2010/02/22 by Bruno Fabre, Fabre, Bruno
Computer Science · Mathematics · #32C15 #32C30 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Coding theory and cryptography #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:32C15 #msc:32C30
paper · pdf · doi:10.48550/arxiv.1002.4211
11 pages
arxiv created 2010/02/22 · openalex publication_date 2010/02/22 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
First we recall the definition of locally residual currents and their basic properties. We prove in this first section a trace theorem, that we use later. Then we define the Abel-Radon transform of a current \cal R(α), on a projective variety X⊂ ¶N, for a family of p-cycles of incidence variety I⊂ T× X, for which p1:I→ T is proper and p2:I→ X is submersive, and a domain U⊂ T. Then we show the following theorem, for a family of sections of X with r-planes (which was proved for the family of lines of X=¶N by the author for p=1, for \cal R(α)=0 and p-planes for any q>0, and by Henkin and Passare for p-planes in ¶N and integration currents α=ω\wedge[Y], with a meromorphic q-form ω, and projective convexity on U): Let α be a locally residual current of bidegree (q+p,p) on U^*, with U^*:=∪t∈ UHt⊂ X, where t× Ht:=p1-1(t). Then \cal R(α) is a meromorphic q-form on U, holomorphic iff α is ∂-closed. Let us assume that α is ∂-closed, and q>0. If \cal R(α) extends meromorphically (resp. holomorphically) to a greater domain U, then α extends in a unique way as a locally residual current (resp. ∂-closed) to the greater domain U^*⊂ X.