A simple construction of Grassmannian polylogarithms

Abstract

The classical n-logarithm is a multivalued analytic function defined inductively: Lin(z):=∫0zLin−1(t)dlogt,Li1(z)=−log(1−z). In this paper we give a simple explicit construction of the Grassmannian n-logarithm, which is a multivalued analytic function on the quotient of the Grassmannian of n-dimensional subspaces in C2n in generic position to the coordinate hyperplanes by the natural action of the torus (C∗)2n. The classical n-logarithm appears at a certain one dimensional boundary stratum.We study Tate iterated integrals, which are homotopy invariant integrals of 1-forms dlogfi where fi are rational functions. We give a simple explicit formula for the Tate iterated integral which describes the Grassmannian n-logarithm.Another example is the Tate iterated integrals for the multiple polylogarithms on the moduli spaces M0,n, calculated in Section 4.4 of Goncharov (2005) [13] using the combinatorics of plane trivalent trees decorated by the arguments of the multiple polylogarithms.Variations of mixed Hodge–Tate structures on X are described by a Hopf algebra H•(X). We upgrade Tate iterated integrals on a (rational) complex variety X to elements of H•(X). The coproducts of these elements are very interesting invariants of the iterated integrals. In general their calculation is a non-trivial problem. We show however, that working modulo the ideal of H•(X) generated by constant variations, there is a simple way to calculate the coproduct.It is a pleasure to dedicate this paper to Andrey Suslin, whose works Suslin (1984) [21] and Suslin (1991) [22] played an essential role in the development of the story.