An algebraic proof of Deligne's regularity criterion for integrable connections

Details An algebraic proof of Deligne's regularity criterion for integrable connections An algebraic proof of Deligne's regularity criterion for integrable connections

2007-01-30

arxiv.org/abs/math/0701895v1

Mathematics

Algebraic Geometry

13 pages. This is a sequel to: [Baldassarri F., Towards an algebraic proof of Deligne's regularity criterion. An informal surv

Scientific paper

Deligne'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.

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