Lectures on the structure of algebraic groups and geometric applications By Michel Brion, Preena Samuel, V. Uma
English | PDF | 2013 | 125 Pages | ISBN : 9380250460 | 9.24 MB
The theory of algebraic groups has chiefly been developed along two distinct directions: linear (or, equivalently, affine) algebraic groups, and abelian varieties (complete, connected algebraic groups). This is made possible by a fundamental theorem of Chevalley: any con- nected algebraic group over an algebraically closed field is an ex- tension of an abelian variety by a connected linear algebraic group, and these are unique.
In these notes, we first expose the above theorem and related structure results about connected algebraic groups that are neither affine nor complete. The class of anti-affine algebraic groups (those having only constant global regular functions) features prominently in these developments. We then present applications to some ques tions of algebraic geometry: the classification of complete homoge- neous varieties, and the structure of homogeneous (or translation- invariant) vector bundles and principal bundles over abelian varieties.

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