2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/123001Deligne's regularity criterion for an integrable connection $\nabla$ on a smooth complex algebraic variety $X$ says that $\nabla$ is regular along the irreducible divisors at infinity in some fixed normal compactification of $X$ if and only if the restriction of $\nabla$ 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. 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.13 pages. This is a sequel to: [Baldassarri F., Towards an algebraic proof of Deligne's regularity criterion. An informal survey of open problems, Milan J. Math. 73 (2005)], and replaces math.AG/0411549. to appear in RIMS Kokyuroku BessatsuAlgebraic GeometryClassical Analysis and ODEs14 (primary), 34 (secondary)text