Integral Hodge conjecture for Fermat varieties

TL;DR

This paper presents an algorithm that verifies the integral Hodge conjecture for Fermat varieties by comparing algebraic and Hodge cycles, confirmed through computer implementation for specific cases.

Contribution

The paper introduces a novel algorithm to verify the integral Hodge conjecture for Fermat varieties using lattice computations, confirmed for quartic and quintic fourfolds.

Findings

Algorithm successfully verifies the conjecture for specific Fermat fourfolds.

Computational approach based on elementary divisors of cycle lattices.

Confirmed the conjecture for quartic and quintic Fermat fourfolds.

Abstract

We describe an algorithm which verifies whether linear algebraic cycles of the Fermat variety generate the lattice of Hodge cycles. A computer implementation of this confirms the integral Hodge conjecture for quartic and quintic Fermat fourfolds. Our algorithm is based on computation of the list of elementary divisors of both the lattice of linear algebraic cycles, and the lattice of Hodge cycles written in terms of vanishing cycles, and observing that these two lists are the same.