Complex Geometry: An Introduction

The book is based on a year course on complex geometry and its interaction with Riemannian geometry. It prepares a basic ground for a study of complex geometry as well as for understanding ideas coming recently from string theory. The book starts with a summary of facts from several complex variables (properties of holomorphic functions, properties of analytic sets, including the Nullstellensatz), algebraic preliminaries on the Grassmann algebra (needed in a study of Kähler manifolds) and properties of the Dolbeault complex. The second chapter treats complex manifolds and holomorphic vector bundles (including a discussion of divisors and global sections of holomorphic line bundles, properties of the complex projective space and its use for a complex surgery and the Newlander-Nierenberg theorem). Kähler manifolds are studied in the third chapter (including the Hodge theory and a discussion of the Hodge conjecture). In the fourth chapter, the author describes basic tools of complex analysis on complex manifolds (connections and their curvature, Chern classes). Central results in complex algebraic geometry (the Hirzebruch-Riemann-Roch theorem, the Kodaira vanishing and embedding theorems) are contained in the fifth chapter. The last chapter introduces local aspects of classification of complex structures on a given smooth manifold and the deformation theory of complex manifolds. The case of Calabi-Yau manifolds is treated using the language of differential algebras. The book is a very good introduction to the subject and can be very useful both for mathematicians and mathematical physicists.

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