Periods of Complete Intersection Algebraic Cycles

In this article we compute periods of complete intersection algebraic cycles inside smooth degree $d$ hypersurfaces of $\mathbb{P}^{n+1}$, of even dimension $n$. This is done by determining the Poincare dual of the given algebraic cycle. As an application, 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. This proves variational Hodge conjecture for those algebraic cycles.