We classify stably simple reducible curve singularities in complex spaces of any dimension. This extends the same classification of irreducible curve singuarities obtained by V. I. Arnold. The proof is essentially based on the method of complete transversals by J. Bruce et al.

A Steiner bundle $E$ on ${\mathbb{P}}^n$ has a linear resolution of the form $0\to \mathcal{O}{\left(-1\right)}^s\to {\mathcal{O}}^t\to E\to 0$. In this paper we prove that a generic Steiner bundle $E$ is simple if and only if $\chi \left(\mathrm{End}E\right)$ is less or equal to 1. In particular we show that either $E$ is exceptional or it satisfies the inequality $t\le \left(\frac{n+1+\sqrt{{\left(n+1\right)}^2-4}}{2}\right)s$.

We give some easy necessary and sufficient criteria for twists of abelian varieties by Artin representations to be simple.

We consider the Springer fiber ${ℬ}_u$ corresponding to a nilpotent endomorphism $u$ of nilpotent order $2$. As a first result, we give a description of the elements of a given component of ${ℬ}_u$ which are fixed by the action of the standard torus relative to some Jordan basis of $u$. By using this result, we establish a necessary and sufficient condition of singularity for the components of ${ℬ}_u$.

We prove a singular version of Beilinson–Bernstein localization for a complex semi-simple Lie algebra following ideas from the positive characteristic case settled by [BMR06]. We apply this theory to translation functors, singular blocks in the Bernstein–Gelfand–Gelfand category O and Whittaker modules.

We define open book structures with singular bindings. Starting with an extension of Milnor’s results on local fibrations for germs with nonisolated singularity, we find classes of genuine real analytic mappings which yield such open book structures.

In the present paper, we give a first general construction of compactified moduli spaces for semistable $G$-bundles on an irreducible complex projective curve $X$ with exactly one node, where $G$ is a semisimple linear algebraic group over the complex numbers.

Soit $P\in ℝ[X_1,...,X_n]$ un polynôme. On appelle série de Dirichlet associée à $P$ la fonction : $s\mapsto Z(P;s)=\sum _{m\in {\mathbb{N}}^{*n}}P(m{)}^{-s}(s\in ℂ)$. Dans cet article nous étudions l’existence et les propriétés du prolongement méromorphe d’une telle série sous l’hypothèse qu’il existe $B\in ]0,1[$ tel que : i) $P\left(x\right)\to +\infty$ quand $\left|\right|x\left|\right|\to +\infty$ et $x\in [B,+\infty {[}^n$ et ii) $d\left(Z\right(P),[B,+\infty {[}^n)\>0$ où $Z\left(P\right)=\{z\in {ℂ}^n|P\left(z\right)=0\}$. Cette hypothèse est probablement optimale et en tout cas contient strictement toutes les classes de polynômes déjà traitées antérieurement. Sous cette hypothèse nos principaux résultats sont : l’existence du prolongement méromorphe au plan...