2 - Complex Manifolds

Summary

In this chapter, we introduce and study the notion of a complex structure on a differentiable or complex manifold. A complex manifold X of (complex) dimension n is a differentiable manifold locally equipped with complex-valued coordinates (called holomorphic coordinates) z₁, …, z_n, such that the diffeomorphisms from an open set of ℂⁿ to an open set of ℂⁿ given by coordinate changes are holomorphic. By the definition of a holomorphic transformation, we then see that the structure of a complex vector space on the tangent space T_X,x given by the identification T_X,x ≅ ℂⁿ induced by the holomorphic coordinates z₁, …, z_n does not depend on the choice of holomorphic coordinates. The tangent bundle T_X of a complex manifold X is thus equipped with the structure of a complex vector bundle. Such a structure is called an almost complex structure.

After some preliminaries on manifolds and vector bundles, we turn to the proof of the Newlander–Nirenberg theorem, which characterises the almost complex structures induced as above by a complex structure. This ‘integrability’ criterion is extremely important in the study of the deformations of the complex structure of a manifold. Indeed, we could describe them as the deformations of the almost complex structure (which are essentially parametrised by a vector space of differentiable sections of a certain bundle over X) satisfying the integrability condition. This will enable us to put the structure of an infinite-dimensional manifold on this space of deformations, whose quotient by the group of diffeomorphisms of X describes the deformations of the complex structure of X up to isomorphism.