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On Calabi-Yau complete intersections in toric varieties. (English) Zbl 0908.14015
Andreatta, Marco (ed.) et al., Higher dimensional complex varieties. Proceedings of the international conference, Trento, Italy, June 15–24, 1994. Berlin: Walter de Gruyter. 39-65 (1996).
Summary: 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 the 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 $$V$$ in arbitrary dimension demands considerations of so-called string-theoretic Hodge numbers $$h_{\text{st}}^{p,q} (V)$$. We restrict ourselves to the string-theoretic Hodge numbers $$h_{\text{st}}^{0,q} (V)$$ and $$h^{1,q}_{\text{st}} (V)$$ $$(0\leq q\leq d-r)$$ 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 $$V$$. 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.
For the entire collection see [Zbl 0859.00021].

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14M10 Complete intersections 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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