Éléments de la théorie des systèmes différentiels géométriques

This book contains written versions of lectures presented at the summer school held at Séville in 1996 on coherent modules over the ring of linear differential operators. The book starts with a description of the inverse image and the local cohomology functors (P. Maisonobe and T. Torrelli). Two shorter papers treat the local duality theorem (L. N. Macarro) and a regular meromorphic extension of an integrable holomorphic connection (J. Briançon). L. N. Macarro and A. R. León describe an elementary proof of the faithful flatness of the sheaf of infinite order linear differential operators over the sheaf of finite order ones. Calculations of classical invariants (characteristic variety and its multiplicity, slopes along a smooth hypersurface) of coherent modules over the sheaf of linear differential operators can be found in the paper by F. J. Castro-Jimenéz and M. Granger. The long paper (almost 150 pages) by Z. Mebkhout is devoted to the positivity theorem, the comparison theorem and the Riemann existence theorem. The irregularity sheaf for a holonomic module and a hypersurface is defined and the fundamental regularity criterion gives its vanishing. The existence theorems of Riemann type and Frobenius type are proved here. The comparison theorem for vanishing cycles is discussed by P. Maisonobe and Z. Mebkhout. The book ends with papers devoted to irregular holonomic modules (B. Malgrange) and geometric irregularity (Y. Laurent). The book offers a very nice and comprehensive survey of the theory.

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