Riemann-Hurwitz formula

For a degree-$d$ holomorphic map, $2g_X-2=d(2g_Y-2)+\sum (e_p-1)$.

Let $f:X\to Y$ be a nonconstant holomorphic map between compact Riemann surfaces that are connected spaces, without boundary. Write $g_X,g_Y$ for their genera and $d\ge 1$ for the degree of a holomorphic map. With analytic ramification index $e_p$ at each $p\in X$, the formula isHere $\chi =2-2g$ is the Euler characteristic. Only finitely many ramification points contribute. The holomorphic map is automatically surjective; the valency theorem gives ${\sum }_{p\in f^{-1}(y)}{e}_p=d$ even at a branch value of a holomorphic map. Constant maps are excluded.^[ref]

The useful interpretation is that total ramification of a holomorphic map measures how much the source Euler characteristic falls short of $d$ copies of the target. The ramification of a power map of the Riemann sphere spends that entire correction at two points; the double cover of the sphere with four branch values spends it at four points and produces a genus-one surface.

Count ramification points upstairs in $X$, weighted by $e_p-1$, rather than counting distinct branch values of a holomorphic map downstairs in $Y$. Grouping by fibres givesThe cardinality counts distinct points; the degree of a holomorphic map counts them with their analytic ramification indices. Thus each summand is the number of missing distinct preimages in that fibre.^[ref]

The phrase branch point varies across conventions: to keep the two surfaces distinct, use ramification point for $p\in X$ and branch value of a holomorphic map for $f(p)\in Y$. The existing name branch point of a holomorphic map here denotes the latter. Several ramification points can share one branch value of a holomorphic map.

For an integer $d\ge 2$, extend $f(z)=z^d$ to a holomorphic map from the Riemann sphere to itself with $f(\infty )=\infty$. A nonzero finite regular value has $d$ distinct preimages, so the degree of a holomorphic map is $d$. At zero the analytic ramification index is $d$. At infinity, use local coordinates $u=1/z$ upstairs and $v=1/f(z)$ downstairs: the same holomorphic map becomes $v=u^d$, so $e_{\infty }=d$ too. Everywhere else the derivative $d{z}^{d-1}$ is nonzero.

There are two distinct ramification points and two distinct branch values of a holomorphic map, butFor $d=3$, the total ramification of a holomorphic map is four, not two. The local-circle panel shows three nearby preimages converging to the single ramification point. For $d=1$ the holomorphic map is the identity and neither zero nor infinity is ramified.

Suppose a nonconstant holomorphic map $f:X\to {\mathbb{P}}^1$ from a connected compact Riemann surface has degree of a holomorphic map two and exactly four distinct branch values of a holomorphic map. The valency theorem forces each of their fibres to contain one ramification point with $e_p=2$, a simple ramification point of a holomorphic map. HenceThis determines the genus without recognizing a torus in a drawing.

A concrete realization is the compact Riemann surface obtained by adding the two points at infinity to the even-degree hyperelliptic model given by the algebraic curveProjection to $z$ has degree of a holomorphic map two. At a simple zero $a$ of $P$, $w$ is a local coordinate and $z-a=w^2/P^{\prime }(a)+O(w^4)$, so $e_p=2$. At infinity set $u=1/z$ and $s=w/z^2$: then $s^2=(1-4u^2)(1-u^2)$. There are two points with $u=0$, $s=\pm 1$, each unramified since $u$ is a local coordinate. The four finite roots are therefore all the branch values of a holomorphic map.^[ref]

Choose a triangulation of a surface on the Riemann sphere with these four branch values of a holomorphic map as vertices, combinatorially the boundary of a tetrahedron. Its counts are $V=4,E=6,F=4$. Upstairs, each vertex has one preimage and each open edge and face has two lifts: $V_X=4,E_X=12,F_X=8$, giving Euler characteristic $4-12+8=0$. This is a cell schematic, not a geometric immersion of $X$. The Triangulation proof of the Riemann-Hurwitz formula applies the same counting idea after refining a general triangulation of a surface around the branch values of a holomorphic map.^[ref]

For a nonconstant holomorphic map of Riemann surfaces, choose local coordinates centred at $p$ and $f(p)$. The map has the formThe positive integer $e_p$ is its ramification index. The point is ramified exactly when $e_p>1$ and contributes $e_p-1$ to the ramification divisor.

A point $p$ is ramified when its local degree $e_p$ is greater than one.

A ramification point is simple when its analytic ramification index is two. Its contribution to total ramification of a holomorphic map is one. In suitable local coordinates, the holomorphic map is $t\mapsto t^2$; two nearby preimages merge there. A ramification point with $e_p=3$ contributes two even though it is just one point.

For a nonconstant rational map, choose local source and target coordinates centred at $p$ and $R(p)$. If the local expression is $a{u}^e+O(u^{e+1})$ with $a\ne 0$, its local degree is $e$, and $p$ is critical exactly when $e\ge 2$. This definition covers poles and infinity; a simple pole is not critical merely because the usual finite-coordinate expression is unbounded. Coordinate changes have nonzero first derivatives and leave $e$ unchanged.

A critical orbit is the forward sequence $p,R(p),R^2(p),\dots$ starting at a critical point of a rational map. The postcritical set records the forward images, usually from the first iterate onward, of all critical points; a single critical orbit and the union of all such images are distinct notions. For quadratic polynomials there is one finite critical point, zero, and its orbit controls connectedness of the filled Julia set and membership in the Mandelbrot set. A bounded orbit can be preperiodic to a repelling cycle, as $0\mapsto i\mapsto -1+i\mapsto -i\mapsto -1+i$ for $z^2+i$; boundedness then does not imply Fatou membership.

Local degrees multiply under composition: $e_p(S\circ R)=e_p(R)e_{R(p)}(S)$, directly by substituting the two local power-series expansions. ConsequentlyIn particular every critical point of $R$ remains critical for each positive iterate, and every critical point of an iterate reaches a critical point of $R$ in fewer than $n$ steps. This covers infinity as well as finite points. For polynomials at finite points the same conclusion follows from $(P^n{)}^{\prime }(z)={\prod }_{j=0}^{n-1}{P}^{\prime }(P^j(z))$.

At a point of local degree $e$, the critical multiplicity is $e-1$. At an ordinary finite point this is the order of vanishing of the derivative. The invariant local-coordinate definition also works at poles and infinity. For $R(z)=z^d$, zero and infinity both have multiplicity $d-1$, so a critical-point count must count multiplicities: there are two distinct critical points but total multiplicity $2d-2$.

A degree-$d$ rational map has total critical multiplicity $2d-2$. Here is an algebraic proof. Critical points are isolated zeros of local derivatives and are finite in number by compactness. Change domain and range Möbius coordinates so that the poles are $d$ simple finite poles and infinity is a regular point with a finite image. Write $R=P/Q$, with $\mathrm{deg}Q=d$, and $R(z)=a+b/z+O(z^{-2})$, $b\ne 0$. Then $W=P^{\prime }Q-P{Q}^{\prime }$ has degree exactly $2d-2$, since $R^{\prime }=W/Q^2\sim -b/z^2$. At a pole, $W=-P{Q}^{\prime }\ne 0$. Thus the roots of $W$ are precisely the critical points and their zero orders are the critical multiplicities. The fundamental theorem of algebra gives the count; the coordinate changes preserve it.

A branch value is the image of a ramification point of a holomorphic map. Away from all branch values, a proper holomorphic map is an ordinary unramified covering.

In the compactified Riemann surface of $w^r=z^n-1$, a positive circuit enclosing the $n$ simple finite zeros sends the sheet label $j$ to $j+n$ modulo $r$. This permutation has $d=\gcd (n,r)$ cycles of length $r/d$. Each cycle fills in one point over infinity, with ramification index $r/d$. Hence infinity is unramified exactly when $r$ divides $n$. Each finite zero has ramification index $r$, so the base sphere has exactly $n$ branch values in that case and $n+1$ otherwise.

Triangulate the target with all branch values among its vertices. For a degree-$d$ map, every edge and face has $d$ lifts. Above a target vertex $v$, the number of lifted vertices isbecause ${\sum }_{p\mapsto v}{e}_p=d$. Thuswhich is equivalent to the Riemann-Hurwitz formula.

Up to invertible affine changes in source and target, every complex cubic polynomial is exactly one ofThe extension to the Riemann sphere has two finite units of ramification. A single finite critical point has ramification index three and gives the first form; two distinct simple critical points can be moved to $\pm 1$, after which integration gives the second.