2018/10/29 by Renjie Lyu, Lyu, Renjie
Mathematics · #14C15 #14C25 #14C30 #14J42 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1810.12394
openalex publication_date 2018/10/29 · openalex created_date 2018/11/09 · openalex updated_date 2026/07/28
Let \(X⊂ ℙn+1\) 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 hyperkähler 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 hyperkähler fourfold. In addition, we confirm the integral Tate conjecture for \(F(X)\) of a smooth cubic fourfold \(X\) over a finitely generated field.