Speaker: Prof. Michel Brion (Institut Fourier)

 

Title: Structure of algebraic groups

 

Abstract: 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 connected algebraic group over an algebraically closed field is an extension of an abelian variety by a connected linear algebraic group, and these are unique.

In these lectures, 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 questions of algebraic geometry: the classification of complete homogeneous varieties, and the structure of connected automorphism groups of complete varieties.

Prerequisites: notions of algebraic geometry over an algebraically closed field. Some familiarity with linear algebraic groups will be useful, but not necessary.

Some references:

M. Brion, P. Samuel, V. Uma:
Lectures on the structure of algebraic groups and geometric applications,
CMI Lecture Series in Mathematics 1, Hindustan Book Agency, 2013.
Available at http://www-fourier.ujf-grenoble.fr/~mbrion/chennai.pdf

J. Milne: A proof of the Barsotti-Chevalley theorem on algebraic groups,
arXiv:1311.6060v2

 

 

 

Date: Ⅰ. 15:00-17:00, November 10, 2014

         Ⅱ. 15:00-17:00, November 11, 2014

         Ⅲ. 13:00-15:00, November 12, 2014

         Ⅳ. 15:30-16:30, November 13, 2014 (Special colloquium)