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On Transportable Forms

Published online by Cambridge University Press:  20 November 2018

Tapio Klemola
Tapio Klemola*
Affiliation:
Université de Montréal, Montréal, Quebec
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In the theory of deformations of compact complex manifolds, the hypothesis of constancy of the dimension of diverse structural cohomology groups pertaining to a fibre plays an important role (see, for instance, [3, Propositions 2.5 and 2.7, Theorems 2.2 and 2.3, and Definition 6.1]). This paper is the first of two devoted to the investigation of conditions under which constancy of the dimension of given cohomology groups is assured, and more generally, to the study of the variation of that dimension.

In [2] Griffiths introduces extendible forms in a holomorphic deformation. We consider in this paper a differentiate family, and besides extendible, also co-extendible and transportable forms (see §5), and deduce from their existence conclusions about the variation of the dimension of the corresponding structural cohomology groups. It is left to a subsequent paper to give more explicit conditions by means of cohomology operations, and to deal with some applications.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 1 , March 1972 , pp. 93 - 101
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Fröhlicher, A., Kobayashi, E. T. and Nijenhuis, A., Deformation theory of complex manifolds, Seminar notes, Univ. of Washington, 1957.Google Scholar
2. Griffiths, Ph. A., The extension problem for compact submanifolds of complex manifolds, Proc. of the Conference on Complex Analysis, Minneapolis, 1964.Google Scholar
3. Kodaira, K. and Spencer, D. C., On deformations of complex analytic structures I and 11 , Ann. of Math. 67 (1958), 328-466.Google Scholar
4. Kodaira, K. and Spencer, D. C., On deformations of complex analytic structures HI, Ann. of Math. 71 (1960), 43-76. Google Scholar