Prym-Tjurin Constructions on Cubic Hypersurfaces

TL;DR

This paper develops a Prym-Tjurin construction to analyze the primitive cohomology and Chow groups of cubic hypersurfaces, revealing a quadratic relation and an isomorphism via the Abel-Jacobi map.

Contribution

It introduces a novel Prym-Tjurin framework for cubic hypersurfaces, linking cohomology and Chow groups through incidence correspondences and quadratic relations.

Findings

Action satisfies a quadratic equation

Isomorphism between primitive cohomology and Prym-Tjurin part

Applicable to Chow groups with rational coefficients

Abstract

In this paper, we give a Prym-Tjurin construction for the cohomology and the Chow groups for a cubic hypersurface. On the space of lines meeting a given rational curve, there is the incidence correspondence. This correspondence induces an action on both of the primitive cohomology and the primitive Chow groups. We first show that this action satisfies a quadratic equation. Then the Abel-Jacobi homomorphism induces an isomorphism between the primitive cohomology of the cubic hypersurface and the Prym-Tjurin part of the above action. This also holds for Chow groups with rational coefficients.