Abel transformation and algebraic differential forms

TL;DR

This paper establishes a criterion linking the algebraicity of a complex analytic set and form in projective space to the algebraicity of its Abel transform, extending classical inverse Abel theorems.

Contribution

It proves a new characterization of algebraic varieties and forms via the Abel transform in projective space, generalizing classical inverse Abel theorems.

Findings

V is contained in an algebraic variety if and only if the Abel transform is algebraic.

The meromorphic 1-form on V extends to an algebraic 1-form on projective space.

The result connects complex analytic sets with algebraic geometry through Abel transforms.

Abstract

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