Geodesic
Conditions for triviality of deformations of complex structures
Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 557-567 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $f\colon X\to S$ be a characteristic, holomorphic mapping of complex spaces (with nilpotent elements). The paper proves that, if $f$ is a flat mapping and all its fibers are equivalent to one and the same compact complex space $X_0$, then, with respect to this mapping, $X$ is equivalent to a holomorphic fibering over $S$ with fiber $X_0$ and structure group $\operatorname{Aut}(X_0)$. It is further proved that, if the base $S$ is reduced, the assertion remains true for any holomorphic mapping $f$, at least in the case when the fiber $X_0$ is an irreducible space. This is a strong generalization of the corresponding result of Fischer and Grauert, in which a similar assertion is proved for the case when $X$ and $S$ are complex manifolds and $f$ is a locally trivial mapping. This paper also proves that, if the compact complex space $X_0$ satisfies the condition $H^1(\Omega,X_0)=0$, where $\Omega$ is the sheaf of germs of holomorphic vector fields on $X_0$, then any locally trivial deformation of the space $X_0$, with arbitrary parameter space, is trivial. This generalizes Kerner's result, in which the parameter space is assumed to be a manifold. Bibliography: 7 titles. 
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     author = {I. F. Donin},
     title = {Conditions for triviality of deformations of complex structures},
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     pages = {557--567},
     year = {1970},
     volume = {10},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_10_4_a5/}
}
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I. F. Donin. Conditions for triviality of deformations of complex structures. Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 557-567. http://geodesic.mathdoc.fr/item/SM_1970_10_4_a5/

[1] W. Fisher, H. Grauert, “Lokal-triviale Familien kompakter komplexer Mänigfaltigkeiten”, Nachr. Akad. Wiss. Göttingen, 1965, no. 6, 89–94 | MR

[2] A. Douady, “Le probleme des modules pour les sous-espaces analytiques compacts d'un espace analytique”, Ann. Inst. Fourier, 16:1 (1966), 1–95 | MR | Zbl

[3] H. Grauert, H. Kerner, “Deformations von Singularitäten komplexer Räume”, Math. Ann., 153:3 (1964), 236–260 | DOI | MR | Zbl

[4] H. Kerner, “Familien kompakter und holornorph-vollständiger komplexer Räume”, Math. Z., 92:3 (1966), 225–233 | DOI | MR | Zbl

[5] W. Kaup, “Infinitesimale Transformationsgruppen komplexer Räume”, Math. Ann., 160:1 (1965), 72–92 | DOI | MR | Zbl

[6] G. Grauert, “Teorema iz teorii analiticheskikh puchkov i prostranstva modulei kompleksnykh struktur”, Kompleksnye prostranstva, Mir, Moskva, 1965, 205–299

[7] I. F. Donin, “Usloviya trivialnosti deformatsii golomorfnykh rassloenii nad kompaktnym kompleksnym prostranstvom”, Matem. sb., 77(119) (1968), 602–623 | MR | Zbl