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The global log canonical threshold, algebraic counterpart to Tian's alpha-invariant, plays an important role when studying the geometry of Fano varieties. In particular, Tian showed that Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In recent years Donaldson drafted a programme to precisely determine when a smooth Fano variety X admits a Kahler-Einstein metric. It was conjectured that the existence of such a metric is equivalent to X being K-stable, an algebraic-geometric property. A crucial step in Donaldson's programme consists on finding a Kahler-Einstein metric with edge singularities of small angle along a smooth anticanonical boundary. Jeffres, Mazzeo and Rubinstein showed that a dynamic version of the alpha-invariant could be used to find such metrics. The global log canonical threshold measures how anticanonical pairs fail to be log canonical. In this thesis we compute the global log canonical threshold of del Pezzo surfaces in various settings. First we extend Cheltsov's computation of the global log canonical threshold of complex del Pezzo surfaces to non-singular del Pezzo surfaces over a ground field which is algebraically closed and has arbitrary characteristic. Then we study which anticanonical pairs fail to be log canonical. In particular, we give a very explicit classifiation of very singular anticanonical pairs for del Pezzo surfaces of degree smaller or equal than 3. We conjecture under which circumstances such a classifcation is plausible for an arbitrary Fano variety and derive several consequences. As an application, we compute the dynamic alpha-invariant on smooth del Pezzo surfaces of small degree, where the boundary is any smooth elliptic curve C. Our main result is a computation of the dynamic alpha-invariant on all smooth del Pezzo surfaces with boundary any smooth elliptic curve C. The values of the alpha-invariant depend on the choice of C. We apply our computation to find Kahler-Einstein metrics with edge singularities of angle β along C.

La thèse consiste de deux parties. Dans la première partie, on montre la pureté des fibres de Springer affines pour $\gl_{4}$ dans le cas non-ramifié. Plus précisément, on construit une famille de pavages non standard en espaces affines de la grassmannienne affine, qui induisent des pavages en espaces affines de la fibre de Springer affine. Dans la deuxième partie, on introduit une notion de $\xi$-stabilité sur la grassmannienne affine $\xx$ pour le groupe $\gl_{d}$, et on calcule le polynôme de Poincaré du quotient $\xx^{\xi}/T$ de la partie $\xi$-stable $\xxs$ par le tore maximal $T$ par une processus analogue de la réduction de Harder-Narasimhan. / This thesis consists of two parts. In the first part, we prove the purity of affine Springer fibers for $\gl_{4}$ in the unramified case. More precisely, we have constructed a family of non standard affine pavings for the affine grassmannian, which induce an affine paving for the affine Springer fiber. In the second part, we introduce a notion of $\xi$-stability on the affine grassmannian $\xx$ for the group $G=\gl_{d}$, and we calculate the Poincaré polynomial of the quotient $\xx^{\xi}/T$ of the stable part $\xxs$ by the maximal torus $T$ by a process analogue to the Harder-Narasimhan reduction.