## Introduction to Hodge TheoryHodge theory originated as an application of harmonic theory to the study of the geometry of compact complex manifolds. The ideas have proved to be quite powerful, leading to fundamentally important results throughout algebraic geometry. This book consists of expositions of various aspects of modern Hodge theory. Its purpose is to provide the nonexpert reader with a precise idea of the current status of the subject. The three chapters develop distinct but closely related subjects: $L^2$ Hodge theory and vanishing theorems; Frobenius and Hodge degeneration; variations of Hodge structures and mirror symmetry.The techniques employed cover a wide range of methods borrowed from the heart of mathematics: elliptic PDE theory, complex differential geometry, algebraic geometry in characteristic $p$, cohomological and sheaf-theoretic methods, deformation theory of complex varieties, Calabi-Yau manifolds, singularity theory, etc. A special effort has been made to approach the various themes from their most natural starting points. Each of the three chapters is supplemented with a detailed introduction and numerous references. The reader will find precise statements of quite a number of open problems that have been the subject of active research in recent years. The reader should have some familiarity with differential and algebraic geometry, with other prerequisites varying by chapter. The book is suitable as an accompaniment to a second course in algebraic geometry. |

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### Contents

Introduction | 5 |

L2 Hodge Theory and Vanishing Theorems | 7 |

L2 Hodge theory | 9 |

Differential operators on vector bundles | 12 |

Fundamental results on elliptic operators I | 14 |

Hodge theory of compact Ricmmanian manifolds | 19 |

Hermitian and Kahler manifolds | 24 |

Fundamental identities of Kiihlerian geometry | 27 |

Smoothness and liftings | 107 |

Frobenius and Cartier isomorphism | 113 |

Derived categories and spectral sequences | 119 |

Decomposition degeneration and vanishing theorems in characteristic p 0 | 124 |

From characteristic p 0 to characteristic zero | 130 |

Recent developments and open problems | 137 |

parallelizability and ordinarity | 144 |

Bibliography | 147 |

The groups HpqX E and Serre duality | 35 |

Comohology of compact Kiihlcr manifolds | 36 |

The HodgeFrolichcr spectral sequence | 43 |

Deformations and the semicontinuity theorem 17 | 46 |

L2 estimates and vanishing theorems | 53 |

Hodge theory of complete Kiihler manifolds | 60 |

Bochner techniques and vanishing theorems | 70 |

L2 estimates and existence theorems | 73 |

Vanishing theorems of Nadel and KawamataViehweg | 75 |

On the conjecture of Fujita | 82 |

An effective version of Matsusakas big theorem | 89 |

Bibliography | 95 |

Frobenius and Hodge Degeneration Luc ILLUSIE | 99 |

Introduction | 101 |

differentials the de Rham complex | 103 |

Variations of Hodge Structure CalabiYau Manifolds and Mirror Symmetry JOSE BERTIN AND CHRIS PETERS | 151 |

Introduction | 153 |

Variations of Hodge structures | 161 |

GaussManin connection | 163 |

Variation of Hodge structures | 172 |

Degenerations | 179 |

Higgs bundles | 187 |

Hodge modules | 188 |

Mirror symmetry and CalabiYau manifolds | 193 |

Cohomology of hypersurfaces | 199 |

PicardFuchs equations | 205 |

CalabiYau threefolds and mirror symmetry | 210 |

Relation with mixed Hodge theory | 222 |

Bibliography | 229 |

### Common terms and phrases

abelian adjoint affine ample analytic associated base change bidegree Calabi-Yau manifolds Calabi-Yau threefolds canonical coefficients coherent sheaf cohomology groups complex manifold components computation consequence coordinates COROLLARY curvature deduced defined definition degeneration degree denotes differential dimension Dolbeault Dolbeault cohomology eigenvalues etale exact sequence example exists fiber finite presentation finite type formula Frobenius Gauss-Manin connection Hermitian metric Hodge decomposition Hodge filtration Hodge numbers Hodge theory hypercohomology hypersurface implies induces integral intersection isomorphism Kahler manifold Kahler metric Kodaira-Spencer Kodaira-Spencer map Lefschetz lemma lifting line bundle locally free Math mirror symmetry mixed Hodge structure modules monodromy morphism neighbourhood normal crossings obtain operator Ox-module parameter Picard-Fuchs equation proof quasi-isomorphism relative resp result Rham cohomology Rham complex Rham spectral sequence satisfies scheme sheaves singular spectral sequence structure of weight trivial vanishing theorem variation of Hodge vector bundle Yukawa coupling zero