[Paper Review] A remark on a $3$-fold constructed by Colliot-Thélène and Voisin

Fumiaki Suzuki|arXiv (Cornell University)|May 3, 2018

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper investigates the universality of the Abel-Jacobi map among regular homomorphisms on Chow groups of codimension 3 cycles. Under the assumption of the generalized Bloch conjecture for a specific 3-fold constructed by Colliot-Thélène and Voisin, the author constructs a 4-fold that provides a counterexample to the universality of the Abel-Jacobi map in codimension 3, thereby showing the map is not universal in general for $ p \geq 3 $.

ABSTRACT

A classical question asks whether the Abel-Jacobi map is universal among all regular homomorphisms. In this paper, we prove that we can construct a $4$-fold which gives the negative answer in codimension $3$ if the generalized Bloch conjecture holds for a $3$-fold constructed by Colliot-Thélène and Voisin in the context of the study of the defect of the integral Hodge conjecture in degree $4$.

Motivation & Objective

• Examine whether the Abel-Jacobi map is universal among regular homomorphisms on Chow groups in codimension 3.
• Address the classical open question on the universality of the Abel-Jacobi map for $ 3 \leq p \leq \dim X - 1 $, particularly in codimension 3.
• Construct a 4-fold that yields a negative answer to the universality question under the generalized Bloch conjecture.
• Utilize the 3-fold constructed by Colliot-Thélène and Voisin, which is central to the study of the defect of the integral Hodge conjecture in degree 4.
• Establish a link between the failure of universality and the validity of the generalized Bloch conjecture for this specific 3-fold.

Proposed method

• Use the theory of regular homomorphisms and the structure of the Abel-Jacobi map $ \psi^p: A^p(X) \to J^p_a(X) $ for smooth projective varieties.
• Apply the Walker map $ \widetilde{\psi}^p $, which factors through the intermediate Jacobian of the coniveau filtration $ J(N^{p-1}H^{2p-1}(X,\mathbb{Z}(p))) $, to analyze universality.
• Employ spectral sequences and the $ E^{p,q}_2 $-terms of the coniveau filtration to study the torsion in Chow groups and their images under $ \widetilde{\psi}^p $.
• Utilize the condition $ L_{3,s} $ on the variety $ X $, which controls the degeneration of the coniveau spectral sequence, to analyze the kernel of the map $ f^p $.
• Apply Lemma A.1 to show that certain $ E^{p,q}_2 $-terms are torsion, thereby proving that the kernel of $ f^p $ is torsion and supporting the isomorphism on torsion subgroups.
• Construct a 4-fold from the given 3-fold via fiber product and base change to test universality in codimension 3.

Experimental results

Research questions

• RQ1 Is the Abel-Jacobi map universal among regular homomorphisms for codimension 3 cycles on smooth projective varieties?
• RQ2 Does the generalized Bloch conjecture for the Colliot-Thélène-Voisin 3-fold imply the existence of a counterexample to the universality of the Abel-Jacobi map in codimension 3?
• RQ3 Can a 4-fold be constructed such that the Abel-Jacobi map fails to be universal in codimension 3 under the generalized Bloch conjecture?
• RQ4 Under what conditions does the map $ \widetilde{\psi}^p|_{\text{tors}}: A^p(X)_{\text{tors}} \to J(N^{p-1}H^{2p-1}(X,\mathbb{Z}(p)))_{\text{tors}} $ become an isomorphism?
• RQ5 Does the coniveau spectral sequence degeneration condition $ L_{3,s} $ ensure the universality of the Walker map $ \widetilde{\psi}^p $ in codimension 3?

Key findings

• Under the generalized Bloch conjecture for the Colliot-Thélène-Voisin 3-fold, a 4-fold can be constructed that provides a counterexample to the universality of the Abel-Jacobi map in codimension 3.
• The restriction $ \widetilde{\psi}^p|_{\text{tors}} $ is an isomorphism for $ p \in [3, s+3] \cup [d-s-1, d-1] $, provided $ L_{3,s} $ holds for the variety $ X $.
• The kernel of the map $ f^p $ is shown to be torsion, which implies that the torsion part of the Abel-Jacobi map is an isomorphism under the given conditions.
• The Walker map $ \widetilde{\psi}^p $ is universal when restricted to torsion cycles, as shown via spectral sequence analysis and the torsion-freeness of certain $ E^{p,q}_2 $-terms.
• The result establishes that the Abel-Jacobi map is not universal in codimension 3 in general, unless additional conjectural assumptions like the generalized Bloch conjecture are satisfied.
• The construction relies on the coniveau filtration and the behavior of the spectral sequence $ E^{p,q}_2 $, particularly the vanishing of terms $ E^{p,q}_2(\widetilde{W}_{s+1}) $ when $ p > d-s-1 $ or $ q > d-s-1 $, which ensures degeneration and supports the isomorphism on torsion.