Regular actions of simple algebraic groups on projective threefolds

Abstract

The purpose of this note is to study regular actions of simple algebraic groups on projective threefolds as an application of the theory of algebraic threefolds, especially Mori Theory and the theory of Fano threefolds (cf. Mori [11], Iskovskih [7, 8]). The motivation for this study is as follows. In a series of papers, Umemura, in part jointly with Mukai, has classified maximal connected algebraic subgroups of the Cremona group of three variables and also constructed minimal rational threefolds which correspond to such subgroups (cf. Umemura [16-19], Mukai-Ume-mura [12]). In particular, Umemura and Mukai studied in [12] the SL(2, C)-equivariant smooth projectivization of SL(2, C)/G, where G is a binary icosahedral or octahedral subgroup of SL(2, C). The study of equivariant smooth projectivization of SL(2, C)/G for any finite subgroup G has been completed along their lines in Nakano [14]. The main trick of these studies is the investigation of equivariant contraction maps of extremal rays in the context of Mori Theory [11]. In this note, we apply a similar idea to projective threefolds with a regular action of a simple algebraic group and determine which simple algebraic groups can act regularly and nontrivially on projective threefolds and in which fashion. We also need some standard (but difficult) facts from the theory of Fano threefolds. For the precise statement, see Theorem 1 in the main text. For the proof of this theorem, we need a classification of closed subgroups of simple algebraic groups of codimension 1 and 2, which could be derived easily from the classical work of Dynkin [4]. However, we shall give a geometric proof independent of [4] which leads up directly to the proof of Theorem 1. On the whole, we shall establish by geometric methods the scarcity of closed subgroups of small codimension in simple algebraic groups, which is implied in Dynkin [4].