The Transport Oka-Grauert principle for simple surfaces

Abstract

This article considers the attenuated transport equation on Riemannian surfaces in the light of a novel twistor correspondence under which matrix attenuations correspond to holomorphic vector bundles on a complex surface. The main result is a transport version of the classical Oka-Grauert principle and states that the twistor space of a simple surface supports no nontrivial holomorphic vector bundles. This solves an open problem on the existence of matrix holomorphic integrating factors on simple surfaces and is applied to give a range characterisation for the non-Abelian X-ray transform. The main theorem is proved using the inverse function theorem of Nash and Moser and the required tame estimates are obtained from recent results on the injectivity of attenuated X-ray transforms and microlocal analysis of the associated normal operators.Résumé (Le principe de transport d'Oka-Grauert pour les surfaces simples)Cet article étudie l'équation de transport atténuée sur les surfaces riemanniennes à la lumière d'une nouvelle correspondance de twisteurs dans laquelle les atténuations de matrice correspondent à des fibrés vectoriels holomorphes sur une surface complexe. Le résultat principal est une version de transport du principe classique d'Oka-Grauert et stipule que l'espace des twisteurs d'une surface simple ne supporte aucun fibré vectoriel holomorphe non trivial. Ceci résout un problème ouvert sur l'existence de facteurs intégrants holomorphes matriciels sur des surfaces simples et est appliqué pour donner une caractérisation du domaine pour la transformation en rayons X non abélienne. Le théorème principal est démontré en utilisant le théorème d'inversion locale de Nash et Moser, et les estimations nécessaires sont obtenues à partir de résultats récents sur l'injectivité des transformées en rayons X atténuées et l'analyse microlocale des opérateurs normaux associés.

Invariant distributions and the transport twistor space of closed surfaces

“…At this point, the main purpose of transport twistor spaces is of conceptual nature. For example, while the main theorem of [2] is equivalent to an Oka–Grauert–type result for the twistor space $$Z$$ and while it was inspired by this twistorial point of view, its proof relied in an essential way on the theory of transport equations and microlocal analysis — the same holds true for the results of this paper. This is in contrast with the main result in [20] where the full force of a geometric result is used via a desingularisation of the twistor space $$\mathcal {J}$$ (namely, that $$\mathbb {CP}^{2}$$ has a unique complex structure up to biholomorphism).…”

“…In this section, we review the twistor space construction from [2]. Here, $$(M,g)$$ may be any oriented Riemannian surface, possibly with non‐empty boundary $$\partial M$$.…”

“…On $$SM\times \mathbb {D}^\circ$$ we define complex 1‐forms $$\tau$$ and $$\gamma$$ by the relations $$$$\begin{equation} \def\eqcellsep{&}\begin{array}{l}\tau (\omega ^2\eta _++\eta _-)=\gamma (\partial _{\bar{\omega }})=1\\[3pt] \tau (\partial _{\bar{\omega }})=\gamma (\omega ^2\eta _++\eta _-)=0 \end{array} ,\qquad \tau,\gamma = 0 \text{ on } \operatorname{span}_\mathbb {C}(\bar{\omega }^2\eta _-+\eta _+,\partial _\omega,\mathbf {V}). \end{equation}$$$$While this forces $$\tau$$ itself to blow up as $$\vert \omega \vert \rightarrow 1$$, both $$(1-\vert \omega \vert ^4)\tau$$ and $$\gamma$$ extend smoothly to $$SM\times \mathbb {D}$$, see also [2, Lemma 4.2]. There it was also shown that …”

“…For any oriented Riemannian surface $$(M,g)$$, possibly with a non‐empty boundary, the transport twistor space is the 4‐manifold $$$$\begin{equation} Z=\lbrace (x,v)\in TM:g(v,v)\leqslant 1\rbrace, \end{equation}$$$$equipped with a natural complex structure that turns the interior $$Z^\circ$$ into a classical complex surface, but which degenerates at $$SM\subset \partial Z$$ in a way that encodes the transport equation () — we refer to Section 3 for precise definitions. In [2] several twistor correspondences were set up that relate the transport equation to complex geometric objects. For example, the algebra of holomorphic functions that are smooth up to the boundary $$\partial Z$$, $$$$\begin{equation} \mathcal {A}(Z)=\lbrace f\in C^\infty (Z): f \text{ holomorphic in } Z^\circ \rbrace, \end{equation}$$$$was shown to be isomorphic to the space of smooth fibrewise holomorphic first integrals of the geodesic flow; this is implemented by the following map: $$$$\begin{equation} \mathcal {A}(Z) \xrightarrow {\sim } \lbrace u\in C^\infty (SM): Xu = 0, \nobreakspace u \text{ is fibrewise holomorphi...$$…”

“…One of the main results of [2] was the classification of holomorphic vector bundles over the twistor space of simple Riemannian surfaces: There, a ‘transport Oka–Grauert principle’ was proved, showing that all holomorphic vector bundles are actually trivial. In our last result, we classify all holomorphic line bundles over $$Z$$ having certain distributional behaviour on the boundary $$\partial Z$$, and up to isomorphism in $$Z^\circ$$, see Theorem 6.7.…”