Generalised connected sums of quaternionic manifolds
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.
scientific article; zbMATH DE number 761181
scientific article