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Integrals of logarithmic forms on semi-algebraic sets and a generalized Cauchy formula Part II: generalized Cauchy formula

Masaki Hanamura, Kenichiro Kimura, Tomohide Terasoma

TL;DR

The paper develops a rigorous framework for integrating logarithmic forms on semi-algebraic, face-stable chains and proves a generalized Cauchy formula that relates boundary integrals to face-contribution corrections. It then leverages this framework to construct a Hodge realization functor for mixed Tate motives by combining the cycle algebra $N$, the admissible chain complex $AC^\bullet$, and bar constructions, yielding an ind-mixed Tate Hodge structure $\mathcal{H}_{Hg}$ and a comparison map between Betti and de Rham realizations. The approach provides an explicit, integral-geometry perspective on periods of motives and aligns with Bloch–Kriz’s program, including a detailed treatment of the dilogarithm. Collectively, these results connect geometric semi-algebraic chains to motivic realizations, suggesting a concrete pathway to compare with existing motivic realizations and to access period computations via logarithmic differential forms. The work advances the understanding of Hodge realizations for mixed Tate motives through an explicit, computable integral formalism that incorporates Thom theory and cubical/differential structures.

Abstract

This paper is the continuation of the paper arXiv:1509.06950, which is Part I under the same title. In this paper, we prove a generalized Cauchy formula for the integrals of logarithmic forms on products of projective lines, and give an application to the construction of Hodge realization of mixed Tate motives.

Integrals of logarithmic forms on semi-algebraic sets and a generalized Cauchy formula Part II: generalized Cauchy formula

TL;DR

The paper develops a rigorous framework for integrating logarithmic forms on semi-algebraic, face-stable chains and proves a generalized Cauchy formula that relates boundary integrals to face-contribution corrections. It then leverages this framework to construct a Hodge realization functor for mixed Tate motives by combining the cycle algebra , the admissible chain complex , and bar constructions, yielding an ind-mixed Tate Hodge structure and a comparison map between Betti and de Rham realizations. The approach provides an explicit, integral-geometry perspective on periods of motives and aligns with Bloch–Kriz’s program, including a detailed treatment of the dilogarithm. Collectively, these results connect geometric semi-algebraic chains to motivic realizations, suggesting a concrete pathway to compare with existing motivic realizations and to access period computations via logarithmic differential forms. The work advances the understanding of Hodge realizations for mixed Tate motives through an explicit, computable integral formalism that incorporates Thom theory and cubical/differential structures.

Abstract

This paper is the continuation of the paper arXiv:1509.06950, which is Part I under the same title. In this paper, we prove a generalized Cauchy formula for the integrals of logarithmic forms on products of projective lines, and give an application to the construction of Hodge realization of mixed Tate motives.

Paper Structure

This paper contains 36 sections, 38 theorems, 208 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Admissible chain complexes
  3. Semi-algebraic triangulation
  4. The complex $AC_{\bullet}(P^n,\bold D^n;{\mathbb{Q}})$
  5. Subdivision and inductive limit
  6. Face map and cubical differential
  7. Thom cocycles
  8. Definition of Thom cocycles
  9. Thom form
  10. Singular Thom cocycle $T^B_{ H_{1,0}}$
  11. The cap product with a Thom cocycle
  12. Simplicial cap product
  13. Independence of $T$ and ordering
  14. Cubical differentials
  15. Compatibility of the face map
  16. ...and 21 more sections

Key Result

Theorem 2.1

Let $P$ be a compact semi-algebraic subset of $\Bbb R^m$. The set $P$ is triangulable, i.e. there exists a finite simplicial complex $K$ and a semi-algebraic homeomorphism $\Phi_K:\,|K|\to P$. Moreover, for a given finite family $S=\{S_j\}_{j=1,\cdots,q}$ of semi-algebraic subsets of $P$, we can cho

Figures (1)

  • Figure :

Theorems & Definitions (91)

  • Theorem 2.1: BCR, Theorem 9.2.1
  • Remark 2.2
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8: Moving lemma
  • Definition 3.1: Good triangulation
  • Remark 3.2
  • Definition 3.3
  • ...and 81 more