Variations of Mixed Hodge Structures of Multiple Polylogarithms

[摘要] It is well known that multiple polylogarithms give rise togood unipotent variations of mixed Hodge-Tate structures.In this paper we shall {em explicitly} determine these structuresrelated to multiple logarithms and some other multiple polylogarithmsof lower weights. The purpose of this explicit constructionis to give some important applications: First we study the limit ofmixed Hodge-Tate structures and make a conjecture relating the variationsof mixed Hodge-Tate structures of multiple logarithms to those ofgeneral multiple {em poly}/logarithms. Then followingDeligne and Beilinson we describe anapproach to defining the single-valuedreal analytic version of the multiple polylogarithms whichgeneralizes the well-known result of Zagier onclassical polylogarithms. In the process we find some interestingidentities relating single-valued multiple polylogarithms of thesame weight $k$ when $k=2$ and 3. At the end of this paper,motivated by Zagier's conjecture we posea problem which relates the special values of multipleDedekind zeta functions of a number field to the single-valuedversion of multiple polylogarithms.

[学科分类] 数学（综合）