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On the failure of the integral Hodge/Tate conjecture for products with projective hypersurfaces

Kees Kok

TL;DR

The paper demonstrates the failure of the integral Hodge/Tate conjecture for products of an Enriques surface with a smooth odd-dimensional hypersurface by translating Shen’s obstruction into refined unramified cohomology. It develops and applies a purely algebraic specialization framework, built on Schreieder’s refined unramified cohomology, to produce nontrivial torsion classes that cannot be lifted to integral cohomology. The approach extends Shen’s results to general algebraically closed fields of characteristic not 2, without recourse to complex geometry. It also connects to Colliot-Thélène’s specialization theory and highlights the role of the étale fundamental group in obstructing algebraicity, offering broad implications for when integral Tate-type statements fail in families of varieties.

Abstract

In this paper we show the failure of the integral Hodge/Tate conjecture for the product of an Enriques surface with a smooth odd-dimensional projective hypersurface. To do this, we use a specialization argument of Colliot-Thélène applied to Schreieder's refined unramified cohomology. The results obtained in this way give an interpretation of Shen's result in terms of refined unramified cohomology. Moreover, using this interpretation, we avoid the need to work over the complex numbers so that we may conclude that Shen's result also holds over general algebraically closed fields of characteristic not 2.

On the failure of the integral Hodge/Tate conjecture for products with projective hypersurfaces

TL;DR

The paper demonstrates the failure of the integral Hodge/Tate conjecture for products of an Enriques surface with a smooth odd-dimensional hypersurface by translating Shen’s obstruction into refined unramified cohomology. It develops and applies a purely algebraic specialization framework, built on Schreieder’s refined unramified cohomology, to produce nontrivial torsion classes that cannot be lifted to integral cohomology. The approach extends Shen’s results to general algebraically closed fields of characteristic not 2, without recourse to complex geometry. It also connects to Colliot-Thélène’s specialization theory and highlights the role of the étale fundamental group in obstructing algebraicity, offering broad implications for when integral Tate-type statements fail in families of varieties.

Abstract

In this paper we show the failure of the integral Hodge/Tate conjecture for the product of an Enriques surface with a smooth odd-dimensional projective hypersurface. To do this, we use a specialization argument of Colliot-Thélène applied to Schreieder's refined unramified cohomology. The results obtained in this way give an interpretation of Shen's result in terms of refined unramified cohomology. Moreover, using this interpretation, we avoid the need to work over the complex numbers so that we may conclude that Shen's result also holds over general algebraically closed fields of characteristic not 2.
Paper Structure (25 sections, 50 theorems, 51 equations)

This paper contains 25 sections, 50 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Étale Cohomology
  4. Verdier Duality
  5. Cup Product
  6. Norm Map
  7. Residue Map
  8. Refined Unramified Cohomology
  9. Specialization Map
  10. The Construction
  11. Compatibility with Finite Extensions
  12. Comparing the Specialization and Cospecialization Map
  13. The Cospecialization map
  14. General Comparison
  15. Comparison for Lefschetz Pencils
  16. ...and 10 more sections

Key Result

Theorem 1.4

Let $\mathcal{X}\to\mathbb{P}^1$ be a Lefschetz pencil of odd-dimensional projective hypersurfaces of degree $d\geq 3$ over a field $k=\bar{k}$ with $\mathrm{char}(k)\neq 2$ and let $S$ be an Enriques surface over the same field $k$. Then the integral Tate conjecture fails for $X_{\bar{\eta}}\times

Theorems & Definitions (103)

  • Conjecture 1.1: Hodge Conjecture
  • Conjecture 1.2: Integral Hodge Conjecture
  • Conjecture 1.3: Integral Tate Conjecture as in Schoen
  • Theorem 1.4: cf. \ref{['failure ITC']}
  • Theorem 1.5: Sch1
  • Theorem 1.6: More precise form of \ref{['intro thm']}
  • Theorem 2.1
  • Proposition 2.2: Poincaré Duality
  • proof
  • Definition 2.3
  • ...and 93 more