= Solution
Let $x\in X$ and choose holomorphic <coordinate charts> centred at $x$ and $f(x)$. A nonconstant <holomorphic map> has a local expansion $w\circ f(z)=c z^m+O(z^{m+1})$, where $c\ne0$ and $m\geq1$. Taking a local holomorphic $m$th root of the nonvanishing factor and changing the source coordinate gives the exact local form $w=z^m$. The integer
$$
\boxed{\nu_f(x)=m}
$$
is the <ramification index of a holomorphic map>. A <ramification point of a holomorphic map> is a source point with $\nu_f(x)>1$, equivalently where its <derivative> vanishes; a <branch value of a holomorphic map>, also called a branch point here, is the image of such a point in $Y$. These are different sets, one in the source and one in the target.
Every <fiber of a function> is finite: locally it is the zero set of a nonzero <holomorphic function>, so its points are isolated, and <compactness> prevents infinitely many such zeros without an accumulation point. The same argument applied to the local <derivatives> makes the set of <ramification points of a holomorphic map> finite. The <open mapping theorem> and <compactness> show that $f(X)$ is both open and closed in the <connected> target $Y$, hence $f$ is <surjective>.
For any $y\in Y$, the <degree of a holomorphic map> is
$$
\boxed{d=\deg f=\sum_{x\in f^{-1}(y)}\nu_f(x).}
$$
To see independence of $y$, choose disjoint small <coordinate charts> about its finitely many preimages. The local form $z\mapsto z^{\nu_f(x)}$ shows that nearby values have $\nu_f(x)$ preimages counted with multiplicity in the corresponding chart. <Compactness> excludes additional preimages outside these charts for sufficiently nearby values: a sequence of such preimages would have a limit in $f^{-1}(y)$. The displayed sum is therefore locally constant, hence constant on <connected> $Y$. Away from <branch values of a holomorphic map> it is the ordinary number of preimages, and $f$ is a $d$-sheeted <covering map> there.
The <Riemann-Hurwitz formula> is
$$
\boxed{\chi(X)=d\chi(Y)-\sum_{x\in X}(\nu_f(x)-1),}
$$
or, in terms of the two <genera>,
$$
\boxed{2g_X-2=d(2g_Y-2)+\sum_{x\in X}(\nu_f(x)-1).}
$$
For its proof, choose a <triangulation of a surface> on $Y$ containing every <branch value of a holomorphic map> among its vertices. Every open edge and open face has $d$ lifts, since its interior avoids <branch values of a holomorphic map>. The lifted cells give a finite <CW complex> structure on $X$. If $V_Y,E_Y,F_Y$ are the target counts and $R=\sum_x(\nu_f(x)-1)$, the source counts are
$$
E_X=dE_Y,\qquad F_X=dF_Y,\qquad
V_X=\sum_{y\text{ vertex}}\#f^{-1}(y)=dV_Y-R.
$$
Indeed, at a target vertex the deficit from $d$ distinct preimages is precisely $\sum_{x\mapsto y}(\nu_f(x)-1)$, and all nonzero contributions occur over vertices. Taking the alternating sum gives $\chi(X)=d(V_Y-E_Y+F_Y)-R$, as required. This is the <Triangulation proof of the Riemann-Hurwitz formula>.
Back to article page
