The generalized Hodge conjecture

Let $X$ be a smooth complex projective variety. Define the coniveau filtration by

$$N^iH^m(X)=\bigcup_{\operatorname{codim} Z\geq i}\operatorname{Im}\bigl(H^m_Z(X)\otimes\mathbb{Q}\longrightarrow H^m(X)_{\mathbb{Q}}\bigr),$$

where $Z\subset X$ ranges over subvarieties, and let $H^{m,i}_{\operatorname{Hdg}}(X)$ be the largest Hodge substructure of $F^iH^m(X)_{\mathbb{C}}\cap H^m(X)_{\mathbb{Q}}$. The generalized Hodge conjecture. For all $m$ and $i$, one has

$$H^{m,i}_{\operatorname{Hdg}}(X)=N^iH^m(X).$$

This extends the classical Hodge conjecture from algebraic classes in even-degree cohomology to the entire cohomology. The source presents it as a conjecture; no resolution is supplied.