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Mirror duality and string-theoretic Hodge numbers. (English) Zbl 0872.14035
Due to the earlier fundamental work of the first author, there is a bijective correspondence between Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties and certain reflexive polyhedra [cf. V. V. Batyrev, J. Algebr. Geom. 3, No. 3, 493-535 (1994; Zbl 0829.14023)]. It was conjectured, in this context, that the polar duality of reflexive polyhedra induces the mirror symmetry for the corresponding Calabi-Yau hypersurfaces. This conjecture has been generalized, by both authors of the present paper, to Calabi-Yau complete intersections in Gorenstein Fano varieties and, in the sequel, to generalized Calabi-Yau manifolds.
As one necessarily has to work with singular Calabi-Yau varieties, the first author was then led to introduce the so-called “string-theoretic Hodge numbers” for such varieties, which coincide with the usual ones in the smooth case [cf. V. V. Batyrev and D. I. Dais, Topology 35, No. 4, 901-929 (1996; Zbl 0864.14022)]. In this framework, the mirror symmetry conjecture (predicted by physicists) has to be modified as follows:
Let \((V,W)\) a mirror pair of singular \(n\)-dimensional Calabi-Yau varieties. Then the string-theoretic Hodge numbers are related by the duality \(h^{p,q}_{\text{string}} (V)= h^{n- p,q}_{\text{string}} (W)\) for \(0\leq p\), \(q\leq n\).
The present paper provides an affirmative answer to this mirror symmetry conjecture for the string-theoretic Hodge numbers of Calabi-Yau complete intersections in Gorenstein toric Fano varieties. – The proof of this important result is based on combinatorial properties of certain polynomials arising from the intersection homology of those varieties and its associated mixed Hodge structure.

14J45 Fano varieties
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14M10 Complete intersections
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
32J81 Applications of compact analytic spaces to the sciences
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