Real analytic varieties with the finiteness property and complex abelian integrals. (English. Russian original) Zbl 0584.32016

Khovanskij, A. G.

Funct. Anal. Appl. 18, 119-127 (1984); translation from Funkts. Anal. Prilozh. 18, No. 2, 40-50 (1984).

Suppose f is a smooth real-valued function on a smooth manifold. A non- singular level surface \(S=\{f=c\}\) then evidently has the property that it is an integral manifold for the distribution df and, moreover, forms the boundary of \(\{f<c\}\) compatible with orientation in that df is positive on outward pointing vectors. This idea may be generalized by allowing a hypersurface S to be an integral manifold for a specified 1- form \(\alpha\) (not necessarily defining an integrable distribution). In this way one may, by taking \(\alpha\) to be real-analytic, construct a class of real-analytic manifolds known as Pfaffian manifolds. The category of Pfaffian manifolds enjoys finiteness properties generalizing the fact that the number of components of a level surface of a real analytic function is locally finite with uniform bound with respect to analytic variation of parameters on a compact set. This theory has application to proving finiteness results on real Abelian integrals.
These results are very pleasing and are well presented and motivated in this paper.

Reviewer: M.G.Eastwood

MSC:

32C05	Real-analytic manifolds, real-analytic spaces
58A17	Pfaffian systems
58A07	Real-analytic and Nash manifolds

Keywords:

real analytic variety; Abelian function; Pfaffian systems; real-analytic manifolds; Pfaffian manifolds; finiteness results on real Abelian integrals

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