1. Elementary Properties of Holomorphic Mapping
Let f : X → Y be a nonconstant holomorphic mapping. Let p ∈ X and q = f (p) in Y.
Suppose (ψ, U ) and (ϕ, V ) are local charts around p and q respectively and assume ψ(p) = 0 and ϕ(f (p)) = 0 for simplicity. Without loss of generality, we can further assume f (U ) ⊂ V.
Set
F = ϕ ◦ f ◦ ψ−1: ψ(U ) ⊂ C → ϕ(V ) ⊂ C.
Since f is holomorphic, F is holomorphic on the open set ψ(U ). Moreover, F (0) = 0 implies that there exists a unique n ≥ 1 and an open subset D of ψ(U ) containing 0 and a unique nonvanishing holomorphic function h(ζ) on D such that F (ζ) = ζnh(ζ). Since h(ζ) 6= 0 on D, we define H(ζ) = exp(ln h(ζ)/n) on D. This gives us
F (ζ) = (ζH(ζ))n, ζ ∈ D.
Define a holomorphic function α : D → C by ζ → ζH(ζ). Then α0(ζ) = H(ζ) + ζH0(ζ).
We obtain α0(0) = H(0) 6= 0. Hence in a neighborhood of 0, α is biholomorphic by inverse function theorem. We may also denote such a neighborhood by D and assume that α : D → α(D) is biholomorphic. Now, F (ζ) = (α(ζ))n. Denote z = α(ζ). Then ζ = α−1(z).
We see that
(F ◦ α−1)(z) = zn
for all z ∈ α(D). Denote eψ : α ◦ ψ : eU → α(D), where eU = (ψ ◦ α)−1(α(D)) is an open neighborhood of p. Then ( eψ, eU ) is a local chart around p with eψ(p) = 0 and
ϕ ◦ f ◦ eψ−1
(z) = zn, z ∈ α(D).
We proved:
Theorem 1.1. (Local Normal Form) Let X and Y be Riemann surfaces and f : X → Y be a nonconstant holomorphic mapping. Suppose p ∈ X and q = f (p) ∈ Y. Then there exists a unique integer n ≥ 1 and local charts (U, ψ) around p and (V, ϕ) around q such that
(1) ψ(p) = 0, ϕ(q) = 0, (2) f (U ) ⊂ V
(3) The map F = ϕ ◦ f ◦ ψ−1 : ψ(U ) → ϕ(V ) is given by F (z) = zn, ∀z ∈ ψ(U ).
We denote the integer n in Theorem 1.1 by ep(f ) called the multiplicity of f at p. We say that p is a ramification point of f if ep(f ) ≥ 2 with ramification index ep(f ). In this case q = f (p) is called a branched point of f. The integer ep(f ) − 1 denoted by bp(f ) is called the branched number of f at p.
A branched covering of a compact Riemann surface Y is a compact Riemann surface X together with a nonconstant holomorphic mapping f : X → Y.
Proposition 1.1. Let f : X → Y be a branched covering of Y. Then there are only finitely many ramification points and branched points.
Proof. Let p be any point on X and q = f (p) in Y. We can choose a local chart (ϕ, V ) around q with ϕ(q) = 0 and a local chart (ψ, V ) of p with ψ(p) = 0 so that f (U ) ⊂ V and F = ϕ ◦ f ◦ ψ−1 : ψ(U ) → ϕ(V ) is given by z 7→ zn for n ≥ 1. Since a holomorphic mapping is always open, we may assume V = f (U ). For z 6= 0 in ψ(U ), F0(z) = nzn−16= 0.
By inverse function theorem, the map F : ψ(U ) \ {0} → ϕ(V ) \ {0} is locally invertible at all points of ψ(U ) \ {0}. This shows that f : U → V is locally invertible possibly except p.
Denote this U by Up. Then {Up : p ∈ X} forms an open covering of X. Since X is compact,
1
2
there exist p1, · · · , pn so that {Upj : 1 ≤ j ≤ n} covers X. Then f : Upj → f (Upj) is locally invertible possibly except pj. This shows that the set of ramification points is a subset of {p1, · · · , pn}. Hence the set of ramification points is a finite set; the set of branched points of f is finite.
Lemma 1.1. Let f : X → Y be a branched covering of Y by X. Then f−1(q) is discrete.
Proof. Let p be a point in f−1(q). Choose a local chart (ψ, U ) around p and a local chart (ϕ, V ) around q with ψ(p) = 0 and ϕ(q) = 0. Assume f (U ) ⊂ V. Then
F = ϕ ◦ f ◦ ψ−1: ψ(U ) → ϕ(V )
is holomorphic sending 0 to 0. Since F (0) = 0, and F is holomorphic, by coincidence principle, we can choose a small neighborhood D of 0 contained in ψ(U ) so that 0 is the only zero of F. (Zeros of a holomorphic function are isolated.) In other word, in the open neighborhood U0 = ψ−1(D), of p, p is the only point in U0 so that f (p) = q. Hence U0∩ f−1(q) = {p}. This shows that f−1(q) is discrete. Proposition 1.2. Let f : X → Y be a branched covering of Y by X. The function
q 7→ X
p∈f−1(q)
ep(f )
is a locally constant function. By connectedness of Y, it is a constant function.
Let f : X → Y be a branched covering of Y by X. Proposition 1.2 allows us to define the degree of f by
deg f = X
p∈f−1(q)
ep(f ),
where q is any point of Y.
Theorem 1.2. Let f : X → Y be a ramified cover of Y by X. Denote B = P
p∈Xbp(f ).
Then we obtain χ(X) = deg f · χ(Y ) + B and hence g(X) = (deg f )(g(Y ) − 1) + 1 + B/2.
Proof. Let S be the set of all branched points of f, i.e. for each q ∈ S, bp(f ) ≥ 1 for p ∈ f−1(q). Since X and Y are compact, S is a finite set. We triangulated Y so that every point of S is a vertex of the triangulation. Suppose that this triangulation has v-vertices, e-edges and f -faces. We can lift the triangulation to a triangulation of X via f. The induce triangulation has nv − B vertices, ne-edges and nf -faces. Here n = deg f. Hence the Euler characteristic of X is given by
χ(X) = nv − B − ne + nf
= n(v − e + f ) − B
= nχ(Y ) − B.
Since χ(X) = 2 − 2g(X) and χ(Y ) = 2 − 2g(Y ), 2 − 2g(X) = n(2 − 2g(Y )) − B. This implies the required formula.
The number B is called the total branching number of f. This Theorem also implies that B is always an even number.
