An Algebraic Proof of Deligne’s Regularity Criterion for Integrable Connections

Deligne’s regularity criterion for an integrable connection ∇ on a smooth complex algebraic variety X says that ∇ is regular along the irreducible divisors at infinity in some fixed normal compactification of X if and only if the restriction of ∇ to every smooth curve on X is fuchsian (i. e. has only regular singularities at infinity). The “only if” part is the difficult 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 surfaces with (possibly irregular) singularities along a divisor with normal crossings.