Uniformization of compact complex manifolds by Anosov homomorphisms

We study uniformization problems for compact manifolds that arise as quotients of domains in complex flag varieties by images of Anosov homomorphisms. We focus on Anosov homomorphisms with “small” limit sets, as measured by the Riemannian Hausdorff codimension in the flag variety. Under such a codim...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 31; no. 4; pp. 815 - 854
Main Authors Dumas, David, Sanders, Andrew
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.08.2021
Springer Nature B.V
Subjects
Analysis
Deformation
Flags
Homomorphisms
Hypotheses
Lie groups
Manifolds
Mathematics
Mathematics and Statistics
Quotients
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Summary:We study uniformization problems for compact manifolds that arise as quotients of domains in complex flag varieties by images of Anosov homomorphisms. We focus on Anosov homomorphisms with “small” limit sets, as measured by the Riemannian Hausdorff codimension in the flag variety. Under such a codimension hypothesis, we show that all first-order deformations of complex structure on the associated compact complex manifolds are realized by deformations of the Anosov homomorphism. With some mild additional hypotheses we show that the character variety maps locally homeomorphically to the (generalized) Teichmüller space of the manifold. In particular this provides a local analogue of the Bers Simultaneous Uniformization Theorem in the setting of Anosov homomorphisms to higher-rank complex semisimple Lie groups.
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-021-00572-6