Families of Calabi-Yau manifolds and mirror symmetry

TL;DR

This paper surveys mirror symmetry of Calabi-Yau manifolds focusing on families, period integrals, hypergeometric series, and moduli spaces, highlighting the role of toric varieties and birational geometry in understanding mirror pairs.

Contribution

It provides a comprehensive overview of mirror symmetry for Calabi-Yau manifolds, emphasizing the construction of moduli spaces and the application of hypergeometric series in the context of toric varieties.

Findings

Mirror symmetry is established for Calabi-Yau pairs associated with reflexive polytopes.

Moduli spaces of Calabi-Yau hypersurfaces are constructed using toric geometry.

Connections between mirror symmetry, birational geometry, and Fourier-Mukai partners are explored.

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.