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Classical Polylogarithms

Richard Hain

TL;DR

The paper develops a comprehensive structure for classical polylogarithms, linking their analytic monodromy, iterated-integral representations, and Dilogarithm/Bloch–Wigner phenomena to regulator maps in K-theory and Deligne cohomology. It constructs and analyzes the polylogarithm local systems as Tate variations of mixed Hodge structure, computes their monodromy, and uses iterated integrals to connect to the regulators $c_2$ and higher Beilinson regulators. A central theme is the Hodge-theoretic and motivic interpretation of these regulators via extensions in mixed Hodge structures and the polylogarithm variation, including a motivic description through the cobar construction and path-space formalisms. Together, these results illuminate the Beilinson–Deligne–Goncharov program by tightly coupling polylogarithms with regulator maps, mixed Hodge structures, and motivic frameworks, providing a unified perspective on how classical polylogarithms encode deep arithmetic and geometric information.

Abstract

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.

Classical Polylogarithms

TL;DR

The paper develops a comprehensive structure for classical polylogarithms, linking their analytic monodromy, iterated-integral representations, and Dilogarithm/Bloch–Wigner phenomena to regulator maps in K-theory and Deligne cohomology. It constructs and analyzes the polylogarithm local systems as Tate variations of mixed Hodge structure, computes their monodromy, and uses iterated integrals to connect to the regulators and higher Beilinson regulators. A central theme is the Hodge-theoretic and motivic interpretation of these regulators via extensions in mixed Hodge structures and the polylogarithm variation, including a motivic description through the cobar construction and path-space formalisms. Together, these results illuminate the Beilinson–Deligne–Goncharov program by tightly coupling polylogarithms with regulator maps, mixed Hodge structures, and motivic frameworks, providing a unified perspective on how classical polylogarithms encode deep arithmetic and geometric information.

Abstract

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.

Paper Structure

This paper contains 11 sections, 35 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Monodromy
  3. The Bloch-Wigner function
  4. The regulator $K_3({ \errmessage{Use \Bbb\space only in math mode} C}) \to { \errmessage{Use \Bbb\space only in math mode} C}/{ \errmessage{Use \Bbb\space only in math mode} Q}(2)$
  5. Iterated integrals
  6. The Regulator $K_2(X) \to H^1(X,{ \errmessage{Use \Bbb\space only in math mode} C}^\ast)$
  7. The polylogarithm variation of mixed Hodge structure
  8. Mixed Hodge structure on $\pi_1$
  9. Hodge theoretic interpretation of regulators
  10. The polylogarithm quotient of $\pi_1({ \errmessage{Use \Bbb\space only in math mode} P}^1-\{0,1,\infty\}$
  11. Motivic Description of the Polylogarithm Variation

Key Result

Proposition 2.1

The function $\Lambda(x)$ is a fundamental solution of (diff-eqn). That is, and $\Lambda(x)$ is non-singular for each $x\in { \errmessage{Use \Bbb\space only in math mode} C}-\{0,1\}$. ${\vcenter{}}{\vcenter{}}{\vcenter{}}{\vcenter{}}$

Theorems & Definitions (36)

  • Proposition 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Proposition 2.5
  • Theorem 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Lemma 4.1
  • Theorem 4.2
  • ...and 26 more