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# Foliations on the moduli space of rank two connections on the projective line minus four points

Abstract : We look at natural foliations on the Painlevé VI moduli space of regular connections of rank $2$ on $\pp ^1 -\{ t_1,t_2,t_3,t_4\}$. These foliations are fibrations, and are interpreted in terms of the nonabelian Hodge filtration, giving a proof of the nonabelian Hodge foliation conjecture in this case. Two basic kinds of fibrations arise: from apparent singularities, and from quasiparabolic bundles. We show that these are transverse. Okamoto's additional symmetry, which may be seen as Katz's middle convolution, exchanges the quasiparabolic and apparent-singularity foliations.
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https://hal.archives-ouvertes.fr/hal-00547446
Contributor : Carlos Simpson <>
Submitted on : Friday, June 10, 2011 - 10:11:31 AM
Last modification on : Tuesday, December 8, 2020 - 9:47:59 AM
Long-term archiving on: : Sunday, September 11, 2011 - 2:21:59 AM

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### Identifiers

• HAL Id : hal-00547446, version 2
• ARXIV : 1012.3612

### Citation

Frank Loray, Masa-Hiko Saito, Carlos Simpson. Foliations on the moduli space of rank two connections on the projective line minus four points. D. Bertrand, Ph. Boalch, J.-M. Couveignes, P. Dèbes. Geometric and differential Galois theories, Société Mathématique de France, pp.115-168, 2013, Séminaires et Congrès n° 27. ⟨hal-00547446v2⟩

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