Universal generation of the cylinder homomorphism of cubic hypersurfaces
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 $X/\mathbb{C}$ 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-K\"ahler variety $F(X)$. In addition, when $X/k$ is a smooth cubic fourfold over a finitely generated field $k$ with $\text{char}(k)\neq 2, 3$, our result enable us to prove the integral analog of the Tate conjecture for $1$-cycles on $F(X)$.