On the automorphism group of a toral variety

Mathematical Communications. 2023. Vol. 28. No. 2. P. 277–291.

Anton Shafarevich, Anton Trushin

Let K be an algebraically closed field of characteristic zero. An affine algebraic variety X over K is toral if it is isomorphic to a closed subvariety of a torus K*. We study the group Aut(X) of regular automorpshims of a toral variety X. We prove that if T is a maximal torus in Aut(X), then X is a direct product YxT, where Y is a toral variety with a trivial maximal torus in the automorphism group. We show that knowing Aut (Y), one can compute Aut(X). In the case when the rank of the group K[Y]^*/K^* is dim Y + 1, the group Aut(Y) can be described explicitly.

Keywords: алгебраическая геометрия, Affine Algebraic Geometry, automorphism group, автоморфизмы, rigid variety, жесткие многообразия