Castling equivalence for logarithmic flat connections
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.