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Deformations of Complex Spaces

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Book cover Several Complex Variables IV

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 10))

Abstract

The origin of deformation theory lies in the problem of moduli, first considered by Riemann. The problem in the theory of moduli can be described thus: to bring together all objects of a single type in analytic geometry, for example, all Riemann surfaces of given genus; to organize them by joining them into a fiber space; to describe the base of this space—the moduli space; to introduce on it an analytic structure; and to study the natural parameters, for example, the periods of holomorphic forms defined on Riemann surfaces. For nonsingular Riemann surfaces the moduli space was constructed in the theory of Teichmüller-Ahlfors-Bers, and the periods of holomorphic forms were described by the billinear relations of Riemann and the theorem of Rauch. But the problem of moduli for multidimensional complex manifolds turned out to be significantly more complicated. Its systematic study began with the work of Kodaira and Spencer in which the basic object of study was the analytic families of complex structures on given smooth manifold. The concept of a complete effective family, due to KodairaSpencer, served as a substitute for the moduli space, but it had various peculiar properties: the structure of the fibers of this family can vary “continuously” along one curve in the base and have “jumps” along another, and the base itself can have singular points (Kuranishi).

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  1. V. P. Palamodov
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  1. A.N. Belozersky Laboratory of Molecular Biology and Bioorganic Chemistry, Moscow State University, GSP-234, 119899, Moscow, USSR

    S. G. Gindikin

  2. Academy of Sciences of the USSR, Central Economic and Mathematical Institute, ul. Krasikova 32, 117418, Moscow, USSR

    G. M. Khenkin

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© 1990 Springer-Verlag Berlin Heidelberg

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Palamodov, V.P. (1990). Deformations of Complex Spaces. In: Gindikin, S.G., Khenkin, G.M. (eds) Several Complex Variables IV. Encyclopaedia of Mathematical Sciences, vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61263-3_3

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  • DOI: https://doi.org/10.1007/978-3-642-61263-3_3

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