Proof. Apply one-variable preparation with \(t\) as the last variable. The finitely many preparation cylinders meeting \(t=0\) have, on their bottom bands, center zero: a nonzero center comparable to \(t\) cannot remain comparable as \(t\) tends to zero with \(v\) fixed. On each such band the prepared function is a rational power of \(t\) times a nonzero function of \(v\) times a unit bounded above and below. There are only finitely many rational exponents. Taking \(N\) larger than all their absolute values absorbs the positive factors depending on fixed \(v\) and proves the first assertion. Apply the same argument to \(1/f\) when necessary.

For the second assertion, restrict to a full-dimensional base cell on which the finitely many coefficient functions in the preparation are analytic and their required nonzero denominators remain nonzero. In the analytic unit choose projective coordinate charts about the limiting arguments. An argument involving a negative power of \(t\) either has identically zero numerator on the chosen base cell or tends to infinity; in the latter case its reciprocal is a valid analytic coordinate and its nonzero coefficient can be inverted. In these charts the arguments are analytic in \(v\) and nonnegative rational powers of \(t\). Clear their denominators. The unit is analytic on a neighborhood of its compact argument set, giving the asserted joint analytic extension. Ordinary convergence of analytic power series on a smaller box gives the derivative assertion. The discarded base cells and coefficient singularities have smaller dimension. ◻

Proof. Track irreducible components on a relative Hilbert or Chow parameter space; for their algebraic existence and projectivity see (Grothendieck 1960--1961, sec. 3, Theorems 3.1–3.2) and (Andreotti and Norguet 1967, sec. 2.4(b), Proposition 4). On the locus of a fixed component type the tracking map is generically finite. Remove its branch and nonflat loci and choose a local branch. Resolve the resulting total family and the graphs of the finitely many evaluation maps by projective modifications. Generic smoothness removes a further proper analytic subset of the complex parameter space. A maximally totally real open box is not contained in a proper complex analytic subset, by the holomorphic identity theorem. Thus these deletions are legitimate at generic real parameters. The selected generic fibres are irreducible, and their smooth models are connected. After shrinking, properness and smoothness preserve connectedness. All constructions are made with finitely many projective equations and modifications; restricting to a relatively compact parameter box supplies the stated bounded data. ◻

Proof. Use the local Hartogs-figure criterion in a coordinate polydisc. A relatively compact Hartogs figure with a neighborhood contained in the kernel is contained, together with that neighborhood, in every \(U_t\) for sufficiently small \(t\). Pseudoconvexity fills its associated polydisc in each \(U_t\). To obtain neighborhood inclusion for a compact subset of the filling, slightly enlarge the figure within the given neighborhood before applying the criterion. The resulting enlarged fillings supply a neighborhood of that compact subset in every small \(U_t\). Thus the filling lies in the kernel. This is precisely the Hartogs-figure criterion for the kernel. The same argument uses the corresponding figures in smoothly varying holomorphic charts; strict compact containment persists under this change. ◻

Proof. The subanalytic assertion follows immediately by writing the upper-limit quantifiers with a positive accuracy and a positive upper bound for \(t\). We prove the stronger assertion in the stated polynomial-in-\(y\) situation. Work on one basic piece, written as \[F_i(b,y)=0,\qquad H_j(b,y)\geq0,\qquad L_k(b,y)>0.\] Finite unions cause no difficulty. First require a common strict slack \(L_k\geq t^A\). This does not change the upper limit when \(A\) is sufficiently large. Indeed, for a proposed limit point, a positive accuracy, and a fixed bound on \(h\), take the supremum of the minimum strict slack among witnesses within that accuracy; cap this supremum by \(1\). Whenever witnesses exist for arbitrarily small \(t\), they exist for every sufficiently small \(t\) by one-dimensional subanalyticity. The positive supremum is definable. Lemma 3 gives one exponent \(A\) for this family, allowing the threshold to depend on the point, accuracy, and bound. A witness with at least half the supremum gives the desired slack after increasing \(A\).

For a second exponent \(B>A\), allow residual errors of size \(t^B\) in the equations and weak tests. Include the slack as an extra variable. On a slightly larger compact original-coordinate box the distance inequality for the resulting closed semianalytic set gives \[\operatorname{dist}((b,y,\sigma),C) \leq c\,\operatorname{residual}(b,y,\sigma)^{\theta}\] for some \(\theta>0\). The same statement with external parameters fixed follows from the fibrewise version described in Remark 4. Choose \(B\) so large that \(B\theta>A+1\) and \(\|R_t^{-1}\|t^{B\theta}\to0\). A point satisfying the relaxed tests can then be corrected to \(C\); the correction is \(o(t^A)\) in slack, \(o(1)\) in \(y\), and \(o(1)\) in the normalized coordinate \(h\). Strict tests still have, for example, half the original slack. Thus these relaxed tests have exactly the original upper limits.

Taylor-expand every coefficient at \(b=x\). Since \(R_t\to0\) with a positive power bound, sufficiently high finite Taylor order makes all remainders \(o(t^B)\) on each fixed bounded \(h\)-box. Truncate the Puiseux series at correspondingly high order. Changing the relaxed error by a fixed factor absorbs these remainders and leaves the same upper limit, by the preceding correction argument. The resulting tests are polynomial in \(h,y\) and in a common root of \(t\), with finitely many negative powers removable by multiplication by a positive power of \(t\). Real quantifier elimination therefore gives a semialgebraic upper limit.

The exponent choices can be made independently of the fixed bound on \(h\): include its reciprocal as another bounded fixed parameter in the slack and distance tests. The constants and the small-\(t\) threshold may deteriorate with that bound; the finitely many power exponents do not. Consequently one finite polynomial description works on the full affine \(h\)-space. A finite projective atlas treats all fibre points. The finite Taylor coefficients and the finitely many choices used above are analytic in external parameters after generic restriction. In particular, apply Lemma 3 jointly to the entries of \(R_t\) and \(R_t^{-1}\) and discard the zero loci of their nonzero leading coefficients. Their relevant positive orders of decay and inverse power bounds are then fixed on the smaller parameter box. This supplies a common Taylor order there, rather than only a separate order for each fixed parameter. ◻

Proof. Pointwise limits exist by one-dimensional definable monotonicity. They have the same Lipschitz bound in \(z\). A finite net in \(K\) thus upgrades convergence at fixed \(x\) to uniform convergence on \(K\). For each positive rational \(\delta\), the supremum of the positive numbers \(a\) for which \[|d_t(x,z)-d_0(x,z)|<\delta \quad(0<t<a, z\in K)\] is a positive definable function of \(x\), after capping it by \(1\). It is continuous on a dense open definable set, and hence locally bounded below there. This controls the error also at moving centers \(x_t\) near a chosen generic \(x\). For each point of a countable dense subset of \(K\), the definable function \(d_0(x,z)\) is likewise continuous at generic \(x\). A finite-net argument and the common Lipschitz constant give continuity in \(x\) uniformly on \(K\) at all points outside the resulting countable union of lower-dimensional subanalytic sets. Combining the two estimates proves that \(\widehat d\) is continuous at every point of \(\{x\}\times K\times\{0\}\) for all such \(x\).

To obtain one open good locus for this fixed compact test, define \[\Sigma=\{x\in U:\text{ for some }z\in K,\ \widehat d\text{ is not continuous at }(x,z,0)\}.\] This is a subanalytic set: both \(d_0\) and the extension are definable, and failure of continuity is expressed by the finite quantifiers \[\exists z\in K\ \exists\epsilon>0\ \forall\delta>0\ \exists(x',z',t'):\quad \begin{cases} x'\in U,\ z'\in K,\ t'\geq0,\\ |x'-x|+|z'-z|+t'<\delta,\\ |\widehat d(x',z',t')-d_0(x,z)|\geq\epsilon. \end{cases}\] All variables are restricted to the domains of the given functions. The preceding argument puts \(\Sigma\) inside a countable union of lower-dimensional subanalytic sets. Their closures are nowhere dense, so the Baire theorem shows that \(\Sigma\) has no interior. Since \(\Sigma\) itself is subanalytic, it has dimension smaller than \(\dim U\), as does its relative closure \(E\).

On \(U\setminus E\) the extension is therefore jointly continuous at every boundary point with \(t=0\). Compactness of \(L\times K\) turns this into uniform convergence on that product. Taking a finite union of the exceptional sets proves the finite-family assertion. Taking a countable union gives precisely the weaker countable-family assertion stated above. ◻

Proof. For the fixed matrix use \(a=\operatorname{Re}b\) as an independent parameter and take the closed-set distance tests after setting \(\operatorname{Im}b=R_tv\). Their generic limit at \(a=x\) is unchanged by replacing \(a\) by \(x+R_tu\), by Lemma 9. This applies to arbitrary bounded \(u,v\) and to their convergent perturbations, which are the quantifiers defining upper limits. Taking complements gives the kernel assertion. The final statement is the same lemma applied after, rather than before, the parameter-dependent normalization. ◻

Proof. Let \(f_0(t,v)=\lim_{e\downarrow0}f(e,t,v)\). For fixed positive \(\delta\), the supremum of those \(a\in(0,1)\) such that \(|f(e,t,v)-f_0(t,v)|<\delta\) for every \(0<e<a\) is positive and definable. Lemma 3, with \((v,\delta)\) held fixed, bounds this threshold below by \(t^{N_0}\) for all sufficiently small \(t\), with \(N_0\) independent of \((v,\delta)\). The asserted diagonal then lies below the threshold. Let \(t\) tend to zero and then let \(\delta\) tend to zero. ◻

Proof. Work over a small Stein coordinate box. The map \(F(b,t)=(b,g(b,t))\) is finite, proper, and open. Put \(J'=F(J)\). Its complement \(\Omega'\) is locally pseudoconvex. Here is the justification, including the ramification locus. Take a plurisubharmonic exhaustion of \(\Omega=(B\times\mathbb C)\setminus J\), which is Stein over the small box by the local Levi criterion. On \(\Omega'\) take the maximum of this exhaustion over all preimages under \(F\). Away from branch values this is a finite maximum of plurisubharmonic functions. Local boundedness above and continuity of the finite fibres extend it plurisubharmonically across branch values. It is an exhaustion: a sequence approaching \(J'\) has at least one preimage approaching \(J\), while properness of \(F\) handles escape to infinity. This proves the claim about \(\Omega'\).

Invert the image line coordinate and adjoin its point at infinity. In the inverse coordinate \(w\), the resulting domain contains \(w=0\) over every base point. It is locally pseudoconvex there by local boundedness of \(J'\). Its largest centered vertical disc has radius \[r(b)=\bigl(\max_{v\in J'_b}|v|\bigr)^{-1},\] where \(r=+\infty\) when the maximum is zero or the fibre is empty. The Hartogs radius criterion says that \(-\log r\) is plurisubharmonic, proving the assertion. Multiplication by a nowhere-zero holomorphic leading coefficient adds a pluriharmonic function.

For the constant test, note first that empty fibres form an open set by properness. Choose, locally on the base, a constant \(a\) outside a disc containing every \(J_b\). The already proved test \(t-a\) has a positive lower bound on every nonempty fibre and is \(-\infty\) on empty fibres. If there is an empty fibre, it is \(-\infty\) on an open base set and therefore identically \(-\infty\) on the connected base box. Thus either every fibre is empty or none is. The test for \(1\) is accordingly constant \(-\infty\) or constant zero. ◻

Proof. At a possible limiting hole choose fibre coordinates \((z,t)\) in which \(D\) is given by a Weierstrass polynomial \(g(s,z,t)\). Choose the vertical disc with boundary disjoint from \(D\) and shrink all other variables. The limit hypothesis confines the holes to a strictly smaller vertical disc. After filling trivially outside this disc, they give a closed set \(J\) as in Lemma 12, with base \((s,z)\).

For fixed \(z\), the subharmonic function \(\psi_g(s,z)\) is bounded above near a compact subinterval of \(I\) and tends locally uniformly to \(-\infty\) there. To see that it is identically \(-\infty\), fix a smaller half-disc. Harmonic measure comparison bounds it by \(C-M\omega(s)\) for every \(M>0\), where \(\omega(s)>0\) is the harmonic measure of a nontrivial subinterval. Let \(M\) tend to infinity. Thus \(g\) vanishes at every hole. Finitely many such cylinders cover the possible holes on compact fibre sets; outside them the limit hypothesis already excludes holes. This proves the assertion, including its locally uniform version. ◻

Proof. Use the same Weierstrass cylinders and polynomial maximum. For fixed remaining fibre coordinates, \(\psi_g\) is \(-\infty\) on the specified nonpluripolar set. A plurisubharmonic function with this property is identically \(-\infty\) on its connected box. For the last stated case this can also be checked without a capacity criterion: straighten the real analytic slice biholomorphically, and apply the one-variable uniqueness principle successively to the real intervals in a real coordinate box. The resulting vanishing on the complex box proves containment. ◻

Proof. Since \(G_0\) is open and \(A\) has no interior, it contains a point outside \(A\). Connect that point to any prescribed point of \(Q_0\setminus A\) by a path and cover the path by finitely many submersion boxes avoiding \(A\) on the central fibre. On a sufficiently small fixed one-sided disc the limit hypothesis and compactness give all these boxes on the good side. In each box choose a disc \(\Delta(c,r)\) in the transverse parameter internally tangent to the arc at \(0\). The full product of this disc with a smaller connected vertical patch belongs to \(G\).

For a vertical point \(z\) let \(R(z)\) be the largest radius of the transverse disc centered at \(c\) contained in \(G\) within this box. The Hartogs radius criterion gives that \(-\log R\) is plurisubharmonic, and \(R\geq r\) everywhere. At the starting patch, openness of \(G\) at the tangency point gives \(R>r\) at some point: the rest of the closed tangent circle is compactly on the good side. If \(R=r\) anywhere in the connected patch, \(-\log R\) attains its maximum there and the strong maximum principle would force \(R=r\) everywhere. Thus \(R>r\) throughout the patch, so the central-fibre points of that patch are in \(G\). Continue through successive overlapping boxes along the path. This reaches the prescribed point. ◻

Proof. The measure hypothesis in the standard analyticity theorem is automatic here. Indeed uniformize the closed subanalytic set near a compact coordinate box by a proper analytic map from a manifold of dimension at most \(2n-2\). The inverse image of the box is compact. A finite covering by coordinate patches makes the map Lipschitz on compact smaller patches, so its image has finite \((2n-2)\)-dimensional Hausdorff measure there. The local Stein complement means exactly that \(A\) is \(1\)-concave in the sense of (Vâjâitu 2000, Definition 1). For \(n>1\), apply (Vâjâitu 2000, Theorem 1) with \(q=1\); it gives analyticity and pure dimension \(n-1\). For \(n=1\), a zero-dimensional closed subanalytic set is locally finite and hence is an analytic divisor directly. For smooth ambient spaces, the thin-pseudoconcave-set method already appears in (Hirschowitz 1973, Theorem 1.4). ◻

Proof. First restrict near a generic parameter at which \(A\to B\) is flat. Since \(q\) is smooth, the exact sequence of its ideal \(\mathcal I_A\) shows that \(\mathcal I_A\) is flat over this smaller base as well. In a relative projective embedding, choose a twist \(\mathcal I_A(k)\) which is generated and has vanishing higher cohomology on the selected fibre (The Stacks Project Authors 2026, Tag 0B5T), using the comparison in (Serre 1956, Theorems 1–3). Upper semicontinuity preserves the vanishing after shrinking; constancy of the Euler characteristic in a proper flat family makes \(h^0\) constant. Grauert’s constant-cohomology theorem therefore gives a locally free direct image and fibrewise base change (Grauert 1960, sec. 7, Sätze 3, 5 and 6); see the correction (Grauert 1963, 36) for the distinct general base-change statement. The evaluation map generates on the selected fibre. Nakayama’s lemma and properness allow a further shrinking on which it generates throughout. We thus have local holomorphic sections of the twisted ideal whose restrictions generate every fibre ideal. Choose one whose restriction at the selected parameter is not identically zero; such a section exists because \(A\) is proper in that fibre. Extend it over a smaller base box. It stays nonzero at a local section of \(q\) through a point where its central restriction is nonzero. Thus its zero divisor is proper in every nearby fibre and contains \(A\). The construction uses one fixed twist, hence has bounded algebraic fibre degree. Generic flatness supplies the initial parameter outside a proper analytic subset; the subsequent choices only shrink its neighbourhood. Such a parameter can be chosen in the maximally totally real box. This argument does not require the general bad base-change locus to be closed. ◻

Proof. Regard the finitely many real polynomial coefficients and the projective fibre equations as independent algebraic parameters. Real algebraic cell decomposition, with these parameters first, expresses each fibre as finitely many Nash pieces of uniformly bounded algebraic complexity. Their algebraic closures, and the complex Zariski closures of their projections into the complex fibre coordinates, consequently have bounded degree by elimination. The complex Zariski closure of the original set is the union of these finitely many closures. When it is proper, a homogeneous polynomial of bounded degree vanishes on it without vanishing identically on the ambient fibre. Its zero divisor is a container.

At a fixed degree the condition that a homogeneous equation vanish on every point of the semialgebraic set, but not on the whole fibre, is a quantified semialgebraic condition on its coefficients. Quantifier elimination and definable choice give the parameter locus and the selection. In a varying smooth projective family, record the chosen zero divisors as cycles in the finite union of relative Chow spaces of these degrees. Their closure is proper over a smaller parameter box. Its limiting cycles retain their fibre dimension, so their supports are proper divisors; a limiting homogeneous equation alone would not suffice, since it could vanish identically on a special fibre.

Across a generic real wall, one-sided subanalytic limits of this bounded cycle data exist and vary real analytically after a stratification and generic restriction. This follows equally by applying Lemma 3 to the normal wall variable in projective coordinate charts. More precisely, restrict to a full-dimensional real analytic stratum of the wall on which the cycle degree and these charts are fixed. Preparation in the one-sided normal variable gives an analytic limit map on that stratum and convergence uniform on each smaller compact wall box. Restrict this map to the real interval of a transverse analytic complex disc. Its real analytic coefficients complexify holomorphically in the disc variable, jointly real analytically in the other wall parameters. The equations defining the relative Chow space vanish on the real interval, so continue to vanish by the holomorphic identity theorem. Pulling back its universal cycle gives a relative divisor on the smooth pulled-back family. The analytic extensions exist on a common small disc after restriction to a compact smaller wall box. This is the claimed locally uniform continued container; no holomorphic extension in all complex base variables at once has been asserted. ◻

Proof. Call a parameter good when its holes have proper complex Zariski closure. Lemma 18 makes the good locus subanalytic. Choose a finite local stratification of it and its complement. We first show that no generic real hypersurface wall can separate an open good region from an open bad region inside \(S_0\).

At a generic point of a proposed wall choose transverse analytic complex discs, with their real arcs in the wall. Concretely, choose a real analytic tangent vector field on the wall which is not in its complex tangent hyperplane, take its local real integral curves, and complexify those curves. Their imaginary directions are transverse to the wall, and the resulting family fills a neighborhood. Lemma 18 gives on each arc a limiting proper divisor from the good side and its holomorphic continuation on the disc. All limiting holes on that side are in its support. The central fibre has a point in \(G\), since the wall lies in \(S\). The complement of a proper divisor in a connected smooth projective variety is connected: a path between two points can be perturbed off its real-codimension-two strata. Lemma 15 therefore puts that entire complement in the wall fibre of \(G\).

This also confines holes on both sides near the limiting divisor. Indeed, any compact subset of the wall fibre outside the divisor has a neighborhood in \(G\); finite coverings and properness make this uniform on smaller arc intervals. We can now choose Weierstrass cylinders for the continued divisor. On the real interval their holes belong to the divisor. Lemma 14, in the transverse disc, confines the holes to that divisor on both sides. Varying the transverse discs through generic wall points yields an open good region on the proposed bad side, a contradiction.

A connected open real manifold cannot be separated by a subanalytic subset of real codimension at least two: a compact path can be perturbed to meet only the hypersurface strata of a finite stratification. Thus the preceding wall argument shows that the open good locus is dense in \(S_0\). Apply the same argument to generic hypersurface strata whose two neighboring open regions are good. It makes their generic parameters good as well. What remains is a subanalytic subset \(E\subset S_0\) of real codimension at least two.

Write \(n=\dim_{\mathbb C}Q\) and \(f=\dim_{\mathbb C}Q_b\). Outside \(E\) the fibre holes have real dimension at most \(2f-2\); over \(E\) they have dimension at most \(2f\). The dimension theorem for subanalytic fibres therefore gives \(\dim_{\mathbb R}J\leq2n-2\). Theorem 16 makes \(J\) an analytic divisor. No fibre is contained in \(J\), because every parameter of \(S_0\) has a point in \(G\). Its intersection with a fibre is consequently a proper analytic subset of a projective variety, and is algebraic by Chow’s theorem. This proves the final assertion. ◻

Proof. Every irreducible component \(H\) of \(J\) dominates \(S_0\). To check this, \(q|_H\) is proper, so its image is analytic. If that image were proper, dimension of \(H\) would force it to be a hypersurface in \(S_0\) with full-dimensional generic fibres in \(Q_b\). Those fibres would equal \(Q_b\), contradicting \(G_b\ne\varnothing\). Thus \(H\) meets the stated open subregion. The local equations of \(D\) vanish on a nonempty open subset of \(H\), and hence on \(H\) by the analytic identity theorem. Apply this to every component. ◻

Proof. The first assertion is immediate from containment of the holes. Since \(S=q(G)\) is open, a boundary point of its component \(S_0\) cannot belong to \(S\): a small connected neighborhood within \(S\) would otherwise join it to \(S_0\). In submersion coordinates near \(\sigma(b_0)\) choose a connected vertical patch disjoint from \(D\). Over \(S_0\) this whole patch is in \(G\), while over the boundary it is disjoint from \(G\).

More explicitly, \(\sigma^{-1}(G)\) is locally pseudoconvex, since holomorphic pullback preserves the local Hartogs criterion, and it is contained in \(S\). In a sufficiently small parameter ball, each component of \(S_0\) intersected with that ball is a component of \(\sigma^{-1}(G)\) there: a path in the latter is a path in \(S\) and hence cannot leave \(S_0\), while every point of \(S_0\) belongs to the pullback. Thus the local pieces of \(S_0\) satisfy the same criterion. This proves the assertion at \(b_0\). ◻

Proof. Apply Lemma 3 to the finitely many coordinate functions of \(f\). On a generic parameter box they have joint convergent Puiseux expansions with a common denominator. Boundedness forces every negative-power coefficient to vanish, giving the analytic map \(a(x,t^{1/q})\). For a positive coordinate, restrict further so that its first nonzero coefficient does not vanish. Positivity then makes that coefficient positive and gives the asserted factorization. The discarded preparation cells and coefficient zero loci come from finitely many data on the bounded boxes in use. Their lower-dimensional relative closures give the closed subanalytic exceptional set in the statement. Restricting the joint analytic extensions to smaller compact parameter boxes gives a common complex neighborhood and bounds for every derivative. ◻

Proof. Apply definable Hardt triviality to the projection of \(D\) onto \((x,t)\), compatibly with the two marked endpoint graphs (Coste 1999, Theorem 5.22). On each base cell the trivialization identifies these sections with fixed points in a model fiber. They lie in the same component; choose a fixed definable path between them and transport it by the continuous trivialization. This gives continuity jointly in \((x,t,a)\) on that cell. The finite base walls have dimension at most \(k\), and their limiting loci at \(t=0\) have dimension less than \(k\). Deleting those loci puts a smaller product \(U\times(0,\epsilon)\) in one cell. Retain the original path parameter \(a\) and let \(T\) be the closure of its graph in \((x,t,a,z)\). This graph has dimension \(k+2\). Over \(t>0\) and interior base parameters its closure is still that graph. Uniformize \(T\) properly, resolve \(t\), and discard components supported at \(t=0\). The maps from the resulting manifold into \((x,t,a,z)\) are analytic.

For every stratum of the normal-crossings divisor, delete the closure of its critical values in \(x\). A stratum that cannot dominate the \(k\)-dimensional base is avoided altogether. At every remaining point over \((x_0,0)\), the original \(x\) coordinates can be retained as tangential coordinates while all divisor coordinates are retained as normal coordinates. Indeed the differential of \(x\) is surjective on their common zero stratum, so the analytic inverse function theorem allows precisely this coordinate change. Absorbing the nonzero monomial unit gives, on each fixed sign piece, \[ t=y_1^{m_1}\cdots y_r^{m_r},\qquad y_i>0. \tag{5}\] The coordinates not displayed are free coordinates in an ordinary box. We use smaller closed boxes inside larger analytic chart boxes. Properness gives finitely many of them that cover every relevant graph point for \(x\) near \(x_0\) and \(t\) small. Their maps have common complex neighborhoods in \(x\) after further shrinking.

Here is a precise gluing argument. At fixed \((x,t)\), the image of a smaller closed chart box with fixed signs and (5) is compact and connected: in logarithmic coordinates its monomial part is an affine hyperplane intersected with coordinate half-spaces, and the free part is a box. The positivity constraints do not spoil compactness: if \(y_h\leq B_h\) and \(t>0\) is fixed, then \(y_i\geq(t/\prod_{h\ne i}B_h^{m_h})^{1/m_i}>0\). These finitely many sets are subsets of the actual continuous path graph for \(t>0\) and cover that graph. The intersection graph of the sets needed to connect its two endpoints therefore has a chain joining them; use a simple chain, whose length is bounded by the number of chart/sign pieces. The existence of each finite chain, its intersection points, and their lifts is a subanalytic condition. Selection followed by a generic restriction fixes one chain and bounded subanalytic endpoint lifts for all \((x,t)\) in a smaller product. Adjacent endpoint images agree exactly. All endpoint lifts remain in the larger analytic boxes. This argument uses closed boxes to retain the interface points and does not assume that open chart images overlap uniformly.

By Lemma 23, all these finitely many endpoint coordinates have analytic power parametrizations with a common denominator. For two positive endpoint coordinates \(A_i(x,t)\) and \(B_i(x,t)\), join them by \[y_i(x,t,s)=A_i(x,t)^{1-s}B_i(x,t)^s, \qquad 0\leq s\leq1.\] The monomial equality is retained, since it holds at both endpoints. The interpolation stays in the coordinate box. Free coordinates are interpolated linearly. If needed, insert one interior waypoint before making these interpolations; this also retains specified strict sign conditions. Every image point with \(t>0\) still belongs to the original graph, and hence its \(z\) coordinate belongs to \(D_{x,t}\).

For completeness, the clock controlling the speed is independent of \(x\). Write \(y_i^0(s)=y_i(x_0,t,s)\). Positive endpoint coordinates have the form \(t^\alpha\) times analytic positive units. On a fixed small complex neighborhood of \(x_0\), logarithms of the unit ratios to their values at \(x_0\) are analytic and uniformly bounded, with bounded derivatives. It follows that \[y_i(x,t,s)=y_i^0(s)E_i(x,t,s),\qquad |E_i|+|\partial_s E_i|\leq C,\] uniformly also for complex \(x\). Thus \[|\partial_s y_i(x,t,s)| \leq C\bigl(|(y_i^0)'(s)|+y_i^0(s)\bigr).\] Each \(y_i^0\) is monotone and bounded. Define the increasing clock \[h_t(s)= \frac{s+\sum_i|y_i^0(s)-y_i^0(0)|} {1+\sum_i|y_i^0(1)-y_i^0(0)|}.\] After \(s=h_t^{-1}(a)\), the displayed derivative estimate, the bounded total variations of the \(y_i^0\), and the elementary estimates for the free coordinates give a common Lipschitz bound. Shrinking the complex \(x\) neighborhood ensures that the complex interpolations remain in the complex analytic chart neighborhoods. Cauchy estimates give the same Lipschitz bounds for every \(x\)-derivative on a smaller neighborhood. Analytic chart maps preserve the estimates. Allocate fixed consecutive subintervals to the finitely many chart segments and concatenate them. The joints agree for real \(x\), and hence also on the complex neighborhood by analytic uniqueness. This proves (4). ◻

Proof of Theorem 22. We first prove the analytic assertion for one controlled path family provided by Lemma 24. Fix its generic real center \(x\) and suppress \(x\) on \(R_t,G\) and \(P\). Suppose \(F_t(x,0)=p\) and the paths are uniformly bounded. Passing to a subsequence, the uniform Lipschitz estimate gives \[F_t(x,\cdot)\longrightarrow F(\cdot) \quad\hbox{in }C^\alpha([0,1]) \quad(0<\alpha<1).\] Here and below this convergence is along the selected subsequence. The derivative bounds in Lemma 24 and \(R_t\to0\) show that on every fixed finite real \(u\) box the actual graphs \[ \Phi_t(u,a)=u+C_t(x,u)F_t(x+R_t(x)u,a) \tag{6}\] converge to \[ \Phi(u,a)=u+G(u)F(a). \tag{7}\] This holds in \(C^\alpha\) in \(a\), with any fixed finite number of \(u\) derivatives, after lowering \(\alpha\) if necessary. Every real point of (6) belongs to \(D_t^x\): in original coordinates it is exactly \[x'=x+R_t(x)u,\qquad x'+R_t(x')F_t(x',a)\in S.\] This identity is why all bounded real displacements, rather than a small fixed collection of orbits, must be retained.

The initial graph has room at every real displacement needed below. Indeed, the initial kernel condition, definable selection, and generic restriction give a real neighborhood \(U'\) and numbers \(r,\epsilon>0\) such that \[ B(p,r)\subset D_t^{x'} \qquad(x'\in U',\ 0<t<\epsilon). \tag{8}\] To see the uniformity, select positive radii and thresholds definably from the initial kernel condition, restrict to a cell on which they are continuous, and shrink to a relatively compact smaller box. For every fixed \(M\), (2) and invertibility give \(c_M>0\) such that \[ B\bigl(u+C_t(x,u)p,c_Mr\bigr)\subset D_t^x \qquad(|u|\leq M,\ 0<t<\epsilon_M). \tag{9}\] One may take \(c_M\) smaller than \(\bigl(2\sup_{|u|\leq M}\|G(u)^{-1}\|\bigr)^{-1}\). Thus the later real corrections are admissible even when they do not stay close to zero.

The triangular equation. We use Bishop’s reduction of the attached-disc condition to a boundary conjugation equation (Bishop 1965); see also (Hill and Taiani 1978, sec. 2, equation (2.1) and Proposition 2.1). The triangular polynomial system and the endpoint estimates needed here are established below. Normalize circle measure to have total mass one, denote its mean by \(\langle\cdot\rangle\), and let \(H\) be the real Hilbert transform with \(H(1)=0\) and \(H(\cos n\theta)=\sin n\theta\). A complex boundary value \(v+iw\) bounds a holomorphic disc precisely when \(v-\langle v\rangle=-Hw\). The operator \(H\) is bounded on \(C^\alpha\) for \(0<\alpha<1\) by the conjugate-function estimate of (Priwaloff 1916, 100–102), and on \(L^q\) for \(1<q<\infty\) by the M. Riesz theorem in the form proved in (Essén 1984, equation (0.1) and §§1–2).

Choose a Lipschitz schedule \(a:\partial\mathbb D\to[0,1]\) that equals zero on a nonempty open arc \(I\). Choose a smooth nonnegative function \(\chi\) supported in \(I\), with \(\langle\chi\rangle>0\). Add a complex parameter \(\zeta_j\chi\) in each block. Write \[V_j(u,\theta,\zeta) =G_j(u_{>j}(\theta))F_j(a(\theta))+\chi(\theta)\zeta_j.\] The real unknowns \(u_j\in C^\alpha(\partial\mathbb D,\mathbb R^{d_j})\) are required to have mean zero. Their equations are exactly \[ u_j=-H\operatorname{Im}V_j -(I-\langle\cdot\rangle)\operatorname{Re}V_j, \qquad j=s,s-1,\ldots,1. \tag{10}\] Starting with block \(s\), whose matrix is the identity, these formulas define a unique solution successively. They give a holomorphic disc \(f_{a,\zeta}\) with boundary \(u+G(u)F(a)+\chi\zeta\). Polynomial composition is continuous on \(C^\alpha\), so every compact family of schedules and sufficiently small perturbations has a uniform \(C^\alpha\) bound on its solutions. In particular it uses just one finite real \(u\) box. No estimate is asserted uniformly over schedules whose transition intervals subsequently shrink to zero.

Its center is \[ C_j(a,\zeta)=\langle G_j(u_{>j})F_j(a)\rangle +\langle\chi\rangle\zeta_j. \tag{11}\] Later blocks do not depend on earlier perturbations. Hence the real differential of \(\zeta\mapsto C(a,\zeta)\) is block triangular, with diagonal \(\langle\chi\rangle I\) on each complex block. It is invertible at every solution. This proves full real rank \(2d\), not only motion in the imaginary center directions.

Persistence and filling. Replace \(G(u)F(a)\) in the equations by \(C_t(x,u)F_t(x+R_t(x)u,a)\). On the Banach space of mean-zero real \(C^\alpha\) functions, the differential of the limiting left-hand system is identity plus a strictly block triangular bounded operator. Its inverse is the finite sum for a nilpotent triangular operator. The convergence after (6) gives convergence of these systems and their first derivatives in operator norm near any compact family of limiting solutions. For example it follows directly from the \(C^2\) bounds in \(u\), the product estimate in \(C^\alpha\), and convergence of the \(a\)-dependent coefficients in a slightly stronger Hölder norm. The implicit function theorem therefore gives nearby solutions, uniformly along the compact family, and convergence of their centers and center differentials. Local uniqueness patches the solutions over the family. To justify a uniform center-image ball, let \(K\) bound the inverse differentials of the limiting center maps at zero perturbation. Compactness and \(C^1\) convergence allow a fixed perturbation radius \(r_0>0\) on which the variation of those differentials, including the small-\(t\) error, is at most \(1/(4K)\), while the small-\(t\) inverse differentials at zero have norm at most \(2K\). The contraction proof of the inverse function theorem then puts a ball of radius \(r_0/(4K)\) about each unperturbed center in its center-map image. One first decreases \(r_0\) further if the start-room restriction below requires it.

Use the homotopy of schedules \(a_\sigma=\sigma a\), \(0\leq\sigma\leq1\); all retain the start arc \(I\). First use a preliminary closed perturbation ball to bound the limiting solutions in a finite \(u\) box. Enlarge that box slightly, compute its start-room constant in (9), and then decrease the perturbation ball so that \(\|\chi\zeta\|_\infty\) is smaller than half that room. Only after these choices take \(t\) small. On the support of \(\chi\), (9) admits all the chosen small complex perturbations. Away from that support the boundary lies on the actual graphs (6). Thus the entire compact homotopy of disc boundaries lies in \(D_t^x\). At \(\sigma=0\) and \(\zeta=0\) the limiting solution is \(u=0\) and the disc is constant \(p\). For small \(t\) and small \(\zeta\), its actual disc lies in the initial ball. The continuity principle in the locally pseudoconvex domain now keeps every filled disc in \(D_t^x\).

There is no issue at the exterior of the original coordinate box: for this fixed schedule the normalized boundary values, and hence the entire discs, are bounded by a fixed constant. Their original coordinates converge uniformly to \(x\in\Omega\). Choose a fixed convex coordinate box \(\Omega_0\) with \(x\in\Omega_0\Subset\Omega\). The intersection \(S\cap\Omega_0\) is locally pseudoconvex at every boundary point in \(\mathbb C^d\), using the given condition on \(S\) and convexity at the artificial boundary. Its components are therefore Stein by the Euclidean Levi theorem. A plurisubharmonic exhaustion of the component in use bounds every filled disc by its boundary maximum; the compact boundary homotopy prevents a first exit. All the discs stay in \(\Omega_0\) by their already established coordinate bounds.

For each \(\sigma\), the uniform center rank supplies a fixed ball \(B(C(a_\sigma,0),\rho)\) contained in \(D_t^x\) along the chosen subsequence, after slightly decreasing \(\rho\). This already implies eventual inclusion for all small \(t\): the set \[\{t>0:B(C(a_\sigma,0),\rho)\subset D_t^x\}\] is a one-dimensional subanalytic set and has \(0\) in its closure. It therefore contains an interval \((0,\epsilon_\sigma)\). Consequently \(C(a_\sigma,0)\) belongs to the full open kernel. The continuous curve of these centers starts at \(p\), so all its points belong to \(P\). This step does not require the replacement paths or the schedule homotopy themselves to be subanalytic.

Approaching the endpoint. For each \(\delta>0\), choose a Lipschitz loop \(a_\delta\) that starts at \(0\), goes to \(1\) and returns, equals \(0\) on a nonempty arc, and equals \(1\) outside a set of measure at most \(\delta\). Apply the preceding argument at this fixed \(\delta\), with zero perturbation. It gives \(C(a_\delta,0)\in P\). Set \(c=F(1)\). Since \(F\) is bounded, \[F(a_\delta)\longrightarrow c\quad\hbox{in }L^q \quad(1<q<\infty).\] We claim that every block of \(u^\delta\) from (10) converges to zero in every such \(L^q\). For the last block this is the boundedness of \(H\) and subtraction of the mean. If it holds for the later blocks, polynomiality gives \(G_j(u_{>j}^\delta)\to I\) in every finite \(L^q\): each monomial is estimated by Hölder’s inequality using the finitely larger exponents required by its degree. Multiplication by the uniformly bounded \(F_j(a_\delta)\) then gives \(V_j\to c_j\) in every finite \(L^q\). Equation (10) proves the induction. Taking means in (11) yields \[C(a_\delta,0)\longrightarrow F(1).\] Thus \(F(1)\in\overline P\). The order of choices was: fix the loop and its nonempty start arc; solve and bound all corrections; choose their finite real box; let \(t\downarrow0\) and obtain a center in \(P\); only then shrink the exceptional arcs. No shrinking-arc estimate is used to justify the finite-\(t\) implicit function theorem.

The generic and uniform quantifiers. It remains to pass from controlled paths to all paths. The family of kernels \(\mathcal K(D_t^x)\) is subanalytic, by its quantified ball definition; so is its component \(P_x\) marked by \(p\). For fixed integer \(L>|p|\) and positive rational \(\eta\), let \(E_{L,\eta}\) be the set of real centers for which arbitrarily small \(t\) admit a point of \(A_t^x(L)\) at distance at least \(\eta\) from \(\overline{P_x}\). This is a subanalytic set, using definability of components.

If \(E_{L,\eta}\) had interior, choose within it a generic smaller real box. One-dimensional definability in \(t\) and selection give endpoints and paths to them for every sufficiently small \(t\), after reducing the box generically. Use a slightly larger fixed spatial box for the paths. Lemma 24 supplies a controlled replacement family there. At a generic fixed center, a subsequence of its endpoints converges, and the analytic argument just proved puts its limit in \(\overline{P_x}\). This contradicts their distance of at least \(\eta\). Thus \(\dim E_{L,\eta}<d\). Delete these exceptional sets for the countably many \(L,\eta\). At every remaining center the negation of membership in \(E_{L,\eta}\) is exactly (3). Increasing integer boxes and decreasing rational accuracies give the stated quantifiers for all finite boxes and all positive accuracies.

Finally, a starting point approaching \(p\) can be joined to \(p\) by a segment in the uniform initial ball. Enlarging the fixed path box by a fixed amount accommodates this segment and proves the last assertion. ◻

Proof. If one inequality failed, a sequence in the bounded box would have a convergent subsequence on which the corresponding least eigenvalue is bounded above by a fixed negative number. Theorem 22 puts its limit in \(\overline P\), contradicting continuity and the semidefinite test there. There are only finitely many tests. ◻

Proof. Choose a real analytic defining function \(\rho\), and let \(\theta\) be its characteristic one-form on \(M\). Write \(D=\ker\theta=TM\cap JTM\). Since the Levi form is semidefinite, its null space agrees with the characteristic distribution \[K=\{v\in D:d\theta(v,D)=0\}.\] This distribution has real rank \(2\ell\) and is \(J\)-invariant. If \(X\) is a section of \(K\) and \(Y\) is a section of \(D\), then \(\theta([X,Y])=-d\theta(X,Y)=0\). Applying the Jacobi identity to two sections of \(K\) and one section of \(D\) shows that their bracket again has this property. Consequently \(K\) is involutive. The analytic Frobenius theorem gives analytic leaves, and their \(J\)-invariant tangent spaces make them complex submanifolds.

In affine algebraic coordinates write the real polynomial defining \(M\) as \[\rho(z,\bar z)=v(z)^*C v(z),\qquad C=C^*,\] where \(v\) is a finite vector of holomorphic monomials; adjoining a constant monomial is harmless. For a parametrization \(f\) of a complex plaque, polarization of the identity \(\rho(f(\zeta),\overline{f(\zeta)})=0\) gives \[v(f(\eta))^*C v(f(\zeta))=0\] for all \(\eta,\zeta\) in a smaller plaque. Indeed the polarized expression is holomorphic in \(\zeta\) and antiholomorphic in \(\eta\), and its restriction to the diagonal determines every coefficient of its convergent series. The complex span \(W\) of the vectors \(v(f(\zeta))\) is therefore totally isotropic for \(C\).

The algebraic set \(A=v^{-1}(W)\) contains the plaque and lies in \(\{\rho=0\}\). Every complex submanifold of \(M\) has tangent space in \(K\): the restriction of \(\rho\) vanishes identically, so its Levi form vanishes there, and semidefiniteness identifies its zero vectors with its kernel. Applying this at regular points of each local component of \(A\) bounds that component’s dimension by \(\ell\). The component containing the plaque thus has dimension exactly \(\ell\), and the plaque is open in it.

Only finitely many evaluation vectors are needed to span \(W\). Choose them by fixed evaluations in analytic plaque coordinates, and restrict to a parameter neighborhood where their rank is maximal and constant. Their span is then an analytic map into a Grassmannian. The equations \(v(z)\in W\) have uniformly bounded degrees, and their projective closures therefore have bounded degree. Track the component containing the marked plaque, using its irreducible germ. The family and its evaluation have the analytic algebraic presentation of Lemma 5. Complexification, a local branch in the component space, relative resolution, and generic smoothness give the stated smooth projective family. The generic restrictions here remove rank-drop and component-change loci; they do not assert uniformity over their closures. ◻

Proof. The quotient of \(M\) by the characteristic plaques is a real analytic contact manifold of dimension \(2r+1\). Indeed the hyperplane field \(D\) descends because \([K,D]\subset D\), and its induced exterior derivative is nondegenerate. Choose Darboux coordinates with contact form \(da-\sum_j v_j\,du_j\). Holding \(v\) fixed and writing \(a=s+v^{\mathsf T}w\), \(u=w\), gives a family on which the pulled-back contact form is \(ds\). Thus the \(w\)-directions form a Legendrian Lagrangian and the \(s\)-direction is transverse. The parameters \(v\) are the external parameters just mentioned.

At a point of a plaque, divide the complex tangent space by its null space. This gives a complex vector space of dimension \(r\) with a positive Levi form. The chosen real Lagrangian is complex independent: if its real span contained both \(a\) and \(Ja\ne0\), its symplectic form would vanish on \((a,Ja)\), contrary to Levi positivity. Its complexified evaluation, the leaf tangent, and the complexified transverse direction therefore span the ambient complex tangent space. This proves the full-rank assertion.

Let \(e_1,\ldots,e_r,f_1,\ldots,f_r\) be a fixed real symplectic frame of the contact quotient, with the \(e_j\) the chosen Lagrangian. Normalize the characteristic form by its value on \(\partial_s\). Its pullback is the fixed contact form, so its exterior derivative on the contact directions is the fixed symplectic form \(\omega\), including as the point moves along the plaque. The infinitesimal evaluations of the frame are holomorphic in the plaque variable. Use the evaluations of the \(e_j\) as a complex frame, and write those of the \(f_j\) as the columns of \(H\). The complexified kernel of evaluation consists of \((-H\xi,\xi)\). It is isotropic for the complexification of \(\omega\), since \(\omega\) has type \((1,1)\). The equation \[\omega((-H\xi,\xi),(-H\eta,\eta)) =\xi^{\mathsf T}(H-H^{\mathsf T})\eta=0\] proves symmetry. Choose the sign of the second symplectic frame so that \(B=\Im H\) is positive definite. For a vector \(a\) in the first real Lagrangian, its \(J\)-multiple in real symplectic coordinates is \[Ja=(-\Re H\,B^{-1}a,B^{-1}a).\] Consequently the metric \(\omega(a,Ja')\) on that Lagrangian has matrix \(B^{-1}\).

Here is the Taylor calculation, including the mixed terms. In the complexified family let \(\widetilde\rho(b,z)\) be the pulled-back defining function, positive on the selected side, and write \(b=x+iy\). On the real parameter family it is identically zero. The Legendrian choice gives \(\widetilde\rho_{y_w}(x,z)=0\) identically there, whereas \(a(z)=\widetilde\rho_{y_s}(x_1,z)>0\) after orientation. Thus every pure \(x_w\) derivative and every \(x_wy_w\) derivative occurring in the weight-two Taylor polynomial vanishes. Dividing by \(a(z)e^2\) gives, uniformly on compact sets in \((s,w,z)\) near the original plaque, \[\frac{\widetilde\rho(x_1+(e^2s,ew),z)}{a(z)e^2} \longrightarrow\Im s-(\Im w)^{\mathsf T}A(z)\Im w.\] The positive matrix \(A\) is the normalized Levi metric on the chosen Lagrangian, up to the fixed positive convention in the Levi form and Taylor coefficient. Rescale the symplectic frames by that constant; the preceding computation then gives \(A=(\Im H)^{-1}\) exactly.

Finally all infinitesimal evaluations come from a family algebraic in the projective fibre variable. Their entries are rational sections on the plaque variety. Taking the indicated frame ratios and pulling to its projective model makes \(H\) meromorphic on \(L\). This also proves the asserted bounded algebraic and analytic parameter dependence. ◻

Proof. Let \(K\) be any compact set in the entire scaled \((s,w)\)-space. For sufficiently small \(e\), the points \(x_1+(e^2s,ew)\) for \((s,w)\in K\), and \(z\) in a fixed smaller plaque patch, lie in the original regular defining chart. The Taylor convergence in the preceding proof holds uniformly there. Strict positive points of \(q_{z_0}\) have neighborhoods eventually in the chosen side; strict negative points have neighborhoods eventually outside it. A zero of \(q_{z_0}\) cannot be in the open kernel, because every neighborhood contains a strict negative point. Since \(K\) was arbitrary, this identifies the whole positive component with \(T_A\). It is connected. If \(r=0\) and both original sides occur, the two halfplanes are distinct fibre components: the real axis consists of the original boundary family and is still forbidden. The component through \(i\) is the upper halfplane in this case as well.

There is an explicit biholomorphism of \(T_A\) with a ball. First set \[S=s+\frac{i}{2}w^{\mathsf T}Aw, \qquad W=\frac1{\sqrt2}A^{1/2}w.\] An immediate expansion gives \(\Im S-|W|^2=\Im s-(\Im w)^{\mathsf T}A\Im w\). The map \[(S,W)\longmapsto \left(\frac{S-i}{S+i},\frac{2W}{S+i}\right)\] then maps the Siegel half-space onto the unit ball in \(\mathbb C^{r+1}\). The complex Jacobian \(g\) of their composition is nowhere zero, and the change-of-variables formula yields \[\int_{T_A}|g|^2 =\operatorname{Vol}(\mathbb B^{r+1})<\infty.\] Thus every point evaluation on \(A^2(T_A)\) is nonzero. If the fibre of \(E\) has other components, extend \(g\) by zero to them; this gives a holomorphic square-integrable function on that same whole fibre. No extension from a proper subdomain is being used. ◻

Proof. We first classify the fibres away from a proper real algebraic subset of \(L\). Refine a Boolean algebraic description of \(V\) using the irreducible real polynomial factors of its boundary equations, and take the factor defining the chosen regular hypersurface patch. Any other irreducible factor cannot vanish on every generic plaque: varying all \(2r+1\) real transverse parameters would make it vanish on an open part of the chosen irreducible hypersurface. The condition that its restriction to a plaque is identically zero is a finite algebraic condition on the bounded-degree family. Exclude its proper parameter locus. For the resulting generic plaque, discard the zeros of the remaining boundary equations, the singular locus of the chosen equation, and the zeros or poles of the frame determinants and normal derivative. These form a proper real algebraic subset of \(L\): none contains the original regular plaque patch.

At any remaining point, the original Boolean description is locally empty, full, or consists of one or both sides of the chosen equation. The identities of the Taylor calculation and of \(A=(\Im H)^{-1}\) continue as identities of real meromorphic functions on \(L\). Accordingly the full kernel fibre there is empty, full, or one or both of the two strict quadratic sides. Each side is connected, being an epigraph or a hypograph of a quadratic form. If such a side meets \(E\), it is wholly in its fibre, since a path in that side stays in the same total kernel component.

Each fibre component is a pseudoconvex tube and hence is convex by Bochner’s tube theorem (Noguchi 2020, Theorem 1.1). Moreover a generic component of \(E\) cannot contain a complex affine line. Indeed, over a small coordinate polydisk \(U\subset L\), the total domain \(E\cap(\mathbb C^{r+1}\times U)\) is pseudoconvex in a Stein coordinate space. Berndtsson’s theorem, with weight zero, says that \[((s,w),z)\longmapsto\log K_{E_z}(s,w)\] is plurisubharmonic, allowing the value \(-\infty\) (Berndtsson 2006, Theorem 1.1). The slices here are the actual unbounded slices; the fibre coordinates have not been truncated. These local assertions therefore agree on overlapping base boxes. If the restricted total domain is disconnected, apply the theorem to its components, or to an exhaustion componentwise. The Bergman kernel at a point only uses its fibre component, so this gives the same function.

A convex domain containing a complex affine line is invariant under translation in that line direction. To see this, take the convex hull of a small ball at any of its points and successively longer segments in the line; openness then gives each fixed translate of a smaller ball. In complex linear coordinates the domain is consequently a product with \(\mathbb C\). Fubini and the fact that an entire square-integrable function on \(\mathbb C\) is zero imply that its Bergman space is zero. An open chamber of such fibres would make the displayed plurisubharmonic function identically \(-\infty\) on a nonempty open subset of connected \(E\). This contradicts Lemma 28.

At a generic point the quadratic matrix is nonsingular. Its epigraph is convex precisely when it is positive semidefinite, and its hypograph is convex precisely when it is negative semidefinite. Hence the only generic nonempty components remaining are the positive definite epigraph, in \(\{\Im s>0\}\), and the negative definite hypograph, in \(\{\Im s<0\}\). No point of \(E\) can have \(\Im s=0\): an open neighborhood of such a point would contain a point over a generic base with that same imaginary scalar coordinate, contrary to this classification. Connectedness and the seed select the positive sign. Thus \(\Im H>0\) on a dense open subset of the projection \(V_2\) of \(E\).

The meromorphic Cayley matrix \[C=(H-iI)(H+iI)^{-1}\] has operator norm less than one on that dense subset. It is bounded where meromorphic and therefore extends holomorphically across its polar locus. Its norm is at most one throughout \(V_2\). Equality on some unit vector would, by the maximum principle applied to a suitable scalar matrix coefficient, give equality on the connected domain, contradicting strict contraction on the original plaque patch. Thus \(\|C\|<1\) everywhere, \(I-C\) is invertible, and \(H=i(I+C)(I-C)^{-1}\) is holomorphic with positive imaginary part. This defines the full epigraph \(F\) on the right of (13). The generic classification and openness imply \(E\subset F\), since the non-strict limiting inequalities cannot have an interior zero, by variation of \(\Im s\).

We give the remaining fullness argument, to keep it distinct from the Bergman argument. At a generic base point \(E_z=F_z\). Stratify the exceptional base set. At a regular codimension-one wall point in \(V_2\), some point of \(F_z\) is in \(E\) by the definition of the projection. Every compact path in the connected slice \(F_z\) is covered by finitely many small coordinate boxes compactly contained in \(F\). On a generic side of the wall all these boxes are available. The tangency propagation lemma, Lemma 15, applied successively to these boxes, propagates the one available central patch along the path. It uses only transverse parameter disks and relatively compact fibre patches, so compactness of the whole tube is not required. We conclude \(E_z=F_z\) at the generic wall points.

The remaining omission \(A=F\setminus E\) is relatively closed and subanalytic of real codimension at least two. Local pseudoconvexity and Theorem 16 make \(A\) complex analytic. It is invariant under all real translations in the fibre coordinates. For each such direction, analyticity continues this invariance to small complex translations: restrict its local defining functions to the complex translation parameter and apply the identity theorem. Iteration along paths in the connected slice \(F_z\) shows that if \(A_z\) is nonempty it is all of \(F_z\). This is impossible for \(z\in V_2=q(E)\). Hence \(A\) is empty. Finally \(V_2\) is the inverse image of \(E\) by \(z\mapsto((i,0),z)\), proving its local pseudoconvexity and the last assertion. ◻

Proof. Put \(v=t^{1/q}\) with a common denominator for the finitely many Puiseux expansions. There are integers \(a>0\) and \(b\ge0\) such that \(R_t=v^a A(v,x)\), all multiplying factors have norm \(O(v^{-b})\), and their analytic parts extend to a common complex neighborhood. Taylor-expand \(f(x+R_tz)\) through order \(m\). On any fixed bounded complex \(z\)-box the remainder after multiplication is \(O(v^{a(m+1)-c})\), where \(c\) is a fixed integer bounding the poles of all factors. Choose \(m\) with \(a(m+1)>c\). The retained part has a convergent Laurent expansion in \(v\), with finitely many negative terms and coefficients polynomial in \(z\). Boundedness on a nonempty complex open set forces the coefficient of the lowest negative power to vanish there and hence identically as a polynomial. Repeating removes all negative powers. The coefficient of \(v^0\) is the claimed polynomial, and the Taylor remainder proves convergence on every bounded complex box. Cauchy estimates give derivative convergence. The same argument with the coefficient parameter \(x\) retained gives the generic local analytic statement.

For the ratio, Taylor-expand the analytic part of \(R_t(x+R_t(x)u)\) in the increment \(R_t(x)u\), and then multiply by \(R_t(x)^{-1}\). Again sufficiently many Taylor terms leave a vanishing remainder. Boundedness for real \(u\) on an open box kills every negative Laurent coefficient, because a polynomial vanishing on a real open box vanishes identically. This proves even locally uniform holomorphic convergence on bounded complex \(u\)-boxes. Finally, at a generic \(x\), \[\det R_t(x)=v^k\bigl(d(x)+O(v)\bigr), \qquad d(x)\ne0.\] Since \(R_t(x)u\to0\), the quotient of these determinants at \(x+R_t(x)u\) and \(x\) tends uniformly to one. Thus \(\det G_x=1\). The adjugate formula gives both its polynomial inverse and uniform boundedness of the inverses on fixed boxes. ◻

Proof. The positive square root and its inverse are subanalytic algebraic operations on positive matrices. Generic joint Puiseux preparation therefore applies to \(T_t\) and \(T_t^{-1}\); positivity of the seed matrix and the nonzero leading coefficients give their power bounds. Every compact subset of \(P_2\) lies in the normalized \(S_2\) for small \(t\), so \(H_t\) maps there into Siegel space and \(H_t(p_2)=iI\). Apply the matrix Cayley transform. Its values lie in the unit matrix ball and its seed value is zero. Montel’s theorem and the maximum principle imply bounds for the inverse Cayley transform on compact subsets of connected \(P_2\): a limiting contraction cannot reach the unit sphere at an interior point while taking zero at \(p_2\). Thus \(H_t\) is locally bounded on a nonempty complex open set.

Lemma 30 now gives the polynomial limit on all bounded boxes, not merely within \(P_2\). Its imaginary part is nonnegative on \(P_2\). For every fixed nonzero complex vector \(v\), the harmonic function \(v^*\Im\tau\,v\) is nonnegative and is \(|v|^2\) at \(p_2\). The strong minimum principle makes it positive everywhere. This proves strict positivity of the matrix. ◻

Proof. We induct on \(\dim Y\). If the boundary has no ordinary real hypersurface patch, semialgebraic stratification shows that its complement has real codimension at least two. Indeed an open exterior would be separated from the nonempty open set \(V\) by a stratum of codimension one. Theorem 16 then makes \(Y\setminus V\) analytic, and hence algebraic by Chow’s theorem. Set \(d=0\), \(Q=Y\), \(P=S\) a point, and take this complement as the container. In a family, Lemma 18 selects a containing proper algebraic set with bounded equations; generic analytic selection of their normalized coefficients gives a real analytic container. Complexify those coefficients and use Lemma 17, shrinking so that the resulting relative divisor is proper on every fibre of the full box. The actual complements need not themselves be selected as an analytic parameter family. The same construction is available as a terminal step as soon as the complement consists only of algebraic holes. All zero-dimensional assertions have their evident empty-matrix meaning.

Composing the projective families. Otherwise choose a regular algebraic boundary patch of constant Levi rank, orient its inward side, and apply Lemmas 26–29. This gives the geometric first kernel \(E\) over the smaller projective model \(L\), with projection \(V_2\subset L\) and \(\dim L=\dim Y-r-1<\dim Y\). Write \(b_1\) for the complex first-family parameters and \(x_1\) for their real center; \(e\) will be their scale. Apply the induction hypothesis to \((L,V_2)\), keeping \(x_1\) and the unused real contact parameters as external parameters. Its output is \[q_2:Q_2\to B_2,\qquad D_2,\qquad S_2,\qquad (P_2,p_2,R_t^2).\] Here \(b_2\in B_2\) is the lower-stage parameter, \(x_2\) its real center, and \(t\) its scale. For now \(b_2\) is left unscaled.

Replace \(D_2\), if necessary, by the full-box proper relative divisor of Lemma 17. This only reduces the available complement and preserves its containment in the open family. Complexifying the analytic dependence on \(x_1\) composes the lower family with the first plaque family. We obtain \[q:Q\longrightarrow B_1\times B_2, \qquad \mathrm{ev}:Q\longrightarrow Y, \qquad G=\mathrm{ev}^{-1}(V),\] where \(q\) is smooth, proper, and projective. The continued divisor, still denoted \(D_2\), is defined on this full box. All resolutions, component choices, and box shrinkings here precede the choice of scales. We must select one component \(S\) of \(q(G)\) retaining the complements of \(D_2\), then identify its first kernel. This will allow the lower scale \(t\) to be introduced without changing the actual domain being normalized.

On a projective fibre of \(Q_2\) over \(S_2\), the pullback of \(H\) is constant. It is meromorphic on that connected projective fibre and lies in Siegel space off \(D_2\). Its Cayley transform is bounded there, extends across \(D_2\), and is constant on a compact connected complex manifold. The resulting symmetric matrix is therefore a meromorphic function \(H_*(b_2)\), with positive imaginary part on \(S_2\); its dependence on the retained first parameters is understood. Constancy along fibres persists meromorphically on the full connected box by uniqueness. Evaluate on a local holomorphic section avoiding indeterminacy to make \(H_*\) holomorphic on a full smaller box about a generic real center \(x_2\). A proper complex analytic polar set cannot contain an open subset of the totally real center box, so this choice is available.

The fixed parameter domain and its whole fibres. We now pass from the limiting fibres of \(E\) to the actual fibres of \(G\). Fix an open chamber compactly inside \(S_2\) and let \(b_1=x_1+(i\eta,0)\), \(\eta>0\). Every point of a central projective fibre off \(D_2\) evaluates into \(V_2\). By (13), a neighborhood of each such point is eventually in the inverse image of \(V\). Properness gives the following useful closed-set formulation: every accumulation point, as \(\eta\downarrow0\), of a hole on a compact parameter subbox belongs to \(D_2\). Otherwise a convergent sequence of holes would end in one of the just-described eventually included neighborhoods.

Here the limiting statement holds for all approaches through the upper half-disk, not just for the vertical ray. Write its scalar coordinate as \(u+i\eta\), recenter the real first parameter at \(u\), and put \(e=\sqrt\eta\). In normalized coordinates the point is again \((i,0)\). On a compact fibre patch outside \(D_2\), this point has a positive distance from the limiting forbidden set. Apply Lemma 9 to the capped distances of the normalized forbidden sets, including the remaining real parameters and the compact fibre variables jointly. Use one fixed bounded seed box and finitely many compact projective coordinate charts. The finite-test part of that lemma supplies one open real center neighborhood on which convergence is uniform on compact parameter and test boxes. Now choose an accuracy smaller than half the positive separation of the patch under consideration. The same separation holds for all sufficiently small \(e\) and nearby real centers. The threshold may depend on the patch; no positive lower bound for the separation near \(D_2\) is required. Every sequence of upper half-disk holes approaching a point off \(D_2\) would violate this separation on a compact patch about its limit. Thus all such accumulation points belong to \(D_2\), as required by the boundary trapping hypothesis. The container has a holomorphic continuation on the full box. Lemma 13, applied to the scalar upper half-disk, therefore traps the holes in that continuation on the indicated side. For the remaining first parameters use Lemma 14: their real boxes are totally real and nonpluripolar. This gives actual containment on a nonempty complex open parameter region. Choose the component \(S\) of \(q(G)\) reached there. Proposition 19 makes the actual holes analytic throughout \(S\), and Corollary 20 keeps them inside this same full-box divisor \(D_2\). Their proper images are precisely \(Z_b\setminus V\); these are compact analytic and hence algebraic. The container is proper fibrewise after shrinking at a generic center. Corollary 21 now gives local pseudoconvexity of \(S\) at its boundary. Since \(q(G)\) is open, a boundary point of one of its components cannot belong to any of its components; this proves the asserted exclusion from \(q(G)\).

For the first-kernel comparison below, make one further full-box restriction. The total evaluation \(Q_2\to L\) is generically a submersion. Choose at the generic center a point off \(D_2\) where its differential has full rank. Smoothness of the projective family gives a holomorphic section \(\sigma\) through this point. Shrink the box so that \(\sigma\) is everywhere off the continued container and the evaluation retains full rank on a fibre-coordinate patch around \(\sigma\). Include the first parameters in this holomorphic choice. The differential includes both base and projective fibre variables; it need not have full rank in the fibre directions alone.

After these box restrictions, retain the selected connected components \(S\) and \(S_2\). Fix \(b_2^0\) in the retained chamber of \(S_2\). For a compact path in \(S_2\) starting at \(b_2^0\), choose local sections of \(Q_2\) off \(D_2\) along a finite covering of the path. Their evaluations lie in \(V_2\). The strict inequalities in (13) and neighborhood eventual inclusion put their small first-scale displacements in \(q(G)\), with overlaps joining them to the region used to choose \(S\). Thus every such displaced path lies in this same \(S\) for all sufficiently small first scales. The threshold may depend on the compact path, but \(S\) is fixed before either scale tends to zero.

The first kernel in parameter space. Let \(\mathcal K_1\) be the full first parameter kernel of this fixed \(S\), with \(b_1=x_1+(e^2s,ew)\) and \(b_2\) unscaled. Its component \(A\) through \((i,0,b_2^0)\) is exactly \[ A=\left\{(s,w,b_2):b_2\in S_2,\quad \Im\begin{pmatrix}s&w^{\mathsf T}\\w&H_*(b_2)\end{pmatrix}>0\right\}. \tag{20}\] The compact-path argument and strict neighborhood inclusion in (13) put the entire displayed domain in this component. For the reverse inclusion, a parameter neighborhood in \(\mathcal K_1\) must give a neighborhood in the geometric kernel \(E\). Use the section \(\sigma\) and evaluation patch prepared above to make this an open evaluation test.

Suppose a neighborhood belongs eventually to the first normalized parameter domain. By (16) every point in this evaluation patch is then in the inverse image of \(V\). The submersion theorem, on a smaller fixed patch, sends these neighborhoods to neighborhoods in the first plaque family. The openness here is uniform after normalization. In local coordinates the normalized evaluation has the form \[(s,w,b_2,v)\longmapsto (s,w,F(x_1+(e^2s,ew),b_2,v)),\] where \(v\) is a projective fibre coordinate. The first coordinates are unchanged, and \(D_{(b_2,v)}F\) has a uniformly bounded right inverse on a smaller patch around \(\sigma\). The remaining derivative block tends to zero because it contains the factors \(e^2,e\). The block triangular differential consequently has a uniformly bounded right inverse on each compact normalized path neighborhood. The quantitative local submersion theorem gives the needed neighborhoods of a fixed positive size. In the limit this is a neighborhood test for membership in the full first geometric kernel. Along a path \((s(a),w(a),b_2(a))\) from \((i,0,b_2^0)\) in the selected component, the tested points \[(s(a),w(a),\mathrm{ev}_2(\sigma(x_1,b_2(a))))\] form one continuous path in that kernel. At the starting point, the section is off \(D_2\) over \(S_2\), so its evaluation is in \(V_2\) and the tested point is in \(E\). The entire tested path therefore stays in \(E\). There are no changes of section in this argument. Its base values stay in \(q_2(\mathrm{ev}_2^{-1}(V_2))\) and, by connection to the seed, in its component \(S_2\). Equation (13) gives precisely the strict matrix test in (20). This proves the reverse inclusion. The same rank argument, followed by the first family’s full-rank evaluation, proves full rank of the final evaluation on a patch meeting every fibre in a smaller box. Its differential is then injective also on the fibre tangent. The irreducible image therefore has dimension \(\dim Q_b=\dim Y-d\); properness of \(Q_b\) gives the remaining fibre-image assertions.

The second specialization and its marked component. We have obtained a fixed domain \(S\) and identified its first marked kernel \(A\). Now normalize \(H_*\) by (14)–(15). In the first block use \(b_1=x_1+(e^2s,eT_tw)\); in later blocks use \(b_2=x_2+R_t^2z\). Congruence of the matrix inequality by \(\operatorname{diag}(1,T_t^{-1})\) identifies its first-stage test with \[\Im\begin{pmatrix}s&w^{\mathsf T}\\w&H_t(z)\end{pmatrix}>0.\] Lemma 31 gives its limit (17). Apply the above \(t\)-dependent affine normalization to \(\mathcal K_1\) and \(A\) to obtain \(\mathcal K_{1,t}\) and \(A_t\). The seed \((i,0,p_2)\) lies in \(A_t\) for every small \(t\), because \(H_t(p_2)=iI\). Consider a compact path from this seed in the marked component of the open kernel of \(\mathcal K_{1,t}\). A compact connected tube of small neighborhoods around this path is eventually contained in \(\mathcal K_{1,t}\). Since that tube contains the seed, it is contained in its component \(A_t\) for all small \(t\). Its projection supplies neighborhoods eventually in the normalized \(S_2\). The projected path consequently belongs to the second open kernel and, by connection to \(p_2\), to \(P_2\). The limit matrix is nonnegative on the tube. With its lower block positive on \(P_2\), varying \(\Im s\) shows that a zero Schur complement cannot be an interior point. Thus the limiting matrix test is strict. Conversely, compact neighborhoods in the strict tower inequalities are eventually in \(A_t\), by kernel inclusion for \(S_2\) and locally uniform convergence of \(H_t\); their paths connect them to the seed. This proves the equality of marked iterated kernels. In particular, other components of the full first kernel cannot enter this marked component when the second scale tends to zero. Only compact-path neighborhood inclusion is used here, not the later accessible-path localization theorem.

A single scale. Apply the closed-set power diagonalization theorem, Lemma 11, to these fixed definable forbidden-set tests and their complements. It supplies an integer \(N\) such that replacing \(e\) by \(t^N\) preserves the iterated open kernel on every fixed bounded box, with the seed component just identified. Accuracy and box radius are fixed parameters in that theorem; the exponent is uniform in those parameters whereas the threshold in \(t\) is allowed to depend on them. Increasing \(N\) preserves this conclusion. We shall do so finitely many times below. The final normalization is \[ R_t=\operatorname{diag} (t^{2N},t^NT_t,R_t^2). \tag{21}\] It tends to zero and has polynomial inverse bounds. A compact ball about \((i,0,p_2)\) supplies the fixed seed neighborhood.

Recentering ratios and localization. We verify the ratio condition, including the square-root order. By induction the later-block ratio \(C_t^2(u_2)\) is bounded and converges polynomially. Replacing the later center by \(x_2'=x_2+R_t^2u_2\) makes its seed evaluation point equal to \[x_2+R_t^2\bigl(u_2+C_t^2(u_2)p_2\bigr).\] This is a bounded normalized point for bounded \(u_2\). Polynomial bounded-box convergence of \(H_t\) therefore bounds \[T_t^{-1}(T_t')^2T_t^{-1} =(T_t^{-1}T_t')(T_t^{-1}T_t')^{\mathsf T}.\] It follows that \(T_t^{-1}T_t'\) is bounded. No commutativity of the positive square roots is asserted or needed.

All functions involved have joint analytic Puiseux expansions near generic centers. Their derivatives of any fixed finite order have power bounds. The shift of \(x_1\) is \(O(t^N\|T_t\|+t^{2N})\) on bounded boxes. Choose \(N\) larger than the finitely many pole orders in the lower-scale data, their inverses, and the Taylor remainders used here. Its effect on those data and on the square-root ratio then tends to zero. The normal-coordinate ratio is identically one, and the tangential ratio has a limit depending only on \(u_2\). Apply Lemma 30 to obtain polynomial convergence with all derivatives. Applying its determinant argument separately to \(T_t\) and to each old block shows that every limiting block has determinant one. The later-variable dependence in older blocks is the induction hypothesis. This proves (18) and all its asserted moderation properties.

Finally use Lemma 9 for the closed forbidden-set tests in the normalized coordinates. At a generic real center the distance limits on each fixed box are unchanged by all centers tending to that center. A separate neighborhood and threshold are permitted for each fixed accuracy. The exact change of center in the original fixed set \(S\) is \[x+R_t(x)\bigl(u+C_t(x,u)Z\bigr) =x+R_t(x)u+R_t(x+R_t(x)u)Z.\] Taking neighborhood limit tests on both sides and using the bounded inverse ratios proves invariance of the full kernel under \(Z\mapsto u+G_x(u)Z\). For small \(u\), the image of the seed stays in the same component; hence this is an automorphism of \(P\). The bounded-path conclusion is now Theorem 22, applied to the fixed pseudoconvex domain \(S\), the seed ball, and the ratios just proved. This completes the induction. Here \(D_t^x=S_{t,x}\) in the notation of the localization theorem. ◻

Proof. At one stage of the tower put \(B(z)=\Im\tau(z)\). The Schur complement rewrites its condition as \[\Im s>(\Im w)^{\mathsf T}B(z)^{-1}\Im w.\] The positive slack \[\lambda=\Im s-(\Im w)^{\mathsf T}B(z)^{-1}\Im w\] gives smooth coordinates \((z,w,\xi,\eta)\) on the whole stage, where \(\xi=\Re s\) and \(\eta=\log\lambda\). Their inverse recovers \(s=\xi+i\bigl(e^\eta+ (\Im w)^{\mathsf T}B(z)^{-1}\Im w\bigr)\). Thus this stage is diffeomorphic to \(P_2\times\mathbb C^r\times\mathbb R^2\). Induction, starting from a point, proves the claimed diffeomorphism and contractibility.

For Steinness, let \(M_j\) be the polynomial matrices in all blocks. The function \[\Phi(Z)=|Z|^2-\sum_j\log\det\Im M_j(Z)\] is strictly plurisubharmonic on \(P\). The matrix entries have harmonic imaginary parts, and differentiating \(-\log\det\) gives a nonnegative Levi form, equivalently the squared matrix norm of the derivative conjugated by the positive inverse square root. At a finite boundary point at least one positive matrix becomes singular, so \(\Phi\to+\infty\). At infinity its negative terms are at worst logarithmic in \(|Z|\), because the \(M_j\) are polynomial, whereas \(|Z|^2\) dominates them. Thus \(\Phi\) is an exhaustion and \(P\) is Stein.

For a bounded realization use one block at a time. Since \(\Im M>0\), the matrix \(K=I-iM\) has Hermitian part greater than \(I\), and hence \(\|K^{-1}\|\le1\). Write its first column as \((a,b)^{\mathsf T}\in\mathbb C\times\mathbb C^r\). The lower-right block \(D=I-i\tau(z)\) is invertible, and the Schur complement formula gives \(a\ne0\). The equations for that column recover the original coordinates holomorphically: \[w=\frac{Db}{ia},\qquad s=\frac{i(1-a+iw^{\mathsf T}b)}{a}.\] Thus \((s,w,z)\mapsto(a,b,\psi_2(z))\), with an inductively obtained bounded realization \(\psi_2\), is injective, bounded, and has a holomorphic inverse on its image. It uses exactly \(r+1\) new coordinates and its image is open. This proves the same-dimensional bounded realization.

The automorphism group of a bounded domain is a real Lie group (Kobayashi 1967, Proposition 3.1 and Theorem 6.2). Differentiate the curves (19) at \(u=0\) to obtain complete real holomorphic vector fields. In the block order of the theorem their values have identity diagonal: differentiation in a coordinate of block \(j\) contributes its coordinate unit vector in that block, nothing in later blocks, and possibly vectors in earlier blocks, because \(G_k\) only depends on blocks later than \(k\). This triangular matrix has determinant one at every point. The resulting fields therefore span the complex tangent space everywhere. ◻

Proof. This is precisely Corollary 2(a) of Ivashkovich (Ivashkovich 2008). Here is also the Hartogs argument in the present projective case. On a Hartogs figure in a coordinate polydisc \(B\subset Y\), the covering map \(p:V\to X^+\) extends meromorphically to \(B\), by the meromorphic Hartogs extension theorem of Ivashkovich (Ivashkovich 1992), in the form of Theorem 4 of (Ivashkovich 2008). It agrees with \(p\) on the component of \(B\cap V\) containing the figure.

If that component does not fill \(B\), choose a straight segment from a point of the figure to a point outside \(V\), and let \(a\) be its first exit. The complex line containing the segment is not contained in the indeterminacy locus, since it meets the figure. Restriction of the meromorphic extension to this line extends holomorphically across its isolated indeterminacies: in projective homogeneous coordinates cancel the common vanishing order. Consequently \(p\) tends to a point \(x\in X^+\) along the segment. An evenly covered neighbourhood of \(x\) contains the image of a connected final subsegment. That subsegment lies in one sheet and converges, by the inverse sheet map, to a point of \(V\). Its limit in the Hausdorff space \(Y\) must also be \(a\), a contradiction. Every such Hartogs figure therefore fills in \(V\), which is the local pseudoconvexity criterion. Pulling back its local plurisubharmonic tests gives the same boundary property for holomorphic inverse images used in the construction. ◻

Proof. The first three assertions are Proposition 33. For the distance assertion, the map \(Z\mapsto(M_1(Z),\ldots,M_k(Z))\) is a closed holomorphic embedding into the product of the corresponding Siegel upper half spaces. Closedness follows because the entries \(s_j,w_j\) recover \(Z\), and a limit in that product still satisfies every strict matrix inequality. The Cayley transform identifies each Siegel upper half space with the unit ball for the operator norm in symmetric matrices. The Kobayashi distance from its origin is \(\operatorname{arctanh}\|W\|\): the radial disc gives the upper bound, and a norming complex linear functional gives the lower bound. Its automorphisms act transitively, so its distance balls are compact. Distance decrease therefore puts a Kobayashi ball of \(P\) inside a compact subset of \(P\). Bounded realization implies nondegeneracy, and coordinate balls give the local upper estimates proving that the Kobayashi distance induces the usual topology. The balls are consequently closed in those compact sets, and hence compact. ◻

Proof. Choose the scale on the ball of radius \(R_i+1\) so that localization gives \(\operatorname{Im}M_j\geq-\eta_i I\), with \(\eta_i\downarrow0\) and \(\eta_i<1/2\), throughout the accessible component. Require also the distance of its points to \(\overline P\), on each previously chosen bounded ball, to tend to zero. This is a diagonal use of fixed-ball estimates.

For a matrix with \(\operatorname{Re}A_j\geq I/2\), the principal matrix logarithm is holomorphic and \(|\operatorname{Im}\operatorname{tr}\log A_j|<n_j\pi/2\), where \(n_j\) is its size. Choose \(0<\alpha<1/(2\sum_jn_j)\). This proves the sector bound. All singular values of \(A_j\) are at least \(1/2\); hence \[|\det A_j|\geq2^{-n_j},\qquad |\det A_j|\geq2^{-(n_j-1)}\|A_j\|.\] Since the matrices contain every coordinate of \(Z\), multiplying these inequalities proves the asserted decay estimate.

Let \(r_i\downarrow0\) bound \(|h|\) near the artificial sphere. Take \(a_iR_i^2\leq1\), and choose \(m_i\to-\infty\) with \(\log r_i+1<m_i\). Let \(H_i\) be the seed component of \(\{u_i>m_i\}\) in \(A_i\). Any connected compact subset of \(P\) containing the seed belongs to \(H_i\) eventually, since \(h\) has a positive minimum modulus there. At a level boundary \(|h|\leq e^{m_i}\); the artificial boundary lies below the level; and the other boundaries admit no incidence with \(V\). Take \(\delta_i=e^{m_i}\). The function \(u_i\) is strictly plurisubharmonic because \(h\) is nowhere zero. ◻

Proof. Fix \(y\in V\). The set of its incident parameters is analytic. A positive-dimensional irreducible component meeting \(H_i\) would have a maximum of \(u_i\) greater than \(m_i\): on the artificial boundary \(u_i<m_i\), while the other parameter boundaries cannot contain an incidence with \(y\). The part above any intermediate level between \(m_i\) and a value on that component is compact. The maximum principle for strictly plurisubharmonic functions on positive-dimensional analytic sets is a contradiction. Thus all incident parameters in \(H_i\) are isolated.

Near any one of these roots, the projection of \(\mathcal I\) to \(V\) is proper and finite after choosing a small parameter neighbourhood whose boundary avoids the root. It is open: each local irreducible component of the finite source has dimension \(n\), so its image germ is the full germ of the smooth \(n\)-dimensional target. Count the roots with their finite-map multiplicities. The function \[ \Phi_i(y)=\sum_{b:y\in Z_b,\ b\in H_i} \max(0,u_i(b)-m_i) \tag{23}\] is bounded and plurisubharmonic on \(V\), with value zero when there are no positive summands. Here and below one may first use a slightly larger high region and put the cutoff strictly inside it. This avoids any finiteness assertion exactly on a level seam. Away from a finite projection’s discriminant the assertion is the sum rule for plurisubharmonic functions on holomorphic branches. Across the discriminant the roots remain in a compact parameter set, so the bounded removable extension gives the same trace, including multiplicities. Across a cutoff seam the summands tend to zero, and the plurisubharmonic pasting lemma applies. No upper part can appear without a limiting incidence. Finally, the subanalytic incidence family has a uniform finite bound on its number of isolated roots on this fixed ball. Multiplicity is bounded by the corresponding number of simple roots at nearby generic target points. This proves boundedness of the trace.

A bounded-above plurisubharmonic function on a Zariski open subset of a connected smooth projective variety extends across its analytic complement by (Demailly 2012, I.5.24) and is constant by compactness. In particular \(\Phi_i\) is constant after pullback to the projective model of any \(Z_b\cap V\), \(b\in S\), or by any map \(g\) in the statement. On a small disc in that model, lift the finite correspondence of high roots to its normalization and make a finite ramified base change. All its branches then become holomorphic. The trace is a sum of subharmonic functions, and each high branch contributes \(u_i\) minus a constant. Since the sum is constant and \(u_i\) is strictly plurisubharmonic, each high parameter branch has zero derivative. This also proves constancy at a multiple root, by continuity.

If high fibres \(Z_b\cap V\) and \(Z_c\cap V\) meet, apply this observation on a projective model of the first fibre. A neighbourhood of the meeting point in that fibre lies in \(Z_c\). Irreducibility and equal dimension give \(Z_b=Z_c\), hence \(b=c\). The finite incidence projection consequently has one point in each fibre over \(U_i\). Its reduced graph is a finite bimeromorphic map onto a normal space, so it is an isomorphism. This proves holomorphy and openness. Since \(\pi_i\circ e=q\) over the high region and \(q\) is a submersion, \(\pi_i\) is a submersion at every point: choose any preimage in \(Q_b\). The complement of a proper analytic subset of the connected smooth \(Q_b\) is connected and maps onto \(Z_b\cap V\); hence that fibre is connected. Finally the argument with \(g\) gives a constant parameter near a meeting point. The inverse image of the closed analytic subset \(Z_b\cap V\) then contains an open set in the connected source \(T\setminus D\), so is the whole source. ◻

Proof. Increase \(\epsilon_i\) by \(1/i\) if necessary so it is positive. First work on a disc compactly contained in \(D\), enlarging it slightly for estimates. Put \(\eta_i=2\epsilon_i\), and extend \(v_i=\max(|h_i|,\eta_i)\) by the constant \(\eta_i\). Pasting gives a uniformly bounded subharmonic function. If \(\chi\) is a smooth cutoff equal to one on a smaller disc, \(\int\chi\,\Delta v_i=\int v_i\Delta\chi\) gives a uniform bound for its Riesz mass there. On \(|h_i|>\eta_i\), \[\Delta|h_i|=\frac{|h_i'|^2}{|h_i|}\] with the usual Euclidean Laplacian. It follows that the energy of \(h_i\) on this set is bounded, and that \[\int_{2\eta_i<|h_i|<4\eta_i}|h_i'|^2=O(\eta_i).\] Choose a smooth scalar cutoff \(\lambda\), zero on \([0,2]\) and one on \([4,\infty)\), and extend \(F_i=\lambda(|h_i|/\eta_i)h_i\) by zero. The extension is locally in \(W^{1,2}\): near every internal boundary the formula vanishes, and the preceding energy bounds control the remaining region. Moreover \[\|F_i\|_{W^{1,2}(D')}\leq C_{D'},\qquad \|\bar\partial F_i\|_{L^2(D')}^2=O(\eta_i),\qquad |F_i-h_i|\leq4\eta_i\] for the zero extension in the last expression. Rellich compactness and the distributional Cauchy–Riemann equation show that a subsequence converges in \(L^2_{\rm loc}\) to a holomorphic function \(h\).

We verify uniform convergence, which does not follow just from this Sobolev bound. Fix an annulus compactly inside the test disc and first pass to a subsequence for which the squared \(L^2\) errors \(\|F_i-h\|^2\) on that annulus are summable. Fubini’s theorem gives \(L^2\) convergence of the traces for almost every radius. The integrals in the radial variable of the tangential \(W^{1,2}\) energies are uniformly bounded. Fatou’s lemma shows that their lower limit is finite for almost every radius. Choose a radius satisfying both conclusions and then a further subsequence with bounded tangential energies on that circle. Compactness of \(W^{1,2}\) of a circle in the continuous functions gives uniform trace convergence; the difference between \(F_i\) and the zero extension of \(h_i\) tends uniformly to zero as well. If \(h(z_0)\ne0\), start with an arbitrarily small annulus about \(z_0\) on which \(h\) takes values in a disc \(B(a,r)\) with \(0\notin\overline{B(a,2r)}\). The preceding radius choice then supplies such a surrounding circle inside this annulus. For large \(i\), the original \(h_i\) is defined near that circle and its values belong to \(B(a,2r)\).

For completeness, the argument principle excludes hidden holes inside this circle. Choose a regular level \(t_i\in(4\eta_i,5\eta_i)\) and apply the argument principle to the part of \(\{|h_i|>t_i\}\) inside the circle. This part has finitely many boundary components after a harmless choice of the level and outer circle: it is compact in \(A_i\), and the level is real analytic and regular. On each inner boundary, the image has modulus \(t_i\), so its winding number about a value \(w\) with \(|w|>t_i\) is zero. For \(w\notin\overline{B(a,2r)}\), the outer winding number is also zero. Thus \(h_i\) cannot take such a value with \(|w|>t_i\) inside. The component adjoining the outer circle cannot reach an inner level boundary: a continuous image from \(B(a,2r)\) to that small circle would pass through an excluded value. It therefore fills the disc, and all its values belong to \(\overline{B(a,2r)}\). Taking arbitrarily small circles and radii proves locally uniform convergence near \(z_0\).

At a zero of \(h\), choose a surrounding circle with no zeros of \(h\) and small maximum modulus, and apply the maximum principle to the pasted \(v_i\). This proves uniform convergence also there. If \(h\equiv0\), the sliced circles and the same maximum principle suffice directly. Hence every sequence has a uniformly convergent subsequence on a smaller disc.

The estimates were uniform for discs of fixed radius lying in a fixed coordinate box. They consequently apply to moving complex lines as well. They imply asymptotic equicontinuity of the zero extensions: if it failed, one could choose indices tending to infinity and two converging points with the same limit and a fixed positive difference of values. Apply the one-variable compactness result on the complex lines through these pairs, using fixed-radius parametrized discs in the box, to obtain a contradiction. The zero extensions can still have small jumps at each fixed index, so apply Arzela–Ascoli to the continuous functions \(F_i=\lambda(|h_i|/\eta_i)h_i\), defined by the same cutoff formula and by zero in all dimensions. Their difference from the zero extensions is at most \(4\eta_i\). Asymptotic equicontinuity, together with continuity of each of the finitely many initial \(F_i\), gives ordinary equicontinuity of this family on compact subsets. Arzela–Ascoli and a countable exhaustion give local uniform convergence of \(F_i\), and hence of the zero extensions. The limit is holomorphic on every coordinate line, by the disc result, hence holomorphic.

Finally \(\operatorname{Re}h\geq0\), and the moving-point condition gives \(h(x)=a\ne0\). The strong minimum principle gives \(\operatorname{Re}h>0\) on connected \(D\). Its modulus has a positive minimum on every compact set. Uniform convergence then excludes points where the zero extension was used. If compact containment failed for the original sequence, a violating subsequence would have a further uniformly convergent subsequence with the same nowhere-zero conclusion. This proves eventual compact containment for the original sequence as well. ◻

Proof. On each connected relatively compact test domain containing \(x\), the maps \(g_i\) are defined for all sufficiently large \(i\). Apply Lemma 38 there to \(h\circ\pi_i\circ g_i\) on \(g_i^{-1}(U_i)\). The boundary estimate is (24), including where the projection chart ceases to exist. At the moving seed the guard tends to \(h(p)\ne0\). A diagonal exhaustion of \(D\) therefore gives eventual inclusion in \(U_i\) and a nowhere-zero guard limit. Its positive minimum modulus on compact sets, together with the decay in Lemma 36, bounds all normalized parameters there. Montel’s theorem gives a holomorphic parameter limit \(F\).

To apply localization, join \(x\) to a prescribed point of \(D\) by a compact path, and precede it by a short path from \(x_i\) to \(x\) in a fixed coordinate ball. Include these paths in one relatively compact test domain. The preceding bounds hold on the whole paths, whose images eventually lie in \(U_i\). Their parameter paths start at points tending to \(p\) and stay in one fixed bounded box. The moving-start assertion of Theorem 22 puts their limiting endpoints in \(\overline P\). Hence \(F(D)\subset\overline P\). For each constant vector \(v\), \(v^*\operatorname{Im}M_j(F)v\) is a nonnegative harmonic function, positive at the seed whenever \(v\ne0\). The minimum principle makes it positive everywhere. Thus all the strict matrix inequalities hold and \(F(D)\subset P\). ◻

Proof. Enlarge \(K\) to a compact set contained in a connected relatively compact domain of \(P\) containing \(p\). Suppose the assertion fails. Choose \(j_\nu\to\infty\) and compact sets \(L_\nu\subset H_{j_\nu}\cap P\) exhausting \(P\): each fixed compact subset of \(P\) belongs to \(L_\nu\) eventually. Failure means that, for each fixed \(j_\nu,L_\nu\), there are arbitrarily large \(i\) for which \(\psi_i(K)\not\subset\varphi_{j_\nu}^{-1}(L_\nu)\). Choose one such \(i_\nu\) so large that \(H_{i_\nu}\) contains the \(\nu\)-th member of a fixed exhaustion of \(P\), that \(b_{i_\nu}\in M_{j_\nu}\), and that \[|\varphi_{j_\nu}(b_{i_\nu})- \varphi_{j_\nu}(b_0)|<1/\nu.\] These last conditions are possible because \(j_\nu\) is fixed when \(i_\nu\) is chosen. The displayed starting parameters therefore tend to \(p\).

Test \(\psi_{i_\nu}\) against the chart \(M_{j_\nu}\) and its guard. Corollary 39 applies on the expanding domains in \(P\). It says that \(\psi_{i_\nu}(K)\subset M_{j_\nu}\) eventually, and that their tested parameters lie in one fixed compact subset \(L\subset P\), after a subsequence. For large \(\nu\), \(L\subset L_\nu\), contradicting the choice of \(i_\nu\). This proves the fixed-chart assertion; control in merely changing charts would not have sufficed. ◻

Proof. Realize \(M\) as a bounded domain. Each coordinate of \(f\) is constant on every compact connected fibre of \(\mu\): a holomorphic function on a compact irreducible complex space is constant, and connectedness makes the constants on its components agree. For completeness, a holomorphic function on the inverse image of a small open set under a proper modification descends on its isomorphism locus and is locally bounded by properness. Normal removability extends it across the omitted analytic subset. Consequently \(\mu_*\mathcal O_V=\mathcal O_{\widetilde X}\), which gives the unique holomorphic descent. Its values lie in \(M\), and uniqueness gives equivariance. Also \(\mu^{-1}(f_X^{-1}(m))=f^{-1}(m)\), which proves connectedness.

On an original high chart, apply the projective ambient resolution map to the compact incidence family. Its proper image in parameter times the normal ambient compactification is analytic and proper over parameters. Its restriction to \(\widetilde X\) is exactly the graph of the single-valued descended parameter: proper surjectivity of \(\mu\) and the preceding fibre identity prove both inclusions. Normalize this compactifying family if necessary; normalization does not change the normal graph portion. The graph portion is open, and its fibre complement is analytic and hence algebraic in each projective fibre. Equivalently, before a deck translation each fibre is the intersection of an algebraic image cycle with the original semialgebraic \(\widetilde X\). This also proves the semialgebraic assertion. Translated charts preserve the biholomorphic assertion, without requiring algebraic deck transformations. ◻

Proof. Choose an abelian subgroup \(A\) of finite index in the deck group. The quotient \(Z_A=W/A\) is a compact Kähler manifold. Pull a Kähler metric on \(Z_A\) back to \(W\), and let \(\Delta=\operatorname{div}\nabla\). The pullback metric is complete. Fix \(o\in W\) and consider the convex set \[\mathcal H=\{h\in C^\infty(W):h>0,\ \Delta h=0,\ h(o)=1\}.\] Local Harnack inequalities (Serrin 1964, Theorems 5–7) and interior elliptic estimates (Nirenberg 1959, Lecture IV, equation (4.2)\('\) and its consequences) make \(\mathcal H\) compact in the topology of smooth convergence on compact subsets. In relatively compact metric charts the scalar Laplacian has smooth uniformly elliptic coefficients; its lower-order terms in divergence form vanish, so these estimates apply without an additive Harnack term. Limits are still strictly positive: they are nonnegative harmonic functions equal to one at \(o\), so the strong maximum principle applies.

Let \(h\) be an extreme point of \(\mathcal H\). Its ray is minimal in the cone of positive harmonic functions. Indeed, if \(0\leq v\leq h\) is harmonic and \(0<v(o)<1\), then \[h=v(o)\frac{v}{v(o)}+(1-v(o))\frac{h-v}{1-v(o)},\] so extremality makes \(v\) a multiple of \(h\). The endpoint cases follow from the maximum principle.

For each \(\alpha\in A\) there is a number \(C_\alpha\) such that \[ h(\alpha x)\leq C_\alpha h(x)\qquad(x\in W) \tag{26}\] for every positive harmonic function \(h\). To see the uniformity, choose a compact set \(K\subset W\) with \(AK=W\). Harnack chains between the two compact sets \(K\) and \(\alpha K\) give one constant for all positive harmonic functions. If \(x=\beta k\), where \(\beta\in A\) and \(k\in K\), apply this inequality to \(h\circ\beta\) and use \(\alpha\beta=\beta\alpha\). This proves (26). Minimality of the ray of \(h\) now gives \(h\circ\alpha=c_\alpha h\) for some \(c_\alpha>0\).

Consequently \(\eta=d\log h\) is \(A\)-invariant and descends to a smooth one-form on \(Z_A\). Its metric dual satisfies \[\operatorname{div}(\eta^\sharp) =\Delta\log h=-|\eta|^2.\] Integrating on \(Z_A\) shows that \(\eta=0\). Notice that the differential descends; a single-valued function \(\log h\) on \(Z_A\) is not required. Thus every extreme point of \(\mathcal H\) is the constant function one. The Krein–Milman theorem implies \(\mathcal H=\{1\}\). Finally, the real and imaginary parts of a holomorphic function are harmonic for a Kähler metric. Adding a constant to each bounded real part makes it positive, so both parts are constant. ◻

Proof. Resolve the graph of \(h\), obtaining a proper modification \(\tau:\widehat Q\to Q\) and a holomorphic map \(\widehat h:\widehat Q\to C\). Put \(\widehat q=q\tau\) and \(A=\widehat h^{-1}(s)\), with its reduced analytic structure. Over \(G_0\), equality of meromorphic maps gives \(\widehat h=h\tau\). In particular every point of \(G_0\cap h^{-1}(s)\) has preimages in \(A\cap\tau^{-1}(G_0)\).

Let \(A_i\) be an irreducible component of \(A\) meeting \(\tau^{-1}(G_0)\). On its nonempty relatively open part in this set, \(\widehat q\) has countable image. A holomorphic map of positive rank on a smooth patch has uncountable image by the holomorphic rank theorem. Thus \(\widehat q\) has rank zero on the regular locus of that open part and is constant on some nonempty relatively open subset of \(A_i\). The identity theorem on an irreducible reduced complex space then makes \(\widehat q\) constant on all of \(A_i\). Equivalently, a fibre of \(\widehat q|_{A_i}\) contains an open subset and hence equals \(A_i\).

For a relatively compact smaller box \(B'\Subset B\), \(\widehat q^{-1}(\overline{B'})\) is compact. The irreducible components of a complex analytic space are locally finite, so only finitely many \(A_i\) meet this compact set. Every attained parameter in \(B'\) is the constant value of one of the components meeting \(\tau^{-1}(G_0)\). There are therefore only finitely many such parameters. Components lying entirely outside the valid open set contribute no parameter and need not be excluded from the compact component count. ◻

Proof. Pull the regular covering \(p\) back along \(F_s\to X^+\). Each connected component of the pullback is a regular cover of \(F_s\) with deck group the image of \(\pi_1(F_s)\) in \(\Gamma\), up to the basepoint identification. That image is virtually abelian by (25). The composite of the lifted map to \(V\) with each bounded coordinate of \(f\) is constant by Lemma 43. Hence \(f\) is constant on each connected component of the pullback. There are at most countably many components, since the fibre of \(p\) is countable. This proves countability of \(D_s\).

Fix a full incidence chart \(q:Q\to B\), \(e:G\to f^{-1}(B)\). On \(Q\setminus D\) consider \[a=p\circ e:Q\setminus D\longrightarrow X^+.\] Replace the containing analytic subset by a reduced divisor if needed. The total space \(Q\) is connected. Its full-rank evaluation locus is a nonempty open set; removing the containing divisor leaves a nonempty open subset on which \(a\) is submersive. Its image contains an open subset of \(X^+\) and thus meets the dense ordinary core locus \(X^\circ\). At a preimage of such a point, the composition with the core has rank \(\dim C\). The extension theorem therefore gives a meromorphic map \[h:Q\dashrightarrow C, \qquad h|_{Q\setminus D}=c\circ r^{-1}\circ p\circ e.\] On the full open incidence locus \(G\), uniqueness of meromorphic continuation identifies this extension with the same meromorphic composition. In particular it is holomorphic and agrees with that composition on \[G_0=e^{-1}\bigl(p^{-1}(X^\circ)\bigr)\subset G.\] This identity also holds at points of \(G_0\cap D\): they have not been removed from the argument.

Surjectivity of \(e\) onto entire leaves gives the exact equality \[ D_s\cap B=q\bigl(G_0\cap h^{-1}(s)\bigr). \tag{27}\] Indeed a point on either side has a representative in the full incidence locus whose image in \(X^\circ\) belongs to the ordinary core fibre at \(s\). Lemma 44, applied to (27), proves local finiteness. Its proof does not require that additional exceptional sets in \(G_0\) extend to the compactification: \(G_0\) is an open set, and every compactified level component meeting it already has constant parameter. Nor does it require evaluation to be submersive at every representative of an attained parameter.

Very general values omit a countable union of proper analytic subsets of \(C\). On the ordinary submersion locus their inverse images have dense complement, by the Baire theorem in coordinate boxes. The ordinary locus is dense in \(X^+\); the covering \(p\) and the submersion \(f\) are open. Their images therefore show that the union of the \(D_s\) is dense in \(M\). When \(C\) is a point use the unique level; the countability argument and connectedness already force \(M\) to be a point. ◻

Proof. The group of automorphisms of a bounded domain is a real Lie group (Kobayashi 1967, Proposition 3.1 and Theorem 6.2) and acts properly (Blanchard and Cartan 1953--1954, sec. 2, Propositions 2–3). Let \(H\) be the closure of \(\Lambda\) in this group. For every very general \(s\), the set \(D_s\) is \(\Lambda\)-invariant: deck transformations preserve \(p\), and \(f\) is equivariant. A locally finite subset of a manifold is closed, so \(D_s\) is also \(H\)-invariant by continuity. The orbit under the connected group \(H^0\) of any point of \(D_s\) is connected and lies in a discrete set. It is a point. Thus \(H^0\) fixes the dense union of the \(D_s\) pointwise and consequently fixes \(M\) pointwise. It is the trivial subgroup. As a closed subgroup of a Lie group, \(H\) is a Lie group (Cartan 1952, II, §III, no. 27); zero-dimensional Lie groups are discrete. This proves the assertion for \(\Lambda\).

The equivariant surjection \(f\) induces a continuous surjection \(X^+=V/\Gamma\to M/\Lambda\). Its source is compact, so the quotient is compact. ◻

Proof. The group \(\Gamma\) is finitely generated, being a quotient of \(\pi_1(X^+)\), and hence so is \(\Lambda\). Let \(\mathfrak g\) be the real Lie algebra of \(\operatorname{Aut}(M)\). Selberg’s lemma (Selberg 1960, Lemma 8), in the matrix-group form recalled in (Wolf 1968, 435), applied to the finitely generated linear group \(\operatorname{Ad}(\Lambda)\subset\operatorname{GL}(\mathfrak g)\) gives a torsion-free subgroup \(L\) of finite index. Set \[\Lambda_1=(\operatorname{Ad}|_\Lambda)^{-1}(L).\] For \(m\in M\), properness and discreteness make \((\Lambda_1)_m\) finite. Its image in the torsion-free group \(L\) is trivial. If \(\lambda\in(\Lambda_1)_m\), therefore, \(\lambda_*\xi=\xi\) for every complete vector field \(\xi\in\mathfrak g\). Evaluating at the fixed point gives \[d\lambda_m\bigl(\xi(m)\bigr)=\xi(m).\] The complex spanning property and complex linearity of the derivative give \(d\lambda_m=1\). A finite-order holomorphic automorphism fixing a point with identity derivative is the identity: if its order is \(k\) and \(z\) is a local coordinate system centered there, then \(k^{-1}\sum_{j=0}^{k-1}z\circ\lambda^j\) is an invariant coordinate system, since its derivative is the identity. The inverse function theorem and analytic continuation prove \(\lambda=1\) on \(M\). Thus all stabilizers are trivial. No faithfulness of the full adjoint representation has been assumed.

The subgroup \(\Lambda_1\) is also torsion-free. Indeed, lift a finite triangulation of the compact smooth quotient \(M/\Lambda_1\) to its contractible universal cover. Its cellular chains form a finite-length free resolution of \(\mathbb Z\) over \(\mathbb Z[\Lambda_1]\). Restriction to a subgroup of prime order would remain a free resolution of finite length, contrary to the nonzero cohomology in arbitrarily high even degrees of a finite cyclic group. Every nontrivial finite-order element has a power of prime order, so no such element exists.

The subgroup \(\rho^{-1}(\Lambda_1)\) has finite index in \(\Gamma\). The corresponding connected topological covering of \(X\) is a finite unramified analytic covering. The proper-base Riemann existence theorem (Grothendieck 2003, XII, Corollary 4.6) algebraizes it. Pullback of an ample bundle under the resulting finite morphism is ample (The Stacks Project Authors 2026, Tag 0B5V), so the cover is projective; its analytic covering charts make it normal. This justifies the simultaneous finite-cover replacement. ◻

Proof. Discreteness and the free finite-index action were just proved. Replace \(\Lambda\) by the subgroup supplied by Lemma 47 and put \(B=M/\Lambda\). The intrinsic Bergman kernel of square-integrable canonical forms on the bounded realization of \(M\) is everywhere positive, and its logarithm has positive definite complex Hessian. The reciprocal kernel is a Hermitian metric on \(K_M\); invariance under biholomorphisms makes it descend to a metric on \(K_B\) with positive curvature. Consequently \(K_B\) is ample by the Kodaira embedding theorem (Kodaira 1954, secs. 1–2; see also §4), \(B\) is projective, and \(c_1(B)<0\). In complex dimension one the same positivity-to-ampleness step follows from Riemann–Roch; the point case is immediate.

The manifold \(B\) is compact and aspherical because its universal cover \(M\) is contractible. The Nadel–Frankel splitting theorem in the aspherical form of (Farb and Weinberger 2008, Theorem 1.10) applies. It gives, on universal covers, a holomorphic decomposition \[M\simeq D_{\mathrm{sym}}\times R,\] where \(D_{\mathrm{sym}}\) is a bounded symmetric domain and \(\operatorname{Aut}(R)\) is discrete. The cited statement is obtained from a product decomposition of a finite cover of \(B\); this does not change its universal cover. Its Hermitian locally symmetric factor is of noncompact type, as its canonical bundle is ample, so its simply connected cover has the bounded symmetric realization used here.

We explain why the identity automorphisms of \(M\) cannot mix these two factors. The Bergman metric of \(M\) descends to compact \(B\), so is complete. On a product, Fubini’s theorem identifies the Hilbert space of square-integrable canonical forms with the Hilbert tensor product of the two factor spaces. Taking product orthonormal bases gives the product formula for the kernels. It follows that \[g_M=g_{D_{\mathrm{sym}}}\oplus g_R.\] Both factor metrics are positive definite, since \(g_M\) is, and complete, since a factor slice is a closed totally geodesic submanifold of this complete product. Both factors are simply connected, since \(M\) is.

There is no Euclidean de Rham factor in \(M\). Otherwise its parallel vector fields span a nonzero parallel real subspace preserved by the parallel complex structure, hence yield a holomorphic Euclidean factor \(\mathbb C^a\), \(a>0\). Restricting a bounded realization of \(M\) to a complex line in this factor contradicts Liouville’s theorem. Uniqueness of the non-Euclidean irreducible de Rham factors (Rham 1952, sec. 7, Théorème III) now implies that every isometry in the identity component preserves each factor distribution, and in particular the two displayed factor foliations. Thus every element of \(\operatorname{Aut}(M)^0\) acts as a product automorphism. Its component on \(R\) is constant as a function of the group element, because \(\operatorname{Aut}(R)\) is discrete; it is the identity. Every vector field from this identity group therefore has zero \(R\)-component. The complex spanning property forces \(\dim R=0\). Hence \(M\simeq D_{\mathrm{sym}}\). ◻

Proof. Choose a smooth projective core model, with \(C\) smooth, \[r:N'\longrightarrow N,\qquad c:N'\longrightarrow C.\] Let \(N^\circ\subset N\) be a Zariski open set on which \(r\) is an isomorphism and the ordinary core is holomorphic and submersive. The composition \(c\circ r^{-1}\circ a\) extends meromorphically to \(\overline U\) by the core extension theorem: the boundary is analytic, it can be included in a reduced divisor, and \(a\) is locally an isomorphism. Resolve this meromorphic map to obtain \[\tau:\widehat U\longrightarrow\overline U, \qquad h:\widehat U\longrightarrow C.\] The modification can be chosen to be an isomorphism above \(a^{-1}(N^\circ)\). Put \(\widehat U_U=\tau^{-1}(U)\). The complement of \(\widehat U_U\) is an analytic subset of the compact smooth projective manifold \(\widehat U\).

Take \(s\in C\) very general, also outside the proper analytic sets required for generic smoothness of \(h\) and \(c\). The compact level \(L=h^{-1}(s)\) has finitely many connected components, each a smooth compact complex manifold. For every component \(L_i\) meeting \(\widehat U_U\), its complement of the analytic boundary is connected. Within this connected complex manifold, the further excluded set \[(a\circ\tau)^{-1}(N\setminus N^\circ)\] is analytic. If the component meets the retained ordinary locus, this is a proper analytic subset and its removal is still connected. Here we use the elementary fact that a proper analytic subset of a connected complex manifold does not disconnect it: a path can be perturbed to avoid its real-codimension-at-least-two stratification. The excluded subset need not extend to all of \(L_i\) for this argument. Components contained in the excluded set are simply discarded.

It follows that the retained ordinary level \[ L\cap\tau^{-1}(a^{-1}(N^\circ)) \simeq a^{-1}(r(F_s^\circ)), \qquad F_s=c^{-1}(s),\quad F_s^\circ=F_s\cap r^{-1}(N^\circ), \tag{28}\] has finitely many connected components. The isomorphism uses the choice that \(\tau\) is an isomorphism on the ordinary locus. The fibre \(F_s\) is smooth, connected, projective and special, and \(F_s^\circ\) is the complement of a proper analytic subset of it.

Fix a point of \(F_s^\circ\) and a lift to \(U\). The monodromy representation of the regular covering \(a\) identifies its fibre with \(K\). If \[H=\operatorname{im}\bigl(\pi_1(F_s^\circ)\longrightarrow K\bigr),\] the connected components of the restricted covering in (28) are the \(H\)-orbits in this fibre, equivalently the cosets of \(H\) in \(K\). Thus \[ [K:H]=\#\pi_0\bigl(a^{-1}(r(F_s^\circ))\bigr)<\infty. \tag{29}\]

There is no increase of the relevant monodromy when the compact fibre is replaced by \(F_s^\circ\). Indeed the map to \(N\) extends as \(r|_{F_s}:F_s\to N\). Moreover \(\pi_1(F_s^\circ)\to\pi_1(F_s)\) is surjective, by perturbing loops off the proper analytic complement. Hence \[H=\operatorname{im}\bigl(\pi_1(F_s)\longrightarrow\pi_1(N) \longrightarrow K\bigr).\] The group \(\pi_1(F_s)\) is virtually abelian by (25); so is its quotient \(H\). An abelian subgroup of finite index in \(H\) also has finite index in \(K\), by (29). This proves the lemma.

If the core is a point, one may instead apply (25) directly to \(N\), since \(N\) is then special and \(K\) is a quotient of \(\pi_1(N)\). This also covers dimension zero. ◻

Proof. Since \(\Lambda\) acts freely, the stabilizer in \(\Gamma\) of \(m\) is exactly \(K\). The equivariant submersion descends to a holomorphic submersion \[g:X^+\longrightarrow B=M/\Lambda.\] It is proper because \(X^+\) is compact. Its fibre \(N_b\), for the image \(b\) of \(m\), is a smooth connected projective manifold. Restricting the covering \(p\) gives \[f^{-1}(m)\longrightarrow N_b\] as a connected regular covering with group \(K\); all identifications follow by lifting to a small evenly covered neighborhood in \(B\). The fibre compactifications in Theorem 42 give the hypothesis of Lemma 49. If the initial compactifying fibre is singular, it is smooth on the open leaf; resolve its boundary while leaving that leaf unchanged. The lemma therefore proves that \(K\) is virtually abelian. Finally the monodromy map \(\pi_1(N_b)\to K\) is surjective, because the restricted covering is connected and regular. A compact manifold has finitely generated fundamental group, so \(K\) is finitely generated. ◻

Proof. Choose a finite-index abelian subgroup of \(K\). It is finitely generated, and hence contains a finite-index torsion-free subgroup \(B\). Put \(d=[K:B]\), and intersect all subgroups of \(K\) of index at most \(d\). There are finitely many such subgroups, since homomorphisms from a finitely generated group to each of the finite symmetric groups form a finite set. Their intersection \(K'\) is characteristic, has finite index, and is contained in \(B\). It is therefore free abelian. Since \(K\) is normal in \(\Gamma\), so is \(K'\).

The map \(\mathcal X\to U/K\) is a finite covering. Thus the stated properness and regularity follow from the projection contract. As \(U\) is simply connected, its quotient by \(K'\) has fundamental group \(K'\). A locally trivial bundle over the contractible manifold \(M\) is a homotopy equivalence on each fibre; alternatively, its homotopy exact sequence gives the displayed isomorphism. The covering induced on a fibre consequently corresponds to the zero subgroup of its fundamental group, and is connected. ◻

Proof. Let \(r:\mathcal Y\to\mathcal X\) be the simultaneous resolution and let \(p^+:\mathcal Y\to M\) be its smooth proper projective structure map. First, for each \(m\), the map \(r_m:Y_m\to N_m\) is surjective on fundamental groups. Indeed, lift it to the connected covering of \(N_m\) corresponding to the image of \(\pi_1(Y_m)\). Each connected point inverse of \(r_m\) maps to one point of the discrete covering fibre. Since a proper surjection is a quotient map, the lift descends to a continuous section of this covering. A connected covering with a section has one sheet. This proves the assertion.

It follows that \[ W_{\mathbb Z}:=R^1p_*\mathbb Z\ \hookrightarrow R^1p^+_*\mathbb Z=:H_{\mathbb Z} \tag{30}\] is an inclusion of local systems, with saturated image. Here the local-system assertion uses the topological local triviality specified above, and its compatibility with \(r\) uses the map of the actual families. Saturation also follows directly: a homomorphism \(\pi_1(Y_m)\to\mathbb Z\) whose nonzero integral multiple vanishes on \(\ker(r_{m*})\) itself vanishes on that kernel.

We use the normal-projective Albanese theorem in its integral form: if \(N\) is normal and projective, then \[H_1(N,\mathbb Z)\longrightarrow H_1(\operatorname{Alb}(N),\mathbb Z)\] is surjective with finite kernel (Claudon et al. 2013, Lemma 4.7). In the present situation \(H_1(N_m,\mathbb Z)=K'\) is torsion-free, so this map is an isomorphism. In particular its rank is \(2a\). Moreover, the image of \(H^1(N_m,\mathbb Q)\) in \(H^1(Y_m,\mathbb Q)\) is a Hodge substructure: it is exactly the pullback from \(\operatorname{Alb}(N_m)\) under the algebraic map \(Y_m\to N_m\to\operatorname{Alb}(N_m)\). This observation also derives the required purity without an assumption about the singularities being rational.

Let \(F^1H_{\mathcal O}\) be the holomorphic Hodge subbundle of the weight-one variation of the smooth projective family \(p^+\). Holomorphicity holds over this analytic base by Griffiths’s period-mapping theorem (Griffiths 1968, II, Theorem (1.1)). The intersection \[W^{1,0}=W_{\mathcal O}\cap F^1H_{\mathcal O}\] is a holomorphic subbundle: it is the kernel of the holomorphic map \(W_{\mathcal O}\to H_{\mathcal O}/F^1H_{\mathcal O}\) and has the constant rank \(a\). Together with \(W_{\mathbb Z}\) it defines a polarizable weight-one variation. The lattice dual to \(W_{\mathbb Z}\) embeds by evaluation in \(E=(W^{1,0})^\vee\), and the quotient gives a smooth family \(E/W_{\mathbb Z}^\vee\) of abelian varieties, by the integral analytic Jacobian construction (Claudon 2018, sec. 2.1). Fibrewise the lattice is a full lattice; continuity in a local basis makes the quotient a holomorphic torus family, with no change of lattice index.

On a small base ball, choose a section of the smooth family \(p^+\). The relative Albanese map of \(\mathcal Y\) based at that section, followed by the quotient specified by \(W\), is a holomorphic map to this torus family; see the smooth construction in (Fujiki 1983, sec. 1.4) and (Griffiths 1968, II, §2(a) and Example (2.15)). Equivalently, integrate the relative holomorphic one-forms in \(W^{1,0}\). Its map on integral first homology is the quotient induced by \(r_m\), so its restriction to every point inverse of \(r_m\) has trivial induced fundamental-group map. Each compact connected point inverse therefore lifts to the vector-space cover of the target torus. Holomorphic functions on a compact connected reduced complex space are constant, so this restriction is constant.

Thus the relative map is constant on the fibres of \(r\). Since \(\mathcal X\) is normal and \(r\) is a proper modification, \(r_*\mathcal O_{\mathcal Y}=\mathcal O_{\mathcal X}\); the map descends holomorphically to \(\mathcal X\). Changing the chosen local section changes only the origin in the torus family. The resulting local maps glue to the family of Albanese torsors \(\mathcal A\) and to \(\alpha\), as in the translation-cocycle description of (Claudon 2018, sec. 2.2, Proposition 2.1). The fibrewise universal property identifies this map with the normal Albanese morphism, which is algebraic on every projective fibre. An element of \(\Gamma/K'\) acts on the intrinsic system \(W_{\mathbb Z}\) and its Hodge subbundle, and thus induces a holomorphic linear map of the torus families. In charts with origins, the additional translation is obtained by evaluating the relative Albanese map at the image of the chosen local section; it is holomorphic. The fibrewise universal property makes these maps agree on overlaps and obey the group law. This gives the asserted \(\Gamma/K'\)-equivariance of the torsor construction.

Properness of \(\alpha\) follows from properness of \(p\): a compact subset of \(\mathcal A\) projects into a compact subset of \(M\), and its inverse image is closed in the compact inverse image under \(p\). The fibrewise fundamental-group assertion follows from the integral Albanese statement and \(\pi_1(N_m)=K'\). The smooth proper torus family \(q\) is topologically locally trivial. Since \(M\) is contractible, its fibres and those of \(p\) induce isomorphisms on fundamental groups. The assertion for \(\alpha\) follows from the commutative diagram of these two fibre inclusions. ◻

Proof. We apply the following precise form of the Kollár–Pardon theorem (Kollár and Pardon 2012, arXiv version 2, Theorem 20): if \(N\) is normal projective, \(B\) is smooth projective with contractible universal cover, and \(g:N\to B\) is a morphism such that the entire pullback to that cover is biholomorphic to a semialgebraic open subset of a projective variety, then \(g\) is a holomorphic fibre bundle. Surjectivity is part of its conclusion; connected fibres are not a hypothesis or an automatic part of the general statement.

Apply this to \(N_m\to\operatorname{Alb}(N_m)\). The target is an abelian variety with universal cover \(\mathbb C^a\). By Lemma 52, the pullback cover is connected and is precisely the universal cover of \(N_m\). The required semialgebraic presentation is therefore the one supplied by the projection construction. Each Albanese map is a holomorphic fibre bundle and is flat. Its fibres are connected: in the homotopy sequence of a bundle, surjectivity on \(\pi_1\) implies that \(\pi_0\) of the fibre is a singleton. Since \(\pi_2\) of a torus is zero and the map on \(\pi_1\) is an isomorphism, the same sequence shows that each fibre is simply connected. Local holomorphic products with a ball and normality of \(N_m\) show that the fibre is reduced and normal. It is projective, being an algebraic fibre of a morphism of projective varieties. These arguments include \(a=0\), when the Albanese map is the map to a point.

The fibrewise flatness criterion now applies to \(\mathcal X\to\mathcal A\to M\): the source is flat over \(M\), the target is smooth over \(M\), and the maps on all fibres over \(M\) are flat. More explicitly, for the analytic local rings \(R\to B\to C\) at \(m\), \(\alpha(x)\) and \(x\), respectively, \(B\) and \(C\) are flat over \(R\), and \(C/\mathfrak m_R C\) is flat over \(B/\mathfrak m_R B\). The fibrewise flatness criterion for Noetherian local rings (The Stacks Project Authors 2026, Tag 00MP) applies with module \(C\), which is finite free over itself, and implies that \(C\) is flat over \(B\). Thus \(\alpha\) is flat at every point.

Let \(T\to\mathcal A\) be its universal covering. The pullback \(\mathcal X\times_{\mathcal A}T\) is connected and simply connected, by the isomorphism on fundamental groups in Lemma 52. It is consequently \(U\), and its projection \(j\) is proper and flat by base change. It is surjective by the fibrewise surjectivity just proved. Its fibres have all the asserted properties.

We explain the structure of \(T\) without requiring the torus family to be isotrivial. On a small contractible ball \(B\subset M\), choose an origin for \(\mathcal A_B\). Relative exponential identifies its fibrewise universal cover with the vector bundle \(E_B\). The inclusion of a fibre in \(\mathcal A\) induces an isomorphism on \(\pi_1\), since \(M\) is contractible. Hence the restriction of the global universal cover \(T\) to \(\mathcal A_B\) is connected and is exactly this cover \(E_B\to\mathcal A_B\). On overlapping balls, changes of origin lift to translations, while the linear terms are the transition maps of the relative Lie bundle \(E\). Thus \(T\to M\) is an affine bundle with linear part \(E\). Its translation cocycle lies in \(H^1(M,\mathcal O(E))=0\) by Cartan’s theorem B (Cartan 1951--1952, Exposé 18, §4). It follows that \(T\simeq E\). The vector bundle \(E\) is topologically trivial because \(M\) is contractible, and holomorphically trivial by the Oka principle (Grauert 1958; Forstnerič and Prezelj 2000). Thus \(T\simeq M\times\mathbb C^a\), which is Stein, contractible and homeomorphic to Euclidean space.

The functorial action on \(\mathcal A\) lifts to \(T\), since \(T\) is its universal cover. Choose a lift \(\gamma_T\) whose value at one point agrees with \(j\circ\gamma\). Then \(\gamma_T\circ j\) and \(j\circ\gamma\) are lifts from the connected space \(U\) of the same map to \(\mathcal A\) and agree at one point, so they agree everywhere. Surjectivity of \(j\) then gives uniqueness of \(\gamma_T\) and the group law. If a compact \(C\subset T\) meets \(\gamma C\), then the compact set \(j^{-1}(C)\) meets its \(\gamma\)-translate. The deck action on \(U\) is proper, so only finitely many \(\gamma\) have this property. This proves properness on \(T\), including finite stabilizers and a possible finite kernel. The induced surjection \(U/\Gamma\to T/\Gamma\) shows that the latter quotient is compact. Finally, pull back an ample line bundle from \(U/\Gamma\). It is \(\Gamma\)-invariant. On a compact fibre its map to \(U/\Gamma\) is finite onto its image, being proper and locally injective, so this pullback is ample on the fibre. ◻

Proof. We give the controlled proper-homotopy argument, including the case of finite stabilizers; compare (Kollár and Pardon 2012, sec. 1, Lemmas 11–12 and Proposition 13). A proper cocompact smooth action admits an invariant complete Riemannian metric. Its distance \(d\) is proper. Fix \(o\in T\) and \(R>0\) such that the balls \(B(\gamma o,R)\) cover \(T\). Choose continuous functions \(\lambda_\gamma:T\to[0,1]\) equal to one on these balls and zero outside \(B(\gamma o,R+1)\), obtained by translating one function of distance. Properness makes their supports locally finite. Cocompactness and properness also give a uniform bound \(N\) on the number of nonzero terms at any point. Indeed translate that point into a fixed compact set and count orbit centres in its compact \((R+1)\)-neighbourhood. Finite stabilizers merely give finite repetitions in this count.

Choose a homotopy \(H:A\times[0,1]\to A\) with \(H_0=\operatorname{id}\) and \(H_1(A)\) contained in a compact subset \(C\subset A\). Such a homotopy is obtained by factoring a homotopy equivalence through a finite CW complex. Order the countable group once and for all. For \(y\in T\), apply, in that order, the finitely many maps \[a\longmapsto\gamma H(\gamma^{-1}a,\lambda_\gamma(y))\] to a fixed point \(a_0\in A\). Call the resulting point \(s(y)\). Local finiteness and \(H_0=\operatorname{id}\) imply that \(s\) is continuous, including where a factor enters or leaves its support.

At least one factor has time one. Immediately after that factor the point lies in \(\gamma C\), at uniformly bounded distance from \(y\). Subsequent factors preserve a uniform distance bound. In detail, if \(d(a,y)\le B\) and \(\lambda_\delta(y)\ne0\), then \(d(\delta^{-1}a,o)\le B+R+1\). The homotopy of the compact set \(A\cap\overline B(o,B+R+1)\) has compact image, so the distance of \(\delta H(\delta^{-1}a,t)\) from \(y\) has a bound depending only on \(B\). Iterating at most \(N\) times gives \[d(s(y),y)\le B_0\] for a fixed \(B_0\). In particular \(s:T\to A\) is proper.

For completeness, bounded distance from the identity gives a proper homotopy in this situation as follows. Choose a contraction \(J:T\times[0,1]\to T\) from the identity to \(o\). Form \(Q(y,x,t)\) by composing the translated contractions \[x\longmapsto\gamma J(\gamma^{-1}x,t\lambda_\gamma(y))\] in the same fixed order. At \(t=1\) a full contraction occurs, so the output is independent of \(x\). If \(d(x,y)\) is bounded, all the intermediate outputs stay at a uniformly bounded distance from \(y\); the compact-set argument above applies now also before the first full contraction. Join \(y\) to \(Q(y,y,1)\) by this construction and return along the construction starting at \(s(y)\). The two middle points agree. This is a continuous proper homotopy from \(\operatorname{id}_T\) to the composite \(T\xrightarrow{s}A\hookrightarrow T\).

If \(b=0\) the conclusion is immediate. Otherwise, if \(x\notin A\), the latter proper map omits \(x\). Its extension to the one-point compactification \(T^+\simeq S^b\) therefore has degree zero, whereas the proper homotopy to the identity gives degree one. This contradiction proves the lemma. No assertion that \(T\to T/\Gamma\) is a covering has been used. ◻

Proof. A proper flat family of pure-dimensional compact subspaces over a reduced complex base defines an analytic family of fundamental cycles; see (Barlet 1999, 27, Remark (iv)). Its map to the projective Chow space is continuous. Indeed, view cycles in \(Z\) as cycles in a fixed smooth projective space. Support and integration continuity (Andreotti and Norguet 1967, sec. 2.4(b), Lemma 4 and §2.5, Theorem 4) identify the usual Chow topology with the analytic cycle topology: the natural map is a continuous bijection from the compact fixed-degree Chow variety onto the corresponding fixed-degree locus in the Hausdorff cycle space, hence a homeomorphism. This comparison uses no smoothness of \(Z\). Apply this to the graph of \(j\) in \(T\times Z\). Every fibre is connected and normal, hence irreducible and reduced, so its fundamental cycle is integral. The degree is locally constant in this family, and hence equals \(d\) on the connected base \(T\).

The integral-cycle locus is constructible. More explicitly, its complement is the finite union, over \(d_1+d_2=d\) with both degrees positive, of the images of the proper addition maps from the corresponding Chow spaces. These images include nonreduced cycles. Containment of the support in \(U\) is a semialgebraic condition by the algebraic universal incidence and real quantifier elimination. Thus \(C\) is semialgebraic.

Any compact irreducible analytic subvariety of \(U\) maps to a point of the Stein manifold \(T\): holomorphic functions are constant on that subvariety, and holomorphic functions on a Stein manifold separate points. An integral \(n\)-cycle in \(U\) therefore lies in a fibre and, by equality of dimension and irreducibility of that fibre, is its whole reduced cycle. This proves that the displayed cycle map is bijective. Its continuity follows from the analytic-family statement. For inverse continuity, if fibre cycles \(c_\nu\) converge to the cycle of \(j^{-1}(t)\), choose a regular point \(x\) of that fibre. Convergence of cycles supplies points \(x_\nu\in|c_\nu|\) tending to \(x\); for example, use a small transverse disk at \(x\), whose local intersection number is one. Then their parameters satisfy \(j(x_\nu)\to j(x)=t\). All spaces involved are metrizable, so this proves inverse continuity. ◻

Proof. Write \(A_Z=\mathcal O_Z(1)|_U\) and choose \(k>0\) for which \(H=L_F^k\) is very ample. If \(g:F\to j^{-1}(t)\) is a polarized isomorphism, its graph in \(F\times Z\), with the product projective embedding defined by \(H\boxtimes\mathcal O_Z(1)\), has degree \[ D=\int_{j^{-1}(t)} \bigl(kc_1(L)+c_1(A_Z)\bigr)^n. \tag{31}\] This integer is independent of \(t\). Here is a direct way to check the constancy, which does not extend \(L\) to \(Z\). Choose a smooth Hermitian metric on \(L\) on the complex space \(U\), and use its curvature form, together with a Fubini–Study form for \(A_Z\). Near any fixed parameter, properness confines the supports of the fibre cycles to a compact subset of \(U\). Integration of the resulting smooth \(2n\)-form over these cycles is continuous in their analytic cycle family. Each value is the integral intersection number in the displayed formula, so it is locally constant. Smooth forms on a complex space here mean forms locally extended from an ambient manifold; a finite partition of unity on the compact set gives the usual continuity-of-integration test. The Chern integral equals the Cartier intersection number also for a singular fibre: pull the line bundles back to a projective resolution and use its degree-one pushforward of the fundamental cycle. The same argument proves constancy of each mixed intersection separately.

Consider the integral-cycle locus in the projective Chow space of \(n\)-cycles of degree \(D\) in \(F\times Z\), and let \(P\) be its locus of cycles \(G\) such that \[|G|\subset F\times U,\qquad p_F:|G|\to F\text{ is bijective},\qquad p_Z:|G|\to U\text{ is injective}.\] All three conditions are first-order conditions on the universal incidence, and hence are semialgebraic. The first projection is a proper quasi-finite algebraic map, hence finite. It is birational, since in characteristic zero a finite map bijective on complex points has degree one. Its target \(F\) is normal, so it is an isomorphism. The second projection has an irreducible compact image of dimension \(n\), which by the preceding lemma is a whole fibre of \(j\). This image is normal, and the same finite birational argument shows that the second projection is an isomorphism onto that fibre. Consequently \(P\) parametrizes exactly the isomorphisms from \(F\) to fibres whose graphs have the specified degree.

The universal incidence over \(P\) maps bijectively and closedly to \(F\times P\): closedness follows by projection along the compact factor \(Z\). It is therefore homeomorphic to \(F\times P\). In particular it gives a continuous evaluation map \[e:F\times P\longrightarrow U.\] Connected semialgebraic sets are path connected, and \(P\) has only finitely many connected components. A path in one such component gives a homotopy of the maps \(e_p:F\to U\). Hence \(c_1(e_p^*L)\in H^2(F,\mathbb Z)\) is constant on each component. This is the integral marking test; neither definability of \(L\) nor an extension of \(L\) to the compactification has been assumed.

For a simply connected normal projective variety \(F\), the map \[c_1:\operatorname{Pic}(F)\longrightarrow H^2(F,\mathbb Z)\] is injective. Indeed \(\operatorname{Pic}^0(F)\) is projective for normal projective \(F\) and is smooth in characteristic zero (Kleiman 2005, Theorem 5.4 and Remark 5.6). GAGA identifies algebraic and analytic line bundles on \(F\) (Serre 1956, Proposition 18). The exponential sequence and \(H^1(F,\mathbb Z)=0\) identify \(\operatorname{Pic}^0(F)\) analytically with the vector group \(H^1(F,\mathcal O_F)\). A compact complex vector group is zero. The kernel of \(c_1\), which is the image of this vector group in the exponential sequence, is therefore zero.

It follows that the condition \(e_p^*L\simeq L_F\) selects a union \(P_L\) of connected components of \(P\), so \(P_L\) is semialgebraic. Taking the second image of a graph gives a semialgebraic map to \(C\). One can see this directly by expressing equality of supports as \[z\in|c|\quad\Longleftrightarrow\quad (\exists x\in F)\ (x,z)\in|G|;\] integrality makes this support determine the cycle. The image of \(P_L\) is exactly the requested polarized isomorphism class, including all its members by the graph-degree calculation. Quantifier elimination proves the assertion. ◻

Proof. By Lemma 56, every polarized isomorphism class is a semialgebraic subset of the semialgebraic model \(C\) of \(T\). It is \(\Gamma\)-invariant, since the polarization is pulled back from \(U/\Gamma\). Its closure in \(C\) is a nonempty closed invariant semialgebraic subset. Compatible semialgebraic triangulation (Łojasiewicz 1964, sec. 3, Theorem 4) gives that closure the homotopy type of a finite CW complex. Under the homeomorphism of Lemma 55, Lemma 54 therefore forces it to be all of \(T\). Thus every class is dense. Two disjoint semialgebraic subsets cannot both be dense in the same semialgebraic space: finite cell decomposition shows that a dense semialgebraic subset contains a dense relatively open subset. All fibres are consequently isomorphic as polarized varieties.

We next justify local holomorphic triviality for these possibly singular fibres. Choose \(r\) so large that \(L_F^r\) is very ample and \(H^i(F,L_F^r)=0\) for \(i>0\). The same holds on every fibre. The constant-cohomology base-change theorem (Grauert 1960, sec. 7, Satz 5) makes \(j_*L^r\) locally free, with its formation commuting with fibres: \(L^r\) is flat over the base and all its fibre cohomology dimensions are constant. After trivializing it on a small open set \(B\subset T\), the evaluation map embeds \(j^{-1}(B)\) in \(B\times\mathbb P^q\) and gives a holomorphic Hilbert map \(h:B\to\operatorname{Hilb}(\mathbb P^q)\). Here one uses the analytic-base formulation: Douady’s universal proper-flat family (Douady 1966, secs. 9.7–9.8, Theorems 1–2) agrees with the analytification of the projective Hilbert scheme (Grothendieck 1960--1961, sec. 3, Theorem 3.2). The natural map between them has the same points and the same local deformation functors, since GAGA applies on every Artinian complex base (Grothendieck 2003, Exposé XII, Theorem 4.4); thus it is an isomorphism of analytic germs. This supplies the holomorphic classifying map also for singular fibres. All its points lie in the single orbit of the embedded \(F\) under \(G_0=\operatorname{PGL}_{q+1}(\mathbb C)\).

This orbit is a locally closed smooth complex subvariety. Since \(B\) is reduced, a holomorphic map with image in it factors holomorphically through it. For its closed algebraic stabilizer \(G\subset G_0\), the quotient map \(G_0\to G_0/G\) has local holomorphic sections, by the holomorphic submersion theorem. Locally lifting \(h\) and applying the inverse projective transformations gives a holomorphic product \(B\times F\). The resulting transition maps take values in the complex linear algebraic group \(G\). Thus \(j\) is the bundle associated to a holomorphic principal \(G\)-bundle on \(T\).

Since \(T\) is contractible and paracompact, that principal bundle is topologically trivial. The Oka principle for complex Lie groups over a Stein base makes it holomorphically trivial; one may use (Forstnerič and Prezelj 2000, Theorem 1.3 and Example (A)) on its continuous trivializing section. If necessary, first reduce to the identity component of \(G\) using the triviality of the discrete covering \(G/G^\circ\) over the simply connected space \(T\). We obtain \(U\simeq T\times F\). Proposition 53 gives the required product, with all the stated properties of \(F\). ◻

Proof. In the Borel embedding, \(D\) is an open orbit in its projective compact dual \(D^\vee\) (Borel 1954, sec. 4, Theorem 2 and §6, Proposition 2); see also (Falbel et al. 2025, sec. 2.1.1, Borel embedding theorem). The acting adjoint real semisimple group is the identity component of the real points of a real algebraic group (Milne 2005, Proposition 1.7); it is semialgebraic, and its action on \(D^\vee\) is algebraic. The image of its orbit map is semialgebraic by real quantifier elimination. Thus \(D\subset D^\vee\) is a semialgebraic open subset. Taking the usual affine chart \(\mathbb C^m\subset\mathbb P^m\) gives \[D\times\mathbb C^m\times F \ \subset\ D^\vee\times\mathbb P^m\times F\] as a semialgebraic open subset of a projective variety. Point factors are allowed. This construction concerns the biholomorphism type of the covering space and imposes no algebraicity condition on deck transformations. ◻