Geometry of Higher Dimensional Algebraic Varieties
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...
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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.
PrefacePt. IGeometry of Rational Curves on VarietiesIntroduction: Why Rational Curves3Lecture IDeformations and Rational Curves6Lecture IIConstruction of Non-Trivial Deformations via Frobenius28Lecture IIIFoliations and Purely Inseparable Coverings52Lecture IVAbundance for Minimal 3-Folds75Lecture VRationally Connected Fibrations and Applications97Pt. IIAn Introduction to the Classification of Higherdimensional Complex VarietiesNNotations1330Surfaces and a first View to Higher Dimensions1351Singularities1372Vanishing Theorems1453The Ample Cone1524The Base Point Free Theorem1555The Cone Theorem1656Fano Manifolds and the Structure of Contractions of Extremal Rays1697Surfaces and Threefolds1788Minimal Models1869A View to Adjunction Theory19610Calabi-Yau Threefolds201Index215
