An algebraic proof of Deligne's regularity criterion

Details An algebraic proof of Deligne's regularity criterion An algebraic proof of Deligne's regularity criterion

2004-11-24

arxiv.org/abs/math/0411549v2

Mathematics

Algebraic Geometry

N. Tsuzuki kindly indicated to us a serious error in section 2 of this paper. We think we know a way out, and are working to a

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 regular ({\it 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 give here an elementary and purely algebraic proof of this implication: it is, as far as we know, the first algebraic proof of Deligne's regularity criterion.

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