Geometry of Higher Dimensional Algebraic Varieties

1. Front Matter

   Pages i-vi

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2. Geometry of Rational Curves on Varieties

   1. Front Matter

      Pages 1-1

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   2. Introduction: Why Rational Curves?

      • Yoichi Miyaoka, Thomas Peternell

      Pages 3-5

   3. Deformations and Rational Curves

      • Yoichi Miyaoka, Thomas Peternell

      Pages 6-27

   4. Construction of Non-Trivial Deformations via Frobenius

      • Yoichi Miyaoka, Thomas Peternell

      Pages 28-51

   5. Foliations and Purely Inseparable Coverings

      • Yoichi Miyaoka, Thomas Peternell

      Pages 52-74

   6. Abundance for Minimal 3-Folds

      • Yoichi Miyaoka, Thomas Peternell

      Pages 75-96

   7. Rationally Connected Fibrations and Applications

      • Yoichi Miyaoka, Thomas Peternell

      Pages 97-120

3. Back Matter

   Pages 121-127

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4. An Introduction to the Classification of Higherdimensional Complex Varieties

   1. Front Matter

      Pages 129-132

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   2. Prerequisites

      • Yoichi Miyaoka, Thomas Peternell

      Pages 133-205

5. Back Matter

   Pages 206-213

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6. Back Matter

   Pages 215-218

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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.

• Algebra
• Complex analysis
• Manifold
• algebraic geometry
• algebraic varieties
• calculus
• complex analyisis
• geometry
• proof
• theorem

"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

• RIMS, Kyoto University, Kyoto, Japan

  Yoichi Miyaoka

• Mathematisches Institut der Universität Bayreuth, Bayreuth, Germany

  Thomas Peternell

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