On Calabi-Yau complete intersections in toric varieties

We investigate Hedge-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 Hedge numbers predicted by mirror symmetry. It is expected that the complete verification of mirror symmetry predictions for singular Calabi-Yau varieties V in arbitrary dimension demands considerations of so called string-theoretic Hedge numbers h(st)(p,q)(V). We restrict ourselves to the string-theoretic Hodge numbers h(st)(0,q)(V) and h(st)(1,q)(V) (0 less than or equal to q less than or equal to d - r) which coincide with the usual Hodge numbers h(0,q)((V) over cap) and h(1,q)((V) over cap) of a MPCP-desingularization (V) over cap of V. We prove the duality for (0, q)-Hodge numbers and for the alternating sum of (1, q)-Hodge numbers. The complete duality for (1,q)-Hodge numbers will be checked only for the case of Calabi-Yau complete intersections in projective spaces.