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Logarithmic Frobenius manifolds, hypergeometric systems and quantum \(\mathcal D\)-modules. (English) Zbl 1349.14139
Summary: We describe mirror symmetry for weak Fano toric manifolds as an equivalence of filtered \( \mathcal {D}\)-modules. We discuss in particular the logarithmic degeneration behavior at the large radius limit point and express the mirror correspondence as an isomorphism of Frobenius manifolds with logarithmic poles. The main tool is an identification of the Gauß-Manin system of the mirror Landau-Ginzburg model with a hypergeometric \( \mathcal {D}\)-module, and a detailed study of a natural filtration defined on this differential system. We obtain a solution of the Birkhoff problem for lattices defined by this filtration and show the existence of a primitive form, which yields the construction of Frobenius structures with logarithmic poles associated to the mirror Laurent polynomial. As a final application, we show the existence of a pure polarized non-commutative Hodge structure on a Zariski open subset of the complexified Kähler moduli space of the variety.

MSC:
14J33 Mirror symmetry (algebro-geometric aspects)
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14J45 Fano varieties
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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