Deforming cohomology classes

Abstract:

Let $\pi :X \to S$ be a flat proper morphism of analytic spaces. $\pi$ may be thought of as providing a family of compact analytic spaces, ${X_s}$, parametrized by the space $S$. Let $\mathcal {F}$ be a coherent sheaf on $X$ flat over $S$. $\mathcal {F}$ may be thought of as a family of coherent sheaves, ${\mathcal {F}_s}$, on the family of spaces ${X_s}$. Let $o \in S$ be a fixed point, ${\xi _o} \in Hq({X_o},{\mathcal {F}_o})$. In this paper, we consider the problem of extending ${\xi _o}$ to a cohomology class $\xi \in Hq({\pi ^{ - 1}}(U),\mathcal {F})$ where $U$ is some neighborhood of $o$ in $S$. Extension problems of this type were first considered by P. A. Griffiths who obtained some results in the case in which the morphism $\pi$ is simple and the sheaf $\mathcal {F}$ 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 ${\xi _o}$; in §3 we show that if no obstructions exist, ${\xi _o}$ may be extended; in §4 we give a useful criterion for no obstructions; and in §5 we discuss some examples.