# Hodge theory for toric varieties

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.

-

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.