2007/01/30 by Yves André, André, Yves
Mathematics · Physics and Astronomy · #14 (primary) #34 (secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Waves and Solitons #math.AG #math.CA #msc:14 #msc:34
paper · pdf · doi:10.48550/arxiv.math/0701895
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
arxiv created 2007/01/30 · openalex publication_date 2007/01/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. 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.