Cylinder maps of algebraic cycles on cubic hypersurfaces

Let be a smooth cubic hypersurface, and let be the variety of lines on . We prove the surjectivity of the cylinder maps on the Chow groups of and if contains a one‐cycle of degree . Mongardi and Ottem previously proved the integral Hodge conjecture for curve classes on hyperkähler manifolds. Using the cylinder maps, we provide an alternative proof for the of a smooth complex cubic fourfold , which is a special hyperkähler fourfold. In addition, we confirm the integral Tate conjecture for of a smooth cubic fourfold over a finitely generated field.