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The cohomological structure of hypersphere arragnements is given. The Gauss-Manin
connections for related hypergeometrtic integrals are given in terms of invariant forms.
They are used to get the explicit differential formula for the volume of a simplex whose
faces are hyperspheres.
We associate to any convenient nondegenerate Laurent polynomial on the complex torus
a canonical Frobenius-Saito structure on the base space of its universal
unfolding. According to the method of K. Saito (primitive forms) and of M. Saito (good
basis of the Gauss-Manin system), the main problem, which is solved in this article, is
the analysis of the Gauss-Manin system of (or its universal unfolding) and of the
corresponding Hodge theory.
We consider a mirror symmetry of simple elliptic singularities. In particular, we construct isomorphisms of Frobenius manifolds among the one from the Gromov–Witten theory of a weighted projective line, the one from the theory of primitive forms for a universal unfolding of a simple elliptic singularity and the one from the invariant theory for an elliptic Weyl group. As a consequence, we give a geometric interpretation of the Fourier coefficients of an eta product considered by K. Saito.
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