Sur les opérations de tores algébriques de complexité un dans les variétés affines

Abstract

This thesis is devoted to the study of geometric properties of affine algebraic varieties endowed with an action of an algebraic torus. It comes from three preprints which correspond to the indicated points (1), (2), (3). Let $X$ be an affine variety equipped with an action of the algebraic torus $\mathbb{T}$. The complexity of the $\mathbb{T}$-action on $X$ is the codimension of the general $\mathbb{T}$-orbits. Under the assumption of normality and when the ground field is algebraically closed of characteristic 0, the variety $X$ admits a combinatorial description in terms of convex geometry. This description obtained by Altmann and Hausen in the year 2006 generalizes the classical one for toric varieties. Our purpose is to investigate new problems on the algebraic and geometric properties of the variety $X$ when the $\mathbb{T}$-action on $X$ is of complexity 1. (1) In the first part, a result gives an effective method to determine the integral closure of any affine variety defined over an algebraically field of characteristic 0 with a $\mathbb{T}$-action of complexity 1 in terms of the combinatorial description of Altmann-Hausen. (2) The calculations of the first part suggest a proof of the validity of the presentation of Altmann-Hausen in the case of complexity 1 over an arbitrary ground field. This is done in the second part. (3) In the third part, when the base field is perfect, we classify all the actions of the additive group on $X$ that are normalized by the $\mathbb{T}$-action of complexity 1. This classification generalizes classical works of Flenner and Zaidenberg in the surface case and of Liendo when the base field is algebraically closed of characteristic 0.