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Geometry of Higher Dimensional Algebraic Varieties : Operator Theory, Advances and Applications - Yoichi Miyaoka

Geometry of Higher Dimensional Algebraic Varieties

Operator Theory, Advances and Applications

Paperback Published: 20th March 1997
ISBN: 9783764354909
Number Of Pages: 218

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This book is based on lecture notes of a seminar of the Deutsche Mathematiker Vereinigung held by the authors at Oberwolfach from April 2 to 8, 1995. It gives an introduction to the classification theory and geometry of higher dimensional complex-algebraic varieties, focusing on the tremendeous developments of the sub­ ject in the last 20 years. The work is in two parts, with each one preceeded by an introduction describing its contents in detail. Here, it will suffice to simply ex­ plain how the subject matter has been divided. Cum grano salis one might say that Part 1 (Miyaoka) is more concerned with the algebraic methods and Part 2 (Peternell) with the more analytic aspects though they have unavoidable overlaps because there is no clearcut distinction between the two methods. Specifically, Part 1 treats the deformation theory, existence and geometry of rational curves via characteristic p, while Part 2 is principally concerned with vanishing theorems and their geometric applications. Part I Geometry of Rational Curves on Varieties Yoichi Miyaoka RIMS Kyoto University 606-01 Kyoto Japan Introduction: Why Rational Curves? This note is based on a series of lectures given at the Mathematisches Forschungsin­ stitut at Oberwolfach, Germany, as a part of the DMV seminar "Mori Theory". The construction of minimal models was discussed by T.

"Contains nice historical notes and exercises... Many examples are given to illustrate the theory... Beautifully written [with] up-to-date references to the subjects... Highly recommended."

--ZAA

Preface
Geometry of Rational Curves on Varieties
Introduction: Why Rational Curvesp. 3
Deformations and Rational Curvesp. 6
Construction of Non-Trivial Deformations via Frobeniusp. 28
Foliations and Purely Inseparable Coveringsp. 52
Abundance for Minimal 3-Foldsp. 75
Rationally Connected Fibrations and Applicationsp. 97
An Introduction to the Classification of Higherdimensional Complex Varieties
Notationsp. 133
Surfaces and a first View to Higher Dimensionsp. 135
Singularitiesp. 137
Vanishing Theoremsp. 145
The Ample Conep. 152
The Base Point Free Theoremp. 155
The Cone Theoremp. 165
Fano Manifolds and the Structure of Contractions of Extremal Raysp. 169
Surfaces and Threefoldsp. 178
Minimal Modelsp. 186
A View to Adjunction Theoryp. 196
Calabi-Yau Threefoldsp. 201
Indexp. 215
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783764354909
ISBN-10: 3764354909
Series: Operator Theory, Advances and Applications
Audience: General
Format: Paperback
Language: English
Number Of Pages: 218
Published: 20th March 1997
Publisher: Birkhauser Verlag AG
Country of Publication: CH
Dimensions (cm): 24.41 x 16.99  x 1.22
Weight (kg): 0.37
Edition Number: 4

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