no. 1
The Riemann-Hilbert mapping for 𝔰𝔩2 systems over genus two curves
Calsamiglia, Gabriel1; Deroin, Bertrand2; Heu, Viktoria3; Loray, Frank4
1 Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis, s/n, Bloco H – Campus do Gragoatá São Domingos-Niterói – RJ – CEP: 24.210-201
2 Laboratoire AGM – CNRS/Université Cergy-Pontoise, 2 av. Adolphe Chauvin, 95302 Cergy-Pontoise, France
3 IRMA, 7 rue René-Descartes, 67084 Strasbourg Cedex, France
4 Univ Rennes, CNRS, IRMAR – UMR 6625, 35000 Rennes, France
Bulletin de la Société Mathématique de France, Volume 147 (2019) no. 1, pp. 159-195

We prove in two different ways that the monodromy map from the space of irreducible 𝔰𝔩2 differential systems on genus two Riemann surfaces, towards the character variety of SL2 representations of the fundamental group, is a local diffeomorphism. We also show that this is no longer true in the higher genus case. Our work is motivated by a question raised by Étienne Ghys about Margulis’ problem: the existence of curves of negative Euler characteristic in compact quotients of SL2(ℂ).

Nous montrons de deux manières différentes que l’application monodromie, depuis l’espace des 𝔰𝔩2 systèmes différentiels irréductibles sur les surfaces de Riemann de genre deux, vers la variété de caractères des SL2 représentations du groupe fondamental, est un difféomorphisme local. Nous montrons aussi que ce n’est plus le cas en genre supérieur. Notre travail est motivé par une question d’Étienne Ghys à propos d’un problème de Margulis  : l’existence de courbes de caractéristique d’Euler négative dans les quotients compacts de SL2(ℂ).

Received:
Revised:
Accepted:
Published online:
DOI: 10.24033/bsmf.2778
Classification: 34Mxx, 14Q10, 32G34, 53A30, 14H15
Keywords: $\mathfrak{sl}_2$ systems over curves, monodromy, Riemann-Hilbert, projective structures, holomorphic connections, foliations
Mots-clés : $\mathfrak{sl}_2$-systèmes sur les courbes, monodromie, Riemann-Hilbert, structures projectives, connexions holomorphes, feuilletages

Calsamiglia, Gabriel 1; Deroin, Bertrand 2; Heu, Viktoria 3; Loray, Frank 4

1 Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis, s/n, Bloco H – Campus do Gragoatá São Domingos-Niterói – RJ – CEP: 24.210-201
2 Laboratoire AGM – CNRS/Université Cergy-Pontoise, 2 av. Adolphe Chauvin, 95302 Cergy-Pontoise, France
3 IRMA, 7 rue René-Descartes, 67084 Strasbourg Cedex, France
4 Univ Rennes, CNRS, IRMAR – UMR 6625, 35000 Rennes, France 
@article{BSMF_2019__147_1_159_0,
     author = {Calsamiglia, Gabriel and Deroin, Bertrand and Heu, Viktoria and Loray, Frank},
     title = {The {Riemann-Hilbert} mapping for $\protect \mathfrak{sl}_2$ systems over genus two curves},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {159--195},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {147},
     number = {1},
     year = {2019},
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     url = {https://www.numdam.org/articles/10.24033/bsmf.2778/}
}
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Calsamiglia, Gabriel; Deroin, Bertrand; Heu, Viktoria; Loray, Frank. The Riemann-Hilbert mapping for $\protect \mathfrak{sl}_2$ systems over genus two curves. Bulletin de la Société Mathématique de France, Volume 147 (2019) no. 1, pp. 159-195. doi: 10.24033/bsmf.2778

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