This article is cited in
3 papers
Conditions for triviality of deformations of complex structures
I. F. Donin
Abstract:
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.
UDC:
513.836+519.46
MSC: 32G05,
58H15,
32H02,
32L05 Received: 10.10.1969
Fulltext:
PDF file (1459 kB)
