Proposition 5.43.13F2BB0D2DE2F4

Proof. We first construct the covering over the regular values, complete it across the critical values, and then extend its component quotient to $X$.

The critical values form a finite set $D\subset C$, by the critical-set argument in the proof of 5.43.7; that argument does not use connectedness of the fibers. Over $C^\circ =C\setminus D$, the map $p$ is a proper submersion. The local construction in lemma 5.43.4 applies there: horizontal transport over a disk has compact inverse image over its closure. Compact fibers have finitely many components, and the resulting local labels are glued by locally constant permutations. We obtain

with $q^\circ $ a holomorphic submersion with connected fibers and $r^\circ $ a finite unramified covering of degree $d$. The complement of a proper analytic subset in a connected complex manifold is connected, so $X^\circ $ and its image $C'^\circ $ are connected. Here only the fiber divisor is removed. Locally, the complement of the zero set of a nonzero holomorphic function in a ball is path connected: join two outside points through an outside intermediate point by two complex-line disks. Each restriction is nonzero because it contains an outside endpoint, and its isolated zeros can be avoided. Applying this local argument along a chain of balls proves the required global connectedness.

For each $s\in D$, take a small coordinate disk centered at $s$ and disjoint from the other critical values. A connected component of the covering over its puncture is a finite connected covering of a punctured disk, of some degree $e$. It has a coordinate $w$ in which $r^\circ (w)=w^ e$: the universal cover of the punctured disk is a half-plane with projection $z=\exp \zeta $, and the index-$e$ subgroup of its deck group is generated by $\zeta \mapsto \zeta +2\pi \mathrm i e$. The coordinate $w=\exp (\zeta /e)$ gives the asserted form. Attach one point $w=0$ to each such component. There are finitely many, with their degrees summing to $d$. Gluing these disks to $C'^\circ $ gives a connected Riemann surface $C'$ and a finite holomorphic map $r:C'\to C$. The new points have disjoint disk neighborhoods, so this gluing is Hausdorff. The inverse image of a small closed base disk is a finite union of closed disks; over the complement of their interiors, the original finite covering is proper. Thus $C'$ is compact.

We extend $q^\circ $. Fix $x\in p^{-1}(s)$, $s\in D$. On a small coordinate ball $B$ about $x$, the image of $p$ lies in the chosen disk. The complement $B\setminus p^{-1}(s)$ is connected. Its image under $q^\circ $ therefore lies in a single punctured disk just attached to $C'$, since those disks are the connected components of $r^{-1}$ of the punctured base disk. In its coordinate $w$, the function $w\circ q^\circ $ is holomorphic and bounded, and

The bounded holomorphic removable-singularity theorem extends it across $p^{-1}(s)\cap B$. To see this without assuming that the divisor is smooth, first extend at its smooth points by the one-variable theorem in transverse coordinates. The singular locus of the reduced divisor has complex codimension at least two in the ambient manifold, so Hartogs extension across that locus completes the extension. In dimension one there are only isolated points. Its $e$th power remains $p-s$ by the identity theorem, and hence its value is zero on that fiber. This defines a holomorphic map from $B$ to the attached disk. The local extensions agree on their dense common regular locus and therefore glue to $q:X\to C'$ with $p=r\circ q$. Its image is closed, by compactness, and contains the dense set $C'^\circ $, so $q$ is surjective.

Every fiber of $q$ is connected. Otherwise write one fiber as a union of two disjoint nonempty compact subsets and put them in disjoint open neighborhoods $U_1,U_2$. Properness gives a disk $V$ about their base point $t$ such that

The map $q$ is open: at any point, a complex line on which the nonconstant germ $q-t$ has its first nonzero term gives an open one-variable restriction. Consequently both $q(U_ i)$ contain a neighborhood of $t$. A nearby point of $C'^\circ $ then has a fiber meeting both neighborhoods, contradicting connectedness of the regular fibers.

The fibers of $p$ are the disjoint union of these connected $q$-fibers over the finitely many points of $r^{-1}(s)$. They are its components. Locally at $t$, the formula for a finite map is $r(w)-s=w^{e_ t}$ after changing the coordinate by a holomorphic root of a nonvanishing factor. Pulling back this equality by $q$ gives the divisor multiplicities stated above. Over $C^\circ $, $r$ is unramified and each sheet is one component, proving the degree assertion.

Finally, any factorization with the stated properties has over $C^\circ $ precisely this covering of fiber components. It is unramified there: a critical point of its finite map, together with surjectivity of its first factor, would make $p$ critical over that base point. Thus the component identification is holomorphic on $C'^\circ $. On every punctured disk it identifies two completions of the same finite covering, and the coordinates $w^ e$ above extend it biholomorphically across the added point. It gives a global isomorphism; density and surjectivity make it unique and give commutation with both maps. $\square$