Generalised connected sums of quaternionic manifolds
type: Item
instance of: scholarly article
title: Generalised connected sums of quaternionic manifolds
zbMATH Open document ID: 0828.53029
DOI: 10.1007/BF00774570
published in: Annals of Global Analysis and Geometry
publication date: 1995-12-18T00:00:00Z
review text: In the paper [Nonlinearity 2, No. 2, 197-239 (1989; Zbl 0671.53029)] \textit{S. Donaldson} and \textit{R. Friedman} have shown the existence of self-dual metrics on the connected sum of an arbitrary number of complex projective planes \(\mathbb{C} P^2 \# \cdots \# \mathbb{C} P^2\). To prove this, they first blew up the corresponding twistor spaces along twistor lines and glued the resulting spaces along the exceptional divisors to obtain a singular model of the twistor space over the connected sum. Then, they showed the existence of a smoothing of this singular space which is a twistor space. In the paper under review the authors want to perform a similar construction in higher dimensions to obtain the existence of new quaternionic manifolds. For that purpose they use Salamon's twistor spaces [\textit{S. M. Salamon}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 31-55 (1986; Zbl 0616.53023)] and introduce the notion of generalized connected sum of quaternionic manifolds of dimension \(8m + 4\) with an embedded copy of \(\mathbb{H} P^m\). In case \(m = 0\) their construction reduces to the usual connected sum. They show that the generalized connected sum of two copies of \(\mathbb{H} P^{2m + 1}\) is again \(\mathbb{H} P^{2m + 1}\). Then they prove along the lines of the Donaldson-Friedman construction the existence of a quaternionic structure on the generalized connected sum of two \((8m + 4)\)-dimensional quaternionic manifolds whose twistor spaces have the property that the second cohomology of the tangent bundle vanishes. They apply this result to obtain the existence of compact quaternionic manifolds having no connected group of quaternionic symmetries. Finally, they consider cocompact actions of \(\mathbb{Z}\) on open sets of \(\mathbb{C} P^{2n + 1}\) to show the existence of compact Salamon twistor spaces of dimension \(2n + 1\) having arbitrary algebraic dimension.
reviewed by: Bernd Kreußler
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C56
Mathematics Subject Classification ID: 32L25
Mathematics Subject Classification ID: 32G99
Mathematics Subject Classification ID: 14J50
zbMATH DE Number: 761181
zbMATH Keywords: deformation theory
zbMATH Keywords: twistor space
zbMATH Keywords: smoothing
zbMATH Keywords: quaternionic manifolds
zbMATH Keywords: generalized connected sum
zbMATH Keywords: algebraic dimension
author: Yat-Sun Poon
author: Andrew Swann
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00774570
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description: scientific article; zbMATH DE number 761181