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

TL;DR

This paper constructs a mixed Tate motive for hyperbolic three-manifolds, linking Chern-Simons invariants to motivic structures and exploring their Hodge realizations and implications for character varieties.

Contribution

It introduces a new motivic framework connecting hyperbolic invariants with mixed Tate motives and character varieties, extending the motivic theory of polylogarithms.

Findings

Constructed a mixed Tate motive over the invariant trace field.

Linked the motive's regulator image to the Chern-Simons invariant and complex volume.

Analyzed Hodge realizations related to 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.