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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 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.

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The authors thank Beatrice Pozzetti and Anna Wienhard for helpful conversations related to this work, and an anonymous referee for helpful suggestions and corrections. The first author was supported by the U.S. National Science Foundation, through award DMS 1709877. The second author was supported by the Deutsche Forschungsgemeinschaft within the RTG 2229 “Asymptotic invariants and limits of groups and spaces” and by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC-2181/1—390900948 (the Heidelberg STRUCTURES Cluster of Excellence).

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The homological algebra appearing in Section 4 is entirely classical, but among geometric topologists interested in Anosov homomorphisms, it is probably less well-known. This appendix provides additional detail on one aspect: the case of the Grothendieck spectral sequence used in the proof of Theorem 4.1. The results here are particular examples of those in the original paper of Grothendieck [Gro57].

Let \(p: X\rightarrow Y\) be a regular covering of complex manifolds with deck group \(\Gamma \) and \(\mathcal {L}\) a \(\Gamma \)-equivariant locally free sheaf on X. There is a canonical locally free sheaf \(\mathcal {L}^{\Gamma }\) on Y such that \(p^{\star }\mathcal {L}^{\Gamma }\simeq \mathcal {L}\); we will call this the descent of \(\mathcal {L}\) to Y.

Next, the Dolbeault complex is the complex

$$\begin{aligned} A^{0}(X, \mathcal {L})\xrightarrow {\overline{\partial }} A^{0,1}(X, \mathcal {L})\rightarrow \cdots \end{aligned}$$

of smooth (0, q)-forms on X with values in the associated holomorphic vector bundle. Since \(\Gamma \) acts holomorphically on X,  the \(\mathbb {C}\)-vector spaces \(A^{0,q}(X, \mathcal {L})\) are \(\Gamma \)-modules via pullback of forms, and thus

$$\begin{aligned} A^{0}(X, \mathcal {L})\xrightarrow {\overline{\partial }} A^{0,1}(X, \mathcal {L})\xrightarrow {\overline{\partial }} \cdots \end{aligned}$$

is an object of the abelian category of complexes of \(\Gamma \)-modules. Taking \(\Gamma \)-invariants defines a functor from this category to the category of complexes of \(\mathbb {C}\)-vector spaces which is left exact, and the right derived functors define the (hyper)-cohomology groups \(\mathbb {H}^{i}(\Gamma , A^{0, *}(X, \mathcal {L})).\)

Theorem A.1

   Let \(\mathcal {L}^{\Gamma }\) be the descent of \(\mathcal {L}\) to Y. Then, there are a canonical isomorphisms \(\mathbb {H}^{k}(\Gamma , A^{0, *}(X, \mathcal {L}))\simeq H^{k}(Y, \mathcal {L}^{\Gamma })\) for every \(k\geqslant 0.\)

Simultaneously taking the complex of inhomogeneous cochains leads to a double complex \(E_{0}^{p,q}:=C^{p}(\Gamma , A^{0,q}(X, \mathcal {L}))\) whose horizontal differentials are given by the differential on inhomogeneous group cochains \(d_{\Gamma }: C^{p}(\Gamma , A^{0,q}(X, \mathcal {L}))\rightarrow C^{p+1}(\Gamma , A^{0,q}(X, \mathcal {L}))\) and vertical differentials are given by \(\overline{\partial }: C^{p}(\Gamma , A^{0,q}(X, \mathcal {L}))\rightarrow C^{p}(\Gamma , A^{0,q+1}(X, \mathcal {L})).\)

Theorem A.2

   The degree k cohomology of the total complex of \(E_{0}^{p,q}:=C^{p}(\Gamma , A^{0,q}(X, \mathcal {L}))\) is canonically isomorphic to \(\mathbb {H}^{k}(\Gamma , A^{0, *}(X, \mathcal {L})).\)

Therefore, there is a spectral sequence with \(E_{2}^{p,q}:=H^{p}(\Gamma , H^{q}(X, \mathcal {L}))\) which converges to

$$\begin{aligned} \mathbb {H}^{p+q}\left( \Gamma , A^{0, *}(X, \mathcal {L})\right) \simeq H^{p+q}\left( Y, \mathcal {L}^{\Gamma }\right) . \end{aligned}$$

For our applications, the regular covering is \(\Omega _{\varrho }^{I}\rightarrow \mathcal {W}_{\varrho }^{I}\), the \(\Gamma \)-equivariant sheaf is \(\Theta _{\Omega _{\varrho }^{I}},\) and its descent is \(\Theta _{\Omega _{\varrho }^{I}}^{\Gamma }=\Theta _{\mathcal {W}_{\varrho }^{I}}.\) Hence, we obtain the spectral sequence with \(E_2\)-page \(H^{p}(\Gamma , H^{q}(\Omega _{\varrho }^{I}, \Theta _{\Omega _{\varrho }^{I}}))\) converging to \(H^{p+q}(\mathcal {W}_{\varrho }^{I}, \Theta _{\mathcal {W}_{\varrho }^{I}})\) that is used in Section 4.

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Dumas, D., Sanders, A. Uniformization of compact complex manifolds by Anosov homomorphisms. Geom. Funct. Anal. 31, 815–854 (2021).

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