Real analytic varieties with the finiteness property and complex abelian integrals
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.
finiteness results on real Abelian integrals
scientific article; zbMATH DE number 3935444