| Summary: | A holomorphic Penrose transform is described for Hitchin's correspondence between complex Einstein-Weyl spaces and "minitwistor" spaces, leading to isomorphisms between the sheaf cohomologies of holomorphic line bundles on a minitwistor space and the solution spaces of some conformally invariant field equations on the corresponding Einstein-Weyl space. The Penrose transforms for complex Euclidean 3-space and complex hyperbolic 3-space, two examples which have preferred Riemannian metrics, are explicitly discussed before the treatment of the general case. The non-holomorphic Penrose transform of Bailey, Eastwood and Singer, which translates holomorphic data on a complex manifold to data on a <I>smooth</I> manifold, using the notion of involutive cohomology, is reviewed and applied to the non-holomorphic twistor correspondences of four homogenous spaces: Euclidean 3-space, hyperbolic 3-space, Euclidean 5-space (considered as the space of trace-free symmetric 3 x 3 matrices) and the space of non-degenerate real conics in complex projective plane. The complexified holomorphic twistor correspondences of the last two cases turn out to be examples of more general correspondence between complex surfaces with rational curves of self-intersection number 4 and their moduli spaces.
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