Complex methods in real integral geometry

summary: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 $M\subset \Omega {\overset \eta\to \leftarrow} \widetilde \Omega @>\tau>> X$ $(\Omega$ 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