Given a formal at 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 André 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 André 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.
Cite this article
Kiran S. Kedlaya, Good Formal Structures for Flat Meromorphic Connections, III: Irregularity and Turning Loci. Publ. Res. Inst. Math. Sci. 57 (2021), no. 3/4, pp. 831–866DOI 10.4171/PRIMS/57-3-4