Introduction page 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 ∂g ∂z = f 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 ∂ and ∂ 2.3.1 Definition 2.3.2 Local exactness 2.3.3 Dolbeault complex of a holomorphic bundle 2.4 Examples of complex manifolds Exercises v vi Contents 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 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.
Dessins d'Enfants are combinatorial objects, namely drawings with vertices and edges on topological surfaces. Their interest lies in their relation with the set of algebraic curves defined over the closure of the rationals, and the corresponding action of the absolute Galois group on them. The study of this group via such realted combinatorial methods as its action on the Dessins and on certain fundamental groups of moduli spaces was initiated by Alexander Grothendieck in his unpublished Esquisse d'un Programme, and developed by many of the mathematicians who have contributed to this volume. The various articles here unite all of the basics of the subject as well as the most recent advances. Researchers in number theory, algebraic geometry or related areas of group theory will find much of interest in this book.
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