Towards an Algebraic Proof of Deligne’s Regularity Criterion. An Informal Survey of Open Problems |
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Authors: | Francesco Baldassarri |
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Institution: | (1) Dipartimento di Matematica P. & A., Università di Padova, Via Belzoni 7, I-35131 Padova, Italy |
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Abstract: | We review the notion of regular singular point of a linear differential equation with meromorphic coefficients, from the viewpoint
of algebraic geometry. We give several equivalent definitions of regularity along a divisor for a meromorphic connection on
a complex algebraic manifold and discuss the global birational theory of fuchsian differential modules over a field of algebraic
functions. We describe the generalized algebraic version of Deligne’s canonical extension, constructed in 1, I.4].
Our main interest lies in the algebraic form of Deligne’s regularity criterion 2, II.4.4 (iii)], asserting that, on a normal
compactification, only one codimensional components of the locus at infinity need to be considered. If one considers the purely
algebraic nature of the statement, it is surprising that the only existing proof of this criterion is the transcendental argument
given by Deligne in his corrigendum to loc. cit. dated April 1971. The algebraic proof given in our book 1, I.5.4] is also incorrect, as J. Bernstein kindly indicated to
us.We introduce some notions of logarithmic geometry to let the reader appreciate Bernstein’s (counter)examples to some statements
in our book 1]. Standard methods of generic projection in projective spaces reduce the question to a two-dimensional puzzle.
We report on ongoing correspondence with Y. André and N. Tsuzuki, leading to partial results and provide examples indicating
the subtlety of the problem.
Lecture held in the Seminario Matematico e Fisico on January 31, 2005 Received: June 2005 |
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Keywords: | Primary 32S40 |
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