Complex methods in real integral geometry. (English) Zbl 0902.53047

Slovák, Jan (ed.), Proceedings of the 16th Winter School on geometry and physics, Srní, Czech Republic, January 13–20, 1996. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 46, 55-71 (1997).
This is an exposition of a general machinery developed by M. G. Eastwood, T. N. Bailey, C. R. Graham which analyses some real integral transforms using complex methods. The machinery deals with double fibrations \(M\subset \Omega {\overset \eta \leftarrow} \widetilde \Omega @>\tau>> X\) \((\Omega\) complex manifold; \(M\) totally real, real-analytic submanifold; \(\widetilde \Omega\) real blow-up of \(\Omega\) along \(M\); \(X\) smooth manifold; \(\tau\) submersion with complex fibers of complex dimension one). The first result relates through an exact sequence the space of sections of a holomorphic vector bundle \(V\) on \(\Omega\), restricted to \(M\), to its Dolbeault cohomology on \(\Omega\), resp. its lift to \(\widetilde \Omega\). The second result proves a spectral sequence relating the involutive cohomology of the lift of \(V\) to its push-down to \(X\). The machinery is illustrated by its application to \(X\)-ray transform.
For the entire collection see [Zbl 0866.00050].


53C65 Integral geometry
44A12 Radon transform
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results