The base of a Kodaira fibration has genus at least two; see [7, Introduction]. Theorem 1.1 therefore applies. ∎

The fibres of $f$ are homologous, so the topological degree of the restriction of $g$ to a fibre of $f$ is independent of the fibre. A holomorphic map of compact curves has degree zero exactly when it is constant. Thus, if $g$ is constant on one fibre of $f$, it is constant on every fibre. Local holomorphic sections of $f$ then give a holomorphic factorization $g=\alpha \circ f$. The map $\alpha :B\to C$ is nonconstant. Over a regular value of $\alpha$, a fibre of $g$ is the disjoint union of $\mathrm{deg}\alpha$ fibres of $f$. Its connectedness forces $\mathrm{deg}\alpha =1$, contrary to inequivalence.

Consequently $g$ is nonconstant on every fibre of $f$. Each such fibre is a smooth connected, hence irreducible, compact curve, so no curve can be contracted by $(f,g)$. Its fibres are therefore zero-dimensional compact analytic sets, hence finite. The map is proper, so it is finite. Its image is a closed analytic subset of dimension two in the connected surface $B\times C$, and is therefore the whole product. ∎

Choose a nonzero meromorphic section $s$ of the line bundle and write $\nabla s=\omega \otimes s$. If $s=z^{m}ue$, where $e$ is a holomorphic frame and $u$ is a holomorphic unit, then

with $\theta$ holomorphic. Thus the poles of $\omega$ have residues equal to the orders of the zeros and poles of $s$. The residue theorem gives

For the second assertion, choose foliation charts $(z,w)$ with $D=\{w=0\}$ and leaves given by $w=\text{constant}$. On a connected overlap the transverse coordinates satisfy $w^{\prime }=\psi (w)$, with $\psi (0)=0$ and ${\psi }^{\prime }(0)\ne 0$. The normal transition functions ${\psi }^{\prime }(0)$ are constant. They define a flat holomorphic connection on ${\mathcal{N}}_{D/X}$, namely the normal Bott connection. The first assertion applies. ∎

In a local coordinate $z$ on $D$, write the two metrics as ${\rho }_u{|dz|}^2$ and ${\rho }_v{|dz|}^2$. At a point with common ramification order $m$, both densities are ${|z|}^{2m}$ times a smooth positive function. Hence

is a smooth real function on all of $D$, independent of the coordinate. Away from the common zeros, the curvature equation gives

where ${\mathrm{\Delta }}_0$ is the Euclidean Laplacian in $z$. Both sides extend smoothly across the zeros. If ${\omega }_u$ and ${\omega }_v$ are the pulled-back area forms, this equation is $i\partial \overline{\partial }w={\omega }_u-{\omega }_v$. Multiplication by $w$ and integration by parts yield

The left-hand side is nonpositive, so $w$ is constant. Since ${\rho }_v$ is positive away from finitely many points, the curvature equation then gives $w=0$.

Where $u$ is locally invertible, $v\circ u^{-1}$ is a holomorphic local hyperbolic isometry. Lift small target charts to the unit disc. Such an isometry agrees locally with the disc automorphism having the same value and differential: isometries with the same first-order data agree along geodesics from that point. For a neighbourhood of a ramification point of $D$, choose a nearby unramified point and the corresponding disc automorphism $A$. The lifted maps satisfy $\tilde{v}=A\circ \tilde{u}$ on a nonempty open set, hence on the whole neighbourhood by the identity theorem. Descending to smaller target charts proves the graph assertion at the ramification point as well. ∎

Let $\nu :\widehat{D}\to X$ be the normalization map of an irreducible component of $R(f,g)$. The cotangent maps induced by $f$ and $g$ are nowhere-zero maps of line bundles into ${\nu }^{\ast }{\mathrm{\Omega }}_X^1$. Their images coincide along $\widehat{D}$, so they differ by a nowhere-zero holomorphic bundle isomorphism. Composing with ${\nu }^{\ast }{\mathrm{\Omega }}_X^1\to {\mathrm{\Omega }}_{\widehat{D}}^1$ shows that $d(f\circ \nu )$ and $d(g\circ \nu )$ are related by the same isomorphism. If either restricted map were constant, both would be constant, contradicting Lemma 2.1. They are therefore nonconstant and have identical ramification divisors.

By Lemma 2.3, each local branch of the ramification image is a graph of a local hyperbolic isometry. Fix a point $p$ of the ramification divisor. All source branches through $p$ give the same graph germ: its value is $(f(p),g(p))$, and its derivative is the unique scalar relating the nonzero covectors $d{f}_p$ and $d{g}_p$ in local target coordinates. To obtain the derivative even when the source branch is singular, take a limit from points of its normalization where the restricted differentials are nonzero. A local hyperbolic isometry is determined by its value and differential, so the graph germs agree.

We now apply this conclusion to the finite germ of $(f,g)$ at $p$. Choose disjoint source neighbourhoods around the finitely many points over $(f(p),g(p))$, and shrink a target bidisc so that its inverse image is contained in their union. The component representing the germ at $p$ is finite over the bidisc, has only $p$ over its centre, and may be chosen connected. After further shrinking, its branch locus is contained in the single graph just identified. This assertion concerns only this source representative; branch images from other points over the centre need not agree with it.

Straighten the graph to $\{v=0\}$. Away from its inverse image, the representative is an unramified cover of $\mathrm{\Delta }\times {\mathrm{\Delta }}^{\ast }$, where $\mathrm{\Delta }$ is a disc and ${\mathrm{\Delta }}^{\ast }=\mathrm{\Delta }\setminus \{0\}$. This cover is connected: a proper analytic divisor in a connected complex manifold does not disconnect it, since paths can be perturbed off its real-codimension-two strata. The fundamental group of $\mathrm{\Delta }\times {\mathrm{\Delta }}^{\ast }$ is $\mathbb{Z}$, so each connected finite cover is isomorphic to

The isomorphism is holomorphic because both covering maps are local biholomorphisms. It extends across the divisor by uniqueness of the normal finite extension: both extensions are obtained by integral closure of the target structure sheaf in the same covering field. Since the source is smooth, it is normal. Thus this is also the local form of the original finite map. Its reduced ramification curve is smooth and maps isomorphically to the local branch graph. In particular, different ramification components cannot meet, and both restrictions to each component are unramified. The local index $n$ is the coefficient of that component in $R(f,g)$ plus one.

Fix a component $D$. Choose a tubular neighbourhood $U$ which deformation retracts onto $D$. The induced image of ${\pi }_1(U)$ under $f$ is the image of ${\pi }_1(D)$ under ${f|}_D$. The covering-space lifting criterion for the unramified covering ${f|}_D:D\to B$ therefore gives a lift $r:U\to D$ with ${r|}_D={\mathrm{id}}_D$. This lift is holomorphic because the covering is locally biholomorphic. The same argument gives a holomorphic lift $s$ of $g$, also restricting to ${\mathrm{id}}_D$. The pair $(r,s)$ is locally a cyclic cover branched over the diagonal; in local coordinates it consequently has the form $r=x$, $s=x+y^n$.

If $x^{\prime }=\varphi (x)$ is a change of coordinate on $D$, the corresponding source coordinates satisfy the exact identities

Writing $y^{\prime }=\lambda (x)y+O(y^{n+1})$ gives $\lambda {(x)}^n={\varphi }^{\prime }(x)$. The normal coordinate changes therefore have $n$th powers equal to the tangent coordinate changes on $D$, proving ${\mathcal{N}}_{D/X}^{\phantom{}\otimes n}\cong TD$. Taking degrees gives (1); here $g(D)\ge 2$ because $D$ covers a hyperbolic curve.

To glue the cyclic action, factor the second identity in (3) as

On each sufficiently small connected overlap, a holomorphic $n$th root of $H$ gives $y^{\prime }=\xi yH{(x,y^n)}^{1/n}$ for a constant $n$th root of unity $\xi$. Every transition thus commutes with $y\mapsto \zeta y$. The coordinate actions glue as germs along $D$, preserve $r,s$, and have order $n$. For completeness, these germs can be realized on an invariant open neighbourhood. Compactness gives representatives of their finitely many powers on a common neighbourhood $V$. Choose a smaller neighbourhood $W$ of $D$ such that all their images lie in $V$ and all group composition identities hold on $W$. The union of these images of $W$ is invariant, because composing any two representatives on $W$ gives another power. Restricting to this union yields the asserted action on an actual neighbourhood. ∎

Suppose $D$ meets a component $E$ of $R_{13}$ at $p$. By Proposition 2.4, ${f_1|}_E$ and ${f_3|}_E$ are unramified coverings, and hence pull back the base metrics to $h_E$. The covectors $d{f}_1$ and $d{f}_3$ are proportional at $p$, so the two rank-one Hermitian forms ${(f_1^{\ast }{h}_{B_1})}_p$ and ${(f_3^{\ast }{h}_{B_3})}_p$ are proportional. Their common nonzero restriction to $T_{p}E$ makes the proportionality factor one.

Since ${f_1|}_D$ is an unramified covering, the restriction of this identity to $T_{p}D$ says that ${f_3|}_D$ attains equality in Schwarz–Pick at $p$. In particular it is nonconstant. The equality case of Schwarz–Pick, applied to its lift between universal covering discs, implies that ${f_3|}_D$ is an unramified covering.

Set $N={\mathcal{N}}_{D/X}$, ${\kappa }_D=2g(D)-2$, and let $n$ be the index of $D$ for the pair $(f_1,f_2)$. The bundle map on ${TX|}_D$ given by

vanishes on $TD$, and therefore induces a holomorphic map $N\to {({f_3|}_D)}^{\ast }T{B}_3$. The bundle of such maps has degree

A line bundle of negative degree has no nonzero holomorphic section, so $\beta =0$. Hence $D\subset R_{13}$, and proportionality with $d{f}_2$ gives $D\subset R_{23}$ as well. The same reasoning, with indices $1$ and $2$ interchanged, applies if $D$ initially meets $R_{23}$. Distinct components of each $R_{ij}$ are disjoint by Proposition 2.4.

In the common-component case, (1) applied to each pair states that $n_{ij}\mathrm{deg}N=\mathrm{deg}TD$. As $\mathrm{deg}N\ne 0$, all three indices $n_{ij}$ are equal. ∎

Use the retraction $r$ and cyclic action associated to $(f_1,f_2)$ by Proposition 2.4. Since $D\cap R_{13}=\mathrm{\varnothing }$, the line bundle $\ker d{f}_3$ is transverse to $\ker dr$ along $D$. After shrinking to an invariant neighbourhood, the restriction of $dr$ to $\ker d{f}_3$ is an isomorphism. Its inverse gives a holomorphic splitting $\sigma :r^{\ast }TD\to TU$.

At any $p\in D$, the differential of $\tau$ has eigenvalue $1$ on $T_{p}D$ and eigenvalue $\zeta \ne 1$ on the vertical line $\ker d{r}_p$. The invariant splitting $A(\sigma )$ therefore has image $T_{p}D$ at $p$. Its image on $U$ is a holomorphic line subbundle of $TU$. A line subbundle is integrable: if $V$ is a local nonvanishing generator, then $[aV,bV]=(aV(b)-bV(a))V$. It thus defines a regular holomorphic foliation with compact leaf $D$. Lemma 2.2 gives $\mathrm{deg}{\mathcal{N}}_{D/X}=0$, contradicting (1). ∎

Lift the three maps on a tubular neighbourhood of $D$ to retractions $r_i:U\to D$ with ${r_i|}_D={\mathrm{id}}_D$, as in the proof of Proposition 2.4. Shrink $U$ so that no other component of any $R_{ij}$ meets it. Let $n\ge 2$ be the common ramification index. Use the cyclic action for $(r_1,r_2)$, and local coordinates with

The derivative $\partial r_3/\partial y$ has order exactly $n-1$ along $D$, and $r_3(x,0)=x$. Consequently

On changing coordinates as in (3), comparison of the coefficients of $y^n$ in $r_3^{\prime }=\varphi (r_3)$ gives $\tilde{a}(\varphi (x)){\varphi }^{\prime }(x)={\varphi }^{\prime }(x)a(x)$, where $\tilde{a}$ denotes the coefficient in the new chart. Thus the coefficients define a holomorphic function on the compact curve $D$, and this function is a constant $a\ne 0$. Moreover,

The common index for $(r_2,r_3)$ is $n$, so $a\ne 1$.

Away from $D$, the line bundles $\ker d{r}_2$ and $\ker d{r}_3$ define splittings ${\sigma }_2,{\sigma }_3$ of $d{r}_1$. They extend meromorphically across $D$, with poles of order at most $n-1$. In the chosen coordinates,

The cyclic action preserves $r_1$ and $r_2$, so it fixes ${\sigma }_2$. Average ${\sigma }_3$ by (4).

Under the pushforward by $\tau (x,y)=(x,\zeta y)$, a vertical monomial $y{}^k\partial {}_y$ is multiplied by ${\zeta }^{1-k}$. An invariant vertical Laurent series can therefore contain this monomial only when $k\equiv 1(modn)$. Among the integers from $1-n$ through $0$, the only such integer is $1-n$. The leading monomial in ${\sigma }_3$ survives averaging, and hence

Since $a$ is a global constant different from $1$, the bundle map

is globally defined. Its pole cancels in every chart, so it is holomorphic across $D$. The coefficients in this affine combination sum to one, giving $d{r}_1\circ \sigma =\mathrm{id}$. It is invariant and has image $TD$ along $D$; equivalently, its local expression is $\sigma ({\partial }_x)={\partial }_x+O(y){\partial }_y$.

The image of $\sigma$ defines a regular holomorphic foliation with compact leaf $D$. Lemma 2.2 contradicts . ∎

Write ${\mathrm{\Gamma }}_i={\pi }_1(B_i)$ and identify ${\pi }_1(X)$ with a finite-index subgroup $\mathrm{\Gamma }\le {\mathrm{\Gamma }}_1\times {\mathrm{\Gamma }}_2$. Set

Each $H_i$ has finite index in ${\mathrm{\Gamma }}_i$, and $H_1\times H_2\subset \mathrm{\Gamma }$. Let $B_i^{\prime }\to B_i$ be the corresponding unramified curve coverings. Covering-space theory gives a finite unramified cover $p:B_1^{\prime }\times B_2^{\prime }\to X$ over $B_1\times B_2$. It is holomorphic because it is locally a lift through a biholomorphic covering chart.

Consider $F=h\circ p:B_1^{\prime }\times B_2^{\prime }\to C$. The restrictions $F_x:B_2^{\prime }\to C$ have a common topological degree. If this degree is zero, every $F_x$ is constant and $F$ depends only on $x\in B_1^{\prime }$. If the degree is positive, differentiation in a vector of $T_x{B}_1^{\prime }$ gives a holomorphic section of $F_x^{\ast }TC$ on $B_2^{\prime }$. Its degree is , so that section vanishes. This holds for every $x$ and every such vector, hence $F$ is independent of the first factor. In either case, $F$ factors through one projection.

Since $p$ is a local biholomorphism, this factorization implies that $dh$ vanishes on $\ker d{f}_i$ for one $i$. The map $h$ is therefore constant on every connected fibre of $f_i$. Local holomorphic sections of $f_i$ give $h=\alpha \circ f_i$ for a holomorphic map $\alpha :B_i\to C$. The map $\alpha$ is nonconstant, and connectedness of the fibres of $h$ forces $\mathrm{deg}\alpha =1$, as in Lemma 2.1. Thus $h$ and $f_i$ are equivalent. ∎

Suppose three pairwise inequivalent smooth fibrations exist. If $R_{12}\ne 0$, choose one of its components $D$. Lemma 3.1 gives two alternatives, excluded by Lemmas 3.2 and 3.3. Hence $R_{12}=0$. By Lemma 2.1, $(f_1,f_2)$ is then a finite unramified cover of $B_1\times B_2$. Lemma 3.4 makes $f_3$ equivalent to one of $f_1,f_2$, a contradiction. ∎