## Complex Differential GeometryThe theory of complex manifolds overlaps with several branches of mathematics, including differential geometry, algebraic geometry, several complex variables, global analysis, topology, algebraic number theory, and mathematical physics. Complex manifolds provide a rich class of geometric objects, for example the (common) zero locus of any generic set of complex polynomials is always a complex manifold. Yet complex manifolds behave differently than generic smooth manifolds; they are more coherent and fragile. The rich yet restrictive character of complex manifolds makes them a special and interesting object of study. |

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### Contents

Metric Connection and Curvature | 29 |

The Geometry of Complete Riemannian Manifolds | 49 |

Complex manifolds and Analytic Varieties | 83 |

Copyright | |

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### Common terms and phrases

admits algebra ample assume ball biholomorphic called Chern Clearly closed cohomology compact complex complex manifold complex structure component connection consider constant contained coordinate covering curve defined definition denote differentiable dimension direct discuss divisor dual elliptic equal equivalent exact example exercise exists fact fiber finite formula frame function geodesic given gives global harmonic hence Hermitian metric holds holomorphic holomorphic map identity implies induced integer intersection irreducible isomorphism lemma line bundle linear locally matrix metric minimal multiple negative neighborhood Note operator orientation particular positive projective proof Prove quotient rank rational readers Recall restriction Ricci curvature Riemannian manifold satisfies says sectional curvature sequence sheaf Show smooth space subgroup subvariety Suppose surface symmetric space tangent tensor theorem topological unique unit vector bundle write zero