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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Deforming cohomology classes

Author: John J. Wavrik
Journal: Trans. Amer. Math. Soc. 181 (1973), 341-350
MSC: Primary 32D15; Secondary 32C35, 32G05
MathSciNet review: 0326002
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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.

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Keywords: Extension problem, deformation theory, families of analytic spaces, obstructions, space of moduli
Article copyright: © Copyright 1973 American Mathematical Society