TITLE: Hodge Numbers and Leray Spectral Sequence
QUESTION [7 upvotes]: Mark Gross' notes survey of SYZ fibrations and toric degenerations begin by explaining why dual torus fibrations interchange Hodge numbers. But he defined the Hodge numbers in an unusual way
$$h^{p,q}(X) = \text{dim}_\mathbb{R}H^p(B,R^qf_\ast \mathbb{R}),$$
where $f: X \to B$ is a torus fibration, $B$ is a real $3$-manifold, and $X$ is a complex Calabi-Yau threefold.
I thought the Hodge numbers were $h^{p,q}(X) = \text{dim}_\mathbb{R}H^p(X,\Omega^q_X).$ Why ought these numbers agree?
Reference. https://arxiv.org/pdf/0802.3407.pdf

REPLY [12 votes]: I don't think I defined the Hodge numbers in this way. Rather, the argument in Section 1 shows that the Hodge numbers agree with the dimensions of
the terms in the $E_2$ page of the Leray spectral sequence. The argument
given there only works in three dimensions (and some strong assumptions
on the fibration) because in the Calabi-Yau 3-fold cases, the Hodge
numbers are in fact topological invariants, i.e., $h^{1,1}=b_2$ and
$h^{1,2}=b_3/2 -1$.
In other work, again with some assumptions, it is shown that the groups
$H^p(B, R^qf_*{\bf Z})$ agree with the graded pieces of the weight filtration for the limiting mixed Hodge structure associated with a toric degeneration.