Algebraic flat connections and o-minimality

TL;DR

This paper characterizes algebraic flat connections with definable flat sections in o-minimal structures, linking regular singularity and unitary monodromy eigenvalues to o-minimality, refining prior results on Gauss-Manin connections.

Contribution

It provides an o-minimal characterization of classical properties of the Gauss-Manin connection, specifically relating definable flat sections to regular singularity and monodromy.

Findings

Algebraic flat connections with definable flat sections are exactly those with regular singularity and unitary monodromy eigenvalues.

Refines previous work by Bakker and Mullane on the o-minimality of flat connections.

Offers a new o-minimal perspective on the classical theory of Gauss-Manin connections.

Abstract

We prove that an algebraic flat connection has definable flat sections in the analytic exponential structure if and only if it is regular singular with unitary monodromy eigenvalues at infinity, refining previous work of Bakker and Mullane. This provides an o minimal characterisation of classical properties of the Gauss-Manin connection. v2: a few typos removed. Appears in the Laumon Volume, Springer Verlag, Simons subseries.