Proper family from disjoint analytic plumbing

Since $\mathrm{\mid }{q}_j\mathrm{\mid }<r_j^0{s}_j^0$, the fixture boundary $\mathrm{\mid }{z}_j\mathrm{\mid }=r_j^0$ has $\mathrm{\mid }{w}_j\mathrm{\mid }=\mathrm{\mid }{q}_j\mathrm{/}{z}_j\mathrm{\mid }<s_j^0$, and the boundary $\mathrm{\mid }{w}_j\mathrm{\mid }=s_j^0$ has $\mathrm{\mid }{z}_j\mathrm{\mid }<r_j^0$. Thus the fixture has two disjoint boundary collars, each attached to the matching boundary of the original complement by its given coordinate. The plumbing neighborhoods of distinct pairs are disjoint, so these finitely many identifications do not overlap or identify points from different pairs.

In the interior of a fixture, eliminating $q_j$ by $q_j=z_j{w}_j$ gives holomorphic coordinates consisting of the base coordinates, $z_j,w_j$, and all other sewing parameters. At the seam $\mathrm{\mid }{z}_j\mathrm{\mid }=r_j^0$, the transition to the original chart is $w_j=q_j\mathrm{/}{z}_j$, holomorphic because $z_j\mathrm{\ne }0$; at the other seam it is $z_j=q_j\mathrm{/}{w}_j$, holomorphic because $w_j\mathrm{\ne }0$. These collar charts extend across the seams. The total space is smooth at the node as well, since $q_j=z_j{w}_j$ is a coordinate on the target rather than a constraint on the total-space coordinates. To check Hausdorffness, form the equivalence relation on the finite disjoint union of the original complement and fixtures. Each seam is a closed embedded boundary with a disjoint collar; the relation is the diagonal together with the finitely many closed graphs of the seam identifications and their reverses. Every equivalence class has at most two points, because distinct pairs have disjoint neighborhoods and the two boundary circles of each fixture are disjoint. Given two distinct classes, choose disjoint neighborhoods of their finitely many representatives in the Hausdorff pieces; if a representative lies on a seam, shrink its neighborhood within that seam's collar and saturate it by the paired collar neighborhood. Disjointness of the collars keeps the resulting saturated neighborhoods disjoint, so their images separate the two classes in the quotient. Thus the finite gluing is Hausdorff. By [F3], the fibers have the asserted annulus and node models.

Let $K\subset B\times Q$ be compact. It is closed because the target is Hausdorff. The original-complement part of its preimage is a closed subset of ${\pi }^{-1}(K_B)\times K_Q$, where $K_B,K_Q$ are the compact projections of $K$; this product is compact by [F1]. Each fixture part of the preimage is closed in $K_B\times {\bar{D}}_{r_j^0}\times {\bar{D}}_{s_j^0}\times K_Q$, hence compact. The full preimage of $K$ is obtained from the finite union of these compact pieces by the collar identifications, so it is a continuous image of a compact space and is compact. This proves properness.

Fix $\mathbf{q}\in Q^{\times }$. The closed fixture in the Statement is parametrized by $z_j$ on the closed annulus $\mathrm{\mid }{q}_j\mathrm{\mid }\mathrm{/}{s}_j^0\le \mathrm{\mid }{z}_j\mathrm{\mid }\le r_j^0$, with $w_j=q_j\mathrm{/}{z}_j$. It is compact as a closed bounded subset of $\mathbb{C}$; F3 supplies the smooth open-annulus model for its interior. Its two boundary circles have disjoint collars by step 1.1. Attaching them to the compact original complements therefore gives a compact fiber without boundary. In the fixture, $q_j\mathrm{\ne }0$ implies $(z_j,w_j)\mathrm{\ne }(0,0)$, so the differential of $(z_j,w_j)\mapsto z_j{w}_j$ is the nonzero functional $w_{j}d{z}_j+z_{j}d{w}_j$. The sewing parameters for different pairs are independent, and outside the fixtures the original family map is a submersion by [F1]. Hence the glued map is a submersion over $B\times Q^{\times }$. It is holomorphic by step 1.2 and proper by step 1.3, proving the claim.