BOUNDEDNESS PROPERTIES OF AUTOMORPHISM GROUPS OF FORMS OF FLAG VARIETIES
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Springer Science+Business Media New York (2020)
BOUNDEDNESS PROPERTIES OF AUTOMORPHISM GROUPS OF FORMS OF FLAG VARIETIES A. GULD∗ R´enyi Alfr´ed Matematikai Kutat´ oint´ezet Re´ altanoda utca 13-15 Budapest, H1053 Hungary [email protected]
Abstract. We call a flag variety admissible if its automorphism group is the projective general linear group. (This holds in most cases.) Let K be a field of characteristic 0, containing all roots of unity. Let the K-variety X be a form of an admissible flag variety. We prove that X is either ruled, or the automorphism group of X is bounded, meaning that there exists a constant C ∈ N such that if G is a finite subgroup of AutK (X), then the cardinality of G is smaller than C.
1. Introduction Before stating our main theorem we need to introduce a couple of definitions and notations. Definition 1 (Definition 2.9 in [Po11]). A group G is called bounded if there exists a constant C ∈ N such that every finite subgroup of G has smaller cardinality than C. Let V be a finite-dimensional vector space (over an arbitrary field). A flag is a strictly increasing sequence of linear subspaces of V (with respect to the the containment order). By Fl(d1 < d2 < · · · < dr , V ) or simply by Fl(d, V ) we denote the flag variety of the sequence of linear subspaces of V (flags) of dimensions determined by the strictly increasing sequence of nonnegative integers d = (d1 , d2 , . . . , dr ), where dr 5 dim V . We also use the notation Fl(d < e, V ) governed by similar logic, using the strictly increasing sequence of nonnegative integers d < e = (d1 , . . . , dp , e1 , . . . , eq ) (eq 5 dim V ). If d1 = n then the notation d − n stands for the strictly increasing sequence of nonnegative integers d − n = (d1 − n, . . . , dr − n). DOI: 10.1007/S00031-020-09569-1 The research was partly supported by the National Research, Development and Innovation Office (NKFIH) Grant No. K120697. The project leading to this application has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 741420). Received August 7, 2018. Accepted July 25, 2019. Corresponding Author: A. Guld, e-mail: [email protected] ∗
A. GULD
If no confusion can arise we omit the specification of the vector space or the strictly increasing sequence of nonnegative integers or both of them. When we say Fl(d, V ) is a flag variety, we implicitly assume that V is a vector space over some field and d = (d1 , . . . , dr ) is a strictly increasing sequence of nonnegative integers, where dr 5 dim V . Definition 2. We call a flag variety admissible, if its automorphism group is the projective general linear group; otherwise we call it non-admissible. Later on we will see that a flag variety is admissible unless it is isomorphic to a flag variety Fl(d1 < · · · < dr , V ), where 0 < d1 , dr < dim V , dim V = 3 and di + dr+1−i = dim V for all i = 1, . . . , r. Notice that the conditions 0 < d1 and dr < dim V are technical assumptions; they d
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