## 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).
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 $${\mathcal D}^*$$.
 32F10 $$q$$-convexity, $$q$$-concavity 32C30 Integration on analytic sets and spaces, currents 44A12 Radon transform