Holomorphic vertical line bundle of the twistor space over a quaternionic manifold (Q1585899)

A result of \textit{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.