Parabolic connections and stack of roots

TL;DR

This paper explores the relationship between strongly parabolic connections and stacks of roots on schemes with normal crossings divisors, establishing a correspondence that allows reconstruction of connections from their direct images.

Contribution

It introduces a new correspondence between strongly parabolic connections and holomorphic connections on stacks of roots, extending current understanding in the field.

Findings

Strongly parabolic connections correspond to holomorphic connections on stacks of roots.

Connections on stacks of roots can be reconstructed from their direct images.

The work generalizes the theory for rational weights with prescribed denominators.

Abstract

Given a scheme over a field endowed with a strict normal crossings divisor, we define strongly parabolic connections, consistently with the current terminology for Higgs bundles. When the weights are rational with prescribed denominators, we show that strongly parabolic connections correspond to holomorphic connections on the corresponding stack of roots. We use this correspondence to establish that a holomorphic connection on a stack of roots can be reconstructed from its direct image to the moduli space.