Jordan groups, conic bundles and abelian varieties

Abstract

A group G is called Jordan if there is a positive integer J = J G such that every finite subgroup B of G contains a commutative subgroup A ⊂ B such that A is normal in B and the index [B : A] is at most J (V. L. Popov). In this paper, we deal with Jordan properties of the groups Bir(X) of birational automorphisms of irreducible smooth projective varieties X over an algebraically closed field of characteristic zero. It is known (Yu. Prokhorov and C. Shramov) that Bir(X) is Jordan if X is non-uniruled. On the other hand, the second-named author proved that Bir(X) is not Jordan if X is birational to a product of the projective line P 1 and a positive-dimensional abelian variety.We prove that Bir(X) is Jordan if (the uniruled variety) X is a conic bundle over a non-uniruled variety Y but is not birational to Y × P 1 . (Such a conic bundle exists if and only if dim(Y ) 2.) When Y is an abelian surface, this Jordan property result gives an answer to a question of Prokhorov and Shramov.

Jordan Properties of Automorphism Groups of Certain Open Algebraic Varieties

“…f: We may choose B in such a way that Z B := Z∩X B meets every fiber P b , b ∈ B, at precisely m(Z) K− points. In particular, Z B is a finite cover of B. g: Every birational map f ∈ Bir(X) is p−fiberwise : we denote by g f ∈ Bir(A) the corresponding automorphism g f : A A (see [BZ2]). Since A is rigid, g f actually belongs to Aut(A).…”

“…(a) G is called bounded [Po2,PS1] if there is a positive integer C = C G such that the order of every finite subgroup of G does not exceed C. (b) G is called quasi-bounded if there is a nonnegative integer a := a(G) such that each finite abelian subgroup of G is generated by at most A elements. (c) G is called strongly Jordan [PS2,BZ2] if it is Jordan and quasi-bounded.…”

“…The case of dim(W ) = 2, dim(A) = 1 was done in [Po1,Za2,BZ1]. The case of dim(W ) = 3 was studied in [Za1,PS0,PS1,BZ2,PS2]. Here is the final answer for Bir(X) [PS2].…”

Fiberwise bimeromorphic maps of conic bundles

“…The following theorem was proved in [BZ17] (see also [SV18,Corollary 4.12] for a little bit more general assertion).…”

Complex Tori, Theta Groups and Their Jordan Properties

“…6] that the algebraic dimension dim a (X) of X is positive if and only if X admits as a quotient-torus a positive-dimensional complex abelian variety. If x ∈ X then we write T x ∈ Aut(X) for the translation map (1) T x : X → X, z → z + x ∀z ∈ X.…”