By Yves André

This is a learn of algebraic differential modules in numerous variables, and of a few in their family with analytic differential modules. allow us to clarify its resource. the assumption of computing the cohomology of a manifold, specifically its Betti numbers, through differential kinds is going again to E. Cartan and G. De Rham. when it comes to a gentle complicated algebraic kind X, there are 3 versions: i) utilizing the De Rham advanced of algebraic differential types on X, ii) utilizing the De Rham advanced of holomorphic differential varieties at the analytic an manifold X underlying X, iii) utilizing the De Rham advanced of Coo advanced differential kinds at the vary entiable manifold Xdlf underlying Xan. those editions tum out to be similar. particularly, one has canonical isomorphisms of hypercohomology: whereas the second one isomorphism is a straightforward sheaf-theoretic end result of the Poincare lemma, which identifies either vector areas with the complicated cohomology H (XtoP, C) of the topological area underlying X, the 1st isomorphism is a deeper results of A. Grothendieck, which exhibits particularly that the Betti numbers could be computed algebraically. This consequence has been generalized through P. Deligne to the case of nonconstant coeffi cients: for any algebraic vector package .M on X endowed with an integrable normal connection, one has canonical isomorphisms The inspiration of standard connection is a better dimensional generalization of the classical proposal of fuchsian differential equations (only commonplace singularities).

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Xd). Then both wtX,Z)/K and nl/K(logZ) are freely generated by {~l , dX2, ... , dXd}, so the first assertion is clear. 1], with the simplification due to the assumption that K is a field of characteristic zero, so that the Models and log schemes 48 divided power calculus is not needed. 5), follows by lack of torsion. 2. With this definition, the functoriality of Diff~x, M)/(y,N) is formal. ~w OX,r,z ~ wlX,Z)/K,r,z ~ wlx,z) /(y,W),r,z ~ O. 3) is dual to (B. 1). 2 Irregularity in several variables Introduction In this chapter, we tackle the study of irregularity in several variables.

In Section 6, we establish, under a technical assumption, a formal decomposition of integrable connections in several variables at a singUlar divisor. The last section contains technical results about cyclic vectors and indicial polynomials which will be crucial in the last chapter. In particular, we give sufficient (and i: v v 51 Irregularity in several variables essentially necessary) conditions for the existence, in the neighborhood of a singular divisor, of a cyclic vector with respect to a transversal derivation.

In the last three sections, which are of a less elementary nature, we give up the birational point of view. We then study the variation of Newton polygons when one restricts the connection to various curves transversal to Z, not necessarily belonging to a family of integral curves. 1). 5). We introduce the (principal) Newton polygon of an integrable connection at a singular divisor and study its behaviour under inverse image. 4). In Section 6, we establish, under a technical assumption, a formal decomposition of integrable connections in several variables at a singUlar divisor.