A new construction of compact 8-manifolds with holonomy Spin(7).
type: Item
instance of: scholarly article
title: A new construction of compact 8-manifolds with holonomy Spin(7).
zbMATH Open document ID: 1040.53062
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9910002
full work available at URL: 1 3.pdf
review text: A Spin(7)-structure on an \(8\)-manifold \(M\) is a principal subbundle \(Q\) of the frame bundle \({\mathcal F}\) with structure group Spin(7). Each Spin(7)-structure \(Q\) induces a \(4\)-form \(\Omega \), a metric \(g\) and an orientation on \(M\). A Spin(7)-manifold is a triple (\(M,\Omega ,g\)) where \(M\) is an \(8\)-manifold and (\(\Omega \), \(g\)) is a torsion-free Spin(7)-structure. The author has already given [Invent. Math. 123, 507--552 (1996; Zbl 0858.53037)] a method for constructing compact \(8\)-manifolds starting with a \(8\)-torus \(T^{8}\) factorized by a finite group of isometries of \(T^{8}\) preserving the standard flat Spin(7)-structure on \(T^{8}\). In this paper the author describes a new method for constructing compact \(8\)-manifolds starting with a Calabi-Yau \(4\)-orbifold \(Y\) with isolated singular points \(p_{1},p_{2}, \dots ,p_{k}\). An antiholomorphic isometric involution \(\sigma \) \(:Y\rightarrow Y\) whose only fixed points are \(p_{1},p_{2}, \dots , p_{k}\) is assumed to exist. The main result consists in proving that given a Calabi-Yau \(4\)-orbifold \(Y\) and an antiholomorphic isometric involution \(\sigma : Y\rightarrow Y\) satisfying certain conditions, a compact \(8\)-manifold \(M\) with a Spin(7)-structure can be constructed. Using algebraic geometry the author gives plenty of suitable complex orbifolds \(Y\). Moreover, several ways of finding orbifolds \(Y\) and involutions \(\sigma :Y\rightarrow Y\) satisfying suitable conditions are show. Also, it is indicated how to calculate the Betti numbers of the resulting \(8\)-manifolds \(M\) with holonomy Spin(7). The last three sections of this paper are concerned with important examples of compact \(8\)-manifolds with Spin(7)-structures, whose Betti numbers are determined.
reviewed by: Ilie Burdujan
Mathematics Subject Classification ID: 53C29
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 32Q25
zbMATH DE Number: 1782618
zbMATH Keywords: Calabi-Yau orbifold
zbMATH Keywords: weighted projective spaces
zbMATH Keywords: asymptotically locally Euclidean manifold
zbMATH Keywords: holonomy group
zbMATH Keywords: orbifolds
Wikidata QID: Q115193610
author: Dominic Joyce
MaRDI profile type: Publication
OpenAlex ID: W1571437197
arXiv ID: math/9910002
DOI: 10.4310/JDG/1214425448
description: scientific article