Castling equivalence for logarithmic flat connections

TL;DR

This paper investigates the extension of flat connections with logarithmic poles across hypersurfaces, providing new proofs for known results in specific cases and exploring the relationship with castling equivalence in higher dimensions.

Contribution

It offers a new proof of Mebkhout's theorem for weighted homogeneous plane curves and links the extension problem to castling equivalence of prehomogeneous vector spaces.

Findings

Extensions always exist for weighted homogeneous plane curves.

Castling equivalent linear free divisors have birationally Morita equivalent fundamental groupoids.

Examples of non-extendable flat connections are generated using this relationship.

Abstract

Let $X$ be a complex manifold containing a hypersurface $D$ and let $D^s$ denote the singular locus. We study the problem of extending a flat connection with logarithmic poles along $D$ from the complement $X \setminus D^s$ to all of $X$. In the setting where $D$ is a weighted homogeneous plane curve, we give a new proof of Mebkhout's theorem that extensions always exist. Our proof makes use of a Jordan decomposition for logarithmic connections as well as a version of Grothendieck's decomposition theorem for vector bundles over the `football' orbifold which is due to Martens and Thaddeus. In higher dimensions, we point out a close relationship between the extension problem and castling equivalence of prehomogeneous vector spaces. In particular, we show that the twisted fundamental groupoids of castling equivalent linear free divisors are `birationally' Morita equivalent and we use this to generate examples of non-extendable flat connections.