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Pré-Publication, Document De Travail Année : 2022

ON THE MONODROMY OF HOLOMORPHIC DIFFERENTIAL SYSTEMS

Indranil Biswas
  • Fonction : Auteur
  • PersonId : 924998
Sorin Dumitrescu
Sebastian Heller
  • Fonction : Auteur
  • PersonId : 1076079
Lynn Heller
  • Fonction : Auteur

Résumé

We explain the strategy of some recent results that construct holomorphic sl(2, C)-differential systems over some Riemann surfaces Σ g of genus g ≥ 2, satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian [BDHH1] or some cocompact Kleinian subgroup Γ ⊂ SL(2, C) [BDHH2]. As a consequence, there exist holomorphic maps from Σ g to the quotient space SL(2, C)/Γ, where Γ ⊂ SL(2, C) is a cocompact lattice, that do not factor through any elliptic curve [BDHH2]. This answers positively a question of Ghys in [Gh]; the question was also raised by Huckleberry and Winkelmann in [HW]. When M is a Riemann surface, a Torelli type theorem holds for the affine group scheme over C obtained from the category of holomorphic connections on étale trivial holomorphic bundles. We explain how to compute in a simple way the holonomy of a holomorphic connection on a free vector bundle. For a compact Kähler manifold M , we investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over M. If (respectively, Θ) stands for the affine group scheme over C obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism v : O(Θ) −→ O() of Hopf algebras. We present a description of the transpose of v in terms of the iterated integrals.
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Dates et versions

hal-03863150 , version 1 (21-11-2022)
hal-03863150 , version 2 (23-11-2022)

Identifiants

  • HAL Id : hal-03863150 , version 2

Citer

Indranil Biswas, Sorin Dumitrescu, Sebastian Heller, Lynn Heller, Joao Pedro dos Santos. ON THE MONODROMY OF HOLOMORPHIC DIFFERENTIAL SYSTEMS. 2022. ⟨hal-03863150v2⟩
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