Good formal structures on flat meromorphic connections, I: Surfaces

Details Good formal structures on flat meromorphic connections, I: Surfaces Good formal structures on flat meromorphic connections, I: Surfaces

2008-11-02

arxiv.org/abs/0811.0190v4

Mathematics

Algebraic Geometry

61 pages; v4: refereed version, completely restructured from v3

Scientific paper

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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