Towards an algebraic proof of Deligne's regularity criterion

The point of the talk is that of providing purely algebraic definitions and proofs of otherwise known results on integrable systems of linear di erential equations with only a finite dimensional space of local solutions over a smooth algebraic manifold. This talk represents work in collaboration with Y. Andr´e and it is based on our joint book: “De Rham Cohomology of Di erential Modules on Algebraic Varieties”, Progress in Mathematics Vol. 189, Birkhaueser (2001). We correct some statements in that book. The result is still incomplete. Summary: 1. Review of the notion of regular singular point for a linear ordinary di erential equation with meromorphic coecients. Global counterpart “Fuchsian equations”) on a Riemann surface. 2. Examples: Hypergeometric equations on the projective complex line, with 3 singular points. 3. Generalisation of the notion of regular singularity along a divisor to integrable overdetermined systems of linear PDE’s over a complex algebraic manifold, and of the notion of fuchsian connection on an algebraic vector bundle. 4. Deligne’s canonical extension of a Fuchsian connection on a smooth complex algebraic variety, as a logarithmic connection with singularities along a divisor with normal crossings on a Hironaka compactification. 5. Deligne’s criterion of regularity on a normal (not necessarily smooth!) compactification: one only needs to consider the behaviour along the divisors at infinity. 6. (Counter)examples of J. Bernstein to some statements in loc. cit.. 6. Reduction process in a purely algebraic proof of 5. 1