Equivariant automorphism group and real forms of complexity-one varieties

2025/07/24 by Giancarlo Lucchini Arteche, Ronan Terpereau, Arteche, Giancarlo Lucchini +1
Computer Science · Mathematics · #14D06 #14F06 #14G27 #14J50 #14M17 #14P99 #Advanced Algebra and Logic #Algebraic Geometry (math.AG) #FOS: Mathematics #Finite Group Theory Research #Primary 14L30 #Rings, Modules, and Algebras #Secondary 14L15

paper · pdf · doi:10.48550/arxiv.2507.18475

openalex publication_date 2025/07/24 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

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.

Citations

• Real forms of minimal SL₂-threefolds
• Smooth projective surfaces with infinitely many real forms
• Smooth complex projective rational surfaces with infinitely many real forms
• A smooth complex rational affine surface with uncountably many real forms
• Forms of almost homogeneous varieties over perfect fields
• On the classification of normal G-varieties with spherical orbits
• Spherical Spaces
• Moduli of affine schemes with reductive group action

Related

• Uniform K-stability of G-varieties of complexity 1
• Equivariant Chow groups of a complexity one T-variety
• Abelian varieties as automorphism groups of smooth projective varieties in arbitrary characteristics
• Restricting Representations from a Complex Group to a Real Form
• Representations of field automorphism groups