Conditions for triviality of deformations of complex structures

I. F. Donin

Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 557-567

Résumé

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.