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Birationality and Landau-Ginzburg models. (English) Zbl 1428.14074
Summary: We introduce a new technique for approaching birationality questions that arise in the mirror symmetry of complete intersections in toric varieties. As an application we answer affirmatively and conclusively the question of V. Batyrev and B. Nill [Contemp. Math. 452, 35–66 (2008; Zbl 1161.14037)]. about the birationality of Calabi-Yau families associated to multiple mirror nef-partitions. This completes the progress in this direction made by Z. Li’s breakthrough [Adv. Math. 299, 71–107 (2016; Zbl 1360.14040)]. In the process, we obtain results in the theory of L. Borisov’s nef-partitions [“Towards the mirror symmetry for Calabi-Yau complete intersections in Gorenstein toric Fano varieties”, Preprint, arXiv:alg-geom/9310001] and provide new insight into the geometric content of the multiple mirror phenomenon.

MSC:
14J33 Mirror symmetry (algebro-geometric aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14E07 Birational automorphisms, Cremona group and generalizations
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
81T13 Yang-Mills and other gauge theories in quantum field theory
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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