## 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].

### MSC:

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