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Real forms of minimal SL$_2$-threefolds

Lucas Moulin

Abstract

We complete the classification of the real forms of almost homogeneous SL$_2$-threefolds. More precisely, we use the Luna-Vust theory to determine the real forms of minimal smooth complete SL$_2$-varieties containing an orbit isomorphic to SL$_2/H$, where $H$ is a finite cyclic subgroup of SL$_2$. Moreover, we study the rationality and the set of real points of those varieties.

Real forms of minimal SL$_2$-threefolds

Abstract

We complete the classification of the real forms of almost homogeneous SL-threefolds. More precisely, we use the Luna-Vust theory to determine the real forms of minimal smooth complete SL-varieties containing an orbit isomorphic to SL, where is a finite cyclic subgroup of SL. Moreover, we study the rationality and the set of real points of those varieties.
Paper Structure (12 sections, 17 theorems, 28 equations, 4 figures)

This paper contains 12 sections, 17 theorems, 28 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Our results
  4. Structure of the article
  5. Acknowledgements
  6. Preliminaries
  7. Real structures and real forms
  8. Luna-Vust Theory for equivariant embeddings of $\boldsymbol{\mathop{\mathrm{SL}}\nolimits_2/H}$
  9. Equivariant automorphism groups
  10. Determination of real forms
  11. Proof of the Main Theorem
  12. Appendix: Diagrams of minimal smooth completions of $\mathop{\mathrm{SL}}\nolimits_2/A_k$

Key Result

Proposition 2.4

(MJT21) Let $(G,\sigma)$ be a complex algebraic group with a real group structure, and let $(X,\mu_0)$ be a homogeneous space with a $(G,\sigma)$-equivariant real structure. There is a bijection of pointed sets

Figures (4)

  • Figure 1: List of diagrams of the minimal smooth completions of $\mathop{\mathrm{SL}}\nolimits_2$.
  • Figure 2: List of diagrams of the minimal smooth completions of $\mathop{\mathrm{PGL}}\nolimits_2$.
  • Figure 3: List of diagrams of the minimal smooth completions of $\mathop{\mathrm{SL}}\nolimits_2/A_k$, where $k \geq 3$ and $k$ is odd.
  • Figure 4: List of diagrams of the minimal smooth completions of $\mathop{\mathrm{SL}}\nolimits_2/A_k$, where $k \geq 4$ and $k$ is even.

Theorems & Definitions (40)

  • Remark
  • Remark
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Theorem 3.1
  • Proposition 3.2
  • proof : Proof of the theorem
  • ...and 30 more