# Article

 Title: Complex methods in real integral geometry (English) Author: Eastwood, Michael Language: English Journal: Proceedings of the 16th Winter School "Geometry and Physics" Volume: Issue: 1996 Year: Pages: [55]-71 . Category: math . 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. (English) MSC: 32L10 MSC: 32L25 MSC: 44A12 MSC: 53C56 MSC: 53C65 idZBL: Zbl 0902.53047 idMR: MR1469021 Note: (with the collaboration of T. N. Bailey and C. R. Graham) (English) . Date available: 2009-07-13T21:37:35Z Last updated: 2012-09-18 Stable URL: http://hdl.handle.net/10338.dmlcz/701595 .

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