Some problems of hypergeometric integrals associated with hypersphere arrangement

The n dimensional hypergeometric integrals associated with a hypersphere arrangement S are formulated by the pairing of n dimensional twisted cohomology [H.sup.n.sub.[nabla]](X, [OMEGA] x (*S)) and its dual. Under the condition of general position there are stated some results and conjectures which concern a representation of the standard form by a special basis of the twisted cohomology, the variational formula of the corresponding integral in terms of special invariant 1-forms using Calyley-Menger minor determinants, a connection relation of the unique twisted n-cycle identified with the unbounded chamber to a special basis of twisted n-cycles identified with bounded chambers. General conjectures are presented under a much weaker assumption. Key words: Hypergeometric integral; hypersphere arrangement; twisted rational de Rham cohomology; Cayley-Menger determinant; contiguity relation; Gauss-Manin connection.