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On the real Section Conjecture in étale homotopy theory

Tim Holzschuh

TL;DR

The paper develops a homotopy-theoretic framework for the Section Conjecture over the real numbers by recasting the conjecture in terms of the profinite étale homotopy type and its sections. It introduces and leverages Σ-profinite homotopy theory, the Sullivan Conjecture in equivariant form, and a pronilpotent analysis to prove a pro-2 real Section Conjecture for equivariantly triangulable varieties, and then extends to the full real Section Conjecture for geometrically étale nilpotent varieties. The approach yields new proofs and generalisations beyond étale K(π,1) varieties, including simply connected and abelian cases, and clarifies the relationship to classical pro-ℓ formulations. The work highlights novel anabelian phenomena where higher étale homotopy data, rather than just π1, drive the Section Conjecture in real settings, with potential broad applicability to higher-dimensional varieties.

Abstract

We study the Section Conjecture in étale homotopy theory for varieties over $\mathbb{R}$. We prove its pro-$2$ variant for equivariantly triangulable varieties. Examples include all smooth varieties as well as all (possibly singular) affine/projective varieties. Building on this, we derive the real Section Conjecture in the geometrically étale nilpotent (e.g. simply connected) case.

On the real Section Conjecture in étale homotopy theory

TL;DR

The paper develops a homotopy-theoretic framework for the Section Conjecture over the real numbers by recasting the conjecture in terms of the profinite étale homotopy type and its sections. It introduces and leverages Σ-profinite homotopy theory, the Sullivan Conjecture in equivariant form, and a pronilpotent analysis to prove a pro-2 real Section Conjecture for equivariantly triangulable varieties, and then extends to the full real Section Conjecture for geometrically étale nilpotent varieties. The approach yields new proofs and generalisations beyond étale K(π,1) varieties, including simply connected and abelian cases, and clarifies the relationship to classical pro-ℓ formulations. The work highlights novel anabelian phenomena where higher étale homotopy data, rather than just π1, drive the Section Conjecture in real settings, with potential broad applicability to higher-dimensional varieties.

Abstract

We study the Section Conjecture in étale homotopy theory for varieties over . We prove its pro- variant for equivariantly triangulable varieties. Examples include all smooth varieties as well as all (possibly singular) affine/projective varieties. Building on this, we derive the real Section Conjecture in the geometrically étale nilpotent (e.g. simply connected) case.
Paper Structure (22 sections, 28 theorems, 53 equations)

This paper contains 22 sections, 28 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. The real Section Conjecture
  3. The Section Conjecture in étale homotopy theory
  4. Overview of results
  5. Related work
  6. Linear overview
  7. $\Sigma$-profinite homotopy theory
  8. $\Sigma$-profinite anima and their homotopy and cohomology groups
  9. Formal mapping anima
  10. Homotopy fixed points and the Sullivan Conjecture
  11. Pronilpotent $\Sigma$-profinite anima
  12. A characterisation of pronilpotent Σ-profinite anima
  13. The Section Conjecture in étale homotopy theory
  14. The étale homotopy type
  15. The Section Conjecture
  16. ...and 7 more sections

Key Result

theorem 1.2

Let $X/\! \mathop{\mathrm{\mathbb{R}}}\nolimits$ be a smooth, geometrically connected curve of arithmetic genus $g \geq 1$. All of the maps are bijective.

Theorems & Definitions (80)

  • theorem 1.2: (pro-$2$) real Section Conjecture
  • Conjecture : Section Conjecture
  • Conjecture : pro-$\ell$ Section Conjecture
  • definition 1.4: equivariantly triangulable, \ref{['def:equivariantly-triangulable-scheme']}
  • remark 1.5
  • remark 1.6
  • definition 2.2: $\Sigma$-finite groups and anima
  • definition 2.3: $\Sigma$-profinite groups and anima
  • remark 2.4
  • theorem 2.6: profinite Whitehead theorem, see SAG
  • ...and 70 more