The Hodge Theory of Projective Manifolds

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ISBN: 9781860948008 | 1860948006
Cover: Hardcover
Copyright: 9/20/2007

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This book is a written-up and expanded version of eight lectures on the Hodge theory of projective manifolds. It assumes very little background and aims at describing how the theory becomes progressively richer and more beautiful as one specializes from Riemannian, to Kalher, to complex projective manifolds.

Preface	p. vii
Calculus on smooth manifolds	p. 1
The Euclidean structure on the exterior algebra	p. 1
The star isomorphism on [Lambda](V)	p. 2
The tangent and cotangent bundles of a smooth manifold	p. 4
The de Rham cohomology groups	p. 6
Riemannian metrics	p. 11
Partitions of unity	p. 12
Orientation and integration	p. 12
The Hodge theory of a smooth, oriented, compact Riemannian manifold	p. 19
The adjoint of d : d	p. 19
The Laplace-Beltrami operator of an oriented Riemannian manifold	p. 21
Harmonic forms and the Hodge Isomorphism Theorem	p. 22
Complex manifolds	p. 27
Conjugations	p. 27
Tangent bundles on a complex manifold	p. 28
Cotangent bundles on complex manifolds	p. 31
The standard orientation of a complex manifold	p. 33
The quasi complex structure	p. 34
Complex-valued forms	p. 37
Dolbeault and Bott-Chern cohomology	p. 40
Hermitean linear algebra	p. 43
The exterior algebra on [Characters not reproducible]	p. 43
Bases	p. 44
Hermitean metrics	p. 45
The inner product and the * operator on the complexified exterior algebra [Lambda subscript c]([Characters not reproducible])	p. 48
The Weil operator	p. 50
The Hodge theory of Hermitean manifolds	p. 51
Hermitean metrics on complex manifolds	p. 51
The Hodge theory of a compact Hermitean manifold	p. 53
Kahler manifolds	p. 57
The Kahler condition	p. 57
The fundamental identities of Kahler geometry	p. 61
The Hodge Decomposition for compact Kahler manifolds	p. 66
Some consequences	p. 69
The Hard Lefschetz Theorem and the Hodge-Riemann Bilinear Relations	p. 71
Hodge structures	p. 72
The cup product with the Chern class of a hyperplane bundle	p. 74
The Hard Lefschetz Theorem and the Hodge-Riemann Bilinear Relations	p. 76
The Weak Lefschetz Theorem	p. 80
Mixed Hodge structures, semi-simplicity and approximability	p. 83
The mixed Hodge structure on the cohomology of complex algebraic varieties	p. 83
The Semi-simplicity Theorem	p. 85
The Leray spectral sequence	p. 87
The Global Invariant Cycle Theorem	p. 88
The Lefschetz Theorems and semi-simplicity	p. 89
Approximability for the space of primitive vectors	p. 93
Bibliography	p. 99
Index	p. 101
Table of Contents provided by Ingram. All Rights Reserved.

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The Hodge Theory of Projective Manifolds

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