Flexibility of affine horospherical varieties of semisimple groups

Flexibility of normal affine horospherical varieties

“…In this paper we prove that any normal complexity-zero horospherical variety X of any connected linear group G is flexible, see Theorem 3. This theorem generalizes both the result of [16] and the result of [5] stating flexibility of toric varieties. Note that we do not use results of [5] and [16] in our work.…”

“…Affine SL 2 -embeddings and smooth affine varieties with locally transitive action of a semisimple group are flexible, see [4]. Affine complexity-zero horospherical varieties of a semisimple group G are flexible, see [16]. Recall that a horospherical variety is an irreducible variety with an action of an algebraic group G such that the stabilizer of a generic point contains a maximal unipotent subgroup of G. A horospherical variety is complexity-zero if G-action on it is locally transitive.…”

Infinite transitivity, finite generation, and Demazure roots

“…Besides the toric affine varieties with no torus factor there are several other interesting classes of flexible affine varieties, see, e.g., [3,4,5,6,17,40,41,42,44,45,47].…”

On orbits of automorphism groups on horospherical varieties

“…Boldyrev and Gaifullin [4] give a criterium for a (not nessesary normal) toric variety to be flexible. Shafarevich [11] proved flexibility of horospherical complexity-zero varieties corresponding to a semisimple group G. Gaifullin and Shafarevich [8] proved flexibility of normal horospherical complexity-zero varieties corresponding to an arbitrary group. So, if we have a normal horospherical complexity-zero variety, all regular points form one Aut(X)orbit.…”