Cylinder maps of algebraic cycles on cubic hypersurfaces

Abstract

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.