Arithmetic of Gysin kernels

Technical Summary: Arithmetic of Gysin Kernels

Problem Statement
The paper investigates the structure of the kernel of the Gysin homomorphism on 0-cycles for a smooth projective surface $X$ over a field $k$. Specifically, given a Lefschetz pencil on $X$ with generic fiber $C$, the authors study the kernel $G_0(k) = \ker(\iota_*: A(k) \to A_0(X))$, where $A = \text{Jac}(C)$ is the Jacobian of the curve and $A_0(X)$ is the Chow group of 0-cycles of degree 0 modulo rational equivalence. The central problem is to determine the arithmetic and geometric structure of this kernel, particularly whether it is countable or contains a large abelian subvariety, and how this relates to the representability of the Chow group $A_0(X)$ and Bloch's conjecture. The study is conducted over fields of arbitrary characteristic, extending previous results known primarily in characteristic 0.

Methodology
The authors employ a synthesis of algebraic geometry, arithmetic geometry, and étale cohomology. Key methodological components include:

1. Suslin-Voevodsky Representability: The paper relies heavily on the representability of 0-cycles by symmetric powers (Theorem 2.1.1), which allows for the rigorous treatment of rational equivalence in arbitrary characteristic. This is used to establish the Mumford-Ro˘ıtman lemma (Section 2.3), proving that rational equivalence is encoded by a countable collection of closed subschemes even in positive characteristic.
2. Countability Lemmas over Uncountable Domains: Working over an uncountable universal domain $\bar{k}$, the authors utilize topological arguments regarding "c-closed" sets (countable unions of Zariski closed sets). They prove that if a subgroup of an abelian variety is c-closed, its irreducible decomposition contains a unique component passing through the origin, which is an abelian subvariety (Lemmas 3.2.4, 3.2.5). This establishes that the Gysin kernel is a union of countably many translates of a specific abelian subvariety $A_0$.
3. Étale Monodromy and Deligne's Irreducibility: The core of the structural analysis involves the action of the étale fundamental group (monodromy) on the cohomology of the vanishing cycles. By spreading the abelian varieties $A_0$ and $A_1$ (the connected component of the kernel of the Albanese map) over a base curve, the authors analyze the monodromy action on the $l$-adic cohomology groups $H^1(A_{\bar{\eta}}, \mathbb{Q}_l)$. They invoke Deligne's irreducibility theorem for the monodromy action on vanishing cycles (Corollary 7.4 in [13]) to force a dichotomy in the structure of the kernel.
4. Field Theory and Transcendence: The proof of the third dichotomy utilizes field-theoretic arguments involving disjoint sets of algebraically independent elements to show that if the kernel is countable, it must be defined over the minimal field of definition of the pencil.

Key Contributions and Results
The paper establishes three "dichotomy theorems" describing the structure of the Gysin kernel $G_0$:

• Theorem A (First Dichotomy): In terms of the geometric generic fiber $\bar{\eta}$, the abelian subvariety $A_{\bar{\eta},0}$ (associated with the Gysin kernel) is either trivial ($0$) or coincides with $A_{\bar{\eta},1}$ (the connected component of the kernel of the map to the Albanese variety). This result holds over fields of arbitrary characteristic and relies on the irreducibility of the monodromy action on vanishing cycles.
• Theorem B (Second Dichotomy): This theorem links the countability of the Gysin kernel to the representability of the Chow group.
  • If $A_0(X_{\bar{k}})$ is not weakly representable (i.e., not finite-dimensional in Mumford's sense), then the kernel $G_0(k(t))$ is countable.
  • Conversely, assuming the Gysin homomorphism on cohomology $H^1(C_{\bar{\eta}}, \mathbb{Q}_l) \to H^3(X_{\bar{\eta}}, \mathbb{Q}_l(1))$ is not injective, if $G_0(k(t))$ is countable, then $A_0(X_{\bar{k}})$ is not weakly representable.
  • This provides a criterion for the failure of weak representability based on the size of the Gysin kernel.
• Theorem C (Third Dichotomy): This theorem addresses the arithmetic nature of the kernel. If the Gysin kernel $G_0(k(t))$ is countable, then $G_0(k(t)) = G_0(k_0(t))$, where $k_0$ is the minimal field of definition of the Lefschetz pencil. This implies that all points in the kernel are algebraic over the minimal field of definition; no "transcendental" points exist in the kernel if it is countable.

Significance and Claims
The authors claim that their arguments are simpler than previous proofs in characteristic 0 (specifically referencing their prior work [2]) and that they successfully extend these structural results to arbitrary characteristic.

• Relation to Bloch's Conjecture: The paper posits that Theorem C has direct implications for Bloch's conjecture. If $H^2(X, \mathbb{Q}_l)$ is algebraic (implying weak representability is equivalent to Bloch's conjecture), proving the conjecture reduces to finding a point of transcendence degree $>0$ in the Gysin kernel. If the kernel is countable, all points are algebraic, suggesting the conjecture holds.
• Applications to K3 Surfaces: The authors note that for K3 surfaces over $\mathbb{Q}$ or $\mathbb{F}_p$, where $A_0(X)$ is known not to be representable, the results imply that all points in the Gysin kernel are rational over the base field of definition (e.g., $\mathbb{Q}(t)$ or $\mathbb{F}_p(t)$).
• Generality: The work removes the restriction to characteristic 0 found in earlier literature (such as [2] and [36]), providing a unified framework for the arithmetic of Gysin kernels using étale monodromy and Suslin-Voevodsky representability.

The paper does not propose new experimental applications or future research directions beyond the immediate logical consequences of the theorems for Bloch's conjecture and the study of K3 surfaces. The focus remains strictly on the structural dichotomy of the kernel and its arithmetic implications.