Bimeromorphic automorphism groups of certain $$\pmb {{\mathbb {P}}^{1}}$$ P 1 -bundles
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Bimeromorphic automorphism groups of certain P1 -bundles Tatiana Bandman1 · Yuri G. Zarhin2
Received: 20 November 2019 / Revised: 18 May 2020 / Accepted: 2 July 2020 © Springer Nature Switzerland AG 2020
Abstract We call a group G very Jordan if it contains a normal abelian subgroup G 0 such that the orders of finite subgroups of the quotient G/G 0 are bounded by a constant depending on G only. Let Y be a complex torus of algebraic dimension 0. We prove that if X is a non-trivial holomorphic P1 -bundle over Y then the group Bim(X ) of its bimeromorphic automorphisms is very Jordan (contrary to the case when Y has positive algebraic dimension). This assertion remains true if Y is any connected compact complex Kähler manifold of algebraic dimension 0 without rational curves or analytic subsets of codimension 1. Keywords Automorphism groups of compact complex manifolds · Algebraic dimension 0 · Complex tori · Conic bundles · Jordan properties of groups Mathematics Subject Classification 14E05 · 14E07 · 14J50 · 32L05 · 32M05 · 32J27 · 32Q15
1 Introduction Let X be a compact complex connected manifold. We denote by Aut(X ) and Bim(X ) the groups of automorphisms and bimeromorphic selfmaps of X , respectively. If X is
The second named author (Yu.Z.) was partially supported by Simons Foundation Collaboration Grant # 585711.
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Yuri G. Zarhin [email protected] Tatiana Bandman [email protected]
1
Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel
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Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
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T. Bandman, Yu.G. Zarhin
projective, Bir(X ) denotes the group of birational automorphisms of X . As usual, Pn stands for the n-dimensional complex projective space; a(X ) stands for the algebraic dimension of X . All manifolds in this paper are assumed to be complex compact and connected unless otherwise stated. Vladimir L. Popov in [24] defined the Jordan property of a group and raised the following question: When the groups Aut(X ) and Bir(X ) are Jordan? Definition 1.1 • A group G is called bounded if the orders of its finite subgroups are bounded by a universal constant that depends only on G [24, Definition 2.9]. • A group G is called Jordan if there is a positive integer J such that every finite subgroup B of G contains an abelian subgroup A that is normal in B and such that the index [B : A] J [24, Definition 2.1]. In this paper we are interested in the following property of groups. Definition 1.2 We call a group G very Jordan if there exist a commutative normal subgroup G 0 of G and a bounded group F that sit in a short exact sequence 1 → G 0 → G → F → 1. Remark 1.3 1. Every finite group is bounded, Jordan, and very Jordan. 2. Every commutative group is Jordan and very Jordan. 3. Every finitely generated commutative group is bounded. 4. A subgroup of a very Jordan group is very Jordan. 5. “Bounded” implies “very Jordan”, “very Jordan” implies “Jordan”. The first goal of the paper is to find complex manifolds with very Jordan group Aut(
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