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Equivariant automorphism group and real forms of complexity-one varieties

Giancarlo Lucchini Arteche, Ronan Terpereau

Abstract

Let G be a connected reductive algebraic group over a perfect field. We study the representability of the equivariant automorphism group of G-varieties. For a broad class of complexity-one G-varieties, we show that this group is representable by a group scheme locally of finite type when the base field has characteristic zero. We also establish representability by a linear algebraic group in the case of almost homogeneous G-varieties of arbitrary complexity. Finally, using an exact sequence description of the equivariant automorphism group, we deduce that complexity-one G-varieties with representable equivariant automorphism group admit only finitely many real forms.

Equivariant automorphism group and real forms of complexity-one varieties

Abstract

Let G be a connected reductive algebraic group over a perfect field. We study the representability of the equivariant automorphism group of G-varieties. For a broad class of complexity-one G-varieties, we show that this group is representable by a group scheme locally of finite type when the base field has characteristic zero. We also establish representability by a linear algebraic group in the case of almost homogeneous G-varieties of arbitrary complexity. Finally, using an exact sequence description of the equivariant automorphism group, we deduce that complexity-one G-varieties with representable equivariant automorphism group admit only finitely many real forms.

Paper Structure

This paper contains 11 sections, 27 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on representability
  3. The case of almost homogeneous varieties
  4. The case of T-torsors
  5. Representability of AutYG(X) in arbitrary complexity
  6. Representability of AutG(X) in complexity one
  7. From $T$-torsors to affine $T$-varieties
  8. From affine $T$-varieties to affine $G$-varieties
  9. From affine $G$-varieties to quasi-affine $G$-varieties
  10. From quasi-affine $G$-varieties to Mori dream spaces
  11. Finiteness of real forms

Key Result

Theorem 1.1

Let $\mathrm{k}$ be a perfect field. Let $G$ be a connected reductive $\mathrm{k}$-group, and let $X$ be an almost homogeneous $G$-variety. Then the group sheaf $\underline{\mathop{\mathrm{Aut}}\nolimits}^G(X)$ is representable by a smooth linear $\mathrm{k}$-group, denoted $\mathop{\mathrm{Aut}}\no

Theorems & Definitions (62)

  • Theorem 1.1: Theorem \ref{['th: aut of an almost homogeneous variety']}
  • Corollary 1.2: Corollary \ref{['cor: forms for almost homog varieties']}
  • Theorem 1.3: Propositions \ref{['prop: affine T-varietes, Aut_C^T']} and \ref{['prop: affine T-varieties, representability']}
  • Proposition 1.4: Proposition \ref{['prop: representability of AutYG(X)']}
  • Theorem 1.5
  • Theorem 1.6: Theorem \ref{['thm: main result on real forms']}
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 52 more