QUATERNIONIC CONNECTIONS, INDUCED HOLOMORPHIC STRUCTURES AND A VANISHING THEOREM

We classify the holomorphic structures of the tangent vertical bundle Θ of the twistor fibration of a quaternionic manifold (M, Q) of dimension 4n ≥ 8. In particular, we show that any self-dual quaternionic connection D of (M, Q) induces an holomorphic structure [Formula: see text] on Θ. We construct a Penrose transform which identifies solutions of the Penrose operator PD on (M, Q) defined by D with the space of [Formula: see text]-holomorphic purely imaginary sections of Θ. We prove that the tensor powers Θs (for any s ∈ ℕ\{0}) have no global non-trivial [Formula: see text]-holomorphic sections, when (M, Q) is compact and has a compatible quaternionic-Kähler metric of negative (respectively, zero) scalar curvature and the quaternionic connection D is closed (respectively, closed but not exact).

On the twistor space of the six-sphere

The set of all complex lines of the right-handed Dirac spinor bundle of a standard six-sphere is the total space of the twistor fibration. The twistor space, endowed with its natural Kähler structure, is recognised to be a six-dimensional complex quadric. The relevant group is Spin(7), which acts transitively on the six-quadric, as a group of fiber-preserving isometries. We use a result due to Berard-Bérgery and Matsuzawa to show the existence of a non-Kähler, non symmetric, Hermitian-Einstein metric on the six-quadric, which is Spin(7)-invariant.