An algebraic proof of Deligne's regularity criterion for integrable connections
Abstract
Description
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.
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 Bessatsu
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 Bessatsu