Real analytic varieties with the finiteness property and complex abelian integrals (Q1070388)

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.