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An application of a Hodge realization of Bloch-Kriz mixed Tate motives

Kenichiro Kimura

TL;DR

This paper provides a concrete Hodge realization framework for Bloch–Kriz’s mixed Tate motives MT_BK over a number field, constructing an explicit complex AC^• to realize periods and regulators. It establishes an equivalence between MT_BK and flat N{1}-connections, builds Betti and de Rham realizations with a comparison isomorphism, and defines a Hodge realization functor that preserves tensor structure. The Polylog motives are realized explicitly, and Abel–Jacobi regulators are computed via explicit chains, yielding period matrices that align with polylogarithmic values. The work advances the Beilinson– Zagier program by validating the structural hypotheses (A)–(E) for MT_BK and its Hodge theory, enabling a rigorous link between motives, regulators, and special values of zeta-like functions.

Abstract

Beilinson and Deligne proved a weak version of Zagier's conjucture on special values of Dedekind zeta functions assuming the existence of a category of mixed Tate motives which has certain properties. We show that Bloch-Kriz category of mixed Tate motives together with a Hodge realization which we constructed has the required properties.

An application of a Hodge realization of Bloch-Kriz mixed Tate motives

TL;DR

This paper provides a concrete Hodge realization framework for Bloch–Kriz’s mixed Tate motives MT_BK over a number field, constructing an explicit complex AC^• to realize periods and regulators. It establishes an equivalence between MT_BK and flat N{1}-connections, builds Betti and de Rham realizations with a comparison isomorphism, and defines a Hodge realization functor that preserves tensor structure. The Polylog motives are realized explicitly, and Abel–Jacobi regulators are computed via explicit chains, yielding period matrices that align with polylogarithmic values. The work advances the Beilinson– Zagier program by validating the structural hypotheses (A)–(E) for MT_BK and its Hodge theory, enabling a rigorous link between motives, regulators, and special values of zeta-like functions.

Abstract

Beilinson and Deligne proved a weak version of Zagier's conjucture on special values of Dedekind zeta functions assuming the existence of a category of mixed Tate motives which has certain properties. We show that Bloch-Kriz category of mixed Tate motives together with a Hodge realization which we constructed has the required properties.

Paper Structure

This paper contains 11 sections, 12 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. The category ${\rm MT_{BK}}$
  3. Cycle complexes and a cdga $N$
  4. The bar complex
  5. The category of flat connections
  6. Hodge realization of ${\rm MT_{BK}}$
  7. The complex $AC^\bullet$
  8. Cauchy-Stokes formula
  9. A Hodge realization of ${\rm MT_{BK}}$.
  10. Hodge realization of the category of flat connections
  11. The Hodge realization of Polylog motives

Key Result

Proposition 2.2

Let $V$ be an object of ${\rm MT_{BK}}$. Then the kernel of the map is equal to $\Delta_V(V)$. Hence it is isomorphic to $V$ as a $\Bbb Q$-vector space by the counitarity of $\Delta_V$.

Theorems & Definitions (28)

  • Definition 2.1: Adams graded $\chi_N$-co-modules, mixed Tate motives, BK
  • Proposition 2.2
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4: Proposition 2.8 part II
  • Theorem 3.5: Theorem 4.1 part II
  • Definition 3.6
  • Theorem 3.7: Cauchy-Stokes formula
  • ...and 18 more