We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface $X$ with the $L^2$ complex relative to a suitable metric on the bundle and a complete metric on the punctured Riemann surface. Applying results of C. Simpson, we show the existence of a harmonic metric on this vector bundle, giving the same $L^2$ complex.

Following the work of Daniel Barlet [Pitman Res. Notes Math. Ser. 366 (1997), 19-59] and Ridha Belgrade [J. Algebra 245 (2001), 193-224], the aim of this article is to study the existence of (a,b)-hermitian forms on regular (a,b)-modules. We show that every regular (a,b)-module E with a non-degenerate bilinear form can be written in a unique way as a direct sum of (a,b)-modules $E_i$ that admit either an (a,b)-hermitian or an (a,b)-anti-hermitian form or both; all three cases are possible, and we give...

Homology with values in a connection with possibly irregular singular points on an algebraic curve is defined, generalizing homology with values in the underlying local system for a connection with regular singular points. Integration defines a perfect pairing between de Rham cohomology with values in the connection and homology with values in the dual connection.

We give an algebraic description of (wave) fronts that appear in strictly hyperbolic Cauchy problems. A concrete form of a defining function of the wave front issued from the initial algebraic variety is obtained with the aid of Gauss-Manin systems satisfied by Leray's residues.

Soit $\omega$ un germe en $0\in {\mathbf{C}}^n$ de 1-forme différentielle holomorphe, satisfaisant la condition d’intégrabilité $\omega \wedge d\omega =0$ et non dicritique, i.e. sur toute surface $Z$ non intégrale de $\omega$, on ne peut tracer, au voisinage de 0, qu’un nombre fini de germes de courbes analytiques $({\Gamma }_i,P_i)$, intégrales de $\omega$, avec $P_i\in Z\cap Sing\omega$. Alors $\omega$ possède un germe d’hypersurface analytique intégrale.