Technical Summary: Zero-Cycles on Quasi-Projective Surfaces over p-adic Fields

1. Problem Statement and Context

The paper addresses the structure of the Chow group of 0-cycles, $CH_0(X)$, for smooth projective varieties $X$ defined over a finite extension $k$ of the $p$-adic field $\mathbb{Q}_p$. A central object of study is the filtration of $CH_0(X)$:
$$0 \subseteq F^2(X) \subseteq F^1(X) \subseteq CH_0(X)$$
where $F^1(X)$ is the kernel of the degree map, and $F^2(X)$ is the kernel of the Albanese map $alb_X: F^1(X) \to Alb_X(k)$.

The authors focus on Colliot-Thélène's Conjecture (Conjecture 1.1), which predicts that for such varieties, the group $F^2(X)$ decomposes as a direct sum of a finite group and a divisible group:
$$F^2(X) \cong F \oplus D$$
While this conjecture is known for curves (where the Albanese map is an isomorphism) and specific cases of products of curves or Kummer surfaces, it remains open for general surfaces. Previous results by Raskind and Spiess ([RS00]) and Saito and Sato ([SS10]) established partial results, particularly regarding the torsion properties of $F^1(X)$ and $F^2(X)$ for products of curves with specific reduction types.

The paper also investigates the quasi-projective analog of this problem. For an open subvariety $U \subset X$, the authors replace $CH_0(X)$ with Suslin's singular homology $H^{sus}_0(U)$ and consider the kernel $F^2(U)$ of the generalized Albanese map $alb_U: F^1(U) \to Alb_U(k)$, where $Alb_U$ is a semi-abelian variety.

2. Methodology

The authors employ a multi-faceted approach combining motivic cohomology, class field theory, and algebraic group theory:

• Motivic Cohomology and Suslin Homology: The authors utilize the identification of Suslin's homology $H^{sus}_0(U)$ with motivic cohomology groups $H^{2d}_{M,c}(U, \mathbb{Z}(d))$ (Theorem 2.8). This allows them to use the machinery of motivic complexes and the six-functor formalism.
• Class Field Theory for Curves: A crucial component is the application of the class field theory for curves over $p$-adic fields developed by Bloch ([Blo81]) and Saito ([Sai85]). Specifically, they analyze the structure of the group $SK_1(C)$ (identified with $CH_2(C,1)$) and its relation to the abelian étale fundamental group and the Tate module of the Jacobian.
• Generalized Albanese Varieties: The paper constructs and analyzes the generalized Albanese variety $Alb_U$ for open subvarieties $U = X - D$, where $D$ is a divisor. This variety is an extension of the standard Albanese variety $Alb_X$ by a torus $T$. The authors study the relationship between the toric part of $Alb_U$ and the structure of $SK_1(D)$ for the divisor $D$.
• Decomposable Cycles: To handle the "toric piece" of the generalized Albanese map, the authors utilize decomposable $(2,1)$-cycles. They show that the image of the map from the Neron-Severi group tensored with $k^\times$ into $SK_1(D)$ is sufficient to annihilate the problematic toric components in the kernel of the map.

3. Key Contributions and Results

A. Equivalence of Projective and Quasi-Projective Conjectures

The central technical result is Theorem 4.5, which establishes an equivalence between the finiteness of the non-divisible part of $F^2$ for a projective surface and its open subvariety.

• Theorem 4.5: Let $X$ be a smooth projective surface and $U$ a dense open subvariety with complement $D$ (either 0-dimensional or a simple normal crossing divisor). Then $F^2(X_L)/F^2(X_L)_{div}$ is finite for all finite extensions $L/k$ if and only if $F^2(U_L)/F^2(U_L)_{div}$ is finite for all $L/k$.
• Corollary 4.9: Consequently, Colliot-Thélène's Conjecture (1.1) holds for all smooth projective surfaces if and only if its quasi-projective analog (Conjecture 4.8) holds for all smooth quasi-projective surfaces.

B. Transfer of Conjectures via Rational Maps

Building on the equivalence above, the authors prove Theorem 1.2 (restated as Theorem 5.2):

• If there exists a generically finite rational map $\pi: X \dashrightarrow Y$ between smooth projective surfaces, and the conjecture holds for $X$ (and its finite extensions), then it holds for $Y$.
• This result generalizes previous push-forward arguments used for Kummer surfaces to a broader class of surfaces, provided one can work with open subvarieties to avoid singularities in the resolution of the map.

C. Verification for New Classes of Surfaces

Using the transfer theorem and the known results for products of curves (Raskind-Spiess), the authors verify Conjecture 1.1 for several new classes of surfaces (Corollary 1.3 and Section 5.2):

1. Geometrically Dominated Surfaces: Any smooth projective surface $X$ that is geometrically dominated by a product of curves $C_1 \times C_2$, provided the Jacobians of $C_1$ and $C_2$ have a mixture of good ordinary and split multiplicative reduction.
2. Isotrivial Fibrations: This includes isotrivial elliptic surfaces and a wide class of K3 surfaces admitting non-symplectic automorphisms (which are often resolutions of quotients of products of curves).
3. Symmetric Squares: The symmetric square $Sym^2(C)$ of a genus 2 curve $C$ (and the associated abelian surface $J_C$) under the reduction assumptions.
4. Fermat Surfaces: Diagonal hypersurfaces $X_m \subset \mathbb{P}^3$ of degree $m \ge 4$, specifically when $p \equiv 1 \pmod m$.
5. Product-Quotient Surfaces: Surfaces arising as resolutions of $(C_1 \times C_2)/G$ for finite groups $G$, extending beyond the specific Kummer case previously known.

D. Structure of Suslin's Homology for Open Varieties

The paper provides detailed structural analysis of $F^2(U)$ for open subvarieties:

• Comparison with $F^2(X)$: When $X-U$ is 0-dimensional, $F^2(U) \cong F^2(X)$. However, when $X-U$ is a divisor, $F^2(U)$ can be significantly larger.
• Toric Contributions: The authors show that the difference between $F^2(U)$ and $F^2(X)$ is governed by the toric part of the generalized Albanese variety.
• Examples:
  • For $U = \mathbb{G}_m \times \mathbb{G}_m$, $F^2(U) \cong K_2^M(k)$, which contains an uncountable divisible group.
  • For $U = C \times \mathbb{G}_m$, $F^2(U) \cong V(C)$, which can have infinite torsion if $C$ is a Mumford curve.
  • Proposition 5.17: In specific geometric settings (e.g., $U = X - s(C)$ where $s$ is a section of a fibration), the equality $F^2(U) = F^2(X)$ can be recovered, implying finiteness even for open varieties.

4. Significance and Claims

The authors claim that their work provides "new evidence" for Colliot-Thélène's Conjecture by significantly expanding the known classes of surfaces for which the conjecture holds.

• Methodological Shift: The paper highlights a shift from relying solely on "simple push-forward" arguments (which work for Kummer surfaces due to simple resolutions) to a more robust method involving open subvarieties and Suslin homology. This allows the authors to handle more complex singularities and quotients (e.g., general product-quotient surfaces).
• Unification: By proving the equivalence between the projective and quasi-projective cases (Theorem 4.5), the authors unify the study of 0-cycles on these different types of varieties, suggesting that the conjecture's validity is a geometric property preserved under open embeddings and dominant rational maps.
• Limitations: The authors modestly note that their results still depend on assumptions regarding the reduction type of the Jacobians (specifically, a mixture of good ordinary and split multiplicative reduction). They acknowledge that removing this assumption (e.g., to include supersingular reduction) remains a subtle and difficult open problem, citing the difficulty even in the case of products of supersingular elliptic curves.

In summary, the paper establishes a bridge between the arithmetic of 0-cycles on projective surfaces and their open counterparts, leveraging this link to prove the Colliot-Thélène conjecture for a broad and diverse collection of surfaces dominated by products of curves.