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Intersection complex of any threefold as a Chow motive

Shruti Rastogi, Vaibhav Vaish

TL;DR

The paper proves that for any irreducible threefold X over characteristic zero, the motivic candidate EM^F_X, constructed via refined Morel weight truncations and punctual gluing, is the motivic intersection complex. It establishes that EM^F_X has Bondarko weight 0, yielding a Chow motive that lifts the motivic intersection complex IC^M_X and satisfies Wildeshaus’s stronger endomorphism criteria. The authors develop a motivic refinement of Morel’s truncations, connect these to mixed realizations, and extend previous Shimura-specific results to arbitrary threefolds. This work provides a canonical, functorial motivic construction of IC objects in DM(X) with a solid weight-theoretic footing, enabling a robust bridge between motivic and classical intersection theory for threefolds.

Abstract

Motivated by the characterization of the intersection complex in terms of S$.$Morel's weight truncations, we introduced an object $EM^{F}_{X}$ in the setting of motivic sheaves for certain schemes $X$ and weight profiles $F$. In this article, we show that when $X$ is any threefold, this object satisfies Wildeshaus's characterization of a motivic intersection complex. In particular, we demonstrate that the construction is a suitably functorial Chow motive lifting the motivic intersection complex for an arbitrary threefold.

Intersection complex of any threefold as a Chow motive

TL;DR

The paper proves that for any irreducible threefold X over characteristic zero, the motivic candidate EM^F_X, constructed via refined Morel weight truncations and punctual gluing, is the motivic intersection complex. It establishes that EM^F_X has Bondarko weight 0, yielding a Chow motive that lifts the motivic intersection complex IC^M_X and satisfies Wildeshaus’s stronger endomorphism criteria. The authors develop a motivic refinement of Morel’s truncations, connect these to mixed realizations, and extend previous Shimura-specific results to arbitrary threefolds. This work provides a canonical, functorial motivic construction of IC objects in DM(X) with a solid weight-theoretic footing, enabling a robust bridge between motivic and classical intersection theory for threefolds.

Abstract

Motivated by the characterization of the intersection complex in terms of SMorel's weight truncations, we introduced an object in the setting of motivic sheaves for certain schemes and weight profiles . In this article, we show that when is any threefold, this object satisfies Wildeshaus's characterization of a motivic intersection complex. In particular, we demonstrate that the construction is a suitably functorial Chow motive lifting the motivic intersection complex for an arbitrary threefold.

Paper Structure

This paper contains 17 sections, 47 theorems, 189 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries on t-structures
  4. Chow motives
  5. Motivic Sheaves
  6. Realizations
  7. Abelian motives
  8. Refined Morel's weight truncations
  9. Morel's weight truncations
  10. Refinement of Morel's truncations, motivically
  11. Refining Morel's truncations in mixed categories.
  12. Geometry
  13. Motivic Intersection complex
  14. Definitions and First results
  15. Existence
  16. ...and 2 more sections

Key Result

Theorem 1.1

(see main theorem) Let $X$ be an irreducible variety of dimension $3$ over a field $k$ of characteristic $0$. Then the following hold: In particular, the construction $EM_X^F$ satisfies Wildeshaus' (stronger) criteria wildeshaus_ic for being a motivic intersection complex.

Figures (1)

  • Figure :

Theorems & Definitions (106)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Definition 2.1
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5: $m$-structures from generators
  • proof
  • Lemma 2.6
  • ...and 96 more