  We describe mirror symmetry for weak toric Fano manifolds as an equivalence
of D-modules equipped with certain filtrations. 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 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 Kaehler moduli space of the variety.
