Chow Polylogarithms and Regulators

where r denotes the cross-ratio. In this note we show that the Bloch-Wigner function can be naturally extended to the (infinite dimensional) variety of all algebraic curves in CP 3 which are in sufficiently general position with respect to a given simplex L. (By definition a simplex in CP 3 is a collection of 4 hyperplanes in generic position). We call the corresponding function the Chow dilogarithm function. When our curve is a straight line we obtain just the Bloch-Wigner function evaluated at the cross-ratio of the 4 intersection points of this line with the faces of the simplex L. It is interesting that even in this case we get a new presentation of L2(z). Any algebraic surface in CP 4 which is in general position with respect to a given simplex produces a 5-term relation for the Chow dilogarithm function. Namely, the intersection of the surface with a codimension 1 face of the simplex provides a curve and a simplex in CP . A simplex in CP 4 has 5 codimension 1 faces. The alternating sum of the corresponding 5 values of the Chow dilogarithm is zero.