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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)].