A note on automorphism groups of algebraic varieties

Abstract

MATSVSAXA has proved [3] that the maximal connected group of automorphisms of a projective variety can be endowed with the structure of an algebraic group. Our aim in this note is to extend this result to arbitrary complete varieties. More generally, we shall show that a "connected" and "finite dimensional" group G of automorphisms (see below for precise definitions) of any algebraic variety X can be endowed with the structure of an algebraic group variety. The main line of argument is similar to the one used by CHEVALLv, Y [2] and SESItADRI [7] in the construction of the Picard variety, but somewhat simpler. We shall prove that the linear map of the Lie algebra of this algebraic group into the space of vector fields on X which associates to any tangent vector at the identity element of G the corresponding "infinitesimal motion" is an injection. It follows easily that G satisfies the universal property for connected algebraic families of automorphisms of X containing the identity, that is, that any algebraic family of automorphisms of an algebraic variety X parametrised by a variety T is induced by a morphism of T into G. As an application, we shall prove that the maximal connected group of automorphisms of a (locally isotrivial) principal fibre space over a complete variety has a structure of a group variety. We have been informed by the referee that this result has also been obtained by H. MATSUMVRA.All varieties will be assumed to be irreducible, and defined over an algebraically closed field K.We shall say that a family {~vt}~ of automorphisms of a variety X, where the p~ametrising set T is also a variety, is an algebraic/amily if the map T × X -~ X given by (t, x) ~ ~v~ (x) is a morphism. It is clear that if ~ : 8 -+ T is a morphism, the family {~(~)}~ s is again algebraic, and that if {~v,}~es is another algebraic family, {~v~ o ~v~}(,,~)cs× ~ is also an algebraic family. Let (J[, p) be the normatisation of X. For every t E T, ~ lifts to a unique automorphism ~vt of J[ such that p o @~ = ~ o p. We shall show that {@~}~T is an algebraic family of automorphisms of J[. Let (~, q) be the normalisation of T. Then (~ × J[, q x p) is the normalisation of T × X, and the morphism T×X£X lifts to a unique morphism @:~× J[-~ such that po = ¢ o (q × p). It follows that for any ~ E ~, the morphism of J~ onto itself * The author wishes to express his gratitude to Professors M. S. NAI~S~m~N and C. S. SESHADBI for many helpful suggestions and discussions.

Cox rings, semigroups and automorphisms of affine algebraic varieties

“…This group is said to be finite-dimensional if Aut(X) has a structure of an affine algebraic group such that the action Aut(X) × X → X is algebraic (an equivalent definition of a finitedimensional group of automorphisms is given in [17]). Proof.…”

Tori in the Cremona groups

“…Although the Cremona groups are infinite dimensional (this has a precise meaning, see [Ra64]), the analogies between them and algebraic groups strike the eye: they have the Zariski topology, algebraic subgroups, tori, roots, the Weyl groups, . .…”

“…"Algebraic families" endow Bir X with the Zariski topology [Ra64], [Bl10,2], [Se10, 1.6]: a subset of Bir X is closed if and only if its inverse image for every algebraic family S → Bir X is closed. For every algebraic subgroup G in Bir X and its subset Z, the closures of Z in this topology and in the Zariski topology of the group G coincide.…”

Торы В Группах Кремоны

“…доказал [4], [5], что всякое алгебраическое действие на A n алгебраического тора размерности не менее n − 1 эквивалентно линейному. Мы распространяем это утверждение на несвязные группы, доказав, что всякое алгебраическое действие на A n либо n-мерной алгебраической группы, связная компонента единицы которой есть тор, либо (n − 1)-мерной диагонализируемой группы эквивалентно линейному (теоремы 11,13).…”