Algebraic Geometry V Fano Varieties

The aim of this survey, written by V.A. Iskovskikh and Yu.G. Prokhorov, is to provide an exposition of the structure theory of Fano varieties, i.e. algebraic vareties with an ample anticanonical divisor. Such varieties naturally appear in the birational classification of varieties of negative Kodaira dimension, and they are very close to rational ones. This EMS volume covers different approaches to the classification of Fano varieties such as the classical Fano-Iskovskikh "double projection" method and its modifications, the vector bundles method due to S. Mukai, and the method of extremal rays. The authors discuss uniruledness and rational connectedness as well as recent progress in rationality problems of Fano varieties. The appendix contains tables of some classes of Fano varieties.

This book will be very useful as a reference and research guide for researchers and graduate students in algebraic geometry.

Introduction.- Preliminaries.- Basic Properties of Fano Varieties.- Del Pezzo Varieties and Fano Varieties of Large Index.- Fano Threefolds with $/rho=1$.- Fano Manifolds of Coindex 3.- Boundedness and Rational Connectedness of Fano Manifolds.- Fano Manifolds with $ rho /ge 2$.- Rationality Questions for Fano Varieties I.- Rationality Questions for Fano Varieties II.- General Constructions of Rationality and Unirationality.- Some Particular Results, Generalizations and Open Problems.- Appendix: Tables.- References.- Index.