T-varietes affines: actións du groupe additif et singularites

Abstract

Let k be an algebraically closed field of characteristic 0. The algebraic torusTn = T of dimension n is the algebraic variety (k*)n with its natural structure of algebraic group. A T-variety is an algebraic variety endowed with an effective action of the torus T. This thesis is devoted to the study of two aspects of normal affine T-varietes: the additive group actions and the characterization of singularities. The introduction is divided in three parts. First, we introduce a combinatorial description of normal affine T-varieties, this corresponds to Chapter 1. We also give a historical overview on the subject. In the second part we prsent the results concerning the additive group actions of affine T-varieties, these results are developed in Chapters 2 and 3. Finally, we expose the results of Chapter 4 abouth the classificatin of singularities on T-varieties. In this introduction all varieties are assumed to be normal.