Let us assume that X is a complex manifold of dimension n

1. Covering Spaces v.s. Manifold

Let X, Y be topological spaces and p : Y → X be a continuous map. We say that p:Y →X is a covering on X if every point x∈X has an open neighborhoodU together with a family of disjoint open sets{V_i :i∈I}inY such thatp⁻¹(U) =`

i∈IV_iand the map p:Vi →U is a homeomorphism. It follows from the definition that a covering p:Y →X is a surjective map.

Theorem 1.1. Letp:Y →X be a covering onX. X is a complex manifold¹ of dimension nif and only if Y is a complex manifold of dimension n.

Proof. Let us assume that X is a complex manifold of dimension n. Let us assume that {(U_α, ϕ_α)} is a (complex) atlas on X. Without loss of generality (using the property that p is a covering), we may assume that p⁻¹(Uα) = `

i∈IVα,i for a disjoint family of open subsets {V_α,i :i∈I} for each α so that p :Vα,i →Uα is a homeomorphism for eachi∈I.

Since p is surjective, {V_α,i : α, i} forms an open covering on Y. Since p : V_α,i → U_α is a homeomorphism, the composition ψα,i =ϕα◦p|_Vα,i :Vα,i→Cⁿ is a homeomorphism from Vα,i to ϕα(Uαi)⊂ Cⁿ.When Vα,i∩Vβ,j is nonempty, ψα,i(Vα,i∩Vβ,j) =ϕα(Uα∩Uβ) and ψ_β,j(V_α,i∩V_β,j) =ϕ_β(U_α∩U_β) are open subsets of Cⁿ.In this case,

ψ_β,j◦ϕ⁻¹_α,i=ϕ_β ◦ϕ⁻¹_α , ψα,i◦ϕ⁻¹_β,j =ϕα◦ϕ⁻¹_β

is biholomorphic from ϕ_α(U_α∩U_β) to ϕ_β(U_α∩U_β).We find that the family {(V_α,i, ψ_α,i)}

gives a complex atlas on Y.We find that Y is also a complex manifold of dimension n.

Let us assume that Y is a complex manifold of dimension n. Since p : Y → X is a covering on X, without loss of generality, we may assume that p⁻¹(X) is a disjoint union of open subsets {V_i : i ∈ I} of Y. Then p : Vi → X is a homeomorphism for each i ∈ I.

Since every open subset of a complex manifold is again a complex manifold,V_i is a complex manifold. Let V =Vi. Then p:V →X is a homeomorphism. If {(W_α, ϕα)} is a complex atlas on V, (here ϕ_α : U_α → Cⁿ) then the family {(p(W_α), ϕ_α ◦p⁻¹)} gives a complex atlas onX. (We leave it to the reader as an exercise.)ThereforeX is a complex manifold of dimensionn.

1The same argument also works for smooth manifold which needs not to be complex.

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