Hodge Theory of Projective Manifolds

M Andrea De Cataldo9781860948008, 1-86094-800-6

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 Kähler, to complex projective manifolds. Though the proof of the Hodge Theorem is omitted, its consequences – topological, geometrical and algebraic – are discussed at some length. The special properties of complex projective manifolds constitute an important body of knowledge and readers are guided through it with the help of selected exercises. Despite starting with very few prerequisites, the concluding chapter works out, in the meaningful special case of surfaces, the proof of a special property of maps between complex projective manifolds, which was discovered only quite recently.

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Table of contents :
Contents……Page 10
Preface……Page 8
1.1 The Euclidean structure on the exterior algebra……Page 12
1.2 The star isomorphism on (V )……Page 13
1.3 The tangent and cotangent bundles of a smooth manifold……Page 15
1.4 The de Rham cohomology groups……Page 17
1.5 Riemannian metrics……Page 22
1.7 Orientation and integration……Page 23
2.1 The adjoint of d : d*……Page 30
2.2 The Laplace-Beltrami operator of an oriented Riemannian manifold……Page 32
2.3 Harmonic forms and the Hodge Isomorphism Theorem……Page 33
3.1 Conjugations……Page 38
3.2 Tangent bundles on a complex manifold……Page 39
3.3 Cotangent bundles on complex manifolds……Page 42
3.4 The standard orientation of a complex manifold……Page 44
3.5 The quasi complex structure……Page 45
3.6 Complex-valued forms……Page 48
3.7 Dolbeault and Bott-Chern cohomology……Page 51
4.1 The exterior algebra on V*c……Page 54
4.2 Bases……Page 55
4.3 Hermitean metrics……Page 56
4.4 The inner product and the * operator on the complexi ed exterior algebra AC (Vc*……Page 59
4.5 The Weil operator……Page 61
5.1 Hermitean metrics on complex manifolds……Page 62
5.2 The Hodge theory of a compact Hermitean manifold……Page 64
6.1 The K ahler condition……Page 68
6.2 The fundamental identities of K ahler geometry……Page 72
6.3 The Hodge Decomposition for compact K ahler manifolds……Page 77
6.4 Some consequences……Page 80
7. The Hard Lefschetz Theorem and the Hodge-Riemann Bilinear Relations……Page 82
7.1 Hodge structures……Page 83
7.2 The cup product with the Chern class of a hyperplane bundle……Page 85
7.3 The Hard Lefschetz Theorem and the Hodge-Riemann Bilinear Relations……Page 87
7.4 The Weak Lefschetz Theorem……Page 91
8.1 The mixed Hodge structure on the cohomology of complex algebraic varieties……Page 94
8.2 The Semi-simplicity Theorem……Page 96
8.3 The Leray spectral sequence……Page 98
8.4 The Global Invariant Cycle Theorem……Page 99
8.5 The Lefschetz Theorems and semi-simplicity……Page 100
8.6 Approximability for the space of primitive vectors……Page 104
Bibliography……Page 110
Index……Page 112