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Deformations of complex structures on the homogeneous spaces of \(\text{SL}(2,\mathbb{C})\). (Déformations des structures complexes sur les espaces homogènes de \(\text{SL}(2,\mathbb{C})\).) (French) Zbl 0868.32023
This paper settles completely the case left open in the following result of M. S. Raghunathan [Osaka Math. J. 3, 243-256 (1966; Zbl 0145.43702)]:
Let \(G\) be a complex semisimple group without any factors locally isomorphic to \(SL(2,\mathbb{C})\) and \( \Gamma\) a discrete cocompact subgroup of \(G\). Then the complex manifold \(M=G/ \Gamma\) is rigid in the sense that every complex structure close to the natural structure is isomorphic to it.
The author studies the rigidity of the spaces \(M=SL(2,\mathbb{C})/ \Gamma\), when \(\Gamma\) is a discrete cocompact subgroup of \(SL(2,\mathbb{C})\) and shows that these spaces do admit nontrivial deformations, and he describes completely all such deformations by determining the Kuranishi space of \(M\).
This article originates from the author’s study of holomorphic dynamical systems of Anasov type [Invent. Math. 119, No. 3, 585-614 (1995; Zbl 0831.58041)].
Reviewer: H.Azad (Dharan)

32G05 Deformations of complex structures
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
37D99 Dynamical systems with hyperbolic behavior
Kuranishi space
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