arXiv subjects
Publications and source records attributed to Richard Hain.
The Hodge de Rham theory of relative Malcev completion is developed in this paper. In the special case where one takes the corresponding reductive group to be trivial, one recovers Chen's de Rham theory of the fundamental group and the corresponding Hodge theory due to Morgan and the author. This work is a principal technical tool in the author's work on the mapping class groups.
In this paper we define real grassmann polylogarithms, which are real single valued analogues of the grassmann polylogarithms (or higher logarithms) defined by Hain and MacPherson. We prove the existence of all such real grassmann polylogs, at least generically. We also prove that the canonical choice of such an m-polylogarithm represents the Beilinson Chern class on the rank m part of the algebraic K-theory of the generic point of every complex algebraic variety. One part of each such grassmann m-polylogarithm is a real, single-valued function defined generically on the grassmannian of m planes in C^{2m}. We prove that this function represents the Borel regulator (up to a factor of 2) on K_{2m-1} of all number fields.
In this paper we prove the existence of all higher logarithms as multivalued and ordinary Deligne cohomology classes.
In this paper we study the proalgebraic completion of mapping class relative to their maps to the symplectic group. The main result is that the natural map from the unipotent (a.k.a. Malcev) completion of the Torelli group to the prounipotent radical of the Sp_g completion of the mapping class group is a non trivial central extension with kernel isomorphic to Q, at least when g \ge 8. The theorem is proved by relating the central extension to the line bundle associated to the archemidean height of the cycle C - C- in the Jacobian of the curve C. We also develop some of the basic theory of relative completions.
This paper is an introduction to classical polylogarithms and is an expanded version of a talk given by the author at the Motives conference. Topics covered include, monodromy; the polylogarithm local systems; Bloch's constructions of regulators using the dilogarithm; polylog locals systems as variations of mixed Hodge structre; the polylogarithm quotient of the fundamental group of C - {0,}. It is intended as background for understanding recent work of Beilinson, Deligne, and Goncharov.