CONDITIONS FOR TRIVIALITY OF DEFORMATIONS OF COMPLEX STRUCTURES

Abstract

Let f : X→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 X0, then, with respect to this mapping, X is equivalent to a holomorphic fibering over S with fiber X0 and structure group Aut(X0). 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 X0 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 X0 satisfies the condition H1(Ω, X0) = 0, where Ω is the sheaf of germs of holomorphic vector fields on X0, then any locally trivial deformation of the space X0, 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.