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On zero-cycles of varieties over Laurent fields

Jan Lange

Abstract

We generalize a recent result of Pavic--Schreieder regarding the surjectivity of the obstruction morphism defined in [PS23]. As a consequence of this result, we show that geometrically (retract) rational varieties over a Laurent field of characteristic 0, which admit a strictly semi-stable model, have trivial Chow group of zero-cycles. Our key new ingredient comes from toric geometry.

On zero-cycles of varieties over Laurent fields

Abstract

We generalize a recent result of Pavic--Schreieder regarding the surjectivity of the obstruction morphism defined in [PS23]. As a consequence of this result, we show that geometrically (retract) rational varieties over a Laurent field of characteristic 0, which admit a strictly semi-stable model, have trivial Chow group of zero-cycles. Our key new ingredient comes from toric geometry.
Paper Structure (16 sections, 14 theorems, 81 equations)

This paper contains 16 sections, 14 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Decomposition of the diagonal
  5. Strictly semi-stable families
  6. Basic constructions in toric geometry
  7. The general setup
  8. Resolution of a toric singularity
  9. Some intersection theory
  10. A particular toric singularity
  11. Chow group of the resolution
  12. Analysis of the base-change
  13. Proof of the main results
  14. Exactness of the complex
  15. Geometrically rational varieties over Laurent fields
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $R$ be a discrete valuation ring with algebraically closed residue field $k$, and let $\mathfrak{X} \to \mathop{\mathrm{Spec}}\nolimits R$ be a strictly semi-stable projective $R$-scheme. Assume that the geometric generic fibre admits a decomposition of the diagonal (e.g. is retract rational). T

Theorems & Definitions (51)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Proposition 2.1: Har01
  • Proposition 3.1: CLS11
  • proof : Sketch of the proof
  • Remark 3.2
  • Lemma 3.3
  • Remark 3.4
  • proof
  • ...and 41 more