Characterization of the currents associated with algebraic cycles by their Chow transform
type: Item
instance of: scholarly article
title: Characterization of the currents associated with algebraic cycles by their Chow transform
zbMATH Open document ID: 0959.32015
DOI: 10.1016/S0021-7824(99)00142-7
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2000-06-22T00:00:00Z
review text: The main result of the paper is the characterization of the divisors of the Grassmannian \(\mathbb G_{q-1,n}\) which are Chow transforms of \((q,q)\)-currents on \(\mathbb P^n.\) This allows to characterize the currents, associated to algebraic cycles, as being those for which the direct image by a Veronese embedding \(\mathbb P_n\to\mathbb P_{\binom{n+ d}{d}-1}\) of degree \(d\geqslant 2\) has a divisor for the Chow transform. The essential ingredients of the proof are the characterization of Chow forms as solutions of a system of differential equations obtained by a direct method and the classical results of integral geometry which express the image of the Chow transformation as the space of solutions of a system of ultra-hyperbolic equations in homogeneous coordinates characterizing the Chow forms, due to that if is managed to simplify the calculations.
Mathematics Subject Classification ID: 32C30
Mathematics Subject Classification ID: 58A25
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 14C99
Mathematics Subject Classification ID: 58A10
zbMATH DE Number: 1463867
zbMATH Keywords: currents
zbMATH Keywords: differential forms
zbMATH Keywords: divisors of the Grassmannian
zbMATH Keywords: Chow transforms
zbMATH Keywords: Chow forms
zbMATH Keywords: Veronese embedding
zbMATH Keywords: ultra-hyperbolic equations
zbMATH Keywords: integral geometry
author: Michel Meo
MaRDI profile type: Publication
full work available at URL: s0021-7824(99)00142-7
OpenAlex ID: W2003526800
description: scientific article; zbMATH DE number 1463867