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The weight filtration on real singular homology is motivic

Raphaël Ruimy

TL;DR

The paper constructs a real motivic realization functor from DM_gm(R, F_2) to D^b(F_2), using Jacobson's theorem and Gersten resolutions to define transfers, and shows it factors through the Sm^cor category, giving a motivic interpretation of Totaro's weight filtration on real singular homology. By employing Bondarko's weight structures and their monoidal enhancements, the authors endow the realization with a canonical weight filtration compatible with real singular homology, hence showing the real weight filtration is a realization of the motivic weight filtration. They further extend to Borel-Moore homology, cohomology, and compact support, and recover McCrory-Parusiński's filtration as a motivic construction. This provides a unifying framework bridging real algebraic geometry, motivic homotopy theory, and weight filtrations.

Abstract

We give an alternative construction of Totaro's weight filtration on singular homology of the real points of a real algebraic variety. Our construction shows that this filtration comes from Bondarko's weight filtration on Voevodsky motives.

The weight filtration on real singular homology is motivic

TL;DR

The paper constructs a real motivic realization functor from DM_gm(R, F_2) to D^b(F_2), using Jacobson's theorem and Gersten resolutions to define transfers, and shows it factors through the Sm^cor category, giving a motivic interpretation of Totaro's weight filtration on real singular homology. By employing Bondarko's weight structures and their monoidal enhancements, the authors endow the realization with a canonical weight filtration compatible with real singular homology, hence showing the real weight filtration is a realization of the motivic weight filtration. They further extend to Borel-Moore homology, cohomology, and compact support, and recover McCrory-Parusiński's filtration as a motivic construction. This provides a unifying framework bridging real algebraic geometry, motivic homotopy theory, and weight filtrations.

Abstract

We give an alternative construction of Totaro's weight filtration on singular homology of the real points of a real algebraic variety. Our construction shows that this filtration comes from Bondarko's weight filtration on Voevodsky motives.
Paper Structure (4 sections, 10 theorems, 59 equations)

This paper contains 4 sections, 10 theorems, 59 equations.

Table of Contents

  1. Higher categorical constructions
  2. The universal property of Voevodsky motives over a perfect field
  3. The real realization
  4. The weight filtration on real singular homology

Key Result

Proposition 1.7

The unit of the adjunction between $\mathcal{P}^\mathrm{rex}_\Sigma$ and the inclusion applied to $\mathrm{A}^\otimes\in \mathrm{MAdd}$ induces the functor $\eta_{\mathrm{A}^\otimes}$ up to a unique equivalence on the target. In particular, if $\mathcal{C}^\otimes$ belongs to $\mathrm{M}\mathrm{Cat}

Theorems & Definitions (35)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.5
  • Remark 1.6
  • Proposition 1.7
  • proof
  • Definition 1.8
  • Definition 1.9
  • Remark 1.10
  • Remark 1.11
  • ...and 25 more