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Sur la conjecture de Tate pour les diviseurs

Bruno Kahn

Abstract

We prove that the Tate conjecture in codimension $1$ over a finitely generated field follows from the same conjecture for surfaces over its prime subfield. In positive characteristic, this is due to de Jong--Morrow over $\mathbf{F}_p$ and to Ambrosi for the reduction to $\mathbf{F}_p$. We give a different proof than Ambrosi's, which also works in characteristic $0$; over $\mathbf{Q}$, the reduction to surfaces follows from a simple argument using Lefschetz's $(1,1)$ theorem.

Sur la conjecture de Tate pour les diviseurs

Abstract

We prove that the Tate conjecture in codimension over a finitely generated field follows from the same conjecture for surfaces over its prime subfield. In positive characteristic, this is due to de Jong--Morrow over and to Ambrosi for the reduction to . We give a different proof than Ambrosi's, which also works in characteristic ; over , the reduction to surfaces follows from a simple argument using Lefschetz's theorem.
Paper Structure (4 sections, 3 theorems, 12 equations)

This paper contains 4 sections, 3 theorems, 12 equations.

Table of Contents

  1. Notations
  2. Une rétraction
  3. Passage aux variétés lisses ouvertes
  4. Changement de corps de base

Key Result

Proposition 1

Pour $X$ projective lisse, $T(X)$ équivaut à $H^2_{\operatorname{tr}}(X,1)^G=0$.

Theorems & Definitions (11)

  • proof
  • proof
  • Proposition 1
  • proof
  • proof
  • Proposition 2
  • proof
  • proof
  • Proposition 3
  • proof
  • ...and 1 more