# Periods of linear algebraic cycles

## TL;DR

This paper computes periods of linear algebraic cycles in Fermat varieties, demonstrating that certain loci of hypersurfaces with specific algebraic cycles form components of the Hodge locus, supporting the variational Hodge conjecture.

## Contribution

It explicitly computes periods of algebraic cycles in Fermat varieties and confirms the variational Hodge conjecture for these cases.

## Key findings

- Loci of hypersurfaces with two linear cycles form reduced components of the Hodge locus.
- Confirmed the Hodge conjecture for algebraic cycles with monodromy in these cases.
- Validated the variational Hodge conjecture for specific algebraic cycles in Fermat varieties.

## Abstract

In this article we use a theorem of Carlson and Griffiths and compute periods of linear algebraic cycles inside the Fermat variety of even dimension $n$ and degree $d$. As an application, for examples of $n$ and $d$ we prove that the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space. We also check the same statement for hypersurfaces containing a complete intersection algebraic cycle. Our result confirms the Hodge conjecture for Hodge cycles obtained by the monodromy of the homology class of such algebraic cycles. This is known as the variational Hodge conjecture.