Holomorphic vertical line bundle of the twistor space over a quaternionic manifold. (English) Zbl 0995.53036

A result of P. Gauduchon [Ann. Sc. Norm. Super. Pisa, Cl. Sci. IV. Ser. 18, No. 4, 563-629 (1991; Zbl 0763.53034)] states that the vertical bundle of the twistor fibration over a self dual \(4\)-dimensional manifold is a holomorphic line bundle. This is related to the fact that the connection induced by the Levi-Civita one is a Chern connection. In the paper under review, the author discusses the same problem for the twistor space of a quaternion manifold \((M,H)\). The main result finds a necessary and sufficient condition for a torsion free connection on \(M\) to induce a Chern connection in the vertical bundle of the twistor fibration. The condition holds, in particular, for the Levi-Civita connection of quaternionic Kähler manifolds and for the Obata connection of a hypercomplex manifold.


53C28 Twistor methods in differential geometry
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C10 \(G\)-structures
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)


Zbl 0763.53034
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