Real analytic varieties with the finiteness property and complex abelian integrals
type: Item
instance of: scholarly article
title: Real analytic varieties with the finiteness property and complex abelian integrals
zbMATH Open document ID: 0584.32016
DOI: 10.1007/BF01077822
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 32C05
Mathematics Subject Classification ID: 58A17
Mathematics Subject Classification ID: 58A07
zbMATH DE Number: 3935444
zbMATH Keywords: real analytic variety
zbMATH Keywords: Abelian function
zbMATH Keywords: Pfaffian systems
zbMATH Keywords: real-analytic manifolds
zbMATH Keywords: Pfaffian manifolds
zbMATH Keywords: finiteness results on real Abelian integrals
author: A. G. Khovanskii
reviewed by: M. Eastwood
MaRDI profile type: Publication
cites work: Q3334683
cites work: Equations différentielles à points singuliers réguliers
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3935444