Deforming cohomology classes

Abstract

Let π : X → S \pi :X \to S be a flat proper morphism of analytic spaces. π \pi may be thought of as providing a family of compact analytic spaces, X s {X_s} , parametrized by the space S S . Let F \mathcal {F} be a coherent sheaf on X X flat over S S . F \mathcal {F} may be thought of as a family of coherent sheaves, F s {\mathcal {F}_s} , on the family of spaces X s {X_s} . Let o ∈ S o \in S be a fixed point, ξ o ∈ H q ( X o , F o ) {\xi _o} \in Hq({X_o},{\mathcal {F}_o}) . In this paper, we consider the problem of extending ξ o {\xi _o} to a cohomology class ξ ∈ H q ( π − 1 ( U ) , F ) \xi \in Hq({\pi ^{ - 1}}(U),\mathcal {F}) where U U is some neighborhood of o o in S S . Extension problems of this type were first considered by P. A. Griffiths who obtained some results in the case in which the morphism π \pi is simple and the sheaf F \mathcal {F} is locally free. We obtain generalizations of these results without the restrictions. Among the applications of these results is a necessary and sufficient condition for the existence of a space of moduli for a compact manifold. This application was discussed in an earlier paper by the author. We use the Grauert “direct image” theorem, the theory of Stein compacta, and a generalization of a result of M. Artin on solutions of analytic equations to reduce the problem to an algebraic problem. In §2 we discuss obstructions to deforming ξ o {\xi _o} ; in §3 we show that if no obstructions exist, ξ o {\xi _o} may be extended; in §4 we give a useful criterion for no obstructions; and in §5 we discuss some examples.