The \(X\)-ray transform for currents.
The X-ray transform, mentioned in the title, was introduced, but not under this name, as early as 1938 by \textit{F. John} [Duke Math. J. 4, 300--322 (1938; Zbl 0019.02404)]. A compactified version associates to each smooth function \(f\) on the three-sphere \(S^3\) a smooth function \(\phi\) on the real Grassmannian \(\text{ Gr}(2,\mathbb R^4)\). The function \(\phi\) is defined by \(\phi(P)={1 \over 2\pi}\oint_{\gamma}f\), where \(P\in \text{Gr}(2,\mathbb R^4)\) is a plane and \(\gamma:= P\cap S^3\). This construction is interesting, because it provides a bijection between smooth even functions \(f\) on \(S^3\) and smooth even solutions \(\phi\) of the ultrahyperbolic wave equation on \(\text{Gr}(2,\mathbb R^4)\). Using ideas from twistor theory, a natural framework for such a correspondence is the following: consider the canonical maps, associated to the flag manifold of lines in planes in four-space \(\mathbb C\mathbb P^3 \leftarrow \mathbb F(1,2) \rightarrow \text{ Gr}(2,\mathbb C^4)\) and restrict it to \(\text{Gr}(2,\mathbb R^4) \subset \text{Gr}(2,\mathbb C^4)\) in order to obtain a pair of morphisms \(\mathbb C\mathbb P^3 \leftarrow F \rightarrow \text{Gr}(2,\mathbb R^4)\). It turns out that \(F\rightarrow \mathbb C\mathbb P^3\) is the real blow-up of the totally real submanifold \(\mathbb R\mathbb P^3\subset \mathbb C\mathbb P^3\). Trying to extend the complex structure from \(\mathbb C\mathbb P^3\setminus\mathbb R\mathbb P^3\) to \(F\) one ends up with an involutive structure on \(F\), which is not a complex structure. Using this set-up, involutive cohomology on \(F\), which is a generalisation of Dolbeault cohomology, provides a close relationship between Dolbeault cohomology on \(\mathbb C\mathbb P^3\) and global sections of higher direct images on the Grassmannian. This relationship is established by a spectral sequence, which arises from a filtration of the complex which defines the involutive cohomology. The filtration encodes the relationship between two involutive structures on \(F\) which are defined with the aid of the two projections. Because the fibres of \(F\rightarrow \text{Gr}(2,\mathbb R^4)\) are complex submanifolds, computations can be made effectively. In this way, spaces of sections of bundles on \(\mathbb R\mathbb P^3\) can be interpreted as kernels or cokernels of certain differential operators on the Grassmannian. In the paper under review, Dolbeault cohomology is replaced by distribution involutive cohomology. To define this type of cohomology, one uses currents instead of smooth forms. Within this framework it is sketched how the existence of a spectral sequence as above can be established. The proof follows the same lines as \textit{M. Eastwood} in [Suppl. Rend. Circ. Mat. Palermo, II. Ser. 46, 55--71 (1997; Zbl 0902.53047)]. The reader might find it illuminating to consult \textit{A. D'Agnolo} and \textit{P. Schapira} [J. Funct. Anal. 139, No. 2, 349--382 (1996; Zbl 0879.32012)].
scientific article; zbMATH DE number 1970793