Hodge Theory and Complex Algebraic Geometry I: Volume 1

基本信息

Series:Cambridge Studies in Advanced Mathematics

Format:Paperback / softback 334 pages, Worked examples or Exercises

Publisher:Cambridge University Press

Imprint:Cambridge University Press

ISBN:9780521718011

Published: 20 Dec 2007

Classifications:Algebraic geometry

Readership:Professional & Vocational

Weight:506g

Dimensions:225 x 152 x 19 (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

两卷中的第一卷提供了对Kaehlerian几何和霍奇结构的现代介绍。书以复变量、复流形、全纯向量束、束束和上同调论的基本材料开始，后者比通常的几何处理方法更加理论化。然后，作者证明了卡勒恒等式，从而引出了hard Lefschetz定理和Hodge指数定理。

这本书以霍奇分解定理而告终。从几个方面研究了这些结果的意义。这本书完全独立，对学生来说是理想的，它的内容讲述了霍奇理论和复杂的代数几何，这是由P. Deligne和S. Bloch发展讨论而得。

文本由练习补充，这些练习在复杂代数几何中提供了有用的结果。

The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem.

The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P.Griffiths and his school, by P. Deligne, and by S. Bloch.

The text is complemented by exercises which provide useful results in complex algebraic geometry.

目录

Introduction

Part I. Preliminaries:

1. Holomorphic functions of many variables

2. Complex manifolds

3. Kähler metrics

4. Sheaves and cohomology

Part II. The Hodge Decomposition:

5. Harmonic forms and cohomology

6. The case of Kähler manifolds

7. Hodge structures and polarisations

8. Holomorphic de Rham complexes and spectral sequences

Part III. Variations of Hodge Structure:

9. Families and deformations

10. Variations of Hodge structure

Part IV. Cycles and Cycle Classes:

11. Hodge classes

12. Deligne-Beilinson cohomology and the Abel-Jacobi map

Bibliography

Index.

媒体推荐

这篇霍奇理论和卡勒几何的导论文章是这门学科的优秀和现代导论，闪烁着全面、严谨、清晰、严谨、主题坚定的目标，激发了人们对当代数学研究这一迷人领域的热情。本书极具启发性、启发性、挑战性、人性化，对复杂几何领域的学生、教师和研究人员都是极有价值的。“——Zentralblatt数学

“我建议任何有兴趣学习复微分或代数几何主题的人阅读Voisin的书。她的工作做得非常好。”——《爱丁堡数学学会学报》

这本书将成为这一领域的一个非常普遍的参考……对学生和研究者都很有用，他们都试图学习这门学科，同时也能以清晰和紧凑的格式找到丰富的结果。阐述非常精确，每一章前的引言有助于读者将注意力集中在文章的中心思想上。“——数学评论

“这本书提供了一个非常令人满意的阐述，本书所有的方法都在研究代数循环和霍奇理论。“——《伦敦数学学会公报》

'This introductory text to Hodge theory and Kahlerian geometry is an excellent and modern introduction to the subject, shining with comprehensiveness, strictness, clarity, rigor, thematic steadfastness of purpose, and catching enthusiasm for this fascinating field of contemporary mathematical research. This book is exceedingly instructive, inspiring, challenging and user-friendly, which makes it truly outstanding and extremely valuable for students, teachers, and researchers in complex geometry.' Zentralblatt MATH

'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

'… this book is going to become a very common reference in this field … useful for both a student trying to learn the subject as well as the researcher that can find a wealth of results in a clear and compact format. The exposition is very precise and the introduction that precedes each chapter helps the reader to focus on the main ideas in the text.' Mathematical Reviews

'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

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