## Mirror Symmetry and Algebraic GeometryMathematicians wanting to get into the field ... will find a very well written and encyclopaedic account of the mathematics which was needed in, and was developed from, what now might be termed classical mirror symmetry. --Bulletin of the LMS The book is highly recommended for everyone who wants to learn about the fascinating recent interplay between physics and mathematics. --Mathematical Reviews Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. This book is a completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. Subjects discussed include toric varieties, Hodge theory, Kahler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem. |

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### Contents

The Quintic Threefold | 15 |

Toric Geometry | 31 |

Mirror Symmetry Constructions | 53 |

Hodge Theory and Yukawa Couplings | 73 |

Moduli Spaces | 113 |

GromovWitten Invariants | 167 |

Quantum Cohomology | 217 |

Localization | 275 |

Conclusion | 397 |

Appendix A Singular Varieties | 407 |

Nonlinear Sigma Models | 419 |

Conformal Field Theories | 423 |

LandauGinzburg Models | 426 |

Gauged Linear Sigma Models | 428 |

Topological Quantum Field Theories | 430 |

Bibliography | 437 |

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### Common terms and phrases

A-model connection A-model correlation function algebraic automorphisms Axiom basis Calabi-Yau manifold Calabi-Yau threefold Chapter compactification complete intersection complex moduli space components compute conjecture construction coordinates correlation function corresponding cup product defined definition degree denote dimension discussed divisor dual equivariant Euler class Example follows formula fundamental class Furthermore genus given Givental2 gives GKZ decomposition Gromov-Witten invariants hand side Hence Hodge structure holomorphic hypersurfaces implies instanton integral isomorphism Kähler cone Kähler moduli space large radius limit Lemma linear map f mathematical maximally unipotent boundary Mg,n mirror map mirror symmetry Mirror Theorem monodromy Msimp notation Note orbifold Picard-Fuchs equations polynomial projective subdivision proof Proposition prove quintic threefold radius limit point rational curves relation Section sigma model simplicial singular small quantum product smooth stable map symplectic toric variety unipotent unipotent boundary point variables variation of Hodge vector virtual fundamental class Yukawa coupling