12-16日，东京大学研究生院数学科学研究生院大讲堂
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发表时间：
2018
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