# Periods of Complete Intersection Algebraic Cycles

## TL;DR

This paper computes periods of complete intersection algebraic cycles on even-dimensional hypersurfaces, analyzes their images under the cycle class map, and applies this to the variational Hodge conjecture, revealing new insights into Hodge loci.

## Contribution

It provides explicit period calculations for complete intersection cycles and links these to the variational Hodge conjecture for non complete intersection cycles.

## Key findings

- Computed periods of all half-dimensional complete intersection cycles.
- Determined cycle class images in De Rham cohomology using Griffiths basis.
- Linked Hodge loci to linear cycles with specific intersection properties.

## Abstract

For every even number $n$, and every $n$-dimensional smooth hypersurface of $\mathbb{P}^{n+1}$ of degree $d$, we compute the periods of all its $\frac{n}{2}$-dimensional complete intersection algebraic cycles. Furthermore, we determine the image of the given algebraic cycle under the cycle class map inside the De Rham cohomology group of the corresponding hypersurface in terms of its Griffiths basis and the polarization. As an application, we use this information to address variational Hodge conjecture for a non complete intersection algebraic cycle. We prove that the locus of general hypersurfaces containing two linear cycles whose intersection is of dimension less than $\frac{n}{2}-\frac{d}{d-2}$, corresponds to the Hodge locus of any integral combination of such linear cycles.