We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We characterize those combinatorial objects corresponding to varieties defined over an arbitrary field k. Then we provide some situations where toric varieties over k are classified by Galois-stable fans, and spherical embeddings over k by Galois-stable colored fans. Moreover, we construct an example of a smooth toric variety under a 3-dimensional nonsplit torus over k whose fan is Galois-stable but which admits no k-form. In the spherical setting, we offer an example of a spherical homogeneous space X0 over R of rank 2 under the action of SU (2, 1) and a smooth embedding of X0 whose fan is Galois-stable but which admits no R-form. Classification of toric varieties over an arbitrary fieldLet k be a field andk a fixed algebraic closure. We denote by T a torus defined over k. By a variety over k we mean a separated geometrically integral scheme of finite type over k. We define toric varieties under the action of T in the following way : Definition 1.1. A toric variety over k under the action of T is a normal T -variety X such that the group T (k) has an open orbit in X(k) in which it acts with trivial isotropy subgroup scheme. A morphism between toric varieties under the action of T is a T -equivariant morphism defined over k.It follows from the definition that a toric variety X under the action of T contains a principal homogeneous space under T as a T -stable open subset. We will denote it by X 0 . Definition 1.2. We will say that X is split if X 0 is isomorphic to T , that is to say, if X 0 has a k-point.Remark 1.3. If X is a split toric variety, then the automorphism group of X is the group T (k).In the rest of this section, we classify the toric varieties under the action of T (up to isomorphism). Assuming first that the torus T is split, we recall how the classification works in terms of combinatorial data named fans (Section 1.1). In Section 1.2, we derive the general case from the split case. We show (Theorem 1.22) that toric varieties under the action of T are, roughly speaking, classified by Galois-stable fans satisfying an additional assumption. In Section 1.3, we provide some situations where this additional assumption is always satisfied, and an example where it is not. The split caseIn this section, we assume that the torus T is split, and give the classification of toric varieties under the action of T . This classification was obtained by Demazure in the case of smooth toric varieties, and by many other people in the general case. See [Dan78] for more details and proofs.Proposition 1.4. Every toric variety under the action of T is split.Proof. By Hilbert's 90 theorem, every principal homogeneous space under T has a k-point.In order to state the main theorem of this section, we need more notations and definitions.
A linear algebraic group G defined over a field k is called special if every G-torsor over every field extension of k is trivial. In 1958 Grothendieck classified special groups in the case where the base field is algebraically closed. In this paper we describe the derived subgroup and the coradical of a special reductive group over an arbitrary field k. We also classify special semisimple groups, special reductive groups of inner type and special quasisplit reductive groups over an arbitrary field k.
We generalize in positive characteristics some results of Bien and Brion on log homogeneous compactifications of a homogeneous space under the action of a connected reductive group. We also construct an explicit smooth log homogeneous compactification of the general linear group by successive blow-ups starting from a grassmannian. By taking fixed points of certain involutions on this compactification, we obtain smooth log homogeneous compactifications of the special orthogonal and the symplectic groups.
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