Weak Lefschetz theorems and the topology of zero loci of ample vector bundles
Weak Lefschetz theorems and the topology of zero loci of ample vector bundles
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DOI：
10.4310/cag.2014.v22.n4.a1
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发表时间：
2014
影响因子：
0.7
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
在过去的几十年里，代数几何中的正性概念和拓扑性质一直是相互滋养的。有一些有趣的结果，如Lefschetz超平面定理、Fulton-Hansen的连通性定理、Barth-Larsen定理等。特别地，根据上下文和目的发展了Lefschetz超平面定理，即光滑射影簇X的同伦群可以与X上的采样线丛的(全纯)截面的零轨迹的同伦群相比较。本文本着Lefschetz超平面定理的精神，致力于研究光滑射影簇上采样向量丛的截面的零轨迹的拓扑。在这个方向上，Sommese给出了以下著名的结果：
Positivity concepts in algebraic geometry and topological properties have been nourishing each other in the last decades. There are interesting results, such as the Lefschetz hyperplane theorem, the connectedness theorem of Fulton–Hansen, the Barth–Larsen theorem, and so on. In particular, the Lefschetz hyperplane theorem has been developed according to the context and the objectives, which says that the homotopy groups of a smooth projective variety X can be compared with those of the zero locus of (holomorphic) sections of ample line bundles on X. This paper is devoted to the study of the topology of the zero locus of sections of ample vector bundles on smooth projective varieties, in the spirit of the Lefschetz hyperplane theorem. In this direction, Sommese gave the following celebrated result: