Le sujet de cet article est le groupe de Picard de la variété de modules $M(r,c_1,c_2)$ des faisceaux algébriques semi-stables de rang $r$ et de classes de Chern $c_1,c_2$ sur ${\mathbf{P}}_2\left(\mathbf{C}\right)$. Le premier résultat est que $M(r,c_1,c_2)$ est localement factorielle, ce qui permet d’identifier Pic$\left(M\right(r,c_1,c_2\left)\right)$ et le groupe des classes d’équivalence linéaire des diviseurs de Weil de $M(r,c_1,c_2))$. Il existe une unique application $\delta :\mathbf{Q}\to \mathbf{Q}$ telle que dim$\left(M\right(r,c_1,c_2\left)\right)\>0$ si et seulement si $(c_2-(r-1)c_1^2/2r)/r\>\delta (c_1/r)$. Si on a égalité, Pic$\left(M\right(r,c_1,c_2\left)\right)$ est isomorphe à $\mathbf{Z}$, et si l’inégalité est stricte, Pic$\left(M\right(r,c_1,c_2\left)\right)$ est isomorphe à ${\mathbf{Z}}^2$. On donne ensuite...

Let M be a manifold with a regular foliation F. We recall the construction of the fundamental groupoid and the homotopy groupoid associated to F. We describe some interesting particular cases and give some glueing techniques. We characterize the cases where these groupoids are Hausdorff spaces.We study in particular both groupoids associated to foliations with Reeb components.

The notion of a $C^{r,s}$-diffeomorphism related to a foliation is introduced. A perfectness theorem for the group of $C^{r,s}$-diffeomorphisms is proved. A remark on $C^{n+1}$-diffeomorphisms is given.

For an exact differential form on a Riemannian manifold to have a primitive bounded by a given function $f$, by Stokes it has to satisfy some weighted isoperimetric inequality. We show the converse up to some constants if $M$ has bounded geometry. For a volume form, it suffices to have the inequality ($\left|\Omega \right|\le {\int }_{\partial \Omega }f\mathrm{d}\sigma$ for every compact domain $\Omega \subset M$). This implies in particular the “well-known” result that if $M$ is the universal covering of a compact Riemannian manifold with non-amenable fundamental group, then the volume...