The vertical bundle of the twistor fibration over a 4-dimensional self-dual manifold is a holomorphic line bundle and plays an important role in a study of the twistor space. On the other hand, the vertical bundle of the twistor space over a quaternionic manifold is not a holomorphic line bundle, in general. We shall give the condition for a vertical bundle to be a holomorphic line bundle.
Toshimasa KOBAYASHI. "Holomorphic vertical line bundle of the twistor space over a quaternionic manifold." J. Math. Soc. Japan 52 (3) 485 - 499, July, 2000. https://doi.org/10.2969/jmsj/05230485