The Riemann-Hilbert mapping for $\mathfrak{sl}_2$ systems over genus two curves

Abstract

We prove in two different ways that the monodromy map from the space of irreducible sl 2 -differential-systems on genus two Riemann surfaces, towards the character variety of SL 2 -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 byÉtienne Ghys about Margulis' problem: the existence of curves of negative Euler characteristic in compact quotients of SL 2 (C). Nous montrons de deux manières différentes que l'application monodromie, depuis l'espace des sl 2 -systèmes différentiels irréductibles sur les surfaces de Riemann de genre deux, vers la variété de caractères des SL 2 -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 Ghysà propos d'un problème de Margulis : l'existence de courbes de caractéristique d'Euler négative dans les quotients compacts de SL 2 (C).

On certain Tannakian categories of integrable connections over Kähler manifolds

“…M [11], [4], [5] (any holomorphic connection on a Riemann surface is automatically integrable). Since the fundamental group of a compact complex torus T is abelian, there is no irreducible integrable connection on a rank two bundle over T. Also, the pullback of a reducible connection is reducible.…”

“…While they can't contain a complex surface [19, p. 239, Theorem 2], it is not known whether they can 2 I. Biswas, J. P. dos Santos, S. Dumitrescu and S. Heller contain compact Riemann surfaces of genus g > 1. A positive answer to Ghys' question would provide a nontrivial holomorphic map from the Riemann surface M to the quotient of SL(2, C) by the cocompact lattice containing the image of the monodromy homomorphism for D. In fact the two problems are equivalent (see [11] for explanations for the origin of Ghys' question).…”

Tame and relatively elliptic $\mathbb{CP}^1$-structures on the thrice-punctured sphere

“…Differentials with zero reduced exponents at all punctures correspond to parabolic projective structures (see [Kra69; Kra71b; Kra71a; DD17; HD19]). Differentials with integer non-zero reduced exponents and trivial holonomy at the punctures (apparent singularities) correspond to branched projective structures (see [Man72;CDF14a;Cal+19;FR21]). The next lemma implies that for structures in P (Σ) the absolute value of the exponent at a puncture coincides with the value of the index, as defined in §3.4.…”

Local deformations of branched projective structures: Schiffer variations and the Teichmüller map

“…A natural question is to better understand the complex geometry of these deformation spaces, which is of course best studied in terms of this projection. This is moreover motivated by the aforementioned correspondence proved in [CDHL19] between sl 2 -systems with irreducible monodromy ρ on a Riemann surface of genus g ≥ 2 and rational curves in M 2g−2,ρ ; more precisely they showed that if π : M 2g−2,ρ → T (S) has a fiber with at least three points, then that fiber is actually a rational curve and ρ is the monodromy of some sl 2 -system.…”

“…Our main motivation for this work lies in a more recent paper by Calsamiglia-Deroin-Heu-Loray (see [CDHL19]) in which they study sl 2 -systems on a closed surface of genus g ≥ 2, i.e. systems of first order ODEs of the form…”