Equivariant automorphism group and real forms of complexity-one varieties

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.

💡 Research Summary

The paper investigates the representability of the equivariant automorphism group of a (G)-variety, where (G) is a connected reductive algebraic group over a perfect field (k). The central object is the sheaf (\operatorname{Aut}_G(X)) on the smooth site of (k), which assigns to any smooth (k)-scheme (S) the group of (G)-equivariant automorphisms of the base‑change (X_S). The authors ask when this sheaf is represented by a smooth group scheme locally of finite type, and what consequences this has for the classification of real forms of (X).

The first technical contribution is a general representability criterion (Lemma 2.1) stating that a subgroup sheaf (H) of a smooth group scheme (G) is represented by a smooth subgroup scheme (G_0) provided (i) (H) commutes with filtered direct limits of (k)-algebras and (ii) for every algebraically closed field extension (K) the (K)-points of (H) coincide with those of a fixed smooth subgroup (G_0). Proposition 2.2 shows how to verify these conditions when (H) is defined by fixing a closed subscheme (F\subset X).

Using this machinery the authors first treat almost homogeneous varieties (Definition 3.1): a (G)-variety with a dense open orbit. Theorem 1.1 (Theorem 3.3) proves that for any almost homogeneous (G)-variety, (\operatorname{Aut}_G(X)) is represented by a smooth linear algebraic group, regardless of the complexity of the action. Consequently, Corollary 1.2 shows that such varieties admit only finitely many ((k,G))-forms.

The main focus then shifts to complexity‑one varieties that are not almost homogeneous. Rosenlicht’s theorem provides a canonical dense open (G)-stable subset (V\subset X) and a surjective (G)-invariant morphism (\theta:V\to C:=V/G) onto a smooth curve (C). This yields an exact sequence of sheaves \