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Geometry of Higher Dimensional Algebraic Varieties / Edition 1

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The subject of this book is the classification theory and geometry of higher dimensional varieties: existence and geometry of rational curves via characteristic p-methods, manifolds with negative Kodaira dimension, vanishing theorems, theory of extremal rays (Mori theory), and minimal models. The book gives a state-of-the-art intro- duction to a difficult and not readily accessible subject which has undergone enormous development in the last two decades. With no loss of precision, the volume focuses on the spread of ideas rather than on a deliberate inclusion of all proofs. The methods presented vary from complex analysis to complex algebraic geometry and algebraic geometry over fields of positive characteristics. The intended audience includes students in algebraic geometry and complex analysis as well as researchers in these fields and experts from other areas who wish to gain an overview of the theory.

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Editorial Reviews

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"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."


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Product Details

  • ISBN-13: 9783764354909
  • Publisher: Birkhauser Basel
  • Publication date: 2/4/2004
  • Series: Oberwolfach Seminars Series , #26
  • Edition description: 1997
  • Edition number: 1
  • Pages: 218
  • Product dimensions: 6.60 (w) x 9.50 (h) x 0.70 (d)

Table of Contents

I Geometry of Rational Curves on Varieties.- Introduction: Why Rational Curves.- Lecture I: Deformations and Rational Curves.- Lecture II: Construction of Non-Trivial Deformations via Frobenius.- Lecture III: Foliations and Purely Inseparable Coverings.- Lecture IV: Abundance for Minimal 3-Folds.- Lecture V: Rationally Connected Fibrations and Applications.- References.- II An Introduction to the Classification of Higherdimensional Complex Varieties.- Preface.- Prerequisites.- N Notations.- 0 Surfaces and a first View to Higher Dimensions.- 1 Sigularities.- 2 Vanishing Theorems.- 3 The Ample Cone.- 4 The Base Point Free Theorem.- 5 The Cone Theorem.- 6 Fano Manifolds and the Structure of Contractions of Extremal Rays.- 7 Surfaces and Threefolds.- 8 Minimal Models.- 9 A View to Adjunction Theory.- 10 Calabi-Yau Threefolds.- References.

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