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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

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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.

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