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This mini-course will present some fundamental results
on the structure of algebraic groups and their actions on
algebraic varieties. Algebraic groups have been chiefly studied
along two distinct directions: linear algebraic groups, and
abelian varieties. The aim of the lectures is to show how these
two directions are combined in the general case.
The prerequisites are some familiarity with algebraic geometry.
No prior knowledge of algebraic group theory will be required.
(1) The introductory Lecture 1 will start with basic definitions and examples, and then present Chevalley's theorem, which asserts that every connected algebraic group over an algebraically closed field is an extension of an abelian variety by a connected linear algebraic group. A "dual" theorem will also be discussed. (2) Lecture 2 will present further applications of these structure theorems; for example, every algebraic group over a finite field is the almost direct product of a linear algebraic group and an abelian variety. The proofs of both theorems will be sketched. (3) In Lecture 3, we will apply the former results to the study of algebraic groups which occur naturally in algebraic geometry: automorphism groups and Picard varieties of proper varieties. (4) Lecture 4 will present basic results on algebraic group actions: linearization of line bundles and equivariant projective embeddings (for actions of linear algebraic groups); the theorem of the square (for general groups). REFERENCES : M. Brion, P. Samuel and V. Uma : Lectures on the structure of algebraic groups and geometric applications, CMI Lecture Series in Mathematics 1, Hindustan Book Agency, 2013. http://www-fourier.ujf-grenoble.fr/~mbrion/chennai.pdf M. Brion, On automorphisms and endomorphisms of algebraic varieties, Automorphisms in Birational and Affine Geometry, Springer Proc. Math. Stat. 79, 59-82, Springer, 2014. arXiv:1304.7472 M. Brion, Which algebraic groups are Picard varieties? Science China Mathematics, 2014. arXiv:1310.0186 M.Brion, On linearization of line bundles, J. Math. Sci. Univ. Tokyo 22 (2015), 1-35. arXiv:1312.6267 |