The Fröhlicher spectral sequence on a twistor space (Q1312142)

Associated to any compact self-dual four-manifold \(M\) is a compact complex three-dimensional manifold \(Z\) known as its twistor space. This construction provides a source of interesting complex three-manifolds. The Penrose-transform interprets the Hodge cohomology \(H^ q(Z,\Omega^ P)\) in terms of differential equations on \(M\). In this way, the complex geometry of \(M\) has differential-geometric consequences on \(M\) and vice versa. Several examples are studied. One of these shows that, for compact complex three-manifolds, the property of non-degeneration of the Fröhlicher spectral sequence is unstable under deformation of complex structure. Another consequence of the general analysis is the discovery of a ``period mapping'' for conformal structures on a compact four- manifold.