Algebraic isomonodromic deformations of logarithmic connections on the Riemann sphere and finite braid group orbits on character varieties

TL;DR

This paper explores algebraic isomonodromic deformations of logarithmic connections on the Riemann sphere, linking algebraizability to finite braid group orbits on character varieties and applying results to Garnier systems and higher-dimensional varieties.

Contribution

It introduces a natural property of algebraizability for universal deformations and relates it to monodromy representations with finite braid group orbits, extending to higher ranks.

Findings

Finite braid group orbits correspond to algebraic deformations.

Established equivalence between algebraizability and finite monodromy orbits under certain conditions.

Developed tools for constructing flat meromorphic connections on high-dimensional varieties.

Abstract

We study algebraic isomonodromic deformations of flat logarithmic connections on the Riemann sphere with $n\geq 4$ poles, for arbitrary rank. We introduce a natural property of algebraizability for the germ of universal deformation of such a connection. We relate this property to a peculiarity of the corresponding monodromy representation: to yield a finite braid group orbit on the appropriate character variety. Under reasonable assumptions on the deformed connection, we may actually establish an equivalence between both properties. We apply this result in the rank two case to relate finite branching and algebraicity for solutions of Garnier systems. For general rank, a byproduct of this work is a tool to produce regular flat meromorphic connections on vector bundles over projective varieties of high dimension.