The \(X\)-ray transform for currents.
type: Item
instance of: scholarly article
title: The \(X\)-ray transform for currents.
zbMATH Open document ID: 1040.32018
DOI: 10.1016/S0926-2245(02)00143-2
author: Liana David
published in: Differential Geometry and its Applications
publication date: 2003-08-25T00:00:00Z
review text: 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 betwee…
reviewed by: Bernd Kreußler
Mathematics Subject Classification ID: 32L25
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C28
zbMATH DE Number: 1970793
zbMATH Keywords: currents
zbMATH Keywords: involutive structure
zbMATH Keywords: twistor methods
zbMATH Keywords: Penrose transform
zbMATH Keywords: D-module
zbMATH Keywords: ultrahyperbolic differential equation
zbMATH Keywords: distribution involutive cohomology
zbMATH Keywords: \(X\)-ray
Wikidata QID: Q115338116
MaRDI profile type: Publication
cites work: Real forms of the Radon-Penrose transform
cites work: Radon-Penrose transform for \({\mathcal D}\)-modules
cites work: Q4000245
cites work: Q4352699
cites work: Moderate and formal cohomology associated with constructible sheaves
description: scientific article; zbMATH DE number 1970793