Tori

Abstract

A complex manifold M is constructed analogously to a smooth manifold. We specify an atlas U = {(U, ø)}, where each chart U ⊂ M is an open set and ø: U → ℂ^m is a homeomorphism of U onto its image that is assumed to be open in ℂ^m. It is assumed that the transition functions ψ o ø ⁻¹ : ø(U ⋂ V) → ψ (U ⋂ V) are holomorphic for any two charts (U, ø) and (V,ψ). A complex Lie group (or complex analytic group) is a Hausdorff topological group that is a complex manifold in which the multiplication and inversion maps G×G → G and G → G are holomorphic. The Lie algebra of a complex Lie group is a complex Lie algebra. For example, GL(n, ℂ) is a complex Lie group.