Good Formal Structures for Flat Meromorphic Connections, Ii: Excellent Schemes

The Hukuhara-Levelt-Turrittin decomposition theorem gives a classification of differential modules over the field C((z)) of formal Laurent series resembling the decomposition of a finite-dimensional vector space equipped with a linear endomorphism into generalized eigenspaces. It implies that after adjoining a suitable root of z, one can express any differential module as a successive extension of onedimensional modules. This classification serves as the basis for the asymptotic analysis of meromorphic connections around a (not necessarily regular) singular point. In particular, it leads to a coherent description of the Stokes phenomenon, i.e., the fact that the asymptotic growth of horizontal sections near a singularity must be described using different asymptotic series depending on the direction along which one approaches the singularity. (See [45] for a beautiful exposition of this material.) In our previous paper [26], we gave an analogue of the Hukuhara-Levelt-Turrittin decomposition for irregular flat formal meromorphic connections on complex analytic or algebraic surfaces. (The regular case is already well understood in all dimensions, by the work of Deligne [12].) The result [26, Theorem 6.4.1] states that given a connection, one can find a blowup of its underlying space and a cover of that blowup ramified along the pole locus of the connection, such that after passing to the formal completion at any point of the cover, the connection admits a good decomposition in the sense of Malgrange [31, §3.2]. This implies that one gets (formally at each point) a successive extension of connections of rank 1; one also has some control over the pole loci of these connections. The precise statement had been conjectured by Sabbah [38, Conjecture 2.5.1] and was proved in the algebraic case by Mochizuki [34, Theorem 1.1]. The methods of [34] and [26] are quite different; Mochizuki uses reduction to positive characteristic and some study of pcurvatures, whereas we use properties of differential modules over one-dimensional nonarchimedean analytic spaces. The purpose of this paper is to extend our previous theorem from surfaces to complex analytic or algebraic varieties of arbitrary dimension. Most of the hard