Hodge Numbers and Leray Spectral Sequence

Question

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

Answer 1

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.