Motivic apsects of a remarkable class of Calabi-Yau threefolds
Gregory Pearlstein, Chris Peters
TL;DR
The paper addresses the motivic structure of the middle cohomology $H^3(X)$ for a broad family of symmetric Calabi–Yau threefolds in weighted projective 4-space with a cyclic group action. It develops a framework combining weighted hypersurface geometry, Chow motives, and cyclic-group representations to produce an isotypical decomposition of $H^3(X,\mathbb{Q})$ and a refined CK-motive decomposition, linking nontrivial summands to Fano quotients $Y_d$ and proving $GHC(1,3)$ for these summands. Through explicit computations for Fermat-type and non-Fermat symmetric CYs, it demonstrates how the transcendental part isolates as a separate motivic summand while the $d\neq m$ components satisfy the generalized Hodge conjecture, with the case $d=2$ giving an isogeny between Abel–Jacobi targets. Appendices provide Egyptian-fraction tables and SAGE code to realize the constructions, highlighting the practical impact for understanding CY motivic structures and their connections to Fano geometry and Abel–Jacobi theory.
Abstract
In this note we consider the motivic aspect of the middle cohomology of more than 200 classes of quasi-smooth Calabi--Yau threefolds inside weighted projective 4-space which come with an action of a cyclic group of even order. The action induces a self-dual Chow--Künneth decomposition. All but one component correspond to Fano threefolds. For these the generalized Hodge conjecture is known, but thanks to the nature of the decomposition we can give a direct proof for one of the components.