The Penrose transform in quaternionic geometry (Q2364526)

This technical paper is about an extension of the Penrose transform for the kernel of the Laplacian on classical spaces to a generalized Penrose transform for so-called quaternionic pluriharmonic sections on so-called quaternionic objects. Recall that the classical \textit{Penrose transform} provides us with an isomorphism between the kernel of the Laplacian on a self-dual \(4\)-manifold \((M,g)\) and the \(1^{\mathrm{st}}\) sheaf cohomology group of a certain holomorphic line bundle on the twistor space \(Z\) (a complex \(3\)-manifold) of \((M,g)\). \(Z\) as a real \(6\)-manifold is a smooth \(S^2\)-bundle over \(M\) and the isomorphism arises by integrating the \((0,1)\)-form representatives of these cohomology classes along the \(S^2\cong{\mathbb C}P^1\)-fibers (called the twistor lines). Keeping this basic pattern in mind one may try to generalize the Penrose transform by dropping the quite restrictive self-duality (and four dimensionality) assumption. Motivated by this, the author introduces the technical concept of a \textit{quaternionic object}, a certain structure on a manifold \(M\), and its \textit{twistor space} \(Z\), a complex manifold \(S^2\)-fibered over \(M\) (see Definition 2.2 in the article). The class of quaternionic objects is rich and includes \(3\)-dimensional Einstein-Weyl, \(4\)-dimensional self-dual spaces as well as quaternionic manifolds. In this setting the Penrose transform can be introduced as above and gives rise to a linear map from the \(1^{\mathrm{st}}\) sheaf cohomology group of an appropriate holomorphic line bundle on \(Z\) into the space of sections, containing the so-called \textit{quaternionic pluriharmonic sections} as well, of a certain bundle on \(M\) (see Definition 3.7 or more generally, Section 3 in the article). The main result is Theorem 4.3 which characterizes the image of this generalized Penrose transform under various technical assumptions on the original quaternionic object structure on \(M\).