Vector bundles over quaternionic Kähler manifolds
type: Item
instance of: scholarly article
title: Vector bundles over quaternionic Kähler manifolds
zbMATH Open document ID: 0663.53055
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1988-01-01T00:00:00Z
review text: The main purpose of the paper is to give a quaternionic Kähler analogue of self-dual and anti-self-dual connections, and to construct a natural correspondence between vector bundles over the corresponding twister space. [See \textit{S. M. Salamon}, Invent. Math. 67, 143-171 (1982; Zbl 0486.53048), \textit{S. Kobayashi}, Differential geometry of complex vector bundles (Japan 1987)] for a detailed introduction to the notions and concepts used in this work.] Let (M,g) be a quaternionic Kähler manifold, then \(\Lambda^ 2T^*M\) can be written as a direct sum \(A_ 2'\oplus A_ 2''\oplus B_ 2\) of holonomy invariant subbundles \((Sp(n)\times Sp(1)/{\mathbb{Z}}_ 2-module\) \(\Lambda^ 2H^ n\) is a direct sum of three irreducible modules). A connection on a vector bundle E over M is called \(A_ 2\)-connection (resp. \(B_ 2\)-connection) if its curvature is an (End E)-valued \(A_ 2\)-form (resp. \(B_ 2\)-form). The following result is proved: All \(A_ 2\)-connections and \(B_ 2\)-connections are Yang-Mills connections (i.e. \(d^{\nabla}*R^{\nabla}=0)\). The Riemannian connection on M is also a Yang-Mills connections. Further, an elliptic complex is associated with a \(B_ 2\)-connection; it allows to analyze the space of infinitesimal deformations of \(B_ 2\)-connections. A vector bundle over M with Hermitian \(B_ 2\)-connection is called a Hermitian pair. Such pairs are characterized in terms of holomorphic vector bundles over a corresponding twistor space with Hermitian connection.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32L25
zbMATH DE Number: 4084519
zbMATH Keywords: anti-self-dual connections
zbMATH Keywords: twister space
zbMATH Keywords: quaternionic Kähler manifold
zbMATH Keywords: \(A_ 2\)-connections
zbMATH Keywords: \(B_ 2\)-connections
zbMATH Keywords: Yang-Mills connections
zbMATH Keywords: elliptic complex
zbMATH Keywords: Hermitian pair
zbMATH Keywords: holomorphic vector bundles
author: Takashi Gyoshin Nitta
reviewed by: M. A. Malakhaltsev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227984
OpenAlex ID: W2000570259
cites work: Self-duality in four-dimensional Riemannian geometry
cites work: Q3995501
cites work: Q5731354
cites work: Contact structures on twistor spaces
cites work: Quaternionic Kaehler manifolds
cites work: Q3317724
DOI: 10.2748/TMJ/1178227984
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description: scientific article; zbMATH DE number 4084519