The kernel of the Gysin homomorphism for positive characteristic

1. Problem Statement

The paper addresses the structure of the Gysin kernel for smooth projective connected surfaces over an uncountable algebraically closed field $k$ of positive characteristic ($p > 0$).

• Context: Let $S$ be such a surface, and let $\Sigma$ be a linear system of a very ample divisor $D$ on $S$. For a generic point $t \in \Sigma$, let $C_t = H_t \cap S$ be a smooth hyperplane section (a curve). The inclusion $r_t: C_t \hookrightarrow S$ induces a Gysin homomorphism on Chow groups of 0-cycles of degree zero:
  $$r_{t*}: CH_0(C_t)_{\deg=0} \to CH_0(S)_{\deg=0}$$
  The Gysin kernel is defined as $G_t = \ker(r_{t*})$.
• The Goal: The authors aim to extend a theorem previously proven over the complex numbers $\mathbb{C}$ (in reference [20]) to fields of positive characteristic. The theorem characterizes the structure of $G_t$ and its relationship with the intermediate Jacobian (or its algebraic analogue) and the vanishing cohomology.

2. Methodology

Since classical Hodge theory (which relies on complex analysis and differential forms) is not available in positive characteristic, the authors employ a strategy combining lifting to characteristic zero, étale cohomology, and monodromy arguments.

A. Lifting and Comparison Theorems

The core strategy involves lifting the surface $S_0$ from characteristic $p$ to characteristic $0$.

1. Witt Vectors: Since $k$ is a perfect field of characteristic $p$, there exists a unique unramified complete discrete valuation ring (CDVR) $W(k)$ (the ring of Witt vectors) with residue field $k$ and fraction field $K$ of characteristic $0$.
2. The Lift: The authors assume the existence of a smooth projective polarized flat family $\mathcal{S}$ over $W(k)$, where the special fiber is $S_0$ and the generic fiber is $S_\eta$ (over $K$).
3. Comparison Theorems:
   • Artin's Comparison Theorem: Relates singular cohomology over $\mathbb{C}$ to étale cohomology.
   • Lift Comparison Theorem: Establishes an isomorphism between the étale cohomology of the special fiber $S_0$ and the generic fiber $S_\eta$ (over the algebraic closure of $K$).
   • This allows the authors to transfer results proven over $\mathbb{C}$ (using Hodge theory) to the generic fiber in characteristic 0, and then specialize them back to the positive characteristic setting via the isomorphism of cohomology groups.

B. Étale Cohomology and Tate's Conjecture

Instead of Hodge structures, the authors utilize:

• Étale Cohomology: Working with $H^1_{\text{ét}}(C_t, \mathbb{Q}_\ell)$ and $H^3_{\text{ét}}(S, \mathbb{Q}_\ell)$.
• Tate's Isogeny Theorem: Used to relate subrepresentations of Galois modules to abelian subvarieties. Specifically, an inclusion of $\ell$-adic cohomology groups implies the existence of an abelian subvariety (up to isogeny).
• Bloch's $\ell$-adic Abel-Jacobi Map: Used to connect Chow groups to cohomology in positive characteristic.

C. Monodromy and Irreducibility

For the second part of the theorem, the authors analyze the action of the fundamental group $\pi_1(U', \eta)$ on the cohomology of the vanishing cycles.

• They utilize the Picard-Lefschetz formula and the irreducibility of the space of vanishing cycles under the monodromy action (a result from Deligne/Lefschetz theory adapted to étale cohomology).
• They show that the image of the relevant cohomology map is a stable subspace under this monodromy action. Since the representation is irreducible, the image must be either trivial or the whole space.

3. Key Contributions and Results

The paper proves the following theorem, extending [20, Theorem 5.1] to positive characteristic:

Theorem (Structure of the Gysin Kernel):
Let $k$ be an uncountable algebraically closed field of characteristic $p > 0$. For a smooth projective surface $S$ and a generic hyperplane section $C_t$:

1. Part (a) (Structure): There exists an abelian subvariety $A_t \subset B_t \subset J_t$ (where $J_t$ is the Jacobian of $C_t$ and $B_t$ corresponds to the vanishing cohomology) such that the Gysin kernel $G_t$ is a countable union of translates of $A_t$:
   $$G_t = \bigcup_{x \in \Xi} (x + A_t)$$
   where $\Xi$ is a countable set.
2. Part (b) (Generic Behavior): For a very general point $t \in U$ (meaning $t$ lies in a countable intersection of Zariski open dense sets), the subvariety $A_t$ is either trivial or the entire vanishing part:
   $$A_t = 0 \quad \text{or} \quad A_t = B_t$$

Technical Nuances Addressed:

• Representability of $CH_0$: The authors rigorously establish that the representability of the Chow group $CH_0(X)_{\deg=0}$ and the properties of regular maps into Chow groups hold in positive characteristic, relying on Roitman's theorems and the countability of algebraic cycles over uncountable fields.
• Countability Lemmas: They prove that over an uncountable field, a variety cannot be written as a countable union of proper closed subsets, which is crucial for showing that the "generic" behavior (Part b) holds on a dense open set.
• Specialization Maps: They define and utilize specialization homomorphisms on Chow groups and symmetric products to relate the generic fiber to the special fiber, ensuring that rational equivalence is preserved during the lift.

4. Significance

• Bridging Characteristic 0 and $p$: This work demonstrates that deep structural results about algebraic cycles (specifically the Gysin kernel), which were previously thought to rely heavily on complex Hodge theory, are actually robust and hold in positive characteristic.
• Alternative to Hodge Theory: It provides a blueprint for proving results in positive characteristic by replacing transcendental methods (Hodge theory) with arithmetic-geometric tools (étale cohomology, lifting, and monodromy).
• Validation of Conjectures: By confirming the structure of the Gysin kernel in positive characteristic, the paper supports the broader conjecture that the geometry of algebraic cycles behaves uniformly across different characteristics, provided the base field is uncountable.
• Foundation for Future Research: The techniques used (lifting to Witt vectors, applying Tate's conjecture for finitely generated fields, and analyzing monodromy on vanishing cycles) offer a toolkit for attacking other problems in the arithmetic of algebraic surfaces in positive characteristic.

In summary, Schoemann and Werner successfully generalize a complex-analytic theorem to the arithmetic setting of positive characteristic, proving that the Gysin kernel is governed by an abelian subvariety that is either zero or maximal for very general curves, using a sophisticated interplay of lifting, étale cohomology, and monodromy.