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Pullback order formula for a branched holomorphic map

Statement

Let $f : X \to Y$ be a nonconstant holomorphic map between Riemann surfaces (Holomorphic maps and meromorphic functions on Riemann surfaces) and let $η$ be a meromorphic differential on $Y$ (Meromorphic differentials, orders and residues). The pullback $f^{*} η$ is the meromorphic differential on $X$ whose expression in charts $φ$ at $x \in X$ and $ψ$ at $f (x)$ is $f^{*} η = (h_{η} (w (z))) w^{'} (z) d z,$ where $w = ψ \circ f \circ φ^{- 1}$ is the chart expression of $f$ and $h_{η} d w$ is the expression of $η$ in the chart $ψ$ ; the family of expressions is compatible with all chart changes of $X$ and $Y$ .

Lemma. Let $x \in X$ , $y = f (x)$ , $e = e_{x} (f)$ the ramification index (Ramification index, ramification order and branch value) and let $η$ be a nonzero meromorphic differential near $y$ . Then ${ord}_{x} (f^{*} η) = e {ord}_{y} (η) + e - 1.$ In particular $e = 1$ reproduces ${ord}_{x} (f^{*} η) = {ord}_{y} (η)$ , and for the power map $w = z^{e}$ and $η = d w$ the formula gives the order $e - 1$ of the ramified pullback. This is the local analytic Riemann–Hurwitz interface; no global nonzero meromorphic differential and no canonical divisor is assumed to exist.

Facts & Assumptions

Given: A nonconstant holomorphic map $f : X \to Y$ of Riemann surfaces, points $x \in X$ , $y = f (x)$ , a nonzero meromorphic differential $η$ on a neighbourhood of $y$ , and the ramification index $e = e_{x} (f)$ .

[F1]

${ord}_{p} (ω)$ and ${Res}_{p} (ω)$ of a nonzero meromorphic differential are defined by the Laurent expansion of any local expression in a centred chart, and are independent of the centred chart; the transition law $h_{ψ} (w) = h_{φ} (z (w)) z^{'} (w)$ holds on overlaps (Meromorphic differentials, orders and residues).

[F2]

In suitable centred charts at $x$ and $y$ the map is the power map $ψ \circ f \circ φ^{- 1} (z) = z^{e}$ with the unique positive integer $e = e_{x} (f)$ ; a nonzero meromorphic germ at $0$ has a factorization $h (w) = w^{k} u (w)$ with $k \in Z$ , $u$ holomorphic near $0$ and $u (0) \neq 0$ , where $k = {ord}_{0} (h)$ is the order of $h$ at $0$ (Local power-map normal form on Riemann surfaces, Ramification index, ramification order and branch value, Isolated singularities: removable, poles, and essential singularities, Laurent expansion on an annulus, The order of a zero is the exponent in its local holomorphic factorization).

[F3]

The chain rule and product rule hold for holomorphic derivatives, and a composition of holomorphic functions on plane domains is holomorphic (The chain rule for complex derivatives, Linearity, product, reciprocal, and quotient rules for complex derivatives).

Proof

technique · direct

1.1F1F3given

(Compatibility of the pullback expressions.) Let $z$ and $z^{'}$ be charts on $X$ and $w, w^{'}$ the corresponding expressions of $f$ ; changing the source chart multiplies $w^{'} (z)$ by the transition derivative, and changing the target chart replaces $h_{η}$ by the transition law of $η$ and $w^{'}$ by the chain rule, so the coefficient transforms exactly as the coefficient of a differential on $X$ : the two ways of computing $f^{*} η$ in overlapping charts agree by [F3] and [F1]. Hence $f^{*} η$ is a meromorphic differential on $X$ ; it is nonzero because on a connected chart domain the product of the coefficient $h_{η} \circ w$ and the derivative $w^{'}$ is not identically zero: $w^{'}$ is not identically zero since the nonconstant holomorphic $w$ has isolated values, $h_{η} \circ w$ is not identically zero since the zeros and poles of the nonzero meromorphic $h_{η}$ are isolated while $w$ is not locally constant, and a product of two functions on a domain, neither identically zero, is not identically zero.

1.2F2F3

(Computation in normal-form coordinates.) Choose centred charts as in [F2], so that $w (z) = z^{e}$ ; write the expression of $η$ in the target chart as $h (w) d w$ with $h (w) = w^{k} u (w)$ , $k = {ord}_{y} (η) \in Z$ and $u$ holomorphic with $u (0) \neq 0$ . Then $w (z)^{k} u (w (z)) w^{'} (z) = z^{e k} u (z^{e}) e z^{e - 1} = e z^{e (k + 1) - 1} u (z^{e})$ , and the coefficient $e u (z^{e})$ is holomorphic near $0$ with value $e u (0) \neq 0$ ; hence the expression of $f^{*} η$ in the source chart has order $e (k + 1) - 1 = e k + e - 1$ at $0$ .

2.1step 1.1step 1.2F1∎

(Conclusion.) By [F1] the order of $f^{*} η$ at $x$ is the order of its expression in any centred source chart, and the order of $η$ at $y$ is the exponent of the factorization in [F2]; step 1.2 computes the former as $e {ord}_{y} (η) + e - 1$ in the normal-form charts, and step 1.1 shows that this is the pullback differential defined by all charts. The special cases $e = 1$ and $η = d w$ follow by substitution.

Remarks

The formula is local: it never chooses a global differential, and the order $e - 1$ is the ramification order of $x$ . It is applied in Riemann–Hurwitz formula for compact Riemann surfaces as one of the two interfaces of that theorem, the other being the Euler-characteristic cell count. The same computation shows that a coordinate change, in which $e = 1$ at every point, preserves orders, which is the invariance already recorded in Meromorphic differentials, orders and residues; the point here is that a genuinely ramified map with $e > 1$ multiplies the target order by $e$ and adds $e - 1$ . Thus a nonzero holomorphic differential of order $k \geq 0$ at the image pulls back to a zero of order $e k + e - 1$ , which equals $e - 1$ exactly when the target differential is nonvanishing there.

Depends on

Used by

Riemann–Hurwitz formula for compact Riemann surfaces Theorem

Dependency tree · two levels

43 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eduard Looijenga, Riemann Surfaces (2007) (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Math 213b course notes (2026) (standard reference, not scraped)