Semi-regular varieties and variational Hodge conjecture

Source abstract

Following [1,2], we know that semi-regular sub-varieties satisfy the variational Hodge conjecture, i.e., given a family of smooth projective varieties π : X → B , a special fiber X o and a semi-regular subvariety Z ⊂ X o , the cohomology class corresponding to Z remains a Hodge class (as 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 P 2 n + 1 of large enough degree.