2023/06/30 by Bischoff, Francis
#14M17 #22A22 #32S25 #34M35 #35Q15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2306.17802
Let X be a complex manifold containing a hypersurface D and let Ds denote the singular locus. We study the problem of extending a flat connection with logarithmic poles along D from the complement X ∖ Ds 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.