Abel transformation and algebraic differential forms

Abstract

We prove in this article that given a linearly concave domain $D$ in the projective space ${\mathbb{CP}}^n$, a 1-dimensional comlex analytic set $V$ in $D$, and a meromorphic 1-form $\varphi$ on $V$, $V$ is a subset of an algebraic variety of ${\mathbb{CP}}^n$ and $\varphi$ is the restriction to $V$ of an algebraic 1-form on ${\mathbb{CP}}^n$ if and only if the Abel transform $A(\varphi \wedge [V])$ of the analytic current $\varphi \wedge [V]$ is an algebraic 1-form on $({\mathbb{CP}}^n)\ast$, where an algebraic 1-form on ${\mathbb{CP}}^n$ is a meromorphic 1-form defined on a ramified analytic covering of ${\mathbb{CP}}^n$. This result has its origin in the general inverse Abel theorems of Lie, Darboux, Saint-Donat, Griffiths and Henkin.

4 pages, proofs to be added