Néron--Severi groups of proper schemes over finite fields

Technical Summary: Néron–Severi Groups of Proper Schemes over Finite Fields

Problem Statement
The paper addresses the characterization of the Néron–Severi group $NS(X)$ for proper reduced schemes $X$ over a finite field $k = \mathbb{F}_q$. In the context of smooth projective complex varieties, the Lefschetz $(1,1)$ theorem identifies $NS(X)$ with the integral $(1,1)$-classes in $H^2(X, \mathbb{Z})$. For singular complex varieties, this identification fails; however, for proper seminormal varieties, the intersection of the Hodge filtration level $F^1$ and the subgroup of Zariski-locally trivial classes correctly recovers $NS(X)$ (a result by Barbieri-Viale, Rosenschon, and Srinivas).

Over finite fields, the analogous problem involves the $\ell$-adic cohomology $H^2_{\text{\'et}}(X, \mathbb{Z}_\ell(1))$ (where $\ell \neq \text{char}(k)$). The authors investigate whether $NS(X) \otimes \mathbb{Z}_\ell$ can be characterized by a single condition on weights within the subgroup of Zariski-locally trivial classes, without assuming the scheme is seminormal or irreducible.

Methodology and Strategy
The proof strategy relies on a descent argument via smooth proper hypercovers, adapted to the arithmetic setting of finite fields.

1. Rational Isomorphism via Zariski Cohomology: The authors define a rational invariant $H^1_{\text{Zar}}(X, H^1)$, where $H^1$ is the Zariski sheaf associated to the presheaf $U \mapsto H^1_{\text{\'et}}(U, \mathbb{Q}_\ell(1))$. They aim to prove an isomorphism:
   $$NS(X) \otimes \mathbb{Q}_\ell \xrightarrow{\sim} H^1_{\text{Zar}}(X, H^1)^{w=0}$$
   where the superscript $w=0$ denotes the subspace on which the geometric Frobenius acts with eigenvalues of absolute value 1 (weight zero).

2. Descent via Hypercovers: The proof proceeds by first establishing the result for smooth proper schemes (where it follows from Bloch–Ogus theory) and then extending it to arbitrary proper reduced schemes via a smooth proper hypercover $\pi_\bullet: X_\bullet \to X$.

   • Simplicial Cycle Class: They construct a simplicial cycle class map and analyze the skeletal spectral sequence associated with the hypercover.
   • Weight Decomposition: A crucial step involves separating weight components. The authors show that the kernel and cokernel of the relevant maps are pure of weight $-2$ or $-1$, ensuring that the weight-zero parts behave well under descent.

3. Topological Tools:

   • sdh-topology: To handle the transition from arbitrary schemes to seminormal ones, the authors utilize the sdh-topology (generated by étale covers and proper morphisms with separable residue field extensions). They use Huber–Kelly's results to show that on a seminormal scheme, the $sdh$-sheafification of units recovers the units of the seminormalization.
   • v-descent: For the descent of line bundles (Picard groups) in characteristic $p$, the authors employ Bhatt–Scholze's $v$-descent for vector bundles. This allows them to establish that the Picard group of a scheme is isomorphic to the equalizer of the Picard groups of a smooth proper hypercover after tensoring with $\mathbb{Q}_\ell$.
   • Seminormalization: The authors reduce the general case to the seminormal case by showing that the pullback along the seminormalization map $\rho: X^{sn} \to X$ induces an isomorphism on $NS(X) \otimes \mathbb{Q}_\ell$. This relies on the fact that the cokernel of the unit sheaf map is annihilated by a power of $p$, which vanishes upon tensoring with $\mathbb{Q}_\ell$ (since $\ell \neq p$).

Key Contributions and Results

• Main Theorem (Theorem 3.4): For a proper reduced scheme $X$ over a finite field $k$, the $\ell$-adic cycle class map induces a $G_k$-equivariant isomorphism:
  $$c_1: NS(X) \otimes \mathbb{Q}_\ell \xrightarrow{\sim} H^1_{\text{Zar}}(X, H^1)^{w=0}$$
  Here, $H^1_{\text{Zar}}(X, H^1)^{w=0}$ is the weight-zero subspace of the Zariski cohomology of the sheaf $H^1$.

• Integral Characterization (Corollary 3.6): The main theorem refines to an integral statement. The first Chern class identifies $NS(X) \otimes \mathbb{Z}_\ell$ with the subgroup of Zariski-locally trivial classes in $H^2_{\text{\'et}}(X, \mathbb{Z}_\ell(1))$ whose image in $H^2_{\text{\'et}}(X, \mathbb{Q}_\ell(1))$ has weight zero:
  $$NS(X) \otimes \mathbb{Z}_\ell \xrightarrow{\sim} \left\{ \alpha \in H^2_{\text{ZL}}(X, \mathbb{Z}_\ell(1)) : \alpha \otimes \mathbb{Q}_\ell \in H^2_{\text{\'et}}(X, \mathbb{Q}_\ell(1))^{w=0} \right\}$$

• Relation to the Tate Conjecture: The paper notes that if the Tate conjecture for divisors holds for smooth proper varieties (identifying $NS(Y) \otimes \mathbb{Q}_\ell$ with the finite-order part of étale cohomology), then the main theorem extends this identification to all proper reduced schemes:
  $$NS(X) \otimes \mathbb{Q}_\ell \cong H^2_{\text{\'et}}(X, \mathbb{Q}_\ell(1))_{\text{fo}}$$

• Necessity of Conditions: The authors provide examples demonstrating that both conditions—being Zariski-locally trivial and having weight zero—are necessary.

  • Without the Zariski-local triviality condition, the weight-zero subspace of $H^2_{\text{\'et}}$ can be strictly larger than $NS(X) \otimes \mathbb{Q}_\ell$ (e.g., for non-supersingular abelian surfaces or K3 surfaces).
  • Without the weight-zero condition, $H^1_{\text{Zar}}(X, H^1)$ can contain non-zero weight $(-1)$ summands even for normal projective surfaces (Example 8.11).

Significance
The paper establishes a finite-field analogue of a theorem by Barbieri-Viale, Rosenschon, and Srinivas for complex varieties. Its primary significance lies in the removal of restrictive hypotheses: unlike the complex case where seminormality is required, this result holds for all proper reduced schemes over finite fields. The authors achieve this by leveraging characteristic $p$ specific tools, such as $v$-descent and the $sdh$-topology, to bypass the need for seminormality. The work provides a precise cohomological characterization of the Néron–Severi group in terms of weight filtrations and local triviality, unifying the description of algebraic cycles across different geometric contexts.