Good formal structures for flat meromorphic connections, III: Irregularity and turning loci

TL;DR

This paper refines the understanding of irregularities in formal flat meromorphic connections by analyzing the turning locus, introducing irregularity sheaves, and establishing functorial resolution methods applicable across various geometric contexts.

Contribution

It introduces a refined analysis of the turning locus, constructs irregularity sheaves and b-divisors, and develops a functorial resolution process for turning points applicable to diverse geometric settings.

Findings

Turning locus is of pure codimension 1 within the polar divisor.

Construction of an irregularity sheaf and associated b-divisor.

Functorial resolution of turning points across different schemes and varieties.

Abstract

Given a formal flat meromorphic connection over an excellent scheme over a field of characteristic zero, in a previous paper we established existence of good formal structures and a good Deligne-Malgrange lattice after suitably blowing up. In this paper, we reinterpret and refine these results by introducing some related structures. We consider the turning locus, which is the set of points at which one cannot achieve a good formal structure without blowing up. We show that when the polar divisor has normal crossings, the turning locus is of pure codimension 1 within the polar divisor, and hence of pure codimension 2 within the full space; this had been previously established by Andre in the case of a smooth polar divisor. We also construct an irregularity sheaf and its associated b-divisor, which measure irregularity along divisors on blowups of the original space; this generalizes another result of Andre on the semicontinuity of irregularity in a curve fibration. One concrete consequence of these refinements is a process for resolution of turning points which is functorial with respect to regular morphisms of excellent schemes; this allows us to transfer the result from schemes to formal schemes, complex analytic varieties, and nonarchimedean analytic varieties.