We show how Chern classes of a Calabi Yau hypersurface in a toric Fano manifold can be expressed in terms of the holomorphic at a maximal degeneracy point period of its mirror. We also consider the relation between the Chern classes and the periods of mirrors for complete intersections in Grassmanian Gr(2,5).
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Anatoly Libgober (1999) studied this question.
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