Hodge Theory Of Projective Manifolds.
Title:
Hodge Theory Of Projective Manifolds.
Author:
de Cataldo, Mark Andrea.
ISBN:
9781860948657
Personal Author:
Physical Description:
1 online resource (113 pages)
Contents:
Contents -- Preface -- 1. Calculus on smooth manifolds -- 1.1 The Euclidean structure on the exterior algebra -- 1.2 The star isomorphism on (V ) -- 1.3 The tangent and cotangent bundles of a smooth manifold -- 1.4 The de Rham cohomology groups -- 1.5 Riemannian metrics -- 1.6 Partitions of unity -- 1.7 Orientation and integration -- 2. The Hodge theory of a smooth, oriented, compact Riemannian manifold -- 2.1 The adjoint of d : d* -- 2.2 The Laplace-Beltrami operator of an oriented Riemannian manifold -- 2.3 Harmonic forms and the Hodge Isomorphism Theorem -- 3. Complex manifolds -- 3.1 Conjugations -- 3.2 Tangent bundles on a complex manifold -- 3.3 Cotangent bundles on complex manifolds -- 3.4 The standard orientation of a complex manifold -- 3.5 The quasi complex structure -- 3.6 Complex-valued forms -- 3.7 Dolbeault and Bott-Chern cohomology -- 4. Hermitean linear algebra -- 4.1 The exterior algebra on V*c -- 4.2 Bases -- 4.3 Hermitean metrics -- 4.4 The inner product and the * operator on the complexi ed exterior algebra AC (Vc* -- 4.5 The Weil operator -- 5. The Hodge theory of Hermitean manifolds -- 5.1 Hermitean metrics on complex manifolds -- 5.2 The Hodge theory of a compact Hermitean manifold -- 6. K ahler manifolds -- 6.1 The K ahler condition -- 6.2 The fundamental identities of K ahler geometry -- 6.3 The Hodge Decomposition for compact K ahler manifolds -- 6.4 Some consequences -- 7. The Hard Lefschetz Theorem and the Hodge-Riemann Bilinear Relations -- 7.1 Hodge structures -- 7.2 The cup product with the Chern class of a hyperplane bundle -- 7.3 The Hard Lefschetz Theorem and the Hodge-Riemann Bilinear Relations -- 7.4 The Weak Lefschetz Theorem -- 8. Mixed Hodge structures, semi-simplicity and approximability -- 8.1 The mixed Hodge structure on the cohomology of complex algebraic varieties -- 8.2 The Semi-simplicity Theorem.
8.3 The Leray spectral sequence -- 8.4 The Global Invariant Cycle Theorem -- 8.5 The Lefschetz Theorems and semi-simplicity -- 8.6 Approximability for the space of primitive vectors -- Bibliography -- Index.
Abstract:
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.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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