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The kernel of the Gysin homomorphism for positive characteristic

Claudia Schoemann, Skylar Werner

TL;DR

This work extends the Gysin-kernel description for hyperplane sections of a smooth projective surface from characteristic zero to positive characteristic by lifting to characteristic zero via Witt vectors and applying comparison theorems to transport cohomological information. The authors show that for a Lefschetz pencil in characteristic $p>0$, the Gysin kernel $G_t$ is a countable union of translates of an abelian subvariety $A_t$ of the vanishing Jacobian, and for very general $t$ this abelian piece is either $0$ or coincides with the vanishing-part $B_t$ of the Jacobian. Central to the argument are lift-and-specialize techniques, étale monodromy of vanishing cycles, and Tate’s/isogeny-type results which ensure that the behavior observed in characteristic zero persists in positive characteristic after spreading out. The paper develops the necessary foundations for regular maps into $CH_0$, their representability, and the countability lemmas that control the structure of $G_t$, tying together Chow groups, Albanese varieties, and étale cohomology in a unified framework. The results deepen our understanding of algebraic cycles in positive characteristic and provide a pathway to transfer complex-analytic techniques into the arithmetic setting via lifting and specialization.

Abstract

Let $k$ be an uncountable algebraically closed field of positive characteristic and let $S$ be a smooth projective connected surface over $k$. We extend the theorem on the Gysin kernel from [20, Theorem 5.1] to also be true over $k$, where it was proved over $\mathbb{C}$. This is done by showing that almost all results still hold true over $k$ via the same argument or by using étale base arguments and then using a lift with the Comparison theorems [16, Theorems 21.1 & 20.5] and Tate's Conjecture for finitely generated fields [27] and [31] as needed.

The kernel of the Gysin homomorphism for positive characteristic

TL;DR

This work extends the Gysin-kernel description for hyperplane sections of a smooth projective surface from characteristic zero to positive characteristic by lifting to characteristic zero via Witt vectors and applying comparison theorems to transport cohomological information. The authors show that for a Lefschetz pencil in characteristic , the Gysin kernel is a countable union of translates of an abelian subvariety of the vanishing Jacobian, and for very general this abelian piece is either or coincides with the vanishing-part of the Jacobian. Central to the argument are lift-and-specialize techniques, étale monodromy of vanishing cycles, and Tate’s/isogeny-type results which ensure that the behavior observed in characteristic zero persists in positive characteristic after spreading out. The paper develops the necessary foundations for regular maps into , their representability, and the countability lemmas that control the structure of , tying together Chow groups, Albanese varieties, and étale cohomology in a unified framework. The results deepen our understanding of algebraic cycles in positive characteristic and provide a pathway to transfer complex-analytic techniques into the arithmetic setting via lifting and specialization.

Abstract

Let be an uncountable algebraically closed field of positive characteristic and let be a smooth projective connected surface over . We extend the theorem on the Gysin kernel from [20, Theorem 5.1] to also be true over , where it was proved over . This is done by showing that almost all results still hold true over via the same argument or by using étale base arguments and then using a lift with the Comparison theorems [16, Theorems 21.1 & 20.5] and Tate's Conjecture for finitely generated fields [27] and [31] as needed.

Paper Structure

This paper contains 16 sections, 32 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. A Brief History
  3. Relation between algebraic, rational and homological equivalence
  4. Rational, Algebraic and Homological Equivalence of Lg-cycles
  5. The Lift and Specialization map
  6. Lift via local fields
  7. The specialization homomorphism
  8. Countability Lemmas
  9. Regular maps and Representability of Lg
  10. Regular maps
  11. Representable
  12. A theorem on the Gysin kernel
  13. Part (a)
  14. Part (b)
  15. Connection between generic geometric point and very general points
  16. ...and 1 more sections

Key Result

Theorem 1

where $A_t$ is the unique component containing $0$ and $B_t$ the abelian subvariety associated with $H^1(C_t, \mathbb{Z})_{van}$.

Theorems & Definitions (51)

  • Theorem : A theorem on the Gysin kernel
  • Theorem : Theorem 21.1, Artin's Comparison Theorem
  • Theorem : Theorem 20.5
  • Theorem
  • proof
  • Theorem
  • Proposition
  • Theorem
  • proof
  • Theorem
  • ...and 41 more