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Good formal structures for flat meromorphic connections, I: Surfaces
Kiran S. Kedlaya
Source: Duke Math. J. Volume 154, Number 2
(2010), 343-418.
Abstract
We give a criterion under which one can obtain a good decomposition (in the sense of Malgrange) of a formal flat connection on a complex analytic or algebraic variety of arbitrary dimension. The criterion is stated in terms of the spectral behavior of differential operators and generalizes Robba's construction of the Hukuhara-Levelt-Turrittin decomposition in the one-dimensional case. As an application, we prove the existence of good formal structures for flat meromorphic connections on surfaces after suitable blowing up; this verifies a conjecture of Sabbah and extends a result of Mochizuki for algebraic connections. Our proof uses a finiteness argument on the valuative tree associated to a point on a surface in order to verify the numerical criterion.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1281963652
Digital Object Identifier: doi:10.1215/00127094-2010-041
Zentralblatt MATH identifier: 05788167
Mathematical Reviews number (MathSciNet): MR2682186
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