Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree holomorphic form $\omega_X\in H^0(X, K_X)$. Here is the question that I'm interested in:
Question. Suppose that $X$ is very general in the moduli. Does it hold that the Hodge coniveau filtration $N^1$ coincide with $N^2$ on $H^n(X, \mathbb Q)$?
Here, $N^iH^n(X,\mathbb Q)$ is the largest sub-Hodge structure of $H^n(X, \mathbb Q)$ of coniveau $\geq i$. This question makes sense to me (and I suspect it give a positive answer) due to the following argument.
Let $\mathrm{Def}(X)$ be the Kuranishi space of $X$ with reference point $o\in \mathrm{Def}(X)$. Let $$ \mathcal D: \mathrm{Def}(X)\to \mathbb PH^n(X,\mathbb C) $$ be the period map defined by sending any point $b\in \mathrm{Def}(X)$ to the one-dimensional subspace $\langle \omega_{X_b}\rangle \subset H^n(X_b,\mathbb C)\cong H^n(X,\mathbb C)$. Griffiths's period map theory implies that the differential of the $\mathcal D$ at $o\in \mathrm{Def}(X)$ is given by the wedge-contraction. Namely, the image of the differential $$ d\mathcal D_o: H^1(X, T_X) = T_{\mathrm{Def}(X), o} \to Hom (H^0(X, \Omega_X^n), H^n(X,\mathbb C)/H^0(X, \Omega_X^n)) $$ is exactly $$Hom (H^0(X, \Omega_X^n), H^1(X, \Omega_X^{n-1}))\subset Hom (H^0(X, \Omega_X^n), H^n(X,\mathbb C)/H^0(X, \Omega_X^n)).$$
It seems to me that the last condition here says that there are enough deformations of $X$ such that the sub-Hodge structure orthogonal to $H^{n,0}(X_b)$ for a very general deformation $X_b$ would not contain $H^{n-1,1}(X_b)$. But I'm not sure if this last sentence really make sense and if so, how to make it rigorous?
Any comments or suggestions are welcome!
