Positive twistor bundle of a Kähler surface (Q698043)

The twistor bundle \(Z(M)\) and other related bundles of an oriented Riemannian \(4\)-manifold \((M,g)\) contain a lot of information on the geometric properties of \((M,g)\). Whereas the author studied the negative twistor bundle \(Z^-(M)\) in earlier work, here he concentrates on the positive twistor bundle \(Z^+(M)\), a sphere bundle over \(\Lambda^+M\), the bundle of self-dual two-forms for the Hodge star operator. Both \(Z^+(M)\) and \(P^+(M)\) (essentially the orthonormal frame bundle of \(\Lambda^+M\)) can be equipped with a one-parameter family of natural metrics \(g_c\), where \(c\) is a positive real number. The main result states that an oriented \(4\)-manifold \((M,g)\) admits a complex structure \(J\) compatible with the orientation and such that \((M,g,J)\) is Kähler if and only if \((Z^+(M),g_c)\) admits a vertical Killing vector field, or equivalently, if and only if \((P^+(M),g_c)\) admits a vertical Killing vector field \(\xi\) which is invariant under the action of the structure group \(SO(3)\) of \(P^+(M)\). Further, the author determines under which conditions \(\xi\) determines a K-contact or a Sasakian structure on \(P^+(M)\) and when \(P^+(M)\) is a circle bundle over the product manifold \(M\times \mathbb{C} P^1\).