TY - JOUR TI - Forms of homogeneous spherical varieties DO - https://doi.org/doi:10.7282/T31V5J4B PY - 2018 AB - Let G be a connected reductive algebraic group, spherical G-varieties are generalizations of symmetric G-spaces bearing nice properties on their compactifications. Over an algebraically closed field of characteristic 0, spherical varieties are classified by the Luna-Vust theory (spherical embeddings) together with combinatorial objects called the Luna data (homogeneous spherical varieties). A homogeneous spherical G-variety X can be determined, up to isomorphisms, by its corresponding Luna datum Λ (G,X) . In the first part of this work, Galois cohomology is used to study the spherical varieties over a general field k of characteristic 0, called k-forms of spherical varieties. We start from a homogeneous spherical G-variety X defined over k, with quasi-split G, then it is proven that there is a one-to-one correspondence between the set of k-forms (G ′ ,X ′ ) with a group G ′ which is quasi-split over k, up to k-isomorphisms, and the (continuous) cocycle classes in the first Galois cohomology of the automorphism group of the Luna datum. As an application, in the second part, the Luna data satisfying the transitivity of the automorphism group action on the set of spherical roots are classified. With the transitivity condition, the k-forms corresponding to the sets of the first Galois cohomology of the automorphism group of these Luna data contains all the spherical varieties over k which is of k-rank 1, according to the main theorem in the first part. KW - Mathematical Sciences LA - eng ER -