Let fλ(X) =∑i=1n+1Xid+hλ(X)
where hλ(X) is the generic form of degree d in n + 1 variables over C. The main
theorem of this paper is that certain exponential integrals associated to the
projective hypersurface Xλ defined by the vanishing of fλ(X) have regular
singularities. The main ingredients in the proof are:
Katz’s identification of certain monomial spaces (first studied by Dwork)
in terms of the middle-dimensional cohomology of a variety related to Xλ
and
Griffiths’ theorem, which states that periods on an algebraic variety have
regular singularities.
By essentially the same methods, an upper bound on the order of logarithmic
growth of the integrals is determined.
Also, an example is given to show the relation of periods on a family of cubic
curves to hypergeometric functions.
