Logarithmic decomposition of connections on a relatively punctured disk

TL;DR

This paper proves a decomposition theorem for connections on modules over a power series ring, showing they can be split into regular and diagonalizable parts, compatible with reductions modulo powers of t.

Contribution

It establishes a logarithmic decomposition for connections over a punctured disk with a formal parameter, extending classical results to a relative setting.

Findings

Connections decompose into regular and diagonalizable parts

Decomposition is compatible with modulo t^k reductions

Extension of Turrittin-Levelt-Jordan form to relative setting

Abstract

Let $R=C[[t]]$ be the ring of power series over an algebraically closed field $C$ of characteristic zero. We show that each connection on a finite flat $R((x))$-module is the sum of a regular singular connection and a diagonalizable $R((x))$-linear endomorphism when it admits a Turrittin-Levelt-Jordan form over $R((x))$. This decomposition is compatible with the limit of the logarithmic decompositions of the connections obtained by the reduction modulo $t^{k}$ of a given connection.