## Hodge Theory and Complex Algebraic Geometry I: Volume 1The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry. |

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### Contents

Part II The Hodge Decomposition | 115 |

Part III Variations of Hodge Structure | 217 |

Part IV Cycles and Cycle Classes | 261 |

315 | |

319 | |

### Common terms and phrases

abelian admits algebraic analytic apply associated assume called canonical chapter Chern coefficients cohomology compact complex manifold complex structure compute condition connection consider constant construct contained continuous coordinates Corollary corresponding covering cycle decomposition defined Definition differential forms dimension equal equation equipped equivalent exact sequence example exists fact fibre filtration form of type formula function functor given gives harmonic Hermitian metric Hodge structures Hodge theory holomorphic identified implies induced injective integral isomorphism K¨ahler manifold lemma locally Moreover morphism natural neighbourhood Note objects obtain open set operator positive projective Proof proposition prove rank relation Remark representative resolution restriction Rham satisfying sheaf sheaves singular smooth space spectral sequence subset sufficiently tangent theorem trivialisation values vanishes zero