Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations

Abstract

We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group $G$. These manifolds are obtained as quotients of the domains of discontinuity in generalized flag varieties $G/P$ constructed by Kapovich-Leeb-Porti (arXiv:1306.3837), and in some cases by Guichard-Wienhard (arXiv:1108.0733). For $G$-Fuchsian representations and their Anosov deformations, where $G$ is simple, we compute the homology of the domains of discontinuity and of the quotient manifolds. For $G$-Fuchsian and $G$-quasi-Fuchsian representations in simple $G$ of rank at least two, we show that the quotient manifolds are not Kähler. We also describe the Picard groups of these quotient manifolds, compute the cohomology of line bundles on them, and show that for $G$ of sufficiently large rank these manifolds admit nonconstant meromorphic functions. In a final section, we apply our topological results to several explicit families of domains and derive closed formulas for topological invariants in some cases. We also show that the quotient manifold for a $G$-Fuchsian representation in ${\mathrm{PSL}}_3(\mathbb{C})$ is a fiber bundle over a surface, and we conjecture that this holds for all simple $G$.

69 pages, 2 figures. v3: Minor corrections and note about recent Pozzetti-Sambarino-Wienhard work. v2: Theorems D,E,F strengthened; other small revisions and corrections throughout

References in corpus (2)

• Higgs bundles and geometric structures on manifolds
• Schottky groups acting on homogeneous rational manifolds

Cited by in corpus (2)

• Anosov representations acting on homogeneous spaces: domains of discontinuity
• Nearly geodesic immersions and domains of discontinuity