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>



Hodge Numbers and Leray Spectral Sequence
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.


