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.

对于度为d的\（{\ mathbb {P}} ^ {n + 1} \）的每个n个偶数n个和每个n维光滑超曲面，我们计算其所有\（\ frac {n} { 2} \）-维完整交集的代数循环。此外，我们根据其格里菲思（Griffiths）基础和极化，在相应超曲面的De Rham同源组内的循环类图下确定给定代数循环的图像。作为应用程序，我们使用此信息来解决非完整相交代数循环的变分Hodge猜想。我们证明了包含两个线性循环的一般超曲面的轨迹，其交点的尺寸小于\（\ frac {n} {2}-\ frac {d} {d-2} \）对应于此类线性循环的任何整数组合的Hodge轨迹。