Hyperlogarithmic expansion and the volume of a hyperbolic simplex

Abstract

0. Introduction. Hyperlogarithmic functions (or higher logarithmic functions) are multivalued analytic functions defined on complex projective varieties, with unipotent monodromy and with regular singularity. It is known that they can be expressed by the use of iterated integrals of suitable logarithmic 1-forms in the sense of K. T. Chen (see [A1], [H1]). Recently these functions have played a considerable role in various problems of geometry and arithmetic (for example, see [H2], [B1], [G2], [V], etc.). These are a special case of hypergeometric functions on a Grassmannian manifold (see [A2], [G1], [V]). However, there are other kinds of hyperlogarithmic functions which are related to the configuration of hyperplanes and a hyperquadric (see [A3]). The volume of a simplex in a hyperbolic space is a hyperlogarithmic function of basic algebraic invariants, as a simple consequence of the Schlafli formula. However, there remains the problem of divergence in the case where the vertices lie on the boundary. In this note we want to derive a modified Schlafli formula in such a degenerate case and give a hyperlogarithmic expansion for the volume, by using a technique developed in [A3]. A similar result has been obtained by Kellerhals [K4]. Her method is to decompose a simplex into several orthoschemes and to obtain an explicit formula for each orthoscheme by using the Lobachevskĭi function L(x). In the appendix we discuss a relation between the volume and Appell’s hypergeometric functions of type F4.