Welcome to Bytelearn!

Let’s check out your problem:

The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.$\newline$$\text{Hdg}^{(k)}(X) = \text{H}^{(2k)}(X,\mathbb{Q}) \cap \text{H}^{(k,k)}(X)$.$\newline$We call this the group of Hodge classes of degree $2k$ on $X$.$\newline$The modern statement of the Hodge conjecture is:$\newline$Let $X$ be a non-singular complex projective manifold. Then every Hodge class on $X$ is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of $X$.