The transcendental lattice of the sextic Fermat surface

TL;DR

This paper proves that the transcendental lattice of the sextic Fermat surface has a decomposable integral polarized Hodge structure, challenging previous conjectures and impacting approaches to cubic fourfold irrationality.

Contribution

It demonstrates the decomposability of the transcendental lattice's Hodge structure, providing a counterexample to Kulikov's conjecture.

Findings

Decomposability of the Hodge structure on the transcendental lattice.

Disproof of Kulikov's conjecture.

Implications for the irrationality problem of cubic fourfolds.

Abstract

We prove that the integral polarized Hodge structure on the transcendental lattice of a sextic Fermat surface is decomposable. This disproves a conjecture of Kulikov related to a Hodge theoretic approach to proving the irrationality of the very general cubic fourfold.