Cylinder maps of algebraic cycles on cubic hypersurfaces

Renjie Lyu

doi:10.1112/blms.70200

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arXiv · 1810.12394

Cylinder maps of algebraic cycles on cubic hypersurfaces

Renjie Lyu

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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.

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BibTeX RIS

Renjie Lyu. 2018-10-29. Cylinder maps of algebraic cycles on cubic hypersurfaces. https://doi.org/10.1112/blms.70200

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