The Abel-Radon transform and several complex variables. (English) Zbl 0848.32012

Bloom, Thomas (ed.) et al., Modern methods in complex analysis. The Princeton conference in honor of Robert C. Gunning and Joseph J. Kohn, Princeton University, Princeton, NJ, USA, Mar. 16-20, 1992. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 137, 223-275 (1995).
The main result of this article is the following generalization of the classical Abel’s theorem.
Let \(V\) be a closed one-dimensional complex submanifold in the linearly concave domain \({\mathcal D} \subset \mathbb{C} \mathbb{P}^n\) with connected dual set \({\mathcal D}^* = \{\xi \in (\mathbb{C} \mathbb{P}^n)^*\mid \mathbb{C} \mathbb{P}^{n - 1}_\xi \subset {\mathcal D}\}\). Let \(\psi\) be a meromorphic 1-form on \(V\). The following statements are equivalent:
a) \(V = \widetilde V \cap {\mathcal D}\) where \(\widetilde V\) is an algebraic subset of \(\mathbb{C} \mathbb{P}^n\) and \(\psi = \widetilde \psi |_V\) where \(\widetilde \psi\) is a rational form,
b) the Abel transform \({\mathcal U} \psi\) is rational in \({\mathcal D}^*\).
For the entire collection see [Zbl 0852.00026].


32F10 \(q\)-convexity, \(q\)-concavity
32C30 Integration on analytic sets and spaces, currents
44A12 Radon transform