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Duality of weights, mirror symmetry and Arnold’s strange duality. (English) Zbl 1152.14040

V. V. Batyrev’s toric mirror symmetry is given in terms of the duality of reflexive polytopes [J. Algebr. Geom. 3, No. 3, 493–535 (1994; Zbl 0829.14023)]. A weight system is introduced by K. Saito in weighted projective spaces [e.g., Adv. Stud. Pure Math. 8, 479–526 (1987; Zbl 0626.14028)]. Duality of weight systems is defined combinatorially in terms of the existence of a primitive weighted magic square.
It was observed in a couple of physics literature [e.g., P. Candelas, M. Lynker and R. Schimmrigk, Nucl. Phys., B 341, No. 2, 383–402 (1990; Zbl 0962.14029); P. Candelas, X. de la Ossa and S. Katz, Nucl. Phys., B 450, 267–290 (1995; Zbl 0896.14023)] that the duality of weight systems yield examples of mirror symmetry for Calabi–Yau manifolds in weighted \(\mathbb P^4\), and that those examples correspond to some reflexive polytopes.
The paper under review investigate further the relation between Batyrev’s mirror symmetry of toric Calabi–Yau hypersurfaces and duality of weight systems in dimension \(\geq 4\).
As an application of this relation, mirror symmetry for \(K3\) toric hypersurfaces is discussed in detail, among them, Arnold’s strange duality for exceptional unimodal singularities.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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