On a Hodge locus

TL;DR

This paper explores instances where the deformation space of a Hodge cycle exceeds that of an algebraic cycle, focusing on cubic hypersurfaces, aiming to verify or challenge the Hodge conjecture through computational experiments.

Contribution

It presents a specific example of this phenomenon in cubic hypersurfaces and proposes it as a computational problem to test the Hodge conjecture.

Findings

Evidence for deformation space discrepancies in cubic hypersurfaces

Proposal of a computational approach to verify the Hodge conjecture

Potential implications for counterexamples to the Hodge conjecture

Abstract

There are many instances such that deformation space of the homology class of an algebraic cycle as a Hodge cycle is larger than its deformation space as algebraic cycle. This phenomena can occur for algebraic cycles inside hypersurfaces, however, we are only able to gather evidences for it by computer experiments. In this article we describe one example of this for cubic hypersurfaces. The verification of the mentioned phenomena in this case is proposed as the first GADEPs problem. The main goal is either to verify the (variational) Hodge conjecture in such a case or gather evidences that it might produce a counterexample to the Hodge conjecture.