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Say we are given a complex smooth projective toric variety $X$. How can one read off hodge theoretic information from combinatorial data? For example I would like to extract dimensions of the various $H^{pq}(X,\mathbb C)$ from the fan or the $sl_2(\mathbb C)$ action on $H^*(X,\mathbb C)$ from the moment polytope etc.

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up vote 3 down vote accepted

Say the toric variety $X$ is defined by the lattice polytope $P$ (dimension = $n$). Let $f_i$ denote the number of $i$-dimensional faces of $P$, and set $h_p = \sum_{i=p}^n (-1)^{i-p} \binom{i}{p} f_i$. Then $h^{p,q}(X) = h_p$ if $p=q$ and $0$ otherwise.

I'm reading this from section 9.4 of a version Cox-Little-Schenk's "Toric Varieties". Haven't read the whole book, so I'm afraid I can't shed much light on why this is true, but a draft used to be available online and I bet you can track down a copy.

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