Cylinder maps of algebraic cycles on cubic hypersurfaces

Let $X\subset {\mathbb{P}}^{n+1}$ be a smooth cubic hypersurface, and let $F(X)$ be the variety of lines on $X$. We prove the surjectivity of the cylinder maps on the Chow groups of $F(X)$ and $X$ if $X$ contains a one-cycle of degree $1$. Mongardi and Ottem previously proved the integral Hodge conjecture for curve classes on hyperk"ahler manifolds. Using the cylinder maps, we provide an alternative proof for the $F(X)$ of a smooth complex cubic fourfold $X$, which is a special hyperk"ahler fourfold. In addition, we confirm the integral Tate conjecture for $F(X)$ of a smooth cubic fourfold $X$ over a finitely generated field.