We describe the structure of singular transversely affine foliations of codimension one on projective manifolds with zero first Betti number. Our result can be rephrased as a theorem on rank two reducible flat meromorphic connections. 2 Transversely affine foliations 988 2.1 Definition 988 2.2 Interpretation in terms of rational 1-forms 989 2.3 Singular divisor and residues 990 2.4 Examples and first properties 992 2.5 Holonomy 996 2.6 Transversely affine foliations as transversely projective foliations 997 3 Cohomology jumping loci for local systems 1001 3.1 Group cohomology 1001 3.2 Cohomology jumping loci for quasi-projective manifolds 1002 4 Factorization of representations 1003 Key words and phrases. Foliations, Transverse Affine Structures. Jorge Vitório Pereira is partially supported by CNPq-Brazil. During preparation of this work, Gaël Cousin was a fellow of CNPq-Brazil.
Abstract. We study algebraic isomonodromic deformations of flat logarithmic connections on the Riemann sphere with n ≥ 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. Logarithmic flat connections and Fundamental group representations
The germ of the universal isomonodromic deformation of a logarithmic connection on a stable
$n$
-pointed genus
$g$
curve always exists in the analytic category. The first part of this article investigates under which conditions it is the analytic germification of an algebraic isomonodromic deformation. Up to some minor technical conditions, this turns out to be the case if and only if the monodromy of the connection has finite orbit under the action of the mapping class group. The second part of this work studies the dynamics of this action in the particular case of reducible rank 2 representations and genus
$g>0$
, allowing to classify all finite orbits. Both of these results extend recent ones concerning the genus 0 case.
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