On Calabi-Yau Complete Intersections in Toric Varieties

We investigate Hodge-theoretic properties of Calabi-Yau complete intersections $V$ of $r$ semi-ample divisors in $d$-dimensional toric Fano varieties having at most Gorenstein singularities. Our main purpose is to show that the combinatorial duality proposed by second author agrees with the duality for Hodge numbers predicted by mirror symmetry. It is expected that the complete verification of mirror symmetry predictions for singular Calabi-Yau varieties $V$ of arbitrary dimension demands considerations of so called {\em string-theoretic Hodge numbers} $h^{p,q}_{\rm st}(V)$. We restrict ourselves to the string-theoretic Hodge numbers $h^{0,q}_{\rm st}(V)$ and $h^{1,q}_{\rm st}(V)$ $(0 \leq q \leq d-r) which coincide with the usual Hodge numbers$h^{0,q}(\widehat{V})$and$h^{1,q}(\widehat{V})$of a$MPCP$-desingularization$\widehat{V}$of$V$.