Let $X$ be a complex smooth projective variety and $D$ a divisor on $X$ with normal crossings. As usual, denote by $D(m)$ the disjoint union of all possible intersections of $m$ irreducible components of $D$. Let $W$ be the weight filtration on the sheaf of logarithmic differentials $\Omega^\bullet_X(\log D)$.
Is the following equality of dimensions true? $$ h^q(X, \mathrm{Gr}^W_m \Omega^p_X(\log D))=h^q(D(m), \Omega^{p-m}_{D(m)}) $$
I have seen that there is an isomorphism of complexes $\mathrm{Gr}_m^W \Omega^\bullet_X(\log D) \simeq (a_m)_\ast \Omega^\bullet_{D(m)}[-m]$, but I am uncertain about the meaning of $[-m]$.
