Universal generation of the cylinder homomorphism of cubic hypersurfaces

Abstract

In this article, we prove that the Chow group of algebraic cycles of a smooth cubic hypersurface X over an arbitrary field k is generated, via the natural cylinder homomorphism, by the algebraic cycles of its Fano variety of lines F (X), under an assumption on the 1-cycles of X/k. As an application, if X/C is a smooth complex cubic fourfold, using the result, we provide a proof of the integral Hodge conjecture for curve classes on the polarized hyper-Kähler variety F (X). In addition, when X/k is a smooth cubic fourfold over a finitely generated field k with char(k) = 2, 3, our result enable us to prove the integral analog of the Tate conjecture for 1-cycles on F (X).