Fibres de Springer et jacobiennes compactifiees

Intersections of Deligne--Lusztig varieties and Springer fibres

In this paper we prove a direct geometric relation between Deligne--Lusztig varieties and Springer fibres in type $\mathsf{A}$: For any rational unipotent element, the Springer fibre cuts out a unique component of a specific Deligne--Lusztig variety; moreover, this component forms an open dense subset of a component of the Springer fibre. This boils down to a map from the unipotent variety to the Weyl group, and combines several constructions with a combinatorial flavour (like Weyr normal forms, Robinson--Schensted correspondence, and Spaltenstein's and Steinberg's labellings); it also provide…

Sur le rang des Jacobiennes sur un corps de fonctions

On the rank of Jacobians over function fields.} Let $f:\mathcal{X}\to C$ be a projective surface fibered over a curve and defined over a number field $k$. We give an interpretation of the rank of the Mordell-Weil group over $k(C)$ of the jacobian of the generic fibre (modulo the constant part) in terms of average of the traces of Frobenius on the fibers of $f$. The results also give a reinterpretation of the Tate conjecture for the surface $\mathcal{X}$ and generalizes results of Nagao, Rosen-Silverman and Wazir.

Sur l'homologie des fibres de Springer affines tronquées

Following Goresky, Kottwitz and MacPherson, we compute the homology of truncated affine Springer fibers in the unramified case but under a purity assumption. We prove this assumption in the equivalued case. The truncation parameter is viewed as a divisor on an l-adic toric variety. For each fiber, we introduce a graded quasi-coherent sheaf on the toric variety whose space of global sections is precisely the l-adic homology of the truncated affine Springer fiber. Moreover, for some families of endoscopic groups, these sheaves show up in an exact sequence. As a consequence we prove Arthur's weig…

Dimension des fibres de Springer affines pour les groupes

Following Steinberg, we construct an adjoint quotient for the Vinberg semi-group and a section to this quotient. Then, after Ngô, we show the existence of a regular centralizer on it and use it to compute the affine Springer fibers for groups.

奇异支集与仿射 Springer 纤维的同调

Support singulier et homologie des fibres de Springer affines

本文构建无穷维叠的奇异支集理论，并用其计算 Grothendieck–Springer 仿射 Springer 层的奇异支集，从而关联 Goresky–Kottwitz–McPherson 的局部常定性猜想。

Irreducible components of exotic Springer fibres

Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson-Schensted bijection, which is a variant of the type C Robinson-Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the …