Abstract
LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL 2(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)).
We also show thatG acts trivially on the coherent cohomology groupsH i(Γ/G, O) for anyi≥0.
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References
Bernstein L N and Kazhdan D A, The one-dimensional cohomology of discrete subgroups,Funct. Anal. Appl. 4 (1970) 1–4
Borel A and Wallach N C, Continuous cohomology, discrete subgroups and representations of reductive groups,Ann. Math. Stud. (Princeton: Univ. Press)94 (1980)
Johnson D and Millson J J, Deformation spaces associated to compact hyperbolic manifolds inDiscrete Groups in Geometry and Analysis; Proceedings of a Conference held at Yale University in honor of G. D. MostowProgress in Mathematics Series (Birkhauser) ed. R Howe (1985)
Kodaira K,Complex manifolds and deformations of complex structure (Berlin: Springer-Verlag) (1986)
Lebesse J P and Schwermer J, On liftings and cusp cohomology of arithmetic gorups,Invent. Math. 83 (1986) 383–401
Millson J J, On the first Betti number of a constant negatively curved manifold,Ann. Math. 104 (1976) 235–247
Millson JJ and Raghunathan M S, Geometric construction of cohomology for arithmetic groups,Proc. Indian Acad. Sci. (Math. Sci.) 90 (1981) 103–123
Raghunathan M S Vanishing theorems for cohomology groups associated to discrete subgroups of semisimple groups,Osaka J. Math. 67 (1966) 243–256
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Rajan, C.S. Deformations of complex structures on Γ/SL 2(C). Proc. Indian Acad. Sci. (Math. Sci.) 104, 389–395 (1994). https://doi.org/10.1007/BF02863419
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DOI: https://doi.org/10.1007/BF02863419