Grothendieck–Hodge conjecture for coniveau

Let $X$ be a proper smooth scheme over $\mathbb C$. For integers $i,j$, define the coniveau filtration by

$$N^iH^j(X)=\ker\left(H^j(X)\to\varinjlim_{\operatorname{codim}_X Z\geq i}H^j(X-Z)\right).$$

The inclusion

$$N^iH^j(X)_{\mathbb Q}\subseteq H^j(X,\mathbb Q)\cap F^iH^j(X)$$

identifies the left-hand side with a subspace of rational Hodge classes. Grothendieck–Hodge conjecture. The space $N^iH^j(X)_{\mathbb Q}$ is the largest subspace of $H^j(X,\mathbb Q)\cap F^iH^j(X)$ whose complexification generates a sub-Hodge structure of $H^j(X,\mathbb C)$. This is a strengthening of the Hodge-coniveau comparison and is open in general.