Cylinder maps of algebraic cycles on cubic hypersurfaces

TL;DR

This paper investigates the properties of cylinder maps on Chow groups of cubic hypersurfaces and their varieties of lines, providing new proofs for conjectures related to hyperk"ahler manifolds and confirming the Tate conjecture in specific cases.

Contribution

It proves the surjectivity of cylinder maps on Chow groups of cubic hypersurfaces and their line varieties, offering alternative proofs for the Hodge conjecture and confirming the Tate conjecture in certain contexts.

Findings

Cylinder maps are surjective on Chow groups when the hypersurface contains a degree 1 cycle.

An alternative proof of the Hodge conjecture for hyperk"ahler fourfolds associated with cubic fourfolds.

Confirmation of the Tate conjecture for the variety of lines on cubic fourfolds over finitely generated fields.

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.