Local Torelli theorem for bundles on manifolds with \(K=0\) (Q1070296)

By the local Torelli theorem we shall mean the injectivity of the differential of the period map. The definition of the period map and the calculation of its differential are contained in a paper by \textit{P. A. Griffiths} [Am. J. Math. 90, 805-865 (1968; Zbl 0183.255)]. Theorem 1. Let \(f: V\to B\) be a surface with a bundle of elliptic curves with nontrivial functional invariant without multiple fibers and suppose \(| K|\) has no fixed components. Then the local Torelli theorem holds for V. Let \(f: V\to B\) be a holomorphic mapping of the compact Kähler manifold V onto the compact Kähler manifold B. Let us assume that the generic fiber of this mapping is a manifold with \(K=0\), which does not have a continuous group of automorphisms, and is such that the corresponding family of fibers is effectively parametrized. Theorem 2. If \(| K|\) has no fixed components, then V satisfies the local Torelli theorem for the map of periods of n-forms, \(n=\dim V\).