Motives of Certain Hyperplane Sections of Milnor Hypersurfaces

TL;DR

This paper studies hyperplane sections of Milnor hypersurfaces linked to regular semisimple endomorphisms, providing a motivic decomposition that reveals their cellular structure and eigenvalue arithmetic without relying on the nilpotence principle.

Contribution

It introduces a motivic decomposition for these hyperplane sections, connecting their geometric cellular structure with eigenvalue arithmetic, avoiding the nilpotence principle.

Findings

Motivic decomposition encodes cellular and eigenvalue data

Decomposition proven without nilpotence principle

Links geometric structure with algebraic eigenvalues

Abstract

We construct a hyperplane section $Y$ of a Milnor hypersurface associated to a regular semisimple endomorphism $\varphi$. Exploiting its structure as a hyperplane section of a projective bundle and its natural torus action, we give a motivic decomposition of $Y$, which encodes both the cellular structure of $Y$ and the arithmetic of the eigenvalues of $\varphi$. This decomposition is proven without using the "nilpotence principle", that is to say there are no "phantoms".