Duke Mathematical Journal

Good formal structures for flat meromorphic connections, I: Surfaces

Kiran S. Kedlaya

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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.

Article information

Source
Duke Math. J., Volume 154, Number 2 (2010), 343-418.

Dates
First available in Project Euclid: 16 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1281963652

Digital Object Identifier
doi:10.1215/00127094-2010-041

Mathematical Reviews number (MathSciNet)
MR2682186

Zentralblatt MATH identifier
1204.14010

Subjects
Primary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials [See also 13Nxx, 32C38]
Secondary: 32C38: Sheaves of differential operators and their modules, D-modules [See also 14F10, 16S32, 35A27, 58J15] 32P05: Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem)

Citation

Kedlaya, Kiran S. Good formal structures for flat meromorphic connections, I: Surfaces. Duke Math. J. 154 (2010), no. 2, 343--418. doi:10.1215/00127094-2010-041. https://projecteuclid.org/euclid.dmj/1281963652


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See also

  • See also: Kiran S. Kedlaya. Errata to “Good formal structures for flat meromorphic connections, I: Surfaces”. Duke Math. J. Vol. 161, No. 4 (2012), 733-734.
    Project Euclid: euclid.dmj/1330610811

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