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Abstract

We survey mirror symmetry of Calabi-Yau manifolds from the perspective of families of Calabi-Yau manifolds and their period integrals. Special emphasis is laid on distinguished properties of the hypergeometric series of Gel'fand, Kapranov, and Zelevinsky that appear in mirror symmetry. After defining mirror symmetry in terms of families of Calabi-Yau manifolds, we summarize a general construction of moduli spaces of Calabi-Yau hypersurfaces (complete intersections) in toric varieties. We review the central charge formula, and assuming it, we show mirror symmetry for the pairs of Calabi-Yau manifolds associated with reflexive polytopes. By describing the moduli spaces globally, we present interesting examples of Calabi-Yau manifolds where birational geometry and geometry of Fourier-Mukai partners of a Calabi-Yau manifold arise from the study of mirror symmetry.