Kähler Manifolds

Abstract

Abstract In this chapter we shall introduce the very rich geometry of complex and Kähler manifolds. Complex manifolds are manifolds with a geometric structure called a complex structure, which gives every tangent space the structure of a complex vector space. They are defined in §4.1, together with complex submanifolds and holomorphic maps. Section 4.2 discusses tensors on complex manifolds, and their decomposition into components using the complex structure, and §4.3 defines holomorphic vector bundles over a complex manifold.