Structure of Algebraic Groups and Geometric Applications

This book originates from a series of 10 lectures given by Michel Brion at the Chennai Mathematical Institute during January 2011. The book presents Chevalley's theorem on the structure of connected algebraic groups, over algebraically closed fields, as the starting point of various other structure results developed in the recent past. Chevalley's structure theorem states that any connected algebraic group over an algebraically closed field is an extension of an abelian variety by a connected affine algebraic group. This theorem forms the foundation for the classification of anti-affine groups which plays a central role in the development of the structure theory of homogeneous bundles over abelian varieties and for the classification of complete homogeneous varieties. All these results are presented in this book. The book begins with an overview of the results, the proofs of which constitute the rest of the book. Various open questions also have been indicated in the course of the exposition. This book assumes certain preliminary knowledge of linear algebraic groups, abelian varieties, and algebraic geometry.