TAME DISCRETE SETS IN ALGEBRAIC GROUPS
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TAME DISCRETE SETS IN ALGEBRAIC GROUPS J. WINKELMANN∗ Lehrstuhl Analysis II IB 3-111 Fakult¨at f¨ ur Mathematik Ruhr-Universit¨at Bochum 44780 Bochum, Germany [email protected]
Abstract. Rosay and Rudin introduced the notion of tame discrete subsets of the affine complex space and investigated their properties. We generalize this theory to the case of a complex linear algebraic group with trivial character group.
1. Introduction For discrete subsets in Cn the notion of being “tame” was defined in the important paper of Rosay and Rudin [10]. A discrete subset D ⊂ Cn is called “tame” if and only if there exists an automorphism φ of Cn such that φ(D) = N × {0}n−1 . (In this paper a subset D of a topological space X is called a “discrete subset” if every point p in X admits an open neighbourhood W such that W ∩ D is finite.) In [14] we introduced a new definition of tameness which applies to arbitrary complex manifolds and agrees with the tameness notion of Rosay and Rudin for the case X ' Cn . Definition 1. Let X be a complex manifold. An infinite discrete subset D is called (weakly) tame if for every exhaustion function ρ : X → R+ and every map ζ : D → R+ there exists an automorphism φ of X such that ρ(φ(x)) ≥ ζ(x) for all x ∈ D. In terms of Nevanlinna theory a similar tameness notion may be formulated as follows: A discrete set is tame if its counting function N (r, D) may be made as small as desired. (See §3 for details.) First results about this notion obtained in [14] suggested that best results are to be expected in the case of complex manifolds whose automorphism group is very large in a certain sense. In particular, in [14] we proved some significant results for the case X = SLn (C) and found some evidence suggesting that our tameness notion might not work very well for non-Stein manifolds like Cn \ {(0, . . . , 0)} or partially hyperbolic complex manifolds like ∆ × C. DOI: 10.1007/S00031-020-09608-x ORCID: 0000-0002-1781-5842. Received March 25, 2019. Accepted January 2020. Corresponding Author: J. Winkelmann, e-mail: [email protected] ∗
J. WINKELMANN
In this paper we show that for complex manifolds biholomorphic to complex linear algebraic groups without non-trivial morphisms to the multiplicative group C∗ we obtain a theory of tame discrete sets essentially as strong as the theory which Rosay and Rudin developed for Cn . Andrist and Ugolini ([1]) have proposed a different notion, namely the following: Definition 2. Let X be a complex manifold. An infinite discrete subset D is called (strongly) tame if for every injective map f : D → D there exists an automorphism φ of X such that φ(x) = f (x) for all x ∈ D. They proved that every complex linear algebraic groups admits a strongly tame discrete subset ([1]). It is easily verified that “strongly tame” implies “weakly tame” (see [14]). For X ' Cn and X ' SLn (C) both tameness notions coincide ([14]). Furthermore, for X = Cn both notions agree with tameness as defined by Rosay and Rudin. However, for arbitrary manifolds “s
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