• Intersection theory
• spherical varieties
• toric varieties

The aim of this thesis is to do intersection theory on spherical varieties, i.e. varieties on which a reductive group $G$ acts with an open
$B$-orbit, where $B$ is a Borel subgroup of $G$. It turns out that on these varieties intersection theory is easier (for a smooth complete spherical
variety over $\mathbb{C}$ it is its simplicial cohomology), and in fact the action
of the group gives us an explicit description of how the Chow ring $A^*(X)$ works.

In the first chapter we will recall what we need for the rest of the thesis about intersection theory: the definition of the intersection product and how
the Chow rings are related to the cohomology, if the field is $\mathbb{C}$.

The second one is really the hearth of the thesis: we give a set of generators for the Chow groups of a spherical variety and we describe how intersection
product works.

In the third chapter we study the varieties of the form $G/P$, where $G$ is a semisimple group and $P$ a parabolic subgroup of it. In this
setting the Bruhat decomposition allows us to calculate the Chow group, and we will find
another description of the Chow ring of $G/P$.

In the fourth chapter we discuss the geometry of spherical varieties: the main results will be that those varieties can be described combinatorially. In fact,
we can associate to a spherical variety a fan, which can be used to describe our variety both locally and globally: we can translate in a combinatorial
language questions such as when this variety is quasiprojective? When it is proper? Moreover, we will describe morphisms between spherical varieties
using these fans.
The main examples of spherical varieties are toric varieties, Grassmannians and complete symmetric varieties.

The fifth chapter is mostly aimed at defining the Halphen ring of a homogeneous space in a more friendly environment (which is more explicit
than the general one). But what is the Halphen ring: the idea is the following.
Assume that we want to do intersection theory on a homogeneous spherical variety, for example $(\mathbb{C}^*)^n$. Then the Chow
ring does not give us a lot of information because it is always 0 but in degree $n$, so we need something else: this somethig else is the Halphen ring.
The idea is that if we consider all the compactifications $p:X\to (\mathbb{C}^*)^n$ where $X$ is a spherical variety,
we can consider $\displaystyle{\lim_{\rightarrow} A^*(X)}$ instead of
simply $A^*((\mathbb{C}^*)^n)$.
This ring is much bigger than $A^*((\mathbb{C}^*)^n)$; it will describe better how ``intersection theory'' works,
and we will give a description
of this ring using the action of $G$. Actually this intersection
theory depends on the action of the group on our spherical variety, see the examples of this section.

In the last chapter we will talk about intersection theory on another particular case of spherical varieties: toric varieties.
We will find ways to compute the cap product with combinatorial methods
and we will give a combinatorial description of both the Chow ring of a complete smooth toric variety, and the Halphen ring of $(\mathbb{G}_m)^n$.