Symmetry of semialgebraic bounded domains with compact quotient

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 [Stacks, Lemma 10.115.4, Tag 00OY] gives a finite algebraic map \pi:V_0\to\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

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,

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

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

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

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

satisfies

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 2.1 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 (2.1) 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

For example, send each bounded coordinate to \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 (2.2) 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

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\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-\eps\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 2.3, M=\max_{x\in L}d_U(x_0,x) is finite. For each fixed \zeta\in\D, the earlier discs take values in U, so Schwarz contraction gives

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_{\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 \C(s_1,\ldots,s_d). In characteristic zero, the dimension of the span of their differentials is

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,\eps_0) on which this holds. Choose a square-free polynomial equation

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

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 3.2 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 3.2 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,

The rational limits therefore extend uniquely to a 1-Lipschitz function d_* on \R^k. A finite rational net in a compact set, together with this inequality, proves Equation (3.4). 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

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 3.1, but not identically on V. Such a polynomial exists because the union is finite and V is irreducible. For a\in\C^N, maximize on the compact set \overline U the upper-semicontinuous function

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

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

and \mathcal M_S is defined by the continuous equality \Phi_a(p)=M(a) on \mathcal E_S. Projection \mathcal E_S\to\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 \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

The inequality \Phi_a(q)\leq\Phi_a(p) gives

with the desired modulus inequality also at P(q)=0.

The affine map from the full critical fiber to the conormal space

is surjective. Indeed, on T_pV,

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_{\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 \C^{n-k}\times\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

Set \lambda=\alpha\lambda_0, where the positive constant \alpha will be fixed sufficiently small. For parameters b,e\in\R^k and \delta\ge0, we seek a holomorphic disc A_{b,e,\delta} whose boundary values X+iY satisfy

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 \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\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.

Solving the boundary equation. Let \mathcal H=H^2(S^1,\R^k) be the real Sobolev space of functions with two square-integrable derivatives. The Fourier-series estimate

gives the continuous embedding \mathcal H\hookrightarrow C^1(S^1). This space is a Banach algebra under multiplication. Let T be the bounded conjugation operator which recovers the real part of a holomorphic boundary value from its imaginary part, with real part of mean zero, and put

Thus \mathcal T is bounded on \mathcal H, and Equation (4.2) is equivalent to

Here is the smallness needed for contraction. The identity f(X,0)=0 gives a real-analytic factorization f(X,u)=u\widetilde f(X,u). On a fixed small \mathcal H-ball of functions X, the product rule through two derivatives, the embedding into C^1, and bounded derivatives of \widetilde f give

For example, both expressions contain the factor \lambda+\delta; derivatives falling on that factor are also O(\alpha+|\delta|). The remaining composition and multiplication operators are uniformly bounded on the chosen ball. For all sufficiently small \alpha and |\delta|, the Lipschitz constant in Equation (4.3) is less than 1/2. On common sufficiently small b,e neighborhoods, its right side maps the chosen ball into itself. The contraction theorem gives a unique solution X.

Smooth composition on H^2, or its direct proof by the same product estimates, shows that this equation is smooth in X,b,e,\delta. Its X-linearization is the identity minus an operator of norm less than 1/2. The implicit function theorem therefore gives smooth parameter dependence in \mathcal H. In particular \partial_e X is uniformly bounded as \alpha decreases; it need not be small. The boundary function X+iY extends holomorphically to the disc and continuously to its closure. All these conclusions hold also with \lambda replaced by \rho\lambda, 0\le\rho\le1. Shrinking the fixed \mathcal H-ball keeps every disc in the prescribed coordinate neighborhood, by the boundary bound and the maximum principle.

The limiting radial derivative. For \delta=0, the function Y vanishes on the fixed arc about \theta=0. Hence A_{b,e,0}(1)=b, and reflection across that arc shows that the radial derivative there is purely imaginary. More explicitly, with Y=Y_{b,e,0}, the Schwarz integral gives, near \zeta=1,

There is no singularity on the support of Y. Expanding the kernel at \zeta=1 gives

The remainder is uniform with its first derivatives in b,e on a fixed smaller parameter neighborhood. This follows from the same integral formula and the smooth \mathcal H-dependence proved above.

Differentiating Equation (4.6) in e gives

The remainder contains \partial_X f_{\lambda(\theta)}(X)\partial_e X. The first factor is O(\alpha) and the second is uniformly bounded. Their product vanishes on the arc where \lambda=0, so the integral has a uniformly bounded kernel on its support. Thus \|R_\alpha\|\le C_1\alpha. The positive constant c_\eta is independent of \alpha. Fix \alpha sufficiently small that \partial_e D(0,0) is invertible, and only then fix the smaller b,e,\delta parameter neighborhoods ensuring all the conclusions above.

Interior discs for positive regularization. For this fixed positive \alpha, on \operatorname{supp}\eta the function \lambda has a positive minimum. As X ranges in a fixed smaller closed real ball compactly contained in the graph chart, \theta ranges in this support, and \delta ranges in a sufficiently small closed interval [0,\delta_0], the points

form a compact subset of \Omega. Consequently, after reducing the allowed size of e, their displacements by i\eta(\theta)e still belong to \Omega, uniformly for 0\le\delta\le\delta_0. Outside this support there is no displacement. Thus for every \delta>0 all boundary values in Equation (4.2) are in \Omega.

For a fixed \delta>0, set e=0 and increase \rho from 0 to 1 in \rho\lambda. At \rho=0 the solution is the constant disc b+i f_\delta(b). All boundary values during this deformation lie on interior graphs, so the continuity principle puts every disc in \Omega. Next increase the displacement from 0 to e, now with \lambda fixed. The same principle applies by the preceding compact interior bound. We have proved

No inclusion is claimed for \delta=0. The final radius allowed for e may depend on the fixed \alpha. No uniform lower bound as \alpha\downarrow0 is needed: the included-ball constants below are fixed only after these choices.

Interior balls. For real parameters (B,e) in a fixed small ball in \R^{2k}, define

Equation (4.5) gives convergence in C^1 on that ball:

The real differential J=D\Psi_0(0,0) is invertible. Smooth parameter dependence and bounded evaluation on the closed disc also give, on a fixed compact parameter set,

Indeed, integrate the uniformly bounded derivatives \partial_\delta A, \partial_b\partial_\delta A, and \partial_e\partial_\delta A over [0,\delta]; their bounds in H^2 control the displayed norm. We may therefore take the explicit positive regularization \delta(t)=t^2 for sufficiently small t. The maps

differ from \Psi_t^0 by O(t) in C^1: the value and e-derivative errors are O(\delta/t), while the B-derivative error is O(\delta). They therefore satisfy the same limit in Equation (4.8). By Equation (4.7), their images lie in t^{-1}\Omega.

For completeness, this C^1 convergence gives included balls, not merely interior limit points. Choose a closed parameter ball \overline B(0,r) so small that \|J^{-1}(D\Psi_0-J)\|<1/4 there. For small t the corresponding bound for \Psi_t is less than 1/2. If

the map x\mapsto x-J^{-1}(\Psi_t(x)-y) is a contraction of \overline B(0,r) into itself. Its fixed point proves that y is in the image of \Psi_t. These images thus contain balls of one fixed positive radius, centered at the bounded points \Psi_t(0). Multiplication by t proves Equation (4.1).□

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 \R^k and p to 0.

Restrict the analytic interior graphs from Proposition 3.3 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

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 4.1 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

Set

Intersecting each complex ball with the real affine slice \operatorname{Re}w_0=\operatorname{Re}w_t gives

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 (4.9) implies

for large j. If v belonged to F, the moving-parameter excluded-offset conclusion of Proposition 3.3 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

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 3.3 says that

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

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

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.

Normalization and forward bounds. Choose a_0\in C and write b_0=(0,ia_0)\in D_0. By Proposition 5.1, for any sufficiently small sequence t_j\downarrow0 the points p_j=\Phi(0,it_ja_0) belong to U. The compact covering set K gives

After extraction, k_j\to k_*\in K.

Each support h_a maps U into the unit disc, satisfies h_a(p)=1, and has h_a(p_j)=1+O(t_j). For a fixed compact subset E\subset U, d_U(q,k_j) is bounded uniformly for q\in E. Schwarz contraction for the chain-of-discs distance implies that h_a(\gamma_jq) is at uniformly bounded hyperbolic distance from h_a(p_j). If two disc points \alpha,\beta have pseudohyperbolic distance at most \rho<1, then

It follows that, uniformly on E,

The peak inequality in Equation (3.5) now gives

Here and below the norm of points in V is the norm in its fixed affine embedding. In particular, the images of E eventually lie in \Omega, and their new coordinates satisfy (z,w)=O_E(\sqrt{t_j}).

The k supports sharpen the bound in all the normal directions. Their real logarithmic differentials annihilate T_pS=\{\operatorname{Im}w=0\} and are independent. Therefore, taking the local logarithms normalized by \log h_a(p)=0, we have

where \ell_1,\ldots,\ell_k are independent real rows, extended complex linearly when applied to w. Indeed a complex linear form whose real part vanishes on all z and real w has no z term and purely imaginary w coefficients. Taylor expansion, the preliminary O_E(\sqrt{t_j}) bound, and Equation (5.5) give

Invertibility of the matrix of these rows gives w=O_E(t_j). Consequently the holomorphic maps

are eventually defined on a neighborhood of every compact subset of U and are locally uniformly bounded. A diagonal normal-family extraction gives a holomorphic limit f:U\to\C^n.

Inverse compactness. For each j the inverse coordinate map

is defined on all of D_{t_j} and satisfies g_j(b_0)=k_j. Every compact subset of D_0 is eventually contained in its domain, by Proposition 5.1. More importantly, for every compact E\subset D_0 there is a compact K_E\subset U containing g_j(E) for all sufficiently large j.

To see this last assertion, cover E by finitely many balls B(x_\nu,r_\nu) with \overline B(x_\nu,2r_\nu)\subset D_0. Connectedness and small balls along paths give a fixed finite disc chain from b_0 to each x_\nu, with all closed disc images in D_0. For x\in B(x_\nu,r_\nu), append the disc

whose parameters 0 and 1/2 join x_\nu to x. All these chains lie in one compact subset of D_0, and their total Poincaré lengths have a common bound. That compact subset is contained in D_{t_j} for all large j, so composition with g_j proves

with M_E independent of j. Since k_j\in K and d_U is proper by Proposition 2.3, this is the claimed compactness.

Montel’s theorem in the bounded ambient affine coordinates, followed by diagonal extraction, now gives a holomorphic map g:D_0\to U with g_j\to g locally uniformly. The compact containment just proved puts the limit in U, rather than merely in its closure. It also allows the composition identities to pass to the limit: on each compact subset of D_0, the points g_j(x) lie in a common compact subset of U, where f_j\to f. Thus

In particular, f has full rank at every point of g(D_0).

This identity has not yet located the rest of f(U) or proved global injectivity. The next step uses the excluded-point conclusion of Proposition 5.1 to settle both issues.

Excluding additional image points. The maps f_j have nonzero Jacobian wherever they are defined. Their local Jacobian determinants converge to that of f. The zero-free limit principle says that a limit of zero-free holomorphic functions is either zero-free or identically zero; in several variables this follows by restriction to complex lines. Thus the full-rank locus and its complement are both open: a zero of a local limiting determinant forces that determinant to vanish throughout its coordinate ball. Equation (5.7) makes the full-rank locus nonempty, so connectedness of U implies that f has nonzero Jacobian everywhere.

Fix q\in U. Local inverse stability gives a neighborhood B of f(q) such that

for all sufficiently large j, where V_q\Subset U is a fixed coordinate neighborhood of q. Here is an explicit justification of the stability assertion. Center source coordinates at q and normalize the target by the inverse of df_q. On a sufficiently small closed source ball the normalized derivative of f differs from the identity by less than 1/4. Local uniform convergence of the holomorphic maps gives the same bound with 1/2 for f_j on that ball. For every target point in a fixed smaller ball about f(q), the equation f_j(x)=y is then solved by a contraction of the closed source ball into itself, since f_j(q)\to f(q). This proves Equation (5.8).

If R(f(q))\in F, assertion (ii) of Proposition 5.1 supplies points outside D_{t_j} converging to f(q), contradicting Equation (5.8). Thus f(U)\subset\mathcal O. The set f(U) is connected and contains D_0 by Equation (5.7), so it lies in that same connected component: f(U)=D_0. Finally, for each q\in U the points f_j(q) eventually lie in a compact neighborhood of f(q) inside D_0. Passing to the limit in g_j(f_j(q))=q yields g(f(q))=q. We have proved that f:U\to D_0 and g:D_0\to U are inverse biholomorphisms.

The Hermitian form and its supports. The real quadratic vector P_2 has a unique decomposition

where H is a vector of Hermitian forms and B a vector of holomorphic homogeneous quadratic polynomials. The triangular holomorphic change

is globally invertible and transforms D_0 into D of Equation (5.3).

The functionals in Equation (5.6) retain their independence. We claim that throughout D they satisfy

for a common c>0. To prove this, fix (z,w)\in D and consider the unscaled point

It lies in U for every sufficiently small t, by Proposition 5.1 applied before the triangular change. The local holomorphic logarithm of a support has expansion

where A_a is holomorphic homogeneous quadratic. Dividing the peak inequality -\log|h_a(q_t)|\ge\lVert q_t-p\rVert^2 by t and passing to the limit gives

The linear map J is injective. Thus, when m>0, its right-hand side is bounded below by c\lVert z\rVert^2 with c>0. When m=0, the same inequality holds with any positive c, since z=0.

For each real \theta, the point (e^{i\theta}z,w) also lies in D, because H(e^{i\theta}z,e^{i\theta}z)=H(z,z). Average Equation (5.10) over \theta. The holomorphic quadratic terms have average zero, whereas \lVert J(e^{i\theta}z)\rVert=\lVert Jz\rVert. This proves Equation (5.9).

Putting z=0 gives \ell_a(y)\ge0 on C. Since C is open and \ell_a\ne0, equality at a point of C would produce a nearby negative value. Hence \ell_a(y)>0 for every y\in C. Next fix y\in C. For every r>0,

Apply Equation (5.9) and let r\downarrow0 to obtain \ell_a(H(z,z))\ge c\lVert z\rVert^2. This proves both parts of Equation (5.4). Conjugating the original group by the resulting biholomorphism transfers proper discontinuity and cocompactness to D, completing the proof.□

Completion of the proof of Theorem 1.1. The zero-dimensional case is a point. Otherwise, Theorem 5.2 identifies U biholomorphically with the domain in Equation (5.3). We check every hypothesis of Theorem 6.1.

First we give an explicit bounded realization in the intrinsic dimension n=m+k. Boundedness of the original affine embedding places U inside \C^N; the realization needed here is as a domain in \C^n. Extend the real forms \ell_1,\ldots,\ell_k complex linearly to \C^k, and write u_j=\ell_j(w). For (z,w)\in D, Equation (5.4) gives

Consequently the map

is holomorphic. Its first k coordinates have modulus less than one. If m>0 and y=\Imop u_1, then

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 \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 6.1, that residual becomes

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 6.1 applies and proves the assertion.□