This master’s thesis is meant to be a good introduction to toric varieties and sheaves theory. We will show some results already present in the literature whose proofs are either incomplete, divided in multiple sources or too complex for a novice. We will synthesize and detail these results while avoiding (as much as possible) required knowledge of the subject. We will finish this thesis with a proof of some vanishing theorems, required to tackle the Grauert-Riemenschneider theorem’s proof (in the toric case). Prerequisite : some basic knowledge of algebraic varieties or homology would help understand some of the concepts we discuss and a few examples we give refer to Riemann surfaces or CW complexes. But no knowledge past undergraduate-level is required.

Let G be a simple, connected, simply connected complex linear algebraic group with parabolic subgroup P G and nilpotent ideal n p. The proper collapsing map G x P n = Gn factors through the normal affine variety N := SpecC [G x P n] which is called a nilpotent variety. Assuming the collapsing is generically finite, we describe the equivariant divisor class group of N using rank 1 reflexive equivariant C[N]-modules. A representative of each class may be chosen as global sections of a line bundle over G x P' n' where G x P' n' = Gn' is a possibly distinct collapsing that factors through the same nilpotent variety. Assuming either G is of type A or the collapsing comes from specific weighted Dynkin diagrams,we showthat each representative arise from a weight that may be chosen dominant. Moreover, if the module represents a torsion element within the class group, then it is Cohen– Macaulay and we deduce a cohomological vanishing theorem.

In this master thesis we study finite Galois extensions of C(x). We prove Riemann existence theorem. The notions of rigidity, weak rigidity, and rationality are developed. We obtain the rigidity criterion which enable us to realise some groups as Galois groups over Q. Many examples of ramification types are constructed.

The aim of this master’s thesis is two-fold. Firstly to develop and interpret the low dimensional Hochschild cohomology of a k-algebra and secondly to establish a lower bound for the Hochschild cohomological dimension of a k-algebra; showing that nearly no commutative k-algebra is quasi-free.

In this thesis, we study the cohomology of line bundles on cotangent bundle of projective varieties. To be more precise, let $G$ be an semisimple algebraic group which is simply connected, $P$ a maximal subgroup and $\omega$ a dominant weight that generates the character group of $P$. Our goal is to understand the cohomology groups $H^i(T^*(G/P),\mathcal{L})$ where $\mathcal{L}$ is the sheaf of sections of a line bundle on $T^*(G/P)$. Under some conditions, we will show that there exists an isomorphism, up to grading, between $H^i(T^*(G/P),\mathcal{L})$ and $H^i(T^*(G/P),\mathcal{L}^{\vee})$. After we worked in a theoretical setting, we will focus on maximal parabolic subgroups related to nilpotent varieties. In this case, the Lie algebra of the unipotent radical of $P$ has a structure of prehomogeneous vector spaces. We will be able to determine which cases verify the hypothesis of the isomorphism by showing the existence of a $P$-covariant $f$ in $\comp[\nLie]$ and by studying its properties. We will be interested by the singularities of the affine variety $V(f)$. We will show that the normalisation of $V(f)$ has rational singularities.

In this thesis, we will study several algebraic, geometrical and topological properties of compact Riemann surfaces. Two principal subjects will be treated. First, using the fact that every compact Riemann surfaces of genus g greater or equal to 2 has a finite number of Weierstrass points, we will be able to prove that those surfaces have a finite number of automorphism. Afterward, we will study the Eichler's trace formula. This formula allow us to find the character of an automorphism acting on the space of holomorphic q-differentials. We will start our study using Klein's quartic curve. We will apply Eichler's formula in this case, which will allow us to familiarize ourselves with the statement of the theorem. Finally, we will demonstrate the Eichler's trace formula, treating the case where the automorphism acts fixed point freely separately from the case where the automorphism has fixed points.

The topological notion of a quotient is fairly simple. Given a topological group $G$ acting on a topological space $X$, one gets the natural application from $X$ to the quotient space $X/G$. In algebraic geometry, unfortunately, it is generally not possible to give the orbit space the structure of an algebraic variety. In the special case of a linearly reductive group acting on a projective variety $X$, the geometric invariant theory allows us to get a morphism of variety from an open $U$ of $X$ to a projective variety $X//G$, which is as close as possible to a quotient map, from a topological point of view. As an example, let $ X\subseteq P^{n}$ be a $k$-projective variety on which acts a linearly reductive group $G$. Suppose further that this action is induced by a linear action of $G$ on $A^{n+1}$ and let $\widehat{X}\subseteq A^{n +1}$ be the affine cone over $X$. By an important theorem of the classical invariants theory, there exist homogeneous invariants $f_{1},..., f_{r}\in C[\widehat{X}]^{G}$ such as $$\C[\widehat{X}]^{G}=\C[f_{1},...,f_{r}].$$ The locus in $X$ of $f_{1},...,f_{r}$ is called the nullcone, noted $N$. Let $Proj(C[\widehat{X}]^{G})$ be the projective spectrum of the invariants ring. The rational map $$\pi:X\dashrightarrow Proj(C[f_{1},...,f_{r}])$$ induced by the inclusion of $C[\widehat{X}]^{G}$ in $C[\widehat{X}] $ is then surjective, constant on the orbits and separates orbits as much as possible, that is, the fibres contains exactly one closed orbit. A regular map is obtained by removing the nullcone; we then get a regular map $$\pi:X \backslash N\rightarrow Proj(C[f_{1},...,f_{r}])$$ which still satisfy the preceding properties. The Hilbert-Mumford criterion, due to Hilbert and revisited by Mumford nearly half-century later, can be used to describe $N$ without knowing the generators of the invariants ring. Since those are rarely known, this criterion had proved to be quite useful. Despite the important applications of this criterion in classical algebraic geometry, the demonstrations found in the literature are usually given trough the difficult theory of schemes. The aim of this master thesis is therefore, among others, to provide a demonstration of this criterion using classical algebraic geometry and of commutative algebra. The version that we demonstrate is somewhat wider than the original version of Hilbert \cite{hilbert}; a schematic proof of this general version is given in \cite{kempf}. Finally, the proof given here is valid for $C$ but could be generalised to a field $k$ of characteristic zero, not necessarily algebraically closed. In the second part of this thesis, we study the relationship between the preceding constructions and those obtained by including covariants in addition to the invariants. We give a Hilbert-Mumford criterion for covariants (Theorem 6.3.2) which is a theorem from Brion for which we prove a slightly more general version. This theorem, together with a simplified proof of a theorem of Grosshans (Theorem 6.1.7), are the elements of this thesis that can't be found in the literature.

Let G be a semisimple algebraic group on a field of characteristic 0. This thesis discusses a vanishing theorem for the higher cohomology of the sheaf D of differential operators on a flag variety of G. We show that if P is a parabolic subgroup of G, then H^i(G/P,D)=0 for all i>0. In fact, we give three independent proofs of this theorem. The first proof, due to Hesselink, only works if the parabolic subgroup P is a Borel subgroup. It uses a spectral sequence argument as well as the Borel-Weil-Bott theorem. The second proof, due to Kempf, only works if the unipotent radical of P acts trivially on its Lie algebra. It only uses the Borel-Weil-Bott theorem. Finally, the third proof, due to Elkik, is valid for any parabolic subgroup. However, it uses the Grauert-Riemenschneider theorem. We also present a detailled construction of the sheaf of differential operators on a variety.

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Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.