Characterization of the currents associated with algebraic cycles by their Chow transform
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.
scientific article; zbMATH DE number 1463867