Quaternionic toric varieties
type: Item
instance of: scholarly article
title: Quaternionic toric varieties
zbMATH Open document ID: 0844.14022
DOI: 10.1215/S0012-7094-95-07817-X
author: Richard A. Scott
published in: Duke Mathematical Journal
publication date: 1996-08-29T00:00:00Z
review text: Let \(H\) be the skew field of (Hamilton's) quaternions, and \(G\) the multiplicative group of quaternions of norm 1 so that \(G\) coincides with the 3-sphere \(S^3\). Among more general ``quaternionic spaces'', the author defines ``quaternionic toric varieties'', which are neither algebraic varieties nor spaces with actions of products of copies of \(G\), but which nevertheless share some of the features of complex (resp. real) toric projective varieties arising out of convex polytopes, where the complex unit circle \(S^1\) (resp. \(S^0 = \{\pm 1\})\) is supposed to play the role of \(G\). More specifically, let \(P\) be an \(n\)-dimensional convex polytope. An \(n\)-dimensional quaternionic toric variety \(X\) is a real \(4n\)-dimensional compact Hausdorff space obtained as a quotient of \(P \times G^n\) with respect to an equivalence relation specified by a ``characteristic function'' \(\lambda\). Roughly speaking, \(\lambda\) assigns for each face \(F\) of the polytope \(P\) a map \(\lambda (F) : G^n \leftarrow G^{\dim F}\) obtained as the composite, with the projection to the first \(\dim F\) factors, of a self-diffeomorphism \(\varphi : G^n \leftarrow G^n\) such that both \(\varphi\) and \(\varphi^{- 1}\) are expressed in terms of \(n\)-tuples of words in the \(n\) coordinates of \(G^n\). If \(F'\) is a face of \(F\), then \(\lambda (F')\) and \(\lambda (F)\) are supposed to satisfy suitable compatibility conditions. Then \((p, s_1, \dots, s_n)\) and \((p, t_1, \dots, t_n)\) in \(F \times G^n\) are defined to be equivalent if \(\lambda (F) (s_1, \dots, s_n) = \lambda (F) (t_1, \dots, t_n)\). The \(n\)-dimensional quaternionic projective space \(HP^n\) is a quaternionic toric variety when \(P\) is the \(n\)-dimensional simplex with a suitable \(\lambda\). \(SO(3)\) acts on \(G\) through inner automorphisms of quaternions, hence acts continuously on quaternionic toric varieties. The author determines when a quaternionic toric variety is nonsingular (i.e., a topo…
reviewed by: Tadao Oda
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 14A22
zbMATH DE Number: 815767
zbMATH Keywords: exotic sphere
zbMATH Keywords: SO(3)-action
zbMATH Keywords: quaternionic toric variety
zbMATH Keywords: cohomology ring
MaRDI profile type: Publication
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full work available at URL: s0012-7094-95-07817-x
OpenAlex ID: W2036923700
description: scientific article; zbMATH DE number 815767