Some remarks on Chow correspondences

Detailed Technical Summary of "Some Remarks on Chow Correspondences" by Pablo Pelaez

Problem Statement
The paper addresses a specific problem within the framework of Voevodsky's triangulated category of motives ($DM_k$): characterizing Chow correspondences $\Lambda \in CH^{d+n}(X \times Y)$ between smooth projective varieties $X$ and $Y$ (over an algebraically closed field $k$) for which a specific induced homomorphism $\psi$ vanishes.

The homomorphism in question, $\psi: CH^{d_Y-n}(Y) \to Alb(X)(k)$, is defined via a Chow correspondence $\Lambda$. It maps a cycle $\gamma$ to the Albanese image of the pushforward of the intersection product, adjusted by degree:
$$\psi(\gamma) = alb_X(\Lambda(\gamma) - \deg(\Lambda(\gamma))[x_0])$$
where $alb_X$ is the Albanese map, $x_0$ is a fixed rational point, and $\Lambda(\gamma)$ is the standard action of the correspondence on cycles. The paper seeks to determine the conditions under which $\psi$ is identically zero, specifically relating this vanishing to the structure of the correspondence within the orthogonal filtration of the category of motives.

Methodology
The author employs the machinery of Voevodsky's triangulated category of motives ($DM_k$) and the theory of adequate equivalence relations. The methodology relies on the following key components:

1. Motivic Representation of Cycles: The paper utilizes the isomorphism $Hom_{DM_k}(M(Z), \mathbb{1}(m)[2m]) \cong CH^m(Z)$ to represent algebraic cycles as morphisms in the motivic category.
2. Orthogonal Filtration: A central tool is the orthogonal filtration of $DM_k$, denoted $DM_k^\perp(m)$. This filtration consists of objects orthogonal to the subcategory of $m$-effective motives. The paper specifically focuses on the orthogonal cover $bc_{\leq -1}$ and its counit $\theta_{-1}^A: bc_{\leq -1}(A) \to A$, where $A$ represents the functor of points of the Albanese variety $Alb(X)$.
3. Factorization Criteria: The core technical approach involves analyzing the factorization of the motivic map associated with the correspondence. Specifically, the paper investigates whether the composition of the motivic map induced by $\Lambda$ and the map to the Albanese variety factors through the orthogonal cover $bc_{\leq -1}(A)$.
4. Regular Homomorphisms: The analysis connects the vanishing of $\psi$ to the theory of cycle-induced regular homomorphisms (in the sense of Lieberman), utilizing the properties of the Albanese map as a regular homomorphism.

Key Contributions and Results

• Characterization of Vanishing Correspondences (Theorem 4.1.4):
  The primary result establishes a necessary and sufficient condition for the homomorphism $\psi$ to be zero. The theorem states that $\psi \equiv 0$ if and only if the motivic map associated with the correspondence $\Lambda$ (specifically the composition $A_f \circ \Lambda_X$) factors through the orthogonal cover $\theta_{-1}^A: bc_{\leq -1}(A) \to A$. In diagrammatic terms, there exists a map $\Lambda': M(Y)(-n)[-2n] \to bc_{\leq -1}(A)$ making the relevant triangle in $DM_k$ commute.

• Detection of Square Equivalence (Theorem 4.3.1):
  The paper demonstrates that the orthogonal cover detects correspondences that are "square equivalent to zero" (denoted $alg^{*2}$). Specifically, if a correspondence $\Lambda$ belongs to the subgroup $CH^{d+n}_{alg^{*2}}(Y \times X)$ (the product of the algebraic equivalence relation with itself), then the induced homomorphism $\psi$ is zero. Consequently, such correspondences satisfy the factorization condition through $bc_{\leq -1}(A)$.

• Detection of H. Saito's Filtration (Theorem 4.3.2):
  In the specific case where the base field is $k = \mathbb{C}$, the result is strengthened to include correspondences in the second step of H. Saito's filtration ($H^{*2}$). If $\Lambda \in CH^{d+n}_{H^{*2}}(Y \times X)$, then $\psi$ vanishes, and the factorization through the orthogonal cover holds.

• Sharpness of Conditions (Remark 4.3.3):
  The author provides a counterexample to show that the condition $\Lambda \in CH^{d+n}_{alg^{*2}}$ is sharp. It is shown that for a correspondence merely in $CH^{d+n}_{alg}$ (algebraic equivalence) but not in the square, the homomorphism $\psi$ need not vanish. This is demonstrated by constructing a correspondence involving a non-torsion element in the Albanese kernel and a cycle not numerically equivalent to zero.

Significance and Claims
The paper claims to provide a precise motivic criterion for the vanishing of cycle-induced regular homomorphisms to the Albanese variety. By linking the vanishing of $\psi$ to the factorization through the orthogonal cover $bc_{\leq -1}$, the work connects classical intersection theory (adequate equivalence relations) with the structural properties of Voevodsky's category of motives.

The significance lies in the ability of the orthogonal filtration to "detect" specific classes of correspondences:

1. It identifies those correspondences that induce the zero map on the Albanese variety.
2. It characterizes correspondences that are square equivalent to zero or lie in the second step of Saito's filtration (over $\mathbb{C}$) as those satisfying this motivic factorization.

The author does not claim to solve the general Bloch-Beilinson conjecture but positions the orthogonal filtration as a tool that naturally captures these specific equivalence relations, offering a motivic perspective on the behavior of algebraic cycles under the Albanese map. The results are presented as characterizations within the existing framework of motives, relying on established properties of the Albanese map and Lieberman's theory of regular homomorphisms.