Semi-regular varieties and variational Hodge conjecture

TL;DR

This paper explores semi-regular sub-varieties within algebraic geometry, demonstrating that any smooth projective variety can be embedded as a semi-regular sub-variety in high-degree hypersurfaces, supporting the variational Hodge conjecture.

Contribution

It proves that all smooth projective varieties of a given dimension can be realized as semi-regular sub-varieties in sufficiently high-degree hypersurfaces.

Findings

Semi-regular sub-varieties satisfy the variational Hodge conjecture.

Any smooth projective variety of dimension n can be embedded as a semi-regular sub-variety in a high-degree hypersurface.

Supports the conjecture that cohomology classes remain algebraic under deformation.

Abstract

We know that semi-regular sub-varieties satisfy the variational Hodge conjecture i.e., given a family of smooth projective varieties $\pi:\mathcal{X} \to B$, a special fiber $\mathcal{X}_o$ and a semi-regular subvariety $Z \subset \mathcal{X}_o$, the cohomology class corresponding to $Z$ remains a Hodge class (as $\mathcal{X}_o$ deforms along $B$) if and only if $Z$ remains an algebraic cycle. In this article, we investigate examples of such sub-varieties. In particular, we prove that any smooth projective variety $Z$ of dimension $n$ is a semi-regular sub-variety of a smooth projective hypersurface in $\mathbb{P}^{2n+1}$ of large enough degree.