We define the zero-th Gauss-Manin stratification of a stratified bundle with respect to a smooth morphism and use it to study the homotopy sequence of stratified fundamental group schemes.

The Leitmotiv of this work is to find suitable notions of dual varieties in a general sense. We develop the basic elements of a duality theory for varieties and complex spaces, by adopting a geometric and a categorical point of view. One main feature is to prove a biduality property for each notion which is achieved in most cases.

Let X be a smooth complex projective variety of dimension n ≥ 3. A notion of geometric genus pg(X,E) for ample vector bundles E of rank r < n on X admitting some regular sections is introduced. The following inequality holds: pg(X,E) ≥ hn-r,0(X). The question of characterizing equality is discussed and the answer is given for E decomposable of corank 2. Some conjectures suggested by the result are formulated.

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...