# Uniformization in Several Complex Variables

• Yum-Tong Siu
Part of the The University Series in Mathematics book series (USMA)

## Abstract

In this chapter we survey the current state of the theory of uniformization in several complex variables. In the case of one complex variable the uniformization theorem says that a simply connected Riemann surface is either the Riemann sphere ℙ1, or the open unit disk Δ or the Gaussian plane ℂ. The theory of uniformization in the higher-dimensional case attempts to find a suitable analog of it in the case of several complex variables. In the case of several complex variables the analog of ℙ1 is the Hermitian symmetric manifold of compact type. That means the complex projective space ℙ n , the hyperquadric$$\sum _{v=0}^{n+1}z_{v}^{2}=0$$in ℙ n+1 the Grassmannians, etc. The analog of Δ is the bounded symmetric domain. That means the open ball in ℂ n , the set of all m × n matrices A with I — AA t negative definite (with no symmetry conditions on A, or with the additional condition of skew-symmetry or symmetry on A), etc. The analog of ℂ is simply ℂ n . The problem is the following: when is a given complex manifold biholomorphic to a Hermitian symmetric manifold of compact type, a bounded symmetric domain, or the complex Euclidean space? In the case of Riemann surface the property of simple connectedness is sufficient to narrow it down to ℙ1, Δ, or ℂ. The topological condition of compactness separates ℙ1 from Δ andℂ. To distinguish between Δ and ℂ, one needs conditions such as curvature assumptions. In the case of higher dimension is it possible to characterize the symmetric spaces by topological or curvature or other conditions? Such a characterization is also known by the name of rigidity or strong rigidity, which means that it is not possible to deform the manifold or find a different one under such conditions.

## Keywords

Line Bundle Chern Class Complex Projective Space Maximum Compact Subgroup Compact Complex Manifold

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