Hodge Theory and Complex Algebraic Geometry I

As the title clearly implies, the book deals with the basics of Hodge theory and its relation with complex algebraic geometry. The first four chapters are devoted to preliminaries. The main result of the first chapter is the Riemann and Hartogs Theorems on extension of holomorphic functions and existence of local solutions to the Cauchy-Riemann equations. The second chapter contains an introduction to (holomorphic) vector bundles and the Dolbeault complex on differentiable manifolds with complex structure (determined from an almost complex structure via Newlander-Nirenberg theorem). The third chapter contains a self-contained introduction to Kähler geometry, Kähler metrics and Chern connections on a holomorphic vector bundle equipped with a Kähler metric. The introductory part ends with the fourth chapter describing theory of sheaves, cohomology of a topological space with values in a sheaf and theory of acyclic resolutions leading to a proof of the de Rham theorem.
The second part of the exposition is devoted to a proof of the Hodge and Lefschetz decomposition theorems of cohomology of a complex manifold. The first two chapters of this part summarize a necessary background from analysis on Hilbert spaces used to define (formal) adjoints of elliptic operators, their Laplace operators and finally, as an application, harmonic forms and their relation to cohomology. The following two chapters give conceptual applications of these results. The notion of a polarized Hodge structure is explained and applied to the Kodaira embedding theorem, which states that a complex manifold is projective if it admits an integral polarization. The next chapter is devoted to the holomorphic de Rham complex and the interpretation of the Hodge Theorem in terms of degeneracy of the Frölicher spectral sequence. The chapter ends with an introduction to holomorphic de Rham complex with logarithmic singularities for quasi-projective smooth varieties with mixed Hodge structure on their cohomology. The third part is devoted to a study of variations of Hodge structures. The notion of a family of complex differentiable manifolds leads to the construction of the Kodaira-Spencer map and Gauss-Manin connection associated to the local system of cohomology of fibers of this family. Using the Kodaira-Spencer map and the cup product in Dolbeault cohomology, the period map and its differential are described. In the last chapter, various “cycle classes” are studied. In particular, Hodge classes arising in the study of morphisms of Hodge structures are described in more detail. The Deligne cohomology and the Abel-Jacobi map are introduced, although they are studied in more detail in the next volume of the series.

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