Jordan groups, conic bundles and abelian varieties

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 P1 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 × 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.