Singular complex-symmetric torus embeddings
type: Item
instance of: scholarly article
title: Singular complex-symmetric torus embeddings
zbMATH Open document ID: 0697.32015
DOI: 10.1007/BF01190083
author: Ralf Lehmann
published in: Archiv der Mathematik
publication date: 1991-01-01T00:00:00Z
review text: Let T be an algebraic torus. A compact torus embedding is by definition a (normal) compactification X of T such that the natural action of T on itself extends to an algebraic action on X. We call X complex-symmetric if each point \(x\in X\) is isolated fixed point of a T-equivariant holomorphic involution which is called a holomorphic symmetry. Here we prove that each singular complex-symmetric torus embedding can be obtained as the quotient of a smooth complex-symmetric torus embedding by a finite subgroup of the torus. Since smooth complex-symmetric torus embeddings are classified in a former paper [Ann. Inst. Fourier 39, No.2, 373-416 (1989; Zbl 0649.32021)] the structure of these varieties is now very well understood. However, by giving a counterexample one shows that such a quotient is not necessarily complex-symmetric. This result also completes the classification of normal complex-symmetric spaces with a reductive automorphism group which was begun in the paper quoted above.
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 32J05
zbMATH DE Number: 4142447
zbMATH Keywords: classification of almost-homogeneous spaces
zbMATH Keywords: classification of almost- homogeneous varieties
zbMATH Keywords: Coxeter groups
zbMATH Keywords: fans
zbMATH Keywords: holomorphic involutions
zbMATH Keywords: toric varieties
zbMATH Keywords: torus embedding
MaRDI profile type: Publication
cites work: Symmetric compact complex spaces
cites work: Q3701635
cites work: Q3846750
cites work: Complex-symmetric spaces
cites work: Q4039830
full work available at URL: https://doi.org/10.1007/bf01190083
OpenAlex ID: W2071502574
description: scientific article; zbMATH DE number 4142447