Good formal structures for flat meromorphic connections, II: Excellent schemes

TL;DR

This paper proves the existence of good formal structures for flat meromorphic connections on excellent schemes after blowups, extending Mochizuki's theorem to a broader geometric context.

Contribution

It extends Mochizuki's theorem by establishing the existence of good formal structures on excellent schemes via blowups, using valuation space analysis.

Findings

Existence of good formal structures after blowups on excellent schemes.

Extension of Mochizuki's theorem to formal completions and complex analytic varieties.

Application of valuation space geometry to formal structure analysis.

Abstract

Given a flat meromorphic connection on an excellent scheme over a field of characteristic zero, we prove existence of good formal structures after blowing up; this extends a theorem of Mochizuki for algebraic varieties. The argument combines a numerical criterion for good formal structures from a previous paper, with an analysis based on the geometry of an associated valuation space (Riemann-Zariski space). We obtain a similar result over the formal completion of an excellent scheme along a closed subscheme. If we replace the excellent scheme by a complex analytic variety, we obtain a similar but weaker result in which the blowup can only be constructed in a small neighborhood of a prescribed point.