Good formal structures for flat meromorphic connections, I: Surfaces

We prove 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 numerical criterion, in terms of spectral behavior of differential operators, under which one can obtain a decomposition of a formal flat connection in arbitrary dimension. This generalizes the usual Turrittin-Levelt decomposition in the one-dimensional case. To ensure satisfaction of the numerical criterion after blowing up, we use compactness of the valuative tree associated to a point on a surface.