Proof. We give the argument at a possibly singular and nonnormal point \(q\). Work first on one irreducible algebraic component \(V_0\) through \(q\), of positive dimension \(d\). Noether normalization [10] gives a finite algebraic map \(\pi:V_0\to\mathbb C^d\). Translate so that \(\pi(q)=0\). By properness and finiteness, we may restrict over a sufficiently small polydisc \(B\) about \(0\) and retain just the part \(\Omega\) near \(q\), with \[\pi:\Omega\longrightarrow B\quad\hbox{proper and finite}, \qquad \pi^{-1}(0)\cap\Omega=\{q\},\qquad \Omega\subset W.\] To make this restriction, take disjoint neighborhoods of the finitely many points over \(0\) and shrink \(B\) until its entire inverse image lies in their union. The part near \(q\) is then both open and closed in that inverse image. Outside a proper algebraic hypersurface \(E\) in \(B\), the restricted map is a covering with a fixed number \(r\geq1\) of holomorphic sheets. This follows also directly from separability of the finite function-field extension: discard the denominators and discriminant of a primitive element.

For \(|h|\leq1\), form the monic polynomial of its sheet values, \[P_h(z,T)=\prod_{x\in\pi^{-1}(z)\cap\Omega}(T-h(x)) =T^r+\sum_{\nu=0}^{r-1}a_{\nu,h}(z)T^\nu, \qquad z\in B\setminus E.\] The coefficients are single-valued holomorphic functions bounded by constants depending only on \(r\). They extend holomorphically across \(E\) by the removable-singularity theorem for bounded holomorphic functions. One elementary proof uses a complex line through the point that is not contained in \(E\), a small circle on that line missing \(E\), and the Cauchy integral on nearby parallel circles. Properness and continuity of \(h\) give \[P_h(0,T)=(T-h(q))^r.\] The polynomial identity \(P_h(\pi(x),h(x))=0\) holds on all of \(\Omega\) by continuity, since the sheet locus is dense. Cauchy estimates for the uniformly bounded coefficients, on a smaller polydisc, therefore yield \[|h(x)-h(q)|^r =|P_h(0,h(x))-P_h(\pi(x),h(x))| \leq C\|\pi(x)\|.\] Here \(C\) is independent of \(h\). This proves equicontinuity at \(q\) on \(V_0\). There are only finitely many components through \(q\); applying the argument to each proves the assertion on their union. A zero-dimensional component requires no argument. Local compactness, a countable compact exhaustion, and the Arzelà–Ascoli theorem now give compact convergence of a subsequence.

For the maximum assertion, use the same chart and consider the sheet average \[b_h(z)=\frac1r\sum_{x\in\pi^{-1}(z)\cap\Omega}h(x).\] It extends holomorphically to \(B\), satisfies \(|b_h|\leq1\), and has \(b_h(0)=h(q)\). If \(h_j\to h_\infty\) locally uniformly and \(h_\infty(q)=1\), ordinary normal-family compactness on \(B\) gives, after another extraction, a holomorphic limit of the sheet averages with value \(1\) at \(0\). The usual maximum principle makes that limit identically \(1\). On every regular fiber, an average of numbers in the closed unit disc can equal \(1\) only if every number equals \(1\). Thus \(h_\infty=1\) on the sheet locus near \(q\), hence everywhere near \(q\) by continuity. Apply this on every component through \(q\). Finally, \(\{h_\infty=1\}\) is both closed and open in \(W\), which proves the connected case. No identification of weakly holomorphic functions with holomorphic functions on a nonnormal space was used. ◻

Proof. Choose a positive-dimensional irreducible component \(V_0\) whose smooth points not shared with another component meet \(U\). Such a component exists: these points are dense in each positive-dimensional component, whereas a zero-dimensional irreducible component is isolated and cannot meet the connected nonsingleton \(U\). There is a polynomial \(P\) vanishing on the other components and on the singular locus of \(V_0\), but not identically on \(V_0\). In particular, \(P\) is nonzero somewhere in \(U\).

Maximize \(\log|P(q)|+\|q\|^2\) on \(\overline U\), with value \(-\infty\) at zeros of \(P\). A maximizer \(p\) has \(P(p)\ne0\), so it is a smooth, unshared point of \(V_0\). It belongs to \(\partial U\): near such a point the function is strictly plurisubharmonic, and hence cannot have an interior maximum. The entire function \[h(q)=\frac{P(q)}{P(p)} \exp\bigl(2\overline p\cdot(q-p)\bigr)\] satisfies \[ |h(q)|\leq \exp\bigl(-\|q-p\|^2\bigr) \quad(q\in\overline U),\qquad h(p)=1. \tag{1}\] Indeed, the logarithm of the left-hand side is at most \(\|p\|^2-\|q\|^2+2\operatorname{Re}(\overline p\cdot(q-p))\).

Choose \(q_j\in U\) tending to \(p\), and write \(q_j=\gamma_j k_j\) with \(k_j\in K\). After extraction, \(k_j\to k\in K\), and Lemma 2 gives locally uniform convergence of \(h\circ\gamma_j\). Equicontinuity at the moving points \(k_j\) shows that the limit has value \(1\) at \(k\). It is therefore identically \(1\) on \(U\). Equation (1) implies that \(\gamma_j\) converges to \(p\) uniformly on every compact subset of \(U\).

For each \(q\in U\), the point \(\gamma_j(q)\) eventually lies in a smooth ambient neighborhood of \(p\). Since a biholomorphism preserves smoothness of the complex space, \(q\) is smooth. A smooth point belongs to exactly one irreducible component of the ambient variety; the component is locally constant on \(U\), and hence constant by connectedness. This gives \(V\). ◻

Proof. Coordinate balls give finite disc chains locally, hence globally by connectedness. Bounded ambient coordinate functions and the Schwarz lemma give a constant \(c>0\) such that \[ d_U(x,y)\geq c\|x-y\|. \tag{2}\] For example, send each bounded coordinate to \(\mathbb D\); the Poincaré distance there dominates a constant times Euclidean distance, and sum along a chain. Conversely, small affine complex discs in a manifold chart give a local upper bound tending to zero with the coordinate separation. Thus \(d_U\) is a distance and induces the original topology.

Choose a relatively compact neighborhood of \(K\) in \(U\). Equation (2) and compactness of \(K\) give an \(r>0\) for which every closed \(r\)-ball centered in \(K\) lies in that neighborhood’s compact closure. Such a ball is closed in the original topology, hence compact. Translating by \(\Gamma\) shows that every closed \(r\)-ball in \(U\) is compact, with the same \(r\).

We spell out why this uniform local statement implies properness. Fix \(x\), and suppose \(\overline B_{d_U}(x,R)\) is compact. Cover it by finitely many open \(r/4\)-balls centered at \(x_1,\ldots,x_l\). For \(d_U(x,y)\leq R+r/4\), choose a disc chain of total Poincaré length less than \(d_U(x,y)+r/4\). Replace each disc link by a Poincaré geodesic segment. If the total length exceeds \(R\), the point at accumulated length \(R\) lies in \(\overline B_{d_U}(x,R)\), and the remaining length is less than \(r/2\). Hence \(y\) belongs to an \(r\)-ball centered at some \(x_i\). If the total length is at most \(R\), the same conclusion is immediate. Consequently \[\overline B_{d_U}(x,R+r/4) \subset\bigcup_{i=1}^l\overline B_{d_U}(x_i,r).\] The set on the left is closed and the union on the right is compact. Starting at \(R=r\) and iterating proves compactness for every radius. ◻

Proof. The set \(T=\{t:A_t(\overline\mathbb D)\subset U\}\) is relatively open in \([0,1]\) by compactness of the closed disc. To see that it is closed, let \(t_j\in T\) tend to \(t_*\). The boundary condition and continuity give a fixed collar \(1-\varepsilon\leq|\zeta|\leq1\) whose images, for \(t\) near \(t_*\), lie in a compact set \(L\subset U\). Fix \(\zeta_0\) in the interior of this collar and \(x_0\in U\). By Proposition 4, \(M=\max_{x\in L}d_U(x_0,x)\) is finite. For each fixed \(\zeta\in\mathbb D\), the earlier discs take values in \(U\), so Schwarz contraction gives \[d_U(x_0,A_{t_j}(\zeta)) \leq M+\rho(\zeta_0,\zeta).\] Properness places these values in a fixed compact subset of \(U\). Their ambient limit \(A_{t_*}(\zeta)\) therefore belongs to \(U\). The boundary already lies in \(U\), so \(t_*\in T\). Since \(T\) is nonempty, open, and closed in \([0,1]\), it is all of \([0,1]\). ◻

Proof. Put \(d=\dim_{\mathbb R}S\), and write the ambient complex coordinate functions of an algebraic analytic parametrization of \(S\) as \(q_1(s),\ldots,q_N(s)\), where \(s=(s_1,\ldots,s_d)\). Complexify the parameters. These germs belong to a common finite algebraic extension of \(\mathbb C(s_1,\ldots,s_d)\). In characteristic zero, the dimension of the span of their differentials is \[\operatorname{trdeg}_{\mathbb C}\mathbb C(q_1,\ldots,q_N).\] Indeed, a transcendence basis among the \(q_j\) has independent differentials, as seen by extending it to a transcendence basis of the ambient field; all remaining \(q_j\) are separably algebraic over it and add no independent differentials. This number is also the generic complex rank of the complexified parametrization and the dimension of its complex Zariski closure. The latter lies in \(V\).

If the rank is less than \(n\), a polynomial vanishing on that closure but not identically on \(V\) vanishes on an open patch of \(S\), hence on all of \(S\) by real-analytic continuation. If the rank is \(n\), its failure locus is defined locally by nontrivial analytic minors and has intrinsic measure zero. The complex image of the differential is exactly \(T_pS+iT_pS\), which proves the alternative. ◻

Proof. Use cylindrical decomposition with \(s\) preceding \(t\). Away from a lower-dimensional subset of \(B\), the region immediately above \(t=0\) lies in a cell on which \(b\) is an algebraic analytic branch. Locally at such an \(s_*\), shrink to a rectangle \(B_*\times(0,\varepsilon_0)\) on which this holds. Choose a square-free polynomial equation \[R(s,t,b(s,t))=0\] with polynomial coefficients in \((s,t)\) and nonzero leading coefficient and discriminant in the last variable. Remove the zero sets of the first nonzero \(t\)-coefficients of these latter two polynomials. After shrinking again, each is a power of \(t\) times a nonvanishing holomorphic function on a complex \(s\)-polydisc times a complex \(t\)-disc.

For \(t\ne0\) the roots therefore form a finite unramified covering of this product with the \(t\)-origin deleted. The \(s\)-polydisc is simply connected, so its monodromy is generated by one finite permutation around \(t=0\). The substitution \(t=u^l\) kills this permutation for some \(l\). The chosen branch becomes holomorphic for \(u\ne0\). The polynomial root bound, applied after division by the leading coefficient, bounds this branch by \(C|u|^{-M}\) on a smaller complex polydisc, for some finite \(M\). Thus it has at worst a pole at \(u=0\), whose negative Laurent coefficients are holomorphic in \(s\). Boundedness of the original function for real \(s\) and positive \(u\) makes those coefficients vanish for every real \(s\) in the rectangle. They vanish identically by holomorphic uniqueness. The branch consequently extends holomorphically through \(u=0\), giving the stated real-analytic extension.

There are only finitely many decomposition branches and exceptional polynomial zero sets. This proves the assertion outside a null set. For finitely many functions take a common multiple of their exponents and shrink the neighborhoods. For countably many, discard the union of their exceptional null sets; the individual conclusions suffice. ◻

Proof. The function \[a(s)=\min\{1,\operatorname{dist}(0,A_s)\},\] with value \(1\) for an empty fiber, is semialgebraic and continuous on an open full-measure subset \(B'\subset B\). It is zero there. In fact, if \(a(s_0)>0\) at a continuity point, it is bounded below near \(s_0\); but the boundary point with parameters \((s_0,0)\) is a limit of points of \(U\), whose parameters \((s_j,v_j)\) satisfy \(s_j\to s_0\) and \(v_j\to0\). This is a contradiction.

On \(B'\times(0,t_0)\) semialgebraic selection therefore supplies \(v(s,t)\in A_s\) with \(|v(s,t)|<t\). Apply Lemma 7 to its finitely many coordinates. At almost every \(s_*\) this gives a jointly analytic \(\psi(s,u)=v(s,u^l)\). The bound \(|\psi(s,u)|<u^l\) proves \(\psi(s,0)=0\) for every \(s\) in its neighborhood. This proves (i).

For (ii), apply Lemma 7 separately to \(d(s,t;v)\) for every \(v\in\mathbb{Q}^k\). Their exceptional sets have a null union. At a remaining \(s_*\), each rational test has a limit along every joint approach \((s,t)\to(s_*,0)\), \(t>0\). For every \(s,t\), \[|d(s,t;v)-d(s,t;v')|\leq |v-v'|.\] The rational limits therefore extend uniquely to a \(1\)-Lipschitz function \(d_*\) on \(\mathbb R^k\). A finite rational net in a compact set, together with this inequality, proves Equation (6). Notice that this argument uses only finitely many test neighborhoods at a time; it does not require a common neighborhood for all rational tests.

The zero set \(F\) is closed and contains \(0\), since \(0\in E(s,t)\). For \(c>0\) the exact identity \[E(s,ct)=c^{-1}E(s,t)\] shows, on taking \(s=s_*\) and \(t\downarrow0\), that the distance from \(v\) to the left-hand side tends to zero if and only if the distance from \(cv\) to \(E(s_*,t)\) does. Truncation at \(1\) does not affect vanishing of a limit. Consequently \(v\in F\) if and only if \(cv\in F\).

The assertion for moving vectors follows from the same Lipschitz inequality. If \(d_*(v)=0\), convergence of the truncated distance to zero is convergence of the untruncated distance to zero. If \(d_*(v)>0\), the distance of \(v\) from \(E(s_j,t_j)\) is eventually at least \(d_*(v)/2\), so any smaller fixed ball is in the scaled interior. ◻

Proof. We vary the real linear term in the weighted maximum used in the smoothness proof. We seek one good point at which the maximum is attained for a positive-measure set of linear terms, measured within the set of all terms making that point critical on its boundary cell. The corresponding conormals will provide \(k\) independent supports.

Choose a polynomial \(P\) vanishing on \(V_{\mathrm{sing}}\) and on the proper complex algebraic subsets containing all cells in the second alternative of Lemma 6, but not identically on \(V\). Such a polynomial exists because the union is finite and \(V\) is irreducible. For \(a\in\mathbb C^N\), maximize on the compact set \(\overline U\) the upper-semicontinuous function \[\Phi_a(q)=\log|P(q)|+|q|^2+\operatorname{Re}(a\cdot q),\] with value \(-\infty\) where \(P=0\). Its maximum \(M(a)\) is finite, because \(P\) is not identically zero on the open set \(U\). Every maximizer has \(P\ne0\), lies in \(V_{\mathrm{reg}}\), and belongs to \(\partial U\): on the smooth locus with \(P\ne0\), strict plurisubharmonicity forbids an interior maximum. Thus every maximizer lies on a retained boundary cell.

Fix one retained cell \(S\), of real dimension \(d\), and restrict to \(S^\circ=S\cap\{P\ne0\}\). Its critical-pair space is \[\mathcal E_S= \{(p,a)\in S^\circ\times\mathbb C^N: d(\Phi_a|_S)_p=0\}.\] Ambient real linear functionals restrict onto \(T_p^*S\), so these are \(d\) independent real affine equations on the \(2N\) real coordinates of \(a\). Therefore \(\mathcal E_S\to S^\circ\) is a smooth affine bundle with fiber dimension \(2N-d\) and total dimension \(2N\).

The subset \(\mathcal M_S\subset\mathcal E_S\) of actual maximizers is Borel. Indeed, \(M(a)\) is Lipschitz in \(a\), since \[|M(a)-M(a')| \leq\bigl(\sup_{q\in\overline U}|q|\bigr)|a-a'|,\] and \(\mathcal M_S\) is defined by the continuous equality \(\Phi_a(p)=M(a)\) on \(\mathcal E_S\). Projection \(\mathcal E_S\to\mathbb C^N\) is a smooth map between manifolds of the same real dimension and sends null sets to null sets: in countably many relatively compact coordinate charts it is Lipschitz. Its restrictions to the finitely many sets \(\mathcal M_S\) cover all of \(\mathbb C^N\). Hence at least one \(\mathcal M_S\) has positive measure in \(\mathcal E_S\).

In local bundle coordinates, the points lying over the nongood null subset of \(S\) form a null set, by Fubini. Applying Fubini again to \(\mathcal M_S\), in countably many bounded bundle charts, gives a good point \(p\) for which the set of actual maximizing parameters in the critical fiber has positive \((2N-d)\)-dimensional measure. Fix this \(p\) and write that subset as \(\mathcal A_p\).

Each \(a\in\mathcal A_p\) supplies the entire function \[h_a(q)=\frac{P(q)}{P(p)} \exp\bigl((2\overline p+a)\cdot(q-p)\bigr).\] The inequality \(\Phi_a(q)\leq\Phi_a(p)\) gives \[\log|h_a(q)| \leq |p|^2-|q|^2 +2\operatorname{Re}\bigl(\overline p\cdot(q-p)\bigr) =-|q-p|^2,\] with the desired modulus inequality also at \(P(q)=0\).

The affine map from the full critical fiber to the conormal space \[a\longmapsto \left.d\log|h_a|\right|_{T_pV} \in N_p^*S :=\{\lambda\in T_p^*V:\lambda|_{T_pS}=0\}\] is surjective. Indeed, on \(T_pV\), \[d\log|h_a|_p =d(\log|P|+|q|^2)_p+\operatorname{Re}(a\cdot dq).\] For any \(\lambda\in N_p^*S\), extend \(\lambda-d(\log|P|+|q|^2)_p\) to an ambient real linear functional \(\operatorname{Re}(a\cdot dq)\). The resulting total differential is \(\lambda\), so its restriction to \(T_pS\) vanishes and \(a\) belongs to the critical fiber. Since \(\dim_{\mathbb R}N_p^*S=k\), the inverse image of any proper linear subspace of \(N_p^*S\) has measure zero in the critical fiber. The positive-measure set \(\mathcal A_p\) cannot be contained in such an inverse image. Selecting successively outside the span already chosen gives \(k\) independent conormals and corresponding functions \(h_1,\ldots,h_k\).

Finally choose the local branches of \(\log h_j\) with value zero at \(p\). Their real differentials vanish on \(\mathbb C^{n-k}\times\mathbb R^k=T_pS\). Complex linearity forces the \(dz\) coefficients to be zero and the \(dw\) coefficients to be purely imaginary. Independence of the real conormals is exactly independence of the resulting real rows \(\beta_j\). ◻

Proof. Write \(f_u(X)=f(X,u)\). Choose nonnegative smooth functions \(\lambda_0\) and \(\eta\) on the unit circle, both zero on a fixed arc about \(1\), with \(\eta\not\equiv0\) and \[\operatorname{supp}\eta\Subset\{\lambda_0>0\}.\] Set \(\lambda=\alpha\lambda_0\), where the positive constant \(\alpha\) will be fixed sufficiently small. For parameters \(b,e\in\mathbb R^k\) and \(\delta\ge0\), we seek a holomorphic disc \(A_{b,e,\delta}\) whose boundary values \(X+iY\) satisfy \[ X(0)=b, \qquad Y(\theta)=f_{\lambda(\theta)+\delta}(X(\theta))+ \eta(\theta)e. \tag{9}\] Here \(X(0)\) means the value at the boundary angle \(\theta=0\), not the value at the center of the disc.

The parameters \(b\) and \(e\) have different geometric roles. When \(\delta=0\), the boundary arc about \(1\) maps into \(\mathbb R^k\), with endpoint \(A_{b,e,0}(1)=b\). Varying \(b\) moves that endpoint in the real directions; the perturbation \(\eta e\), supported away from \(1\), will change the inward radial derivative in all imaginary directions. For \(B\in\mathbb R^k\), evaluation at \(1-t\) with \(b=tB\) will therefore give independent displacements of order \(t\) in all \(2k\) real directions. We first prove this derivative assertion and then use a positive \(\delta\) to put the evaluated discs inside \(\Omega\).

Proof. Keep the old coordinates \((z_0,w_0)\) and fix the complex slice \(z_0=z_0(p)\). The graph of \(S\) in this slice is totally real and real analytic. Its parametrization by \(\operatorname{Re}w_0\) therefore has invertible complexified derivative, and its complexification provides a local biholomorphic change of coordinates sending this graph to \(\mathbb R^k\) and \(p\) to \(0\).

Restrict the analytic interior graphs from Proposition 8 to this slice. In the new coordinates their real-part projection is the identity at \(u=0\). The real-analytic implicit function theorem consequently writes them, on a common neighborhood, as \[X+i f_u(X),\qquad f_0(X)=0.\] The identity at \(u=0\) holds for every \(X\) in this neighborhood, not just for \(X=0\). For \(u>0\) these graphs lie in the slice of \(U\). The continuity principle in the ambient smooth chart applies to discs in this fixed slice, and is unchanged by its biholomorphic coordinate change. Lemma 10 now gives slice balls of radii \(ct\) with centers \(O(t)\) from \(p\).

Return to the \(w_0\)-coordinates. Uniform bounds for the coordinate change and its inverse, after one fixed shrinking, give constants \(c',C'>0\) and centers \(w_t\) such that \[B(w_t,c't)\subset\{w_0:(z_0(p),w_0)\in U\}, \qquad |w_t-w_0(p)|\le C't.\] Set \[s_t=(z_0(p),\operatorname{Re}w_t), \qquad v_t=\operatorname{Im}w_t-\phi(s_t).\] Intersecting each complex ball with the real affine slice \(\operatorname{Re}w_0=\operatorname{Re}w_t\) gives \[ B(v_t,c't)\subset A_{s_t}, \qquad s_t\longrightarrow s_*, \qquad |v_t|=O(t). \tag{16}\] The last estimate uses the smoothness of \(\phi\) and the fact that \(p\) lies on its graph.

Choose a sequence \(t_j\downarrow0\) for which \(v_{t_j}/t_j\to v\). Equation (16) implies \[\operatorname{dist}\bigl(v,\mathbb R^k\setminus t_j^{-1}A_{s_{t_j}}\bigr) \ge c'-|v-v_{t_j}/t_j|\ge c'/2\] for large \(j\). If \(v\) belonged to \(F\), the moving-parameter excluded-offset conclusion of Proposition 8 would force this distance to tend to zero. Thus \(v\notin F\), as required. ◻

Proof. Regard the old offset \(v\) as a real analytic function of the new coordinates. Its differential has kernel \(T_pS\), so \(dv_0(z,w)=L(\operatorname{Im}w)\) with \(L\) invertible. Taylor expansion therefore gives \[ R_t(Z,W):=t^{-1}v\bigl(\Phi(\sqrt t\,Z,tW)\bigr) \longrightarrow L(\operatorname{Im}W)-Q(Z) \tag{17}\] uniformly on compact sets. Here \(-Q\) is the part of the quadratic Taylor term involving only \(z\); the mixed and normal quadratic terms disappear after division by \(t\). The old parameter \(s_t(Z,W)=s(\Phi(\sqrt t\,Z,tW))\) tends uniformly on compact sets to \(s_*\).

For clarity, the distance convergence in Proposition 8 says that \[d(s,t;u)=\min\{1,\operatorname{dist} (u,\mathbb R^k\setminus t^{-1}A_s)\} \longrightarrow d_*(u) \quad\text{as }(s,t)\longrightarrow(s_*,0^+),\] locally uniformly in \(u\), with \(F=\{d_*=0\}\). In particular it applies to moving vectors \(u_j\to u\), as well as to moving parameters \(s_j\). If assertion (i) failed, a compactness argument would give \(\xi_j\to\xi\in\mathcal O\) and \(t_j\downarrow0\) with \(\xi_j\notin D_{t_j}\). The corresponding offsets \(R_{t_j}(\xi_j)\) are excluded from \(t_j^{-1}A_{s_{t_j}(\xi_j)}\). Their distances to the excluded sets are thus zero, whereas their limit has distance value \(d_*(R(\xi))>0\). This is a contradiction.

For assertion (ii), set \(q_j=\Phi(\delta_{t_j}\xi)\) and retain its old parameter \(s_j=s(q_j)\). Since \(R_{t_j}(\xi)\to R(\xi)\in F\), the same joint convergence supplies \[u_j\in\mathbb R^k\setminus t_j^{-1}A_{s_j}, \qquad u_j-R_{t_j}(\xi)\longrightarrow0.\] In the old chart change only the imaginary normal coordinate, replacing the offset \(v(q_j)\) by \(t_ju_j\). The resulting point \(q'_j\) is outside \(U\), has the same \(s_j\), and differs from \(q_j\) by \(o(t_j)\) in old coordinates. It remains in \(\Omega\) for large \(j\). The fixed coordinate change is smooth, so the change is also \(o(t_j)\) in the new coordinates. Consequently \[\xi_j:=\delta_{t_j}^{-1}\Phi^{-1}(q'_j) \longrightarrow\xi, \qquad \xi_j\notin D_{t_j}.\] Indeed an \(o(t_j)\) change becomes \(o(\sqrt{t_j})\) in the \(Z\) coordinates and \(o(1)\) in the \(W\) coordinates. All points used here approach \(p\), so the local cutoffs in the definition of \(A_s\) do not affect the argument. ◻

Proof. We first prove \(U\simeq D_0\), and then remove the pure quadratic terms and pass the support inequalities to the limit.

Completion of the proof of Theorem 1. The zero-dimensional case is a point. Otherwise, Theorem 13 identifies \(U\) biholomorphically with the domain in Equation (19). We check every hypothesis of Theorem 14.

First we give an explicit bounded realization in the intrinsic dimension \(n=m+k\). Boundedness of the original affine embedding places \(U\) inside \(\mathbb C^N\); the realization needed here is as a domain in \(\mathbb C^n\). Extend the real forms \(\ell_1,\ldots,\ell_k\) complex linearly to \(\mathbb C^k\), and write \(u_j=\ell_j(w)\). For \((z,w)\in D\), Equation (20) gives \[\mathop{\mathrm{Im}}u_j =\ell_j\bigl(\mathop{\mathrm{Im}}w-H(z,z)\bigr)+\ell_j\bigl(H(z,z)\bigr) >c\lVert z\rVert^{2}\geq0.\] Consequently the map \[ \mathcal B(z,w)=\left( \frac{u_1-i}{u_1+i},\ldots,\frac{u_k-i}{u_k+i}, \frac{z}{u_1+i}\right) \tag{27}\] is holomorphic. Its first \(k\) coordinates have modulus less than one. If \(m>0\) and \(y=\mathop{\mathrm{Im}}u_1\), then \[\lVert \frac{z}{u_1+i}\rVert \leq\frac{\sqrt{y/c}}{1+y} \leq\frac{1}{2\sqrt c}.\] Here \(|u_1+i|\geq1+y\), and \(2\sqrt y\leq1+y\). Thus \(\mathcal B(D)\) is bounded. Inverting the Cayley coordinates recovers each \(u_j\) holomorphically. Independence of the \(\ell_j\) then recovers \(w\), and multiplication by \(u_1+i\) recovers \(z\). These formulas give a holomorphic local inverse on an open neighborhood of every image point, so \(\mathcal B\) is a biholomorphism onto a bounded domain in \(\mathbb C^{k+m}\). When \(m=0\), omit the final coordinate block and its estimate.

Choose \(a_0\in C\). Then \((0,ia_0)\in D\). Real translations of \(w\) preserve the defining residual, as do scalar unit rotations of \(z\), because \(H\) is Hermitian. Under the positive dilation in Theorem 14, that residual becomes \[\mathop{\mathrm{Im}}(rw)-H(r^{1/2}z,r^{1/2}z) =r\bigl(\mathop{\mathrm{Im}}w-H(z,z)\bigr).\] Since \(rC=C\) for every \(r>0\), these maps preserve \(D\) in both directions.

Finally, conjugate the given action on \(U\) by the biholomorphism \(U\simeq D\). Proper discontinuity and the existence of a compact covering set are preserved by conjugacy. This is exactly divisibility in Vey’s definition, including when point stabilizers are nontrivial. In particular, we do not replace the action by a free action or assume that its quotient is a manifold. Theorem 14 applies and proves the assertion. ◻