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Chern-Simons invariants of hyperbolic three-manifolds, mixed Tate motives, and motivic path torsor of augmented character varieties

Dong Uk Lee

Abstract

For any complete hyperbolic three-manifold of finite volume, we construct a mixed Tate motive defined over the invariant trace field whose image under Beilinson regulator equals the PSL2(C)-Chern-Simons invariant, thus equals the complex volume of the manifold, up to constant. Further, we show that when M has single torus boundary, under some assumption on asymptotic behaviour of the Chern-Simons invariant near an ideal point, its Hodge realization is a quotient of the mixed Hodge structure on the path torsor of the smooth locus of the canonical curve component of the augmented character variety of the three-manifold between a geometric point (giving the complete hyperbolic structure) and some tangential base point at an ideal point whose existence is asserted by the assumption. We explain its motivic implication. In the appendix, we verify some cases of the assumption. The theory developed here is parallel to the motivic theory of polylogarithms.

Chern-Simons invariants of hyperbolic three-manifolds, mixed Tate motives, and motivic path torsor of augmented character varieties

Abstract

For any complete hyperbolic three-manifold of finite volume, we construct a mixed Tate motive defined over the invariant trace field whose image under Beilinson regulator equals the PSL2(C)-Chern-Simons invariant, thus equals the complex volume of the manifold, up to constant. Further, we show that when M has single torus boundary, under some assumption on asymptotic behaviour of the Chern-Simons invariant near an ideal point, its Hodge realization is a quotient of the mixed Hodge structure on the path torsor of the smooth locus of the canonical curve component of the augmented character variety of the three-manifold between a geometric point (giving the complete hyperbolic structure) and some tangential base point at an ideal point whose existence is asserted by the assumption. We explain its motivic implication. In the appendix, we verify some cases of the assumption. The theory developed here is parallel to the motivic theory of polylogarithms.

Paper Structure

This paper contains 32 sections, 52 theorems, 306 equations, 1 table.

Table of Contents

  1. Introduction
  2. Chern-Simons invariant of hyperbolic three-manifolds and Bloch regulator
  3. $\mathrm{SO}(3)$ Chern-Simons invariant of hyperbolic $3$-manifolds
  4. Complex volume of hyperbolic $3$-manifolds as $\mathrm{PSL}_2(\mathbb C)$ Chern-Simons invariant
  5. Ideal triangulation of cusped hyperbolic three-manifolds
  6. Complex volume of hyperbolic $3$-manifolds as Bloch regulator
  7. Big and skew-symmetric periods of framed mixed Tate motives
  8. Period matrix of splitted Hodge-Tate structure
  9. Period of framed Hodge-Tate structure: Big period and skew-symmetric period
  10. Example: Polylogarithm Hodge-Tate structure BeilinsonDeligne94
  11. Hodge-Tate Hopf algebra
  12. Chern-Simons mixed Tate motive
  13. Mixed Tate motives
  14. Polylogarithm mixed Tate motive
  15. Comparison with Goncharov's mixed Tate motives
  16. ...and 17 more sections

Key Result

Theorem 1

Goncharov99 With every hyperbolic manifold of odd dimension $m=2n-1$, there exists a mixed Tate motive defined over $\overline{\mathbb Q}\subset\mathbb C$ which is a simple extension of $\mathbb Q(0)$ by $\mathbb Q(n)$ in the abelian category $\mathrm{MT}(\overline{\mathbb Q})$ of mixed Tate motives

Theorems & Definitions (108)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Definition 2.1.4
  • Theorem 2.1.5
  • Remark 2.1.6
  • proof
  • ...and 98 more