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Toric Representation and Positive Cone of Picard Group and Deformation Space in Mirror Symmetry of Calabi-Yau Hypersurfaces in Toric Varieties

We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry is established for a pair of Calabi-Yau (CY) ${\sf n}$-spaces in toric varieties defined by reflexive polytopes for an arbitrary dimension ${\sf n}$. We further identify the Kahler cone of the toric variety and degeneration cone of CY hypersurfaces, by which the Kahler cone and degeneration cone for a mirror CY pair are interchangeable under mirror symmetry. In particular, different degeneration cones of a CY 3-fold are corresponding to flops of its mirror 3-fold.