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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.

MSC:
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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