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Forms of almost homogeneous varieties over perfect fields

Lucy Moser-Jauslin, Ronan Terpereau

Abstract

We study the k-forms of almost homogeneous varieties over perfect base fields k. First, we discuss criteria for the existence of k-forms in the homogeneous case. Then, we extend the Luna-Vust theory from algebraically closed fields to perfect fields to determine when a given k-form of the open orbit of an almost homogeneous variety extends to a k-form of the entire variety. Finally, in the last section, we apply these results to determine the real forms of complex almost homogeneous SL(2)-threefolds.

Forms of almost homogeneous varieties over perfect fields

Abstract

We study the k-forms of almost homogeneous varieties over perfect base fields k. First, we discuss criteria for the existence of k-forms in the homogeneous case. Then, we extend the Luna-Vust theory from algebraically closed fields to perfect fields to determine when a given k-form of the open orbit of an almost homogeneous variety extends to a k-form of the entire variety. Finally, in the last section, we apply these results to determine the real forms of complex almost homogeneous SL(2)-threefolds.

Paper Structure

This paper contains 32 sections, 27 theorems, 87 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Overview of the article
  4. Foreword
  5. Preliminaries
  6. Notation
  7. Forms, descent data, and inner twists for algebraic groups
  8. Forms and descent data for algebraic varieties with group action
  9. Forms of homogeneous spaces
  10. Finiteness of the number of forms for homogeneous spaces
  11. A cohomological invariant
  12. Existence of forms for homogeneous spaces
  13. Luna-Vust theory over perfect fields
  14. Recollections on Luna-Vust theory over algebraically closed fields
  15. Galois actions on the colored equipment
  16. ...and 17 more sections

Key Result

Proposition \oldthetheorem

(§ sec:Existence of forms) Let $G$ be a connected linear algebraic group over $\overline{k}$, and let $F$ be a $k$-form of $G$ corresponding to the descent datum $\rho$ on $G$. Let $H \subseteq G$ be an algebraic subgroup. The homogeneous space $X=G/H$ admits a $(k,F)$-form if and only if there exis If item1-prop:A-item2-prop:A are satisfied, then a $(G,\rho)$-equivariant descent datum on $X$ is g

Figures (6)

  • Figure 1: Diagram for Example \ref{['ex:baby example']}.
  • Figure 2: Diagram for Example \ref{['ex:structure extends but the quotient is not a scheme']}.
  • Figure 3: Diagram for Example \ref{['ex:first bis example']}.
  • Figure 4: Diagrams for the minimal smooth completions of $G/H$ when $H=E_6$, $E_7$, and $E_8$.
  • Figure 5: Diagrams for the minimal smooth completions of $G/H$ when $H=D_n$ ($n \geq 4$).
  • ...and 1 more figures

Theorems & Definitions (79)

  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • ...and 69 more