Hodge Theory and Complex Algebraic Geometry II: Volume 2

基本信息

Series:Cambridge Studies in Advanced Mathematics

Format:Paperback / softback 362 pages, Worked examples or Exercises; 4 Line drawings, unspecified

Publisher:Cambridge University Press

Imprint:Cambridge University Press

ISBN:9780521718028

Published: 20 Dec 2007

Classifications:Algebraic geometry

Readership:Professional & Vocational

Weight:542g

Dimensions:230 x 155 x 22 (mm)

Pub. Country:United Kingdom

Language:English

页面参数仅供参考，具体以实物为准

书籍简介

2003年第二卷关于Kaehlerian几何学和Hodge理论的论述从代数簇的拓扑开始。超平面截面上的Lefschetz定理的证明，Lefschetz铅笔的Picard-Lefschetz研究，以及关于Leray谱序列和全局不变环的退化的Deligne定理。第二部分的主要结果是广义诺瑟尔-左撇子定理，阿贝尔-雅可比图的一般平凡性，以及最重要的诺瑟尔-左撇子连通定理。

本书的最后一部分致力于霍奇理论和代数循环之间的关系。本书以阿贝尔变种的循环为例作结，其中举例说明了布洛赫和比维尔的一些结果。文本由练习补充，在复杂的代数几何中给出有用的结果。

它将受到代数和微分几何研究人员的欢迎。

The 2003 second volume of this account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. Proofs of the Lefschetz theorem on hyperplane sections, the Picard-Lefschetz study of Lefschetz pencils, and Deligne theorems on the degeneration of the Leray spectral sequence and the global invariant cycles follow. The main results of the second part are the generalized Noether-Lefschetz theorems, the generic triviality of the Abel-Jacobi maps, and most importantly Nori's connectivity theorem, which generalizes the above.

The last part of the book is devoted to the relationships between Hodge theory and algebraic cycles. The book concludes with the example of cycles on abelian varieties, where some results of Bloch and Beauville, for example, are expounded. The text is complemented by exercises giving useful results in complex algebraic geometry.

It will be welcomed by researchers in both algebraic and differential geometry.

目录

Introduction. Part I. The Topology of Algebraic Varieties:

1. The Lefschetz theorem on hyperplane sections

2. Lefschetz pencils

3. Monodromy

4. The Leray spectral sequence

Part II. Variations of Hodge Structure:

5. Transversality and applications

6. Hodge filtration of hypersurfaces

7. Normal functions and infinitesimal invariants

8. Nori's work

Part III. Algebraic Cycles:

9. Chow groups

10. Mumford' theorem and its generalisations

11. The Bloch conjecture and its generalisations

References

Index.

媒体推荐

正如所料，作者在第二卷中也一直保持着她精湛的写作风格。整个两卷本的文本具有很高的教育意义，鼓舞人心，对读者友好，总体上是优秀的。毫无疑问，这两卷必须被视为是一个不可缺少的标准文本，尤其是对于超越代数几何的高级学生，教师，以及在这个当代数学领域的研究人员而言。作者同时以独特的方式提供了完整的说明性阐述和主题的全新呈现，这在教科书文献中仍然是一个相当特殊的特征。“——Zentralblatt数学

“这本书提供了一个非常令人满意的阐述，其中所有的方法，研究代数循环，已经产生了霍奇理论。”《伦敦数学学会公报》

“我建议任何对学习复微分或代数几何感兴趣的人阅读Voisin的书。她的工作做得非常好。”——《爱丁堡数学学会学报》

'All together, the author has maintained her masterly style also throughout this second, much more advanced volume, just as expected. The entire two-volume text is highly instructive, inspiring, reader-friendly and generally outstanding. Without any doubt, these two volumes must be seen as an indispensible standard text on transcendental algebraic geometry for advanced students, teachers, and also researchers in this contemporary field of mathematics. The author provides, simultaneously and in a unique manner, both a complete didactic exposition and an up-to-date presentation of the subject, which is still a rather exceptional feature in the textbook literature.' Zentralblatt MATH

'The book provides a very satisfying exposition of all the methods of studying algebraic cycles that have come out of Hodge theory.' Bulletin of the London Mathematical Society

'I would recommend anyone interested in learning about a topic in complex differential or algebraic geometry to read Voisin's volumes. She has done a remarkably good job.' Proceedings of the Edinburgh Mathematical Society

作者简介

克莱尔·沃辛(Claire Voisin)是法国高等科学研究学院的教授

Claire Voisin is a Professor at the Institut des Hautes Études Scientifiques, France