On the canonical real structure on wonderful varieties
type: Item
instance of: scholarly article
title: On the canonical real structure on wonderful varieties
zbMATH Open document ID: 1314.14101
author: Dmitri Akhiezer
author: Stéphanie Cupit-Foutou
publication date: 2014-09-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.6607
Mathematics Subject Classification ID: 14M27
Mathematics Subject Classification ID: 32M10
zbMATH DE Number: 6339936
zbMATH Keywords: wonderful varieties
zbMATH Keywords: real structure
MaRDI profile type: Publication
OpenAlex ID: W2963839753
arXiv ID: 1202.6607
DOI: 10.1515/CRELLE-2012-0101
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published in: Journal für die Reine und Angewandte Mathematik
review text: Let \(G\) be a connected reductive linear algebraic group over the complex numbers. Let \(X\) be a complex algebraic variety endowed with an action of \(G\). A real structure on \(X\) is said to be canonical if it is equivariant with respect to the involution defining the split real form of \(G\). The authors study canonical real structures on certain spherical varieties. Recall that in general a spherical variety is a normal algebraic variety endowed with an action of \(G\) admitting an open dense orbit for a Borel subgroup of \(G\).NEWLINENEWLINEMore precisely, they consider two classes of smooth spherical varieties: homogeneous spherical varieties with self-normalizing isotropy groups, and their wonderful embeddings, that is, smooth compactifications satisfying further regularity properties as defined in [\textit{C. De Concini} and \textit{C. Procesi}, Lect. Notes Math. 996, 1--44 (1983; Zbl 0581.14041)]. They show that in both cases a canonical real structure exists and it is unique.NEWLINENEWLINEFurthermore, they give an estimate of the number of orbits for the real form of \(G\) on the set of real points of a wonderful embedding \(X\), provided all isotropy groups of \(X\) are self-normalizing.
reviewed by: Paolo Bravi
description: scientific article; zbMATH DE number 6339936