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Abel transform and algebraic differential forms. (Transformation d’Abel et formes différentielles algébriques.) (French. Abridged English version) Zbl 0874.32003
The author’s abstract: “We prove that given a linearly concave domain \(D\) in the projective space \(CP^n\), a 1-dimensional complex analytic set \(V\) in \(D\), and a meromorphic 1-form \(\varphi\) on \(V\), \(V\) is a part of an algebraic variety of \(CP^n\) and \(\varphi\) the restriction of \(V\) of an algebraic 1-form on \(CP^n\) if and only if the Abel transform \(A(\varphi\wedge [V])\) of the analytic current \(\varphi\wedge[V]\) is an algebraic 1-form on the dual \((CP^n)^*\), where an algebraic 1-form on \(CP^n\) is a meromorphic 1-form defined on a ramified analytic covering of \(CP^n\). This result has its origin in the general inverse Abel theorems of Lie, Darboux, Saint-Donat, Griffiths and Henkin”.

32C25 Analytic subsets and submanifolds
32C30 Integration on analytic sets and spaces, currents
14H05 Algebraic functions and function fields in algebraic geometry
32D15 Continuation of analytic objects in several complex variables