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Abelian fiber spaces and their Tate-Shafarevich twists with sections

János Kollár

TL;DR

The paper develops a framework to construct Tate-Shafarevich twists of Abelian fiber spaces that admit a rational section. It combines local-to-global patching via translation-automorphisms, Néron models, and the notion of rational multisections to produce translation-twisted forms $\pi_\eta: \mathbb{T}_\eta(X/Z)\to Z$ carrying a rational section $s_\eta$. A central result asserts the existence of these canonical twists under mild hypotheses, with the construction commuting with smooth base change and yielding a robust orbit structure on rational sections. The work also analyzes non-flat bases via Chow fibers, develops a notion of rational homotopy between sections, and shows that twists preserve many geometric and cohomological properties, linking to deformations relevant in hyperkähler and related settings. Collectively, this provides a systematic method to produce twists with sections in higher-dimensional Abelian fibrations and suggests connections to degenerate twistor-type deformations in special geometries.

Abstract

Starting with an Abelian fiber space, the aim is to construct a Tate-Shafarevich twist that has a rational section.

Abelian fiber spaces and their Tate-Shafarevich twists with sections

TL;DR

The paper develops a framework to construct Tate-Shafarevich twists of Abelian fiber spaces that admit a rational section. It combines local-to-global patching via translation-automorphisms, Néron models, and the notion of rational multisections to produce translation-twisted forms carrying a rational section . A central result asserts the existence of these canonical twists under mild hypotheses, with the construction commuting with smooth base change and yielding a robust orbit structure on rational sections. The work also analyzes non-flat bases via Chow fibers, develops a notion of rational homotopy between sections, and shows that twists preserve many geometric and cohomological properties, linking to deformations relevant in hyperkähler and related settings. Collectively, this provides a systematic method to produce twists with sections in higher-dimensional Abelian fibrations and suggests connections to degenerate twistor-type deformations in special geometries.

Abstract

Starting with an Abelian fiber space, the aim is to construct a Tate-Shafarevich twist that has a rational section.
Paper Structure (6 sections, 10 theorems, 34 equations)

This paper contains 6 sections, 10 theorems, 34 equations.

Table of Contents

  1. Examples
  2. Néron models
  3. Rational homotopies
  4. Non-flat case
  5. Main theorem
  6. Twisted forms

Key Result

Theorem 4

Let $\pi:X\to Z$ be an Abelian fiber space satisfying Assumption . Let $\eta\in H^{2d}(X, {\mathbb Z})$ be a cohomology class such that $\eta\cap [\hbox{general fiber}]=1$. Then there is a canonical construction yielding a twisted form The construction commutes with smooth base changes.

Theorems & Definitions (24)

  • Definition 1: Abelian torsors and fiber spaces
  • Definition 2: Twisted forms
  • Theorem 4: Untwisting of Abelian fiber spaces
  • Definition 6: Automorphisms
  • Definition 7: Translation-automorphisms and twists
  • Theorem 8
  • Corollary 11
  • Example 15
  • Example 16
  • Example 17
  • ...and 14 more