Good formal structures on flat meromorphic connections, I: Surfaces

TL;DR

This paper establishes a criterion for good formal decompositions of flat connections on surfaces, generalizing one-dimensional theories and confirming a conjecture by Sabbah through a valuative tree approach.

Contribution

It introduces a spectral criterion for formal decompositions of flat connections on surfaces, extending known one-dimensional results and proving a conjecture of Sabbah.

Findings

Existence of good formal structures after blowing up surfaces

Generalization of Hukuhara-Levelt-Turrittin decomposition to higher dimensions

Verification of Sabbah's conjecture for surfaces

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.