AN ALGEBRAIC PROOF OF DELIGNE’S REGULARITY CRITERION

ABSTRACT. Deligne’s regularity criterion for an integrable connection r on a smooth complex algebraic variety X says thatr is regular along the irreducible divisors at innity in some xed normal compactication of X if and only if the restriction of r to every smooth curve on X is fuchsian (i. e. has only regular singularities at innity). The only if part is the difcult implication. Deligne’s proof is transcendental and uses Hironaka’s resolution of singularities. Following [1], we present a purely algebraic proof of this implication which does not use resolution beyond the case of plane curves. It relies upon a study of the formal structure of integrable connections on sur-faces with (possibly irregular) singularities along a divisor with normal crossings