Vol. 71, No. 1, 1977

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Integral representations of algebraic cohomology classes on hypersurfaces

Edith Twining Stevenson

Vol. 71 (1977), No. 1, 197–212

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:

  1. 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
  2. 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.

Mathematical Subject Classification 2000
Primary: 14C30
Secondary: 14F25
Received: 5 October 1976
Published: 1 July 1977
Edith Twining Stevenson