Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- 0 Introduction -- Part I Preliminaries -- 1 Holomorphic Functions of Many Variables -- 1.1 Holomorphic functions of one variable -- 1.1.1 Definition and basic properties -- 1.1.2 Background on Stokes' formula -- 1.1.3 Cauchy's formula -- 1.2 Holomorphic functions of several variables -- 1.2.1 Cauchy's formula and analyticity -- 1.2.2 Applications of Cauchy's formula -- 1.3 The equation… -- Exercises -- 2 Complex Manifolds -- 2.1 Manifolds and vector bundles -- 2.1.1 Definitions -- 2.1.2 The tangent bundle -- 2.1.3 Complex manifolds -- 2.2 Integrability of almost complex structures -- 2.2.1 Tangent bundle of a complex manifold -- 2.2.2 The Frobenius theorem -- 2.2.3 The Newlander-Nirenberg theorem -- 2.3 The operators… -- 2.3.1 Definition -- 2.3.2 Local exactness -- 2.3.3 Dolbeault complex of a holomorphic bundle -- 2.4 Examples of complex manifolds -- Riemann surfaces -- Complex projective space -- Complex tori -- Exercises -- 3 Kähler Metrics -- 3.1 Definition and basic properties -- 3.1.1 Hermitian geometry -- 3.1.2 Hermitian and Kähler metrics -- 3.1.3 Basic properties -- Volume form -- Submanifolds -- 3.2 Characterisations of Kähler metrics -- 3.2.1 Background on connections -- 3.2.2 Kähler metrics and connections -- 3.3 Examples of Kähler manifolds -- 3.3.1 Chern form of line bundles -- 3.3.2 Fubini-Study metric -- 3.3.3 Blowups -- Exercises -- 4 Sheaves and Cohomology -- 4.1 Sheaves -- 4.1.1 Definitions, examples -- 4.1.2 Stalks, kernels, images -- 4.1.3 Resolutions -- The Cech resolution -- The de Rham resolution -- The Dolbeault resolution -- 4.2 Functors and derived functors -- 4.2.1 Abelian categories -- 4.2.2 Injective resolutions -- 4.2.3 Derived functors -- 4.3 Sheaf cohomology -- 4.3.1 Acyclic resolutions -- 4.3.2 The de Rham theorems.

4.3.3 Interpretations of the group H -- Exercises -- Part II The Hodge Decomposition -- 5 Harmonic Forms and Cohomology -- 5.1 Laplacians -- 5.1.1 The L metric -- 5.1.2 Formal adjoint operators -- 5.1.3 Adjoints of the operators… -- 5.1.4 Laplacians -- 5.2 Elliptic differential operators -- 5.2.1 Symbols of differential operators -- 5.2.2 Symbol of the Laplacian -- 5.2.3 The fundamental theorem -- 5.3 Applications -- 5.3.1 Cohomology and harmonic forms -- 5.3.2 Duality theorems -- Exercises -- 6 The Case of Kähler Manifolds -- 6.1 The Hodge decomposition -- 6.1.1 Kähler identities -- 6.1.2 Comparison of the Laplacians -- 6.1.3 Other applications -- 6.2 Lefschetz decomposition -- 6.2.1 Commutators -- 6.2.2 Lefschetz decomposition on forms -- 6.2.3 Lefschetz decomposition on the cohomology -- 6.3 The Hodge index theorem -- 6.3.1 Other Hermitian identities -- 6.3.2 The Hodge index theorem -- Exercises -- 7 Hodge Structures and Polarisations -- 7.1 Definitions, basic properties -- 7.1.1 Hodge structure -- 7.1.2 Polarisation -- 7.1.3 Polarised varieties -- 7.2 Examples -- 7.2.1 Projective space -- 7.2.2 Hodge structures of weight 1 and abelian varieties -- 7.2.3 Hodge structures of weight 2 -- 7.3 Functoriality -- 7.3.1 Morphisms of Hodge structures -- 7.3.2 The pullback and the Gysin morphism -- 7.3.3 Hodge structure of a blowup -- Exercises -- 8 Holomorphic de Rham Complexes and Spectral Sequences -- 8.1 Hypercohomology -- 8.1.1 Resolutions of complexes -- 8.1.2 Derived functors -- 8.1.3 Composed functors -- Application: Proof of the Leray-Hirsch theorem 7.33 -- 8.2 Holomorphic de Rham complexes -- 8.2.1 Holomorphic de Rham resolutions -- 8.2.2 The logarithmic case -- 8.2.3 Cohomology of the logarithmic complex -- 8.3 Filtrations and spectral sequences -- 8.3.1 Filtered complexes -- 8.3.2 Spectral sequences -- 8.3.3 The Frölicher spectral sequence.

8.4 Hodge theory of open manifolds -- 8.4.1 Filtrations on the logarithmic complex -- 8.4.2 First terms of the spectral sequence -- 8.4.3 Deligne's theorem -- Exercises -- Part III Variations of Hodge Structure -- 9 Families and Deformations -- 9.1 Families of manifolds -- 9.1.1 Trivialisations -- 9.1.2 The Kodaira-Spencer map -- 9.2 The Gauss-Manin connection -- 9.2.1 Local systems and flat connections -- 9.2.2 The Cartan-Lie formula -- 9.3 The Kähler case -- 9.3.1 Semicontinuity theorems -- 9.3.2 The Hodge numbers are constant -- 9.3.3 Stability of Kähler manifolds -- 10 Variations of Hodge Structure -- 10.1 period domain and period map -- 10.1.1 Grassmannians -- 10.1.2 The period map -- 10.1.3 The period domain -- 10.2 Variations of Hodge structure -- 10.2.1 Hodge bundles -- 10.2.2 Transversality -- 10.2.3 Computation of the differential -- 10.3 Applications -- 10.3.1 Curves -- 10.3.2 Calabi-Yau manifolds -- Exercises -- Part IV Cycles and Cycle Classes -- 11 Hodge Classes -- 11.1 Cycle class -- 11.1.1 Analytic subsets -- 11.1.2 Cohomology class -- 11.1.3 The Kähler case -- 11.1.4 Other approaches -- 11.2 Chern classes -- 11.2.1 Construction -- 11.2.2 The Kähler case -- 11.3 Hodge classes -- 11.3.1 Definitions and examples -- 11.3.2 The Hodge conjecture -- 11.3.3 Correspondences -- Exercises -- 12 Deligne-Beilinson Cohomology and the Abel-Jacobi map -- 12.1 The Abel-Jacobi map -- 12.1.1 Intermediate Jacobians -- 12.1.2 The Abel-Jacobi map -- 12.1.3 Picard and Albanese varieties -- 12.2 Properties -- 12.2.1 Correspondences -- 12.2.2 Some results -- 12.3 Deligne cohomology -- 12.3.1 The Deligne complex -- 12.3.2 Differential characters -- 12.3.3 Cycle class -- Exercises -- Bibliography -- Index.