Skip to main content

Uniformization of compact complex manifolds by Anosov homomorphisms

Abstract

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.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. R. J. Baston and M. G. Eastwood. The Penrose transform: Its interaction with representation theory. Oxford Mathematical Monographs. Oxford University Press, New York, 1989.

    MATH  Google Scholar 

  2. I. Bauer and F. Catanese. On rigid compact complex surfaces and manifolds. Adv. Math., 333:620–669, 2018. https://doi.org/10.1016/j.aim.2018.05.041.

    MathSciNet  Article  MATH  Google Scholar 

  3. I. Bauer and C. Gleissner. Towards a Classification of Rigid Product Quotient Varieties of Kodaira Dimension 0. Preprint, 2021. Arxiv:2101.06925.

  4. I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand. Schubert cells, and the cohomology of the spaces G/P. In Representation theory, volume 69 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge-New York, 1982.

  5. L. Bers. Simultaneous uniformization. Bull. Amer. Math. Soc., 66:94–97, 1960. https://doi.org/10.1090/S0002-9904-1960-10413-2.

    Article  Google Scholar 

  6. A. Björner and F. Brenti. Combinatorics of Coxeter groups, volume 231 of Graduate Texts in Mathematics. Springer, New York, 2005.

  7. J. Bochi, R. Potrie, and A. Sambarino. Anosov representations and dominated splittings. J. Eur. Math. Soc., 21(11):3343–3414, 2019. https://doi.org/10.4171/JEMS/905.

    MathSciNet  Article  MATH  Google Scholar 

  8. A. Borel. Linear algebraic groups, volume 126 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991.

  9. R. Bott. Homogeneous vector bundles. Ann. of Math. (2), 66:203–248, 1957. https://doi.org/10.2307/1969996.

    Article  Google Scholar 

  10. E. Calabi and E. Vesentini. Sur les variétés complexes compactes localement symétriques. Bull. Soc. Math. France, 87:311–317, 1959.

    MathSciNet  Article  Google Scholar 

  11. F. Catanese. A superficial working guide to deformations and moduli. In Handbook of moduli I, volume 24 of Advanced Lectures in Mathematics (ALM), pages 161–215. Int. Press, Somerville, MA, 2013.

  12. E. M. Chirka. On the removable singularities for meromorphic mappings. Publ. Mat., 40(1):229–232, 1996. https://doi.org/10.5565/PUBLMAT_40196_15.

    MathSciNet  Article  MATH  Google Scholar 

  13. S. Choi, G.-S. Lee, and L. Marquis. Deformations of convex real projective manifolds and orbifolds. In Handbook of group actions III, volume 40 of Advanced Lectures in Mathematics (ALM), pages 263–310. Int. Press, Somerville, MA, 2018.

  14. D. Dumas and A. Sanders. Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations. Geom. Topol., 24(4):1615–1693, 2020. https://doi.org/10.2140/gt.2020.24.1615.

    MathSciNet  Article  MATH  Google Scholar 

  15. G. Fischer. Complex analytic geometry. Volume 538 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1976.

  16. V. Fock and A. Goncharov. Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci., (103):1–211, 2006. https://doi.org/10.1007/s10240-006-0039-4.

    Article  MATH  Google Scholar 

  17. E. Ghys. Déformations des structures complexes sur les espaces homogènes de \({\rm SL}_{(2,\bf C)}\). J. Reine Angew. Math., 468:113–138, 1995. https://doi.org/10.1515/crll.1995.468.113.

    MathSciNet  Article  MATH  Google Scholar 

  18. W. Goldman. The symplectic nature of fundamental groups of surfaces. Adv. in Math., 54(2):200–225, 1984. https://doi.org/10.1016/0001-8708(84)90040-9.

    MathSciNet  Article  MATH  Google Scholar 

  19. H. Grauert. Der Satz von Kuranishi für kompakte komplexe Räume. Invent. Math., 25:107–142, 1974. https://doi.org/10.1007/BF01390171.

    MathSciNet  Article  MATH  Google Scholar 

  20. H. Grauert and R. Remmert. Coherent analytic sheaves. Springer-Verlag, Berlin, 1984.

    Book  Google Scholar 

  21. A. Grothendieck. Sur quelques points d’algèbre homologique. Tohoku Math. J. (2), 9:119–221, 1957. https://doi.org/10.2748/tmj/1178244839.

    Article  MATH  Google Scholar 

  22. F. Guéritaud, O. Guichard, F. Kassel, and A. Wienhard. Anosov representations and proper actions. Geom. Topol., 21(1):485–584, 2017. https://doi.org/10.2140/gt.2017.21.485.

    MathSciNet  Article  MATH  Google Scholar 

  23. O. Guichard and A. Wienhard. Anosov representations: Domains of discontinuity and applications. Invent. Math., 190(2):357–438, 2012. https://doi.org/10.1007/s00222-012-0382-7.

    MathSciNet  Article  MATH  Google Scholar 

  24. R. Harvey. Removable singularities of cohomology classes in several complex variables. Amer. J. Math., 96:498–504, 1974. https://doi.org/10.2307/2373557.

    MathSciNet  Article  MATH  Google Scholar 

  25. S. Helgason. Differential geometry, Lie groups, and symmetric spaces, volume 34 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001.

  26. N. Hitchin. Lie groups and Teichmüller space. Topology, 31(3):449–473, 1992. https://doi.org/10.1016/0040-9383(92)90044-I.

    MathSciNet  Article  MATH  Google Scholar 

  27. J. E. Humphreys. Linear algebraic groups. Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21.

  28. D. Johnson and J. Millson. Deformation spaces associated to compact hyperbolic manifolds. In Discrete groups in geometry and analysis (New Haven, Conn., 1984), volume 67 of Progress in Mathematics, pages 48–106, Birkhäuser, Boston, MA, 1987.

  29. M. Kapovich. Kleinian groups in higher dimensions. In Geometry and dynamics of groups and spaces, volume 265 of Progress in Mathematics, pages 487–564. Birkhäuser, Basel, 2008.

  30. M. Kapovich, B. Leeb, and J. Porti. Morse actions of discrete groups on symmetric space. Preprint, 2014. Arxiv:1403.7671.

  31. M. Kapovich, B. Leeb, and J. Porti. Dynamics on flag manifolds: domains of proper discontinuity and cocompactness. Geom. Topol., 22(1):157–234, 2018. https://doi.org/10.2140/gt.2018.22.157.

    MathSciNet  Article  MATH  Google Scholar 

  32. M. Kapovich and J. J. Millson. On representation varieties of 3-manifold groups. Geom. Topol., 21(4):1931–1968, 2017. https://doi.org/10.2140/gt.2017.21.1931.

    MathSciNet  Article  MATH  Google Scholar 

  33. B. Kaup. Äquivalenzrelationen auf allgemeinen komplexen Räumen. Schr. Math. Inst. Univ. Münster, 39, 1968.

  34. K. Kodaira and D. C. Spencer. On deformations of complex analytic structures. I, II. Ann. of Math. (2), 67:328–466, 1958.

  35. B. Kostant. The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math., 81:973–1032, 1959. https://doi.org/10.2307/2372999.

    MathSciNet  Article  MATH  Google Scholar 

  36. M. Kuranishi. On the locally complete families of complex analytic structures. Ann. of Math. (2), 75:536–577, 1962. https://doi.org/10.2307/1970211.

    MathSciNet  Article  MATH  Google Scholar 

  37. F. Labourie. Anosov flows, surface groups and curves in projective space. Invent. Math., 165(1):51–114, 2006. https://doi.org/10.1007/s00222-005-0487-3.

    MathSciNet  Article  MATH  Google Scholar 

  38. B. McKay. Extension phenomena for holomorphic geometric structures. SIGMA: Symmetry Integrability Geom. Methods Appl., 5:058, 2009. https://doi.org/10.3842/SIGMA.2009.058.

  39. L. Meersseman. The Teichmüller and Riemann moduli stacks. J. Éc. polytech. Math., 6:879–945, 2019. https://doi.org/10.5802/jep.108.

    MathSciNet  Article  MATH  Google Scholar 

  40. A. Neeman. Algebraic and analytic geometry, volume 345 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2007. https://doi.org/10.1017/CBO9780511800443.

  41. V. P. Palamodov. Deformations of Complex Spaces. In S. G. Gindikin and G. M. Khenkin, editors, Several Complex Variables IV: Algebraic Aspects of Complex Analysis, pages 105–194. Springer, Berlin, 1990. https://doi.org/10.1007/978-3-642-61263-3_3.

  42. J. Porti. Local and infinitesimal rigidity of representations of hyperbolic three manifolds. RIMS Kokyuroku, 1836:154–177, 2013.

    Google Scholar 

  43. B. Pozzetti, A. Sambarino, and A. Wienhard. Anosov representations with Lipschitz limit set. Preprint, 2019. Arxiv:1910.06627.

  44. M. B. Pozzetti, A. Sambarino, and A. Wienhard. Conformality for a robust class of non-conformal attractors. J. Reine Angew. Math., 774:1–51, 2021. https://doi.org/10.1515/crelle-2020-0029.

    MathSciNet  Article  MATH  Google Scholar 

  45. M. S. Raghunathan. Vanishing theorems for cohomology groups associated to discrete subgroups of semisimple Lie groups. Osaka Math. J., 3:243–256, 1966. http://projecteuclid.org/euclid.ojm/1200691729.

  46. H. Seppänen and V. Tsanov. Geometric invariant theory for principal three-dimensional subgroups acting on flag varieties. In Representation theory—current trends and perspectives, EMS Ser. Congr. Rep., pages 637–663. Eur. Math. Soc., Zürich, 2017. https://doi.org/10.4171/171-1/22.

  47. B. Shiffman. On the removal of singularities of analytic sets. Michigan Math. J., 15:111–120, 1968. https://doi.org/10.1307/mmj/1028999912.

    MathSciNet  Article  MATH  Google Scholar 

  48. A. Sikora. Character varieties. Trans. Amer. Math. Soc., 364(10):5173–5208, 2012. https://doi.org/10.1090/S0002-9947-2012-05448-1.

    MathSciNet  Article  MATH  Google Scholar 

  49. T. E. Venkata Balaji. An introduction to families, deformations and moduli. Universitätsdrucke Göttingen, Göttingen, 2010. https://doi.org/10.17875/gup2010-64.

  50. C. Voisin. On the homotopy types of Kähler manifolds and the birational Kodaira problem. J. Differential Geom., 72(1):43–71, 2006. https://doi.org/10.4310/jdg/1143593125.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to David Dumas.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dumas, D., Sanders, A. Uniformization of compact complex manifolds by Anosov homomorphisms. Geom. Funct. Anal. 31, 815–854 (2021). https://doi.org/10.1007/s00039-021-00572-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-021-00572-6