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

Eastwood, Michael

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].

Reviewer: E.Teufel (Stuttgart)

MSC:

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

Keywords:

double fibration; complex manifold; \(X\)-ray transform; Dolbeault cohomology; spectral sequence