Necessity in Theorem 1. The union of the finitely many compact images is contained in \(\mathop{\mathrm{int}}B^{2n}(\sigma)\) for some \(\sigma<R\). Volume preservation gives \(\sum_iR_i^n\le\sigma^n<R^n\). Gromov’s two-ball obstruction [13], in the capacity convention of [23], gives \(R_i+R_j\le\sigma<R\) for every pair. ◻

Proof. Continuity follows from the minimum-over-intervals formula. At a slice of length tending to zero, uniform continuity of \(F\) gives the same conclusion. For convexity, choose minimizing intervals of lengths \(h_0,h_1\) on two slices. Their convex combination is an allowed interval of length \((1-t)h_0+th_1\) on the intermediate slice. Convexity of \(F\) bounds its two endpoint values by the corresponding convex combinations, which proves convexity of the rearrangement.

The minimum formula is nondecreasing in the interval length. If \(F\) is nondecreasing in another coordinate, lowering that coordinate preserves every allowed interval, by the nonnegative coefficients in (2), and can only lower its endpoint values. This proves preservation of the previous monotonicity properties.

On each slice, increasing rearrangement preserves the distribution of values, including flat sublevels, and hence the integral. Integrate this identity in the other coordinates. Order and translation invariance follow directly from the minimum formula. Applying these to \(G-\left\lVert F-G\right\rVert_\infty\le F\le G+\left\lVert F-G\right\rVert_\infty\) proves nonexpansiveness. ◻

Proof of Lemma 2. The continuous convex functions on \(\Delta\) form a complete metric space in the uniform norm. Since \(-P\) is convex, Lemma 3 implies that, for \(0<\lambda<1\), the map \[F\longmapsto R(\lambda F-P)\] is a contraction of this space, with constant \(\lambda\). Let \(K_\lambda\) be its fixed point. Integral preservation gives \[ \overline K_\lambda =-\frac{\overline P}{1-\lambda}. \tag{6}\] The function \(K_\lambda\) is coordinatewise nondecreasing; in particular \(K_\lambda(0)=\min_\Delta K_\lambda\).

Choose \[0<d<\min\left\{\overline P,\, \inf_{\Delta\setminus\tau\Delta}P\right\}.\] We claim that, for \(\lambda\) sufficiently close to one, \[ M_\lambda:=\max_\Delta K_\lambda <-\frac d{1-\lambda}. \tag{7}\] Suppose otherwise for such a \(\lambda\). Then \(\lambda M_\lambda\le M_\lambda+d\), and consequently \(\lambda K_\lambda-P<M_\lambda\) on the outer region. For \(p\in\tau\Delta\), write \(p=\tau v\) with \(v\in\Delta\). Convexity, \(K_\lambda(0)\le\overline K_\lambda\), and (6) give \[K_\lambda(p) \le \tau M_\lambda- \frac{(1-\tau)\overline P}{1-\lambda} \le M_\lambda- \frac{(1-\tau)(\overline P-d)}{1-\lambda}.\] It follows that \[\lambda K_\lambda(p)-P(p) \le M_\lambda+d -\frac{\lambda(1-\tau)(\overline P-d)}{1-\lambda} -\min_\Delta P<M_\lambda\] when \(\lambda\) is close enough to one. The choices here depend only on the displayed constants, so the bound is uniform on both regions. We have shown \(\max_\Delta(\lambda K_\lambda-P)<M_\lambda\), contradicting the fixed-point equation and \(\max RF\le\max F\). This proves (7).

Fix such a \(\lambda\). Since \((1-\lambda)K_\lambda\le-d\), order preservation and translation invariance give \[ R(K_\lambda-P) \le R(\lambda K_\lambda-P)-d=K_\lambda-d. \tag{8}\] Set \(F_0=K_\lambda-P\) and \(F_j=R_jF_{j-1}\). All these functions are continuous and convex. Use Lemma 4 to choose \(q_j\) with \[F_{j-1}\circ q_j\le F_j+\varepsilon_j, \qquad \sum_j\varepsilon_j<d/2.\] In the order \(q=q_1\circ\cdots\circ q_m\), successive substitution of these inequalities yields \[(K_\lambda-P)\circ q \le R(K_\lambda-P)+\sum_j\varepsilon_j <K_\lambda-d/2.\] Each constituent isotopy preserves \(V\) and preserves every outer-face moment on a sufficiently small collar. Choose common threshold collars \(c_\nu-\epsilon_\nu< b_\nu\cdot p\le c_\nu\) for the finitely many maps. Such collars are invariant because the moment is constant along each flow there. Run the isotopy for \(q_m\) first, then that for \(q_{m-1}\) on its output, and continue in this order. Each stage uses one of the original generating Hamiltonians and retains its commutation with every face moment. Reparametrization makes the concatenated path smooth and preserves these commutation relations.

Finally approximate \(K_\lambda\) uniformly on \(\Delta\) by a smooth function \(K(p)\), defined on a neighborhood. Polynomial approximation is sufficient. An approximation error smaller than \(d/4\) changes the last comparison by less than \(d/2\), leaving \((K-P)\circ q<K\). This proves the lemma. ◻

Proof. First construct ordinary orientation-preserving diffeomorphisms with these properties. At a fixed parameter \(y_0\), a disk embedding can be shrunk within its image, straightened while small, moved through the ambient disk, and expanded to another prescribed disk. The velocity along its moving boundary extends to a vector field in a tubular neighborhood and then, by a cutoff, to a compactly supported ambient field. The resulting flow extends the disk isotopy. Apply this construction first to the outermost boundary and then to each successive boundary inside its already positioned outer disk. This gives a compactly supported diffeomorphism \(\phi_{y_0}\), isotopic to the identity, carrying the centered disks to the prescribed ones.

For other parameters, follow the disk boundaries along the segment \(y_0+t(y-y_0)\). Their velocities extend in disjoint tubular neighborhoods of the finitely many boundaries. These extensions can be chosen smoothly in \((t,y)\), by local extensions and a partition of unity. Compactness and strict nesting give uniform collars and support away from the ambient boundary. Integrating the fields produces diffeomorphisms \(\phi_y\), with ordinary isotopies smooth in \(y\), carrying all the centered disks to their prescribed images.

Let \(\omega\) be the standard area form and \(\eta_y=\phi_y^*\omega-\omega\). This is a compactly supported two-form of integral zero. Compactly supported primitives can be chosen linearly, and hence smoothly in \(y\). For completeness, identify the open ambient disk smoothly with \(\mathbb R^2\), write \(\eta=f(x,t)\,\mathrm dx\wedge\mathrm dt\), and choose a fixed compactly supported function \(\rho\) with \(\int\rho=1\). Put \[g(t)=\int_\mathbb Rf(x,t)\,\mathrm dx,\quad F(x,t)=\int_{-\infty}^x\bigl(f(v,t)-\rho(v)g(t)\bigr)\,\mathrm dv, \quad G(t)=\int_{-\infty}^t g(v)\,\mathrm dv.\] Then \[\alpha=F\,\mathrm dt-\rho(x)G(t)\,\mathrm dx,\qquad \mathrm d\alpha=\eta.\] The zero marginal and zero total integral give compact support, with a common support for compact families.

For every centered disk \(D_j\) of area \(a_j\), \[\int_{D_j}\eta_y =\operatorname{area}E_j(y)-a_j=0.\] Thus the restriction of \(\alpha_y\) to \(\partial D_j\) is exact. Subtract the differentials of functions extending these circle primitives into disjoint annular collars. The resulting primitive \(\widehat\alpha_y\) vanishes on the tangent line of each circle. Apply Moser’s method [18] to \(\omega+t\eta_y\), \(0\le t\le1\), using \(\widehat\alpha_y\). This path consists of positive area forms. Its vector fields are tangent to the designated circles, since \(\widehat\alpha_y\) vanishes on their tangents. The resulting correction \(\chi_y\) preserves every \(D_j\) and satisfies \(\chi_y^*\phi_y^*\omega=\omega\). Hence \(\psi_y=\phi_y\circ\chi_y\) is area preserving and has the required disk images.

It remains to provide a symplectic isotopy with this endpoint. The construction has already supplied an ordinary compactly supported isotopy \(\psi_{y,t}\) from the identity to \(\psi_y\). For each \((y,t)\), apply the same Moser correction, now without circle conditions, to \[\omega+s(\psi_{y,t}^*\omega-\omega),\qquad 0\le s\le1.\] Use the linear primitive operator above. When \(\psi_{y,t}^*\omega=\omega\), that primitive and the correction are zero and the identity, respectively. In particular, the corrections are the identity at both endpoints \(t=0,1\). The corrected isotopy is therefore symplectic, has endpoint \(\psi_y\), and is smooth in the parameters. On a disk, a compactly supported symplectic vector field has a Hamiltonian normalized to vanish on the complementary component adjacent to the ambient boundary. These Hamiltonians depend smoothly on the parameters and are supported in a common larger compact subset of the disk. This yields the required Hamiltonian isotopies. ◻

Proof. The minimum formula (5) shows that \(g_y(h)\) is jointly continuous in \((y,h)\).

Choose a small \(\varepsilon'>0\) and a sufficiently fine grid \(0<h_1<\cdots<h_l<1\), including points sufficiently close to both endpoints. The open intervals \[I_j(y)= \{v\in(0,1):f_y(v)<g_y(h_j)+\varepsilon'\}\] are nested in \(j\), and each has length greater than \(h_j\). Indeed, the closed sublevel at \(g_y(h_j)\) contains an interval of length \(h_j\), and the strict enlargement gives extra room. Joint continuity and compactness make this room uniform for the finite grid.

Choose increasing numbers \(w_j<1\), sufficiently close to one, so that \(h_j/w_j\) is strictly increasing in \(j\) and is smaller than the available length of \(I_j(y)\) for every \(y\). There are strictly nested closed intervals \(J_j(y)\), of lengths \(h_j/w_j\), contained in \(I_j(y)\). To choose them, first choose the smallest interval and enlarge successively inside the next available open interval. The admissible centers satisfy strict affine inequalities expressing containment and nesting. Locally constant admissible centers remain admissible near a given parameter. A smooth partition of unity preserves these convex constraints, and therefore gives centers smooth on a neighborhood of the parameter set.

In the open polar chart \((v,\theta)\in(0,1)^2\), take the rectangles \[J_j(y)\times \left[\frac{1-w_j}{2},\frac{1+w_j}{2}\right].\] Their areas are \(h_j\). Round the corners smoothly and adjust the width in the \(v\)-direction slightly to restore the area exactly. For example, superellipses of a common sufficiently large even exponent approximate all these rectangles, and their areas are corrected by one additional dilation in \(v\). The strict margins ensure that the resulting smooth closed disks \(E_j(y)\) remain nested and lie in \(I_j(y)\times(0,1)\). All choices are uniform because the parameter set and grid are compact and finite.

Apply Lemma 5 to these disks. If \(v\le h_j\), the image of a point of radial area \(v\) lies in \(E_j(y)\), and therefore its \(f_y\)-value is below \(g_y(h_j)+\varepsilon'\). Use the next larger grid point and uniform continuity of \(g_y\) to bound this by \(g_y(v)+2\varepsilon'\). For \(v>h_l\), use \(f_y\le\max f_y=g_y(1)\) and choose \(h_l\) close enough to one. Taking \(\varepsilon'\) small proves the assertion. The Hamiltonians extend by zero across the ambient boundary, so the same inequality holds on the closed disk. ◻

Proof of Lemma 4. First suppose that a slice has a smooth positive area scale \(\ell(y)\). Apply Lemma 6 to \(f_y(v)=F(y,\ell(y)v)\). If \(B_t(y,z)\) generates the resulting maps on \(D(1)\), then the Hamiltonian \[\ell(y)B_t\bigl(y,z_j/\sqrt{\ell(y)}\bigr)\] on the whole fiber produces the desired scaled disk motion. The other moments remain fixed because this Hamiltonian has no dependence on their angles. Their angles may change, which does not affect a toric function.

To treat the actual endpoint \(L\), choose a small \(\eta>0\) and work first on the compact convex parameter set \[Y_\eta=\{y:L(y)\ge\eta\}.\] Choose a smooth function \(\ell\) strictly below \(L\), uniformly close to it on \(Y_\eta\), and with \(\ell\ge\eta/2\). For instance, a smooth soft minimum of the finitely many affine expressions defining \(L\), shifted downward by a small positive constant, has these properties when its approximation error is small. The planar construction and its Hamiltonians extend smoothly to a neighborhood of \(Y_\eta\).

For \(h\le\ell(y)\), any interval of length \(h\) in \([0,L(y)]\) can be shifted into \([0,\ell(y)]\) by a distance at most \(L(y)-\ell(y)\). Hence, for a uniform modulus of continuity \(\omega_F\), \[ R_{[0,\ell]}F(h) \le R_{[0,L]}F(h)+\omega_F(L-\ell). \tag{9}\] For \(\ell<h\le L\), let \([a,a+h]\) be a minimizing interval for the right-hand rearrangement. Since \(a\le L-h<L-\ell\), \[ F(y,h)\le R_{[0,L]}F(h)+\omega_F(L-\ell). \tag{10}\] Thus the identity map in this outer strip has the required one-sided estimate. The planar Hamiltonians vanish near their rescaled boundary, so this identity extension is smooth.

Choose a smooth cutoff \(0\le\chi(y)\le1\), supported where \(L>\eta\) and equal to one where \(L\ge2\eta\), with its support contained in the neighborhood where the preceding choices are defined. The full lifted Hamiltonian is \[ \chi(y)\ell(y) B_{\chi(y)t}\bigl(y,z_j/\sqrt{\ell(y)}\bigr), \qquad 0\le t\le1, \tag{11}\] extended by zero elsewhere. At fixed \(y\), it runs the scaled disk isotopy to time \(\chi(y)\). On slices of length at most \(2\eta\), any slice-preserving motion incurs error at most \(\omega_F(2\eta)\), since all slice values of \(F\) have that oscillation bound. Elsewhere \(\chi=1\), and (9)–(10), together with the planar estimate, apply. Choose \(\eta\), the scale error, and the planar error in this order to make the total error less than \(\varepsilon\).

All lifts are smooth at the other coordinate axes because the other moments are smooth functions of the complex variables. The cutoff removes the possible degeneracy at \(L=0\). To check the outer faces, suppose first that \(b_{\nu j}>0\). Equality on that face forces \(p_j=L(y)\). On active slices the common compact planar support satisfies \[p_j\le(1-\delta)\ell(y)\] for some \(\delta>0\), while \(\ell(y)\ge\eta/2\). Writing \(L_\nu(y)\) for this face’s affine endpoint, its support therefore satisfies \[c_\nu-b_\nu\cdot p =b_{\nu j}(L_\nu(y)-p_j) \ge b_{\nu j}\delta\ell(y) \ge b_{\nu j}\delta\eta/2>0.\] Thus the Hamiltonian vanishes on a genuine uniform neighborhood of that face. If \(b_{\nu j}=0\), the Hamiltonian commutes with \(b_\nu\cdot p\) everywhere. These statements imply that the flow preserves all defining inequalities of \(V\) and gives the stated face property. ◻

Proof. In a coordinate disk identified with \(\mathbb R^2\), write \(\eta=h(x,y)\,\mathrm dx\wedge\mathrm dy\), with \(h\) compactly supported and of integral zero. Fix \(\rho\in C_c^\infty(\mathbb R)\) of integral one, and set \[\begin{split} h_0(y)&=\int_\mathbb Rh(x,y)\,\mathrm dx,\\ A(x,y)&=\int_{-\infty}^x \bigl(h(v,y)-\rho(v)h_0(y)\bigr)\,\mathrm dv,\qquad B(y)=\int_{-\infty}^y h_0(v)\,\mathrm dv. \end{split}\] Then \(\alpha=A\,\mathrm dy-\rho(x)B(y)\,\mathrm dx\) is compactly supported and \(\mathrm d\alpha=\eta\). For forms supported in a fixed compact set its support can be fixed in advance. This is a linear operator preserving smooth parameter dependence.

Cover the given compact support by finitely many coordinate disks and use a fixed partition of unity. From each summand subtract its integral times a fixed unit-integral bump in the same disk; the local construction treats the resulting zero-integral forms. Join these disks to one fixed disk by finitely many chains of overlapping coordinate disks in \(\mathop{\mathrm{int}}(\Sigma)\). Unit-integral bumps in successive overlaps express the difference between each original bump and a common bump as a sum of zero-integral forms in individual disks. Apply the local operator to these differences. The coefficient of the common bump is zero because \(\int_\Sigma\eta=0\). All choices are fixed, proving the support and parameter assertions. ◻

Proof. Solve \(\iota_{Y_\lambda}\Omega_\lambda=-\alpha_\lambda\). Horizontality gives \(\alpha_\lambda(X_{\mu_\nu})=0\), so evaluation on \(X_{\mu_\nu}\) gives \(\mathrm d\mu_\nu(Y_\lambda)=0\) near that face. The field is zero in the base boundary collar. Trajectories therefore cannot cross any defining outer level in either time direction. This applies first to interior starting points and then to boundary points by continuity. All inequalities are treated simultaneously, including at nonsimple intersections of faces; no smooth-corner assumption is needed.

The fields extend smoothly to a common neighborhood by compactness and nondegeneracy. Trajectories starting in the compact region stay there and exist throughout the parameter interval. Continuous dependence and compactness give flows on a neighborhood as well. For the flow \(\psi_\lambda\), \[\frac{\mathrm d}{\mathrm d\lambda}\psi_\lambda^*\Omega_\lambda =\psi_\lambda^*\bigl(\mathrm d\alpha_\lambda+ \mathrm d\iota_{Y_\lambda}\Omega_\lambda\bigr)=0.\] The inverse flow has the same preservation properties, giving preservation of the interior. ◻

Proof. We carry out the construction in five steps.

Step 1: retain the comparison gap on a finite surface. By Lemma 2, there are a smooth toric \(K\) and a Hamiltonian isotopy \(q_\xi\), \(0\leq\xi\leq1\), with \(q_0=\mathrm{id}\), \(q_1=q\), preserving \(V\), such that \((K-P)\circ q<K\). Its generators commute with every \(\mu_\nu\) near the corresponding outer face, uniformly in time after shrinking the collars. Write \[g=\min_{v\in V}\bigl(K(q^{-1}v)-K(v)+H(p(v))-C_r(p(v))\bigr)>0\] and choose \(0<\varepsilon<\frac18\min(g,\min_\Delta H)\). Take a sufficiently large \(\Sigma=\Sigma_\ell\) so that \[ \|A_\ell-H\|_\infty<\varepsilon \tag{13}\] and \(\mathop{\mathrm{int}}(\Sigma)\) contains an embedded torus with a disk removed. Write \(\Gamma_0=\Gamma_\ell\), \(\kappa=\kappa_\ell\), and \(\Omega_0=\Omega_\ell\). We use the comparison obtained from \(H\); no concavity of the finite area function \(A_\ell-C_r\) is required.

Step 2: place the base area on an annulus. Inside the handle choose annuli \(A\Subset A^+\Subset\mathop{\mathrm{int}}(\Sigma)\) around one essential circle, and a transverse annulus \(B\) around a circle intersecting it once. There are positive coordinates \((s,t)\) on \(A^+\), with \(t\in\mathbb R/\mathbb Z\), and a smooth closed one-form \(\beta\), supported in \(B\), such that \[ \beta|_{A^+}=f(t)\,\mathrm dt,\qquad f\geq0,\qquad \int_{\mathbb R/\mathbb Z}f(t)\,\mathrm dt=1, \tag{14}\] where \(f\) is positive on one open interval and zero outside its closure. Use the two coordinate bands on a torus, remove a disk away from them, and take a bump form in the transverse coordinate of \(B\). All choices can have collars inside the handle.

Choose a smooth nondecreasing \(S(s)\), zero near and below the lower end of \(A\), and one near and above its upper end. Its derivative is supported in \(\mathop{\mathrm{int}}(A)\). The two-form \[\chi_A=S'(s)\,\mathrm ds\wedge\beta\] extends by zero to \(\Sigma\), is nonnegative, and has integral one.

Choose a positive area form \(\kappa_{\rm sm}(p)\) agreeing with \(\kappa(p)\) in a thin base boundary collar, independent of \(p\) on \(A^+\), and satisfying \[ \sup_{p\in\Delta}\int_\Sigma\kappa_{\rm sm}(p) <\varepsilon. \tag{15}\] To do this, fix a positive area form \(\lambda\) on \(\Sigma\) and write \(\kappa(p)=a(y,p)\lambda\). If \(\zeta\) is one near the boundary and zero off a thin collar disjoint from \(A^+\), put \(\kappa_{\rm sm}=(\zeta a+(1-\zeta)e)\lambda\). Uniform boundedness of \(a\) allows the collar integral to be made arbitrarily small; then take \(e>0\) sufficiently small. The construction is smooth on neighborhoods and equals \(e\lambda\) on \(A^+\).

Set \[ H_*(p)=\int_\Sigma(\kappa(p)-\kappa_{\rm sm}(p)),\qquad \kappa_1(p)=\kappa_{\rm sm}(p)+H_*(p)\chi_A. \tag{16}\] Then \(H_*>0\), \(\|H_*-H\|_\infty<2\varepsilon\), and \(\kappa_1>0\). Moreover \(\kappa_1-\kappa\) has zero integral and compact interior support. Lemma 7 supplies a toric horizontal \(\alpha\) with \(\mathrm d_C\alpha=\kappa_1-\kappa\) and compact interior base support. Initially put \(\widetilde\Gamma_1=\Gamma_0+\alpha\).

Normalize the primitive on a neighborhood of \(\overline A\). Choose a one-form \(\gamma\) on \(A^+\), independent of \(p\), with \(\mathrm d\gamma=e\lambda\); a primitive exists on the annulus. The form \[\delta(p)=\widetilde\Gamma_1-\gamma-SH_*(p)\beta\] is base-closed on \(A^+\). Let \(c(p)\) be its period around the annulus. Since \(\beta\) has period one, \(\delta-c(p)\beta\) has zero period, and equals \(\mathrm d_C F\) for a smooth function \(F(y,p)\) on \(A^+\). Smooth parameter dependence follows by fixing a base point and integrating the zero-period form along paths. Choose \(\zeta_A\in C_c^\infty(A^+)\) equal to one near \(\overline A\), and define globally \[\Gamma_1=\widetilde\Gamma_1-c(p)\beta-\mathrm d_C(\zeta_AF),\] where the last term is extended by zero. This is horizontal and toric, agrees with \(\Gamma_0\) near \(\partial\Sigma\), and satisfies \[ \mathrm d_C\Gamma_1=\kappa_1,\qquad \Gamma_1|_A=\gamma+S(s)H_*(p)\beta . \tag{17}\] The use of \(\mathrm d_C\) in the normalization retains horizontality. All changes of the two-form are exact on the total space. Linear interpolation from \(\Gamma_0\) to \(\Gamma_1\) is symplectic by (12), because its base curvature interpolates between \(\kappa\) and \(\kappa_1\).

Step 3: construct a positive exact path with packing layers. Choose smooth positive toric functions \(h_i\) satisfying \[ h_i(p)>(r_i-\textstyle\sum_jp_j)_+,\qquad 0<\sum_i h_i-C_r<\varepsilon. \tag{18}\] For instance, use sufficiently close smooth upper approximations \(\frac12(x+\sqrt{x^2+\eta_i^2})\) to \(x_+\). Put \[J=K-H_*,\qquad R_*(v)=K(q^{-1}v)-J(v)-\sum_i h_i(p(v)).\] Equations (13)–(18) give \[ R_*(v)\geq g-3\varepsilon>0. \tag{19}\]

The gap says that a coefficient \(G(s,v)\) of \(\beta\), nondecreasing in \(s\), can run from \(J(v)\) to \(K(q^{-1}v)\), with increments \(h_i(p(v))\) and the positive residual \(R_*(v)\). We will change fiber coordinates from the identity at the lower end of \(A\) to \(q\) at the upper end. After pullback these boundary values become \(J\) and \(K=H_*+J\), so both ends match \(\Gamma_1\) after the same global correction \(J\beta\). Since \(J\) is independent of the base and \(\beta\) is closed, this correction leaves the base curvature unchanged and changes the total form by the exact form \(\mathrm d(J\beta)\). The two paths below first insert the fiber change and then arrange the increments as disjoint layers.

With the convention \(\iota_{X_Q}\omega_V=-\mathrm dQ\), write \(\partial_\xi q_\xi=X_{Q_\xi}\circ q_\xi\). For \(0\leq\lambda\leq1\), define on \(A\times V\) \[q_s^\lambda=q_{\lambda S(s)},\qquad D_\lambda(s,t,u)=(s,t,q_s^\lambda u),\qquad G_\lambda(s,v)=S(s)H_*(q_\lambda^{-1}v).\] The \(s\)-generator is \(Q_s^\lambda=\lambda S'(s)Q_{\lambda S(s)}\), and vanishes on the end collars. This is the Hamiltonian lifting mechanism used in symplectic folding; compare [20]. With our sign convention, the suspension identity is \[ D_\lambda^*\omega_V =\omega_V+\mathrm d\bigl((Q_s^\lambda\circ q_s^\lambda)\,\mathrm ds\bigr). \tag{20}\] On a pair \((\partial_s,w)\) with \(w\) vertical, both added terms equal \(-\mathrm d_u(Q_s^\lambda\circ q_s^\lambda)(w)\); on vertical pairs the identity follows from symplecticity of \(q_s^\lambda\).

Use \[ \Omega_\lambda^A =\mathrm d\gamma+ D_\lambda^*\bigl(\omega_V+\mathrm d(G_\lambda\beta)\bigr) \tag{21}\] on \(A\times V\), retaining \(\omega_V+\mathrm d\Gamma_1\) outside. These forms glue. On the lower collar, \(q_s^\lambda=\mathrm{id}\) and \(G_\lambda=0\); on the upper collar, \(q_s^\lambda=q_\lambda\) and \(G_\lambda\circ q_\lambda=H_*\). The suspension term is zero on both collars. Indeed, (21) has horizontal primitive \[\gamma+(Q_s^\lambda\circ q_s^\lambda)\,\mathrm ds +(G_\lambda\circ q_s^\lambda)\beta\] after subtracting \(\omega_V\), and this equals \(\Gamma_1\) on the collars. Its change therefore extends by zero as a global horizontal primitive. At \(\lambda=0\) it is the form from Step 2.

For any \(t\)-independent function \(G(s,v)\), \[ \bigl(\mathrm d\gamma+\omega_V+\mathrm d(G\beta)\bigr)^{m+1} =(m+1)\omega_V^m\wedge \bigl(\mathrm d\gamma+\partial_sG\,\mathrm ds\wedge\beta\bigr). \tag{22}\] Every mixed term contains \(\beta\), so products of two mixed terms vanish. Since \(\partial_sG_\lambda=S'H_*\circ q_\lambda^{-1}\geq0\), the forms (21) are symplectic.

Now keep \(D=D_1\) fixed. Choose successively ordered, pairwise disjoint closed subintervals \(I_1,\ldots,I_k,I_*\) in the interior of the \(s\)-interval of \(A\). Choose smooth nondecreasing functions \(\rho_i,\rho_*\), each zero below its interval, one above it, and constant near its endpoints. Define \[ G_{\rm f}(s,v) =J(v)+\sum_{i=1}^k\rho_i(s)h_i(p(v))+\rho_*(s)R_*(v). \tag{23}\] Its \(s\)-derivative is nonnegative. Its lower and upper values are \(J\) and \(K\circ q^{-1}=(H_*+J)\circ q^{-1}\). On the \(i\)th interval it has the toric expression \[ G_{\rm f}(s,v)=a_i(p(v))+\rho_i(s)h_i(p(v)),\qquad a_i=J+\sum_{j<i}h_j . \tag{24}\] The possibly nontoric contribution to \(\partial_sG_{\rm f}\) is supported in \(I_*\); the residual term may persist above \(I_*\). All preceding packing layers are toric.

Interpolate linearly, with \(0\leq\tau\leq1\), between \(G_1=SH_*\circ q^{-1}\) and \(G_{\rm f}\): \(\widehat G_\tau=(1-\tau)G_1+\tau G_{\rm f}\). On \(A\times V\) use \[\widehat\Omega_\tau=\mathrm d\gamma+ D^*(\omega_V+\mathrm d(\widehat G_\tau\beta)),\] and outside use \(\omega_V+\mathrm d(\Gamma_1+\tau J\beta)\). At the lower end the inside primitive changes by \(\tau J\beta\); at the upper end its pullback changes by the same \(\tau J\beta\). Thus the primitives themselves glue. The \(s\)-derivative of \(\widehat G_\tau\) is nonnegative, and (22) proves positivity inside \(A\). Outside, the primitive is toric and its base curvature is unchanged, since \(\mathrm d_C(J\beta)=0\). This second exact symplectic path ends in the form \(\Omega_{\rm f}\).

Step 4: verify the relative Moser conditions. Every variation primitive is horizontal and supported away from the base boundary: its support lies in the fixed compact interior support from Step 2 or in \(A\cup\mathop{\mathrm{supp}}\beta\). To check the outer faces, fix \(\mu=\mu_\nu\). The comparison isotopy preserves \(\mu\) and intertwines \(X_\mu\) on a common sufficiently thin collar. This follows from \(\{Q_\xi,\mu\}=0\) there; points in a smaller collar stay in it because \(\mu\) is constant along their trajectories. Consequently \(Q_s^\lambda\circ q_s^\lambda\), \(H_*\circ q_\lambda^{-1}\), \(K\circ q^{-1}\), and their indicated pullbacks are invariant under \(X_\mu\) there. The other functions are toric. For every global horizontal primitive \(\Gamma\) along the paths, we therefore have \[\iota_{X_\mu}\Gamma=0,\qquad \mathcal L_{X_\mu}\Gamma=0,\qquad \iota_{X_\mu}(\omega_V+\mathrm d\Gamma)=-\mathrm d\mu\] near the face. Lemma 8 applies. Reparametrize each stage to be constant near its parameter endpoints to obtain a smooth concatenated path. There is a diffeomorphism \(\psi\), defined on a neighborhood and preserving \(\Sigma\times V\) and its interior, with \(\psi^*\Omega_{\rm f}=\Omega_0\). The final packing will return to the original model via \(\psi^{-1}\).

Step 5: embed a closed ball in each layer. Use the \(D\)-coordinates \((s,t,v)\) and the \(i\)th interval. On the open \(t\)-interval where \(f>0\), the coordinate \(T\) with \(\mathrm dT=\beta\), normalized using (14), runs through \((0,1)\). Write \[\mathrm d\gamma=\ell(s,T)\,\mathrm ds\wedge\mathrm dT,\qquad L(s,T)=\int_{s_i^-}^{s}\ell(\sigma,T)\,\mathrm d\sigma ,\] where \(s_i^-\) is the lower endpoint of \(I_i\). Then \(\ell>0\), \(\partial_sL>0\), and \(\mathrm dL\wedge\mathrm dT=\mathrm d\gamma\). Set \[ x=L+G_{\rm f}-a_i=L+\rho_i h_i . \tag{25}\] At fixed \((T,v)\) this is strictly increasing in \(s\). The form becomes \[\omega_V+\mathrm dx\wedge\mathrm dT+\mathrm da_i\wedge\mathrm dT.\] Set \(v'=\phi_{a_i}^{T}v\), using Hamiltonian time \(T\). The result is the product form \(\omega_V(v')+\mathrm dx\wedge\mathrm dT\). The sign follows directly from the convention: since \(a_i\) is toric, \(\theta_j'=\theta_j+T\partial_{p_j}a_i\), and \[ \omega_V(v')=\omega_V(v)+\mathrm da_i\wedge\mathrm dT . \tag{26}\] The flow preserves the moments and is smooth across all axes.

The lower value of \(x\) is zero and the upper value is \(L(s_i^+,T)+h_i(p)>h_i(p)\). The chart therefore contains \[ v'\in\mathop{\mathrm{int}}(V),\qquad 0<T<1,\qquad 0<x<h_i(p(v')). \tag{27}\] Its inverse is smooth by \(\partial_sx>0\) and the implicit function theorem, also at fiber axes.

We use an elementary consequence of Lemma 5; related radial-area control appears in [22]. For any \(A>0\) and \(\delta>0\), a neighborhood of the closed planar disk of area \(A\) admits an area-preserving embedding into the \((x,T)\)-plane such that at radial area \(w=\pi|z|^2\), \[ 0<x<w+\delta,\qquad 0<T<1. \tag{28}\] Take a finite increasing area grid starting below \(\delta/4\), ending above \(A\), and having mesh less than \(\delta/4\). At each grid value \(h\), choose a smooth disk of area \(h\) inside \(0<x<h+\delta/2\), \(0<T<1\), with the disks strictly nested. These can be rounded rectangles: take their heights strictly increasing and sufficiently close to one, and horizontal intervals strictly nested with positive left endpoints close to zero. The finite nesting margins persist after rounding and the small area corrections. Put these disks and the source concentric disks in a sufficiently large open ambient disk and apply Lemma 5. For a point of area \(w\), the next larger grid value \(h\) gives \(x<h+\delta/2<w+\delta\). The last value exceeds \(A\), so the map is defined beyond the closed disk used here.

Choose \(0<\delta<r_i-r_i'\) and use (28), with \(A>r_i'\), for the last complex coordinate of \(B^{2m+2}(r_i')\). Use the first \(m\) coordinates unchanged as \(v'\). On the closed ball, \[w+\sum_jp_j(v')\leq r_i', \qquad x<w+\delta<r_i-\sum_jp_j(v')<h_i(p(v')).\] The fiber simplex is separated from the outer faces, and the planar image is compactly contained in \(0<T<1\), \(x>0\). The upper inequality has uniform positive slack. Thus the product embedding lands compactly in (27) and extends to an open neighborhood of the whole closed ball. The inverse coordinate changes give a symplectic embedding into the \(i\)th layer for \(\Omega_{\rm f}\). The layers have disjoint base intervals, so their compact images are disjoint. Applying \(\psi^{-1}\) from Step 4 gives the required neighborhood embeddings in the original model. ◻

Proof. Put \(\omega_t=(1-t)\omega_0+t\omega_1\). Each \(\omega_t\) is Kähler. Choose a smooth real one-form \(\alpha\) such that \(\mathrm d\alpha=\omega_1-\omega_0\). Since \(S_j\) is complex, its tangent bundle is symplectic for \(\omega_t\), and \[TX|_{S_j}=TS_j\oplus N_{j,t},\qquad N_{j,t}=(TS_j)^{\omega_t}.\] These splittings depend smoothly on \(t\). Along \(S_j\), prescribe the first jet of a function \(f_t\) by requiring \[f_t|_{S_j}=0,\qquad \mathrm df_t|_{TS_j}=0,\qquad \mathrm df_t|_{N_{j,t}}=-\alpha|_{N_{j,t}}.\] The direct-sum decomposition makes this a well-defined smooth covector field. It is realized by a smooth function in a fixed tubular neighborhood: evaluate the prescribed covector on the normal variable and multiply by a cutoff equal to one near the zero section. Compactness of the parameter interval permits fixed neighborhoods and cutoffs. The neighborhoods for different \(S_j\) may be chosen disjoint. Adding the resulting functions produces a global smooth family \(f_t\).

Set \(\alpha_t=\alpha+\mathrm df_t\), and solve \[\iota_{X_t}\omega_t=-\alpha_t.\] Along \(S_j\), the right side annihilates \(N_{j,t}\), so \(X_t\) is tangent to \(S_j\). Its flow exists throughout \(0\le t\le1\), because \(X\) is compact, and the usual Moser calculation gives \[\frac{\mathrm d}{\mathrm dt}(\psi_t^*\omega_t) =\psi_t^*\bigl(\mathrm d\alpha+\mathrm d\iota_{X_t}\omega_t\bigr)=0.\] This proves the first assertion.

In the equivariant case, first average \(\alpha\) over \(G\). The normal splittings are invariant. Averaging \(f_t\) therefore preserves its prescribed first jets, and makes \(X_t\) invariant. The flow is equivariant. An invariant vector field is tangent to the fixed locus, and its flow, starting at the identity, preserves each fixed component. This also explains why no disjointness condition between a fixed component and the \(S_j\) is needed. ◻

Proof. First clear denominators in the polarization. For large divisible \(N\), relative Serre vanishing gives \(R^1f_*(\mathcal I_Y\otimes\mathcal L^N)=0\); see [24], or the relative ampleness and vanishing theorem [15]. Apply \(f_*\) to \[0\longrightarrow\mathcal I_Y\otimes\mathcal L^N \longrightarrow\mathcal L^N \longrightarrow i_*(\mathcal L^N|_Y)\longrightarrow0,\] where \(i:Y\hookrightarrow\mathcal X\). For a union of pairwise disjoint components, \(\mathcal I_Y\) is the ideal of the whole union and the section space is the direct sum of the component section spaces. The same lifting and local transport arguments apply to that union. The resulting surjection extends every section on \(Y\) near \(0\); here \(f_*i_*(\mathcal L^N|_Y)\) is supported at \(0\), with fiber \(H^0(Y,\mathcal L^N|_Y)\). Take an affine neighborhood of \(0\), so these lifts may all be represented by sections over that neighborhood. Choose lifts \(s_0,\ldots,s_r\) of the desired basis. Choose also sections \(t_0,\ldots,t_M\) of the same power which define a relative closed immersion; eventual relative very ampleness over the affine base follows from [24] and properness. Express each \(t_i|_Y\) in the chosen basis, and subtract the corresponding linear combination of the \(s_j\). The modified sections \(t_i'\) vanish on \(Y\). The span of the \(s_j,t_i'\) contains the original \(t_i\), so this enlarged system still defines a relative closed immersion. On \(Y\), the resulting Fubini–Study potential is precisely \(N^{-1}\log\sum_j|s_j|^2\) in a local frame. For a linearized complex torus action, choose a basis of weight vectors. Its compact torus characters preserve the standard diagonal Hermitian norm, making this expression invariant under the compact torus.

The relative embedding into \(\mathbb P^{r+M+1}\times B\) supplies a closed real \((1,1)\)-form \(\Omega\), equal to the projective pullback divided by \(N\), with an arbitrary positive base form added. On the smooth locus it is Kähler, and its restriction to each smooth fiber represents the designated polarization. Adding the base form changes neither fiber forms nor the horizontal lifts used below.

In the submersion locus define a lift of a base tangent vector by requiring it to be \(\Omega\)-orthogonal to the fiber tangent space. The fiber restriction of \(\Omega\) is symplectic, so the lift exists uniquely. Lift a sufficiently short path from \(0\) to \(t\). Compactness of \(K\) gives existence of its flow for a uniform time, on a neighborhood of \(K\). Since \(\Omega\) is closed and the contraction of the horizontal lift with \(\Omega\) vanishes on fiber tangent vectors, this flow preserves the fiber forms. It gives the claimed symplectic embedding. If \(K\) is disjoint from a closed set, it has a neighborhood disjoint from that set; shorten the transport uniformly so that its image remains in this neighborhood.

For the last assertion, invariance of \(\Omega\) and uniqueness of the horizontal lift imply equivariance. Write \(\Phi_t\) for the transport and \(X\) for a circle generator. If \(\iota_X\omega_t=-\mathrm d\mu_t\), then \[\mathrm d(\mu_t\circ\Phi_t) =-\Phi_t^*(\iota_X\omega_t) =-\iota_X\omega_0 =\mathrm d\mu_0.\] The difference is constant on a connected domain, and a single normalizing point determines it. When the circle acts on the whole family, an invariant total form can be obtained by averaging. If its restriction to \(Y\) was already invariant, this averaging leaves that restriction unchanged. ◻

Proof. Form the deformation to the normal cone [11], \[\mathcal X=\mathop{\mathrm{Bl}}_{Z\times0}(T\times\mathbb A^1) \longrightarrow\mathbb A^1.\] Its central fiber is the reduced union of \(T'\) and \(Y\). The normal bundle of \(Z\times0\) in \(T\times\mathbb A^1\) is \(N_{Z/T}\oplus\mathbf1\), giving the displayed description of \(Y\). In the convention of lines, \(\mathcal O_{\mathcal X}(Y)|_Y=\mathcal O_Y(-1)\). Thus the rational polarization \(\mathcal L=L-d[Y]\) restricts to the two classes in the statement. The components meet along the exceptional divisor of \(T'\), identified in \(Y\) with \(D_\infty\). One can check the geometry in local coordinates \((y,x_1,\ldots,x_r,t)\), with \(Z=\{x_1=\cdots=x_r=0\}\), by blowing up the ideal \((x_1,\ldots,x_r,t)\). In the \(t\)-chart, \(x_i=tv_i\); the exceptional component is \(t=0\), and the family is a submersion there. In the \(x_1\)-chart, write \(x_i=x_1v_i\) for \(i\ge2\) and \(t=x_1s\). The central fiber is the reduced crossing \(\{x_1s=0\}\), and its double locus is \(\{x_1=s=0\}\). The other charts are identical. They show that the total space is smooth, the family is flat, and the only nonsubmersion points of the central fiber lie at this double locus. The exceptional normal bundle and the restriction sign also follow from the tautological line in the blowup construction; see [24].

Ampleness on the two components implies ampleness on the central fiber: the map from their disjoint union to that fiber is finite and surjective, and ampleness descends under such maps. Openness of relative ampleness then makes \(\mathcal L\) relatively ample after shrinking about zero. Apply these statements to a divisible integral multiple of \(\mathcal L\); precise forms are [24], and ampleness on components and openness are also given by [15]. The nonzero fibers are canonically \(T\), with polarization \(L\). The total space is smooth, and the family is a submersion along \(Y\setminus D_\infty\).

The scalar action on \(Y\) linearizes on its polarization. By Lemma 11, choose a relative projective form whose restriction \(\widehat\omega_Y\) is invariant. Apply Lemma 10 on \(Y\) to pass from \(\omega_Y\) to \(\widehat\omega_Y\), preserving the indicated vertical submanifolds. They are scalar-invariant. Moreover \(D_\infty\) is a union of fixed components of scalar rotation and is automatically preserved by the equivariant isotopy, including at its intersections with the vertical submanifolds.

The image of the given compact set is now a compact subset of the submersion locus, disjoint from the inverse image of \(F\) under the natural map \(\mathcal X\to T\). Lemma 11 transports a neighborhood into a nearby \(T\setminus F\), with a Kähler form \(\widehat\omega_T\) in class \(L\). Finally apply Lemma 10 on \(T\), preserving the components of \(F\), to identify \(\widehat\omega_T\) with \(\omega_T\). Composing these maps proves the proposition. All maps are defined on neighborhoods of the relevant compact sets; these neighborhoods can be shrunk at each composition. ◻

Proof. On the affine chart \(N\subset\mathbb P(\mathbf1\oplus N)\), let \(w_1,\ldots,w_\ell\) be the normal coordinates, with their induced Hermitian norms. Set \[\omega_Y=\sigma+ \mathrm d\mathrm d^c\left[d\log\left(1+\sum_j\lVert w_j\rVert^2\right)\right].\] The expression in brackets is the local weight of the dual tautological metric, multiplied by \(d\), so its curvature extends globally and represents \(dU\). The moments of the normal rotations are \[ p_j=\frac{d\lVert w_j\rVert^2} {1+\sum_l\lVert w_l\rVert^2}. \tag{30}\] Keeping the phases and using radii \(\sqrt{p_j/\pi}\) identifies each affine fiber with the standard open ball \(\sum_jp_j<d\). The inverse radial change is smooth away from the upper boundary, including all zero coordinates. Writing the metric connection in these coordinates yields (29). Equivalently, differentiating the potential gives \(\sum_jp_j\alpha_j\). In a unitary local frame, \(\alpha_j=\mathrm d\theta_j+A_j\), and \(p_j\mathrm d\theta_j\) is the standard smooth Liouville form of the complex coordinate, while \(p_jA_j\) is plainly smooth.

At any base point choose a holomorphic Chern frame whose metric has zero first derivative at that point. The form then splits into a positive vertical Fubini–Study form and the horizontal form \[\left(1-b\sum_jp_j\right)\sigma.\] This remains valid on the projective infinity chart, where \(\sum_jp_j=d\). The horizontal factor is bounded below by \(1-bd>0\), so the extended form is Kähler. For rational \(d\), a divisible integral multiple is the curvature of a positive Hermitian line bundle, which is ample by the Kodaira embedding theorem [15].

For the curve assertion, every complex line bundle on a bordered surface is smoothly trivial. Choose unitary frames on a slightly larger bordered neighborhood of \(\Sigma\), so \(\alpha_j=\mathrm d\theta_j+A_j\) there. Choose a base primitive \(\gamma\) of \(\sigma\). Then \[\omega_Y=\sum_j\mathrm dp_j\wedge\mathrm d\theta_j+ \mathrm d\Gamma_0,\qquad \Gamma_0=\gamma+\sum_jp_jA_j,\] where \(\Gamma_0\) is horizontal and toric. Its base differential is \((1-b\sum_jp_j)\sigma\). The assertion about integrated areas follows from \(\int_\Sigma\sigma\to\int_C\sigma=\deg A\). The moment coefficient is continuous and bounded on the fixed compact polytope, so convergence is uniform. ◻

Proof. Choose homogeneous coordinates with \(o=[1:0:\cdots:0]\) and \(H_\infty=\{z_0=0\}\). On \(\mathbb P^n\times\mathbb C\) let the circle act by \[([z_0:z'],v)\longmapsto ([z_0:e^{2\pi it}z'],e^{-2\pi it}v).\] For the product of unit Fubini–Study and the standard form on \(\mathbb C\), its Hamiltonian is \[F=\mu-\pi|v|^2,\qquad \mu=\frac{|z'|^2}{|z_0|^2+|z'|^2}.\] The level \(F=a\) is compact, regular, and the circle acts freely there. Indeed, if \(v\ne0\) the action is free on that coordinate; if \(v=0\), then \(\mu=a\in(0,1)\), where the projective scalar action is free. Reduction therefore gives a smooth compact Kähler manifold. This is Lerman’s symplectic cut [16]; its Kähler interpretation is described in [5].

For completeness, its complex model can be identified directly. On the stable set \[z'\ne0,\qquad (z_0,v)\ne(0,0),\] the holomorphic map \[([z_0:z'],v)\longmapsto \bigl([z_0:vz'],[z']\bigr)\] takes values in the graph resolution of the projection from \(o\), which is \(\mathop{\mathrm{Bl}}_o\mathbb P^n\), and its fibers are the free complex scalar orbits. On each such orbit write the positive real scalar as \(\lambda\). The function \[\frac{\lambda^2|z'|^2}{|z_0|^2+\lambda^2|z'|^2} -\frac{\pi|v|^2}{\lambda^2}\] is strictly increasing from a value at most zero, or from \(-\infty\), to the limit one. Consequently it takes the value \(a\) exactly once. Each complex orbit meets the level in one circle, and the Kähler quotient has the indicated complex blowup structure.

On the retained open region \(\mu>a\), choose the representative \[v=\sqrt{(\mu-a)/\pi}>0.\] The auxiliary complex-line form pulls back to zero on this real slice, so the original Fubini–Study form is unchanged. At \(v=0\), reduction of the level \(\mu=a\) gives \(E\simeq \mathbb P^{n-1}\) with line area \(a\), as is seen from its moment simplex \(\sum_jp_j=a\). The proper transform of \(H_\infty\) is retained in the open region and has unit line area. These two restrictions determine the class \(H-aE\): \(H|_E=0\) and \(E|_E=-c_1(\mathcal O_E(1))\). Removing \(E\) and \(H_\infty\), and then applying (31), gives precisely the stated open shell.

For any other Kähler representative of \(H-aE\), Lemma 10 gives an identification preserving the disjoint complex divisors \(E,H_\infty\), and therefore their complement.

Finally the graph model is a closed subvariety of \(\mathbb P^n\times\mathbb P^{n-1}\). The pullbacks of the two hyperplane classes are \(H\) and \(H-E\), respectively, since projection from \(o\) is defined by hyperplanes through \(o\). The restriction of a positive rational product polarization is ample. The identity \[AH-BE=(A-B)H+B(H-E)\] therefore proves the last assertion. ◻

Proof. The quadratic Veronese system restricts to a generated linear system defining a closed immersion on every smooth intermediate intersection, and on its intersection with \(F\). Bertini’s theorem [24], applied successively to both, therefore gives \(N-1\) quadrics whose intersection is a smooth curve transverse to \(F\). The Koszul resolution of its structure sheaf has terms that are sums of \(\mathcal O_{\mathbb P^N}(-2j)\), \(1\le j\le N-1\). The vanishings \(H^i(\mathbb P^N,\mathcal O(t))=0\) for \(0<i<N\) [24], together with the absence of global sections of these negative twists, give \(H^0(C,\mathcal O_C)=\mathbb C\). In particular, \(C\) is connected.

The intersection class is \((2H)^{N-1}\), so intersection with one further hyperplane gives \(\deg A=2^{N-1}\). The defining equations form a regular sequence; hence their classes identify the conormal bundle with \((A^{-2})^{\oplus(N-1)}\). This proves the assertion about the normal bundle. Taking determinants in the tangent-normal exact sequence gives \[K_C=K_{\mathbb P^N}|_C\otimes\det N_{C/\mathbb P^N} =A^{\otimes(N-3)}.\] Consequently \(2g(C)-2=(N-3)2^{N-1}\), proving (32). These uses of Bertini, the Koszul resolution, and adjunction are in their usual smooth projective form [14]; the regular-sequence and conormal statements are also given in [24].

The quadric equations generate \(\mathcal I_C(2)\), and hence give a surjection of graded sheaves of algebras \[\mathcal O_{\mathbb P^N}[T_1,\ldots,T_{N-1}] \longrightarrow\bigoplus_{j\ge0}\mathcal I_C^j(2j).\] The relative Proj of the target is \(\mathop{\mathrm{Bl}}_C\mathbb P^N\): twisting the degree-\(j\) part of the Rees algebra by \(\mathcal O(2j)\) leaves its relative Proj unchanged. Thus this surjection gives a closed embedding \[\mathop{\mathrm{Bl}}_C\mathbb P^N\hookrightarrow\mathbb P^N\times\mathbb P^{N-2}.\] The hyperplane classes of the factors pull back to \(H\) and \(2H-E_C\). For rational \(0<d<1/2\), the identity \[H-dE_C=(1-2d)H+d(2H-E_C)\] expresses this class as the restriction of a positive rational product polarization. After clearing denominators, a Segre–Veronese embedding shows that it is ample.

Finally, choose a positive curvature form \(\sigma\) for \(A\). Lemma 13, with normal exponent two, gives a Kähler form in \(A+dU\): its horizontal coefficient is \(1-2\sum p_j\ge1-2d>0\), also at infinity. A rational class admitting a Kähler representative is ample by the Kodaira criterion, after clearing denominators. ◻

Proof. Set \(m=n-1\). The displayed inequalities are strict and there are finitely many of them. By continuity and density of the rationals, choose rational numbers \(r_i>\rho_i\) such that \[ r_*:=\max_i r_i<\tfrac12,\qquad \sum_i r_i^n<1. \tag{34}\] For \(0<h<1/2\), formula (33) gives \[ J_m(h):=\int_{\{t\le h\}}2^m(1-2t)\,dp =\frac{2^mh^m}{m!}\left(1-\frac{2mh}{m+1}\right) \longrightarrow\frac1{n!} \quad(h\uparrow\tfrac12). \tag{35}\] It follows from (34) that we can choose rational parameters satisfying \[ r_*<h<d<\tfrac12,\qquad J_m(h)>\frac1{n!}\sum_i r_i^n. \tag{36}\] Indeed, choose \(h\) sufficiently close to \(1/2\) and then choose \(d\) strictly between \(h\) and \(1/2\); each requirement is open.

Apply Lemma 15 in \(\mathbb P^n\), with the forbidden hyperplane \(H_\infty\). Deform to the normal cone of the resulting curve \(C\), using the class \(H-d[Y]\). On the two central components the polarizations are \(H-dE_C\) and \(A+dU\), respectively. Both are ample by that lemma, so Proposition 12 applies. On the normal component use the form of Lemma 13. Its affine part, truncated at \(t\le h\), is \[ \sigma+\mathrm d\sum_{j=1}^m p_j\alpha_j, \qquad \mathrm d\alpha_j=-2\sigma,\qquad \int_C\sigma=2^m. \tag{37}\] The strict inequality \(h<d\) supplies a neighborhood of the entire truncated region in the affine part.

Let \(C^o=C\setminus(C\cap H_\infty)\). The removed set is finite, and \(C^o\) has the positive genus calculated in (32). Exhaust it by compact connected smooth bordered surfaces \(\Sigma_\ell\), with collars contained in \(C^o\). For example, remove progressively smaller disjoint coordinate disks about the punctures. Each such surface has the same genus as \(C\). Since \(H^2(\Sigma_\ell;\mathbb Z)=0\), the normal line bundles admit smooth unitary frames on a slightly larger bordered surface. Write in those frames \(\alpha_j=\mathrm d\theta_j+A_j\), where \(\mathrm dA_j=-2\sigma\), and choose \(\gamma_\ell\) with \(\mathrm d\gamma_\ell=\sigma\). The form (37) becomes \[\omega_V+\mathrm d\Gamma_{0,\ell},\qquad \Gamma_{0,\ell}=\gamma_\ell+\sum_jp_jA_j,\qquad \mathrm d_C\Gamma_{0,\ell}=(1-2t)\sigma>0\] on a neighborhood of \(\Sigma_\ell\times V\), for \(\Delta=\{p\ge0:t\le h\}\). The polar notation is smooth at every axis: if \(z_j=x_j+iy_j\), then \(p_j\mathrm d\theta_j=(x_j\mathrm dy_j-y_j\mathrm dx_j)/2\). The integrated horizontal forms converge uniformly on \(\Delta\) to \[H_0(p)=2^m(1-2t),\] since \(\int_{\Sigma_\ell}\sigma\to\int_C\sigma\) and \(1-2t\) is bounded there.

We verify all the remaining hypotheses of the surface packing lemma. Each auxiliary simplex \(\{t\le r_i\}\) lies strictly inside the outer face of \(\Delta\), by (36). The function \[P(p)=H_0(p)-\sum_i(r_i-t)_+\] is continuous and concave, being affine minus a sum of convex functions. Choose \(r_*/h<\tau<1\). On \(\Delta\setminus\tau\Delta\) every cap vanishes, and hence \[P(p)=H_0(p)\ge2^m(1-2h)>0.\] Finally, (33) and (36) give \[\int_\Delta P\,dp=J_m(h)-\frac1{n!}\sum_i r_i^n>0.\] Thus \(P\) has positive mean and meets the hypotheses of Lemma 2. All the hypotheses of Lemma 9 now hold. Since \(\rho_i<r_i\), that lemma provides a compact packing of the requested closed balls in the interior of a sufficiently large surface model.

This packing lies away from the infinity locus of the normal component and from its fibers over \(C\cap H_\infty\). These forbidden fibers are smooth and pairwise disjoint. Proposition 12 therefore transfers the packing into \(\mathbb P^n\setminus H_\infty\) with the standard unit Fubini–Study form; the prescribed-form conclusion uses Lemma 10 preserving \(H_\infty\). The usual projective moment coordinates identify this complement with \(\mathop{\mathrm{int}}B^{2n}(1)\). All transfers are defined on neighborhoods of the compact packing. Together with \(h<d\) and the capacity slack \(\rho_i<r_i\), this proves the asserted neighborhood statement. ◻

Proof. The case of no remaining balls is immediate, so suppose \(k\ge1\). Continuity of the finitely many strict inequalities allows rational parameters \(a,r_1,\ldots,r_k\) with \[ \tfrac12<a<1,\qquad A<a,\qquad \rho_i<r_i<s:=1-a, \qquad\sum_i r_i^n<1-a^n. \tag{38}\] For completeness, the point \((A,\rho_1,\ldots,\rho_k)\) has a neighborhood on which the volume and pairwise inequalities remain strict. Choose \(a>A\) in that neighborhood, also \(a>1/2\), and then choose the \(r_i>\rho_i\) within it. Rational choices exist because all increments may be taken arbitrarily small. In particular \(0<s<1/2\).

Lemma 14 lets us reserve the central ball of capacity \(a\) and seek the remaining packing in its exterior shell, whose volume is \((1-a^n)/n!\). We construct surface models of this shell in two stages: first a normal line over a hyperplane, then a normal model of a curve inside that hyperplane. The second model will be embedded symplectically in the base of the first; pulling back the first normal line then gives a model over the curve.

Put \(q=n-2\) and \(r_*:=\max_i r_i\). For now take rational parameters with \[r_*<h_1<d_1<s,\qquad r_*<h_2<d_2<\tfrac12.\] The construction below works for every such choice and every sufficiently large bordered piece of the curve. We will fix the parameters after computing the model’s volume.

Proof. Projection from \(o\) on the quadric becomes a morphism after blowing up \(o\). It contracts the strict transforms of the two ruling lines through \(o\), which are disjoint curves of self-intersection \(-1\). One can see both the morphism and its inverse in the equation \[x_0x_1=x_2x_3,\qquad o=[1:0:0:0].\] Projection is to \([x_1:x_2:x_3]\); the inverse is the quadratic map \[[u:v:w]\longmapsto[vw:u^2:uv:uw],\] whose two simple base points are \([0:1:0]\) and \([0:0:1]\). Its resolution identifies \(D\) with the stated two-point blowup. The inverse quadratic system gives the first equality in (50); the exceptional curve over \(o\) is the strict transform of \(u=0\), giving the second equality. Since \([D]=2H-E\), restriction gives (51).

The strict component class is ample because \[a-c>0,\qquad (1-2c)-(a-c)=s-c>0.\] The restrictions of the class on \(Y\) to its zero and infinity sections are \(L|_D\) and \((L-cD)|_D\), and its fiber degree is \(c>0\). These conditions imply ampleness here. Indeed, subtract a sufficiently small ample class on \(Y\); the two section restrictions remain ample and the fiber degree remains positive. By Hirschowitz’s invariant-cycle argument, recalled in [12], every curve is rationally, hence numerically, equivalent to an effective cycle invariant under scalar multiplication in the normal line. Its irreducible components lie in the fixed sections or in fibers. The perturbed class is therefore nef. Adding back the small ample class proves ampleness by [15].

Finally, \(E\) and \(H_\infty\) are disjoint, and their restrictions to a general \(D\) are smooth. Their classes in (50) identify their plane images as the line and conic asserted. The conic is smooth for a general choice of the quadric. Their strict transforms are disjoint; equivalently, their only plane intersections are the two marked points, with distinct tangent directions there. ◻

Proof. The circle is Hamiltonian: \(H^1(\widehat V;\mathbb R)=0\), since the projective bundle over \(\mathbb P^2\) has vanishing first cohomology and point blowups preserve it. Thus contraction of the invariant form with its generating vector field is exact. The bottom and upper sections are fixed. At either point before blowing up, the tangent weights are \(0,0,1\), the first two being the plane directions. Thus on the exceptional \(\mathbb P^2\) the fixed loci are the projectivized base directions, a line belonging to the bottom section, and the pure vertical point. A line connecting these loci has area \(s\) and weight one. Its moment difference is therefore \(s\). An ordinary projective fiber has area \(d_1\) and weight one at the bottom. It follows that the fixed levels are exactly \[ 0,\quad s,\quad d_1. \tag{54}\]

The attracting basin \(\mathcal B\) of the bottom section for complex scalar contraction is the total space of \[N'=\mathcal O_D(3M-e_1-e_2).\] To verify this including the exceptional directions, use base coordinates \(y_1,y_2\) at a marked point and an original line coordinate \(z\). In the blowup chart with nonzero first base direction, \[y_2=y_1v,\qquad z=y_1w.\] The contraction acts by \(w\mapsto\lambda w\). On the other base chart \(w'=w/v\), so this coordinate is a fiber coordinate in \(\pi^*\mathcal O(3)\otimes\mathcal O_D(-e_1-e_2)\). The charts with nonzero base direction, together with the original charts away from the marked points, give its entire total space. The omitted pure vertical direction does not contract to the bottom. This proves the description of \(\mathcal B\).

We emphasize that \(\mathcal B\) is an attracting basin, not a moment sublevel. It contains \(\{p_1<s\}\). Indeed, contraction decreases the moment strictly along every nonconstant complex orbit. Its limit exists on the projective variety and is fixed. For a point of moment less than \(s\), (54) forces this limit to lie in the bottom section. Along a nonzero fiber of \(N'\), the opposite limit is at level \(s\) or \(d_1\).

Form the Kähler reduction of \(\widehat V\times\mathbb C\) by the diagonal circle at \[ p_1+\pi|v|^2=c. \tag{55}\] Every point of this level lies over \(\mathcal B\), since its first moment is at most \(c<s\). In \(N'\oplus\mathbf1\), each nonzero pair \((w,v)\) has a unique circle in its complex scalar orbit satisfying (55). At contraction the total moment tends to zero. At expansion it tends to infinity if \(v\ne0\), and to \(s\) or \(d_1\) if \(v=0\); all these limits exceed \(c\). Strict moment increase proves the uniqueness. The circle acts freely there because the fiber weights are one.

The holomorphic quotient of the nonzero pairs is \(\mathbb P_D(N'\oplus\mathbf1)\). The preceding orbit description identifies it with the reduction, with its Kähler complex structure. This is also the usual holomorphic-quotient description of Kähler cutting; see [5]. For completeness, the quotient map is holomorphic and constant on complex scalar orbits. The complex orthogonal complement of an orbit at the level identifies its complex tangent quotient with the tangent space of this projectivization. Positivity of the ambient Kähler form therefore gives precisely the asserted Kähler structure.

The map \[x\longmapsto \bigl[x,\sqrt{(c-p_1(x))/\pi}\,\bigr],\qquad p_1(x)<c,\] uses a positive real additional coordinate and is symplectic: the pullback of its standard area form vanishes. It preserves the base point in \(D\), although it is not in general holomorphic. The quotient class restricts on the bottom to \(3bM-s(e_1+e_2)\). Its projective fiber area is \(c\), from the residual weight-one moment interval \([0,c]\). These two data determine the class on this projective bundle. The base-preserving description also proves the last assertion. ◻

Proof. The class on \(W'\) is a positive product class. On \(W\) use the two algebraic scalar factors, with the second factor lifted to the first bundle using its linearization on \(U_2\). An ordinary surface fiber over \(C\) is \(\mathbb P_{\mathbb P^1}(\mathbf1\oplus\mathcal O_{\mathbb P^1}(1))\). Its toric diagram is \[ \mathcal Q=\{p_1,p_2\ge0:\ p_1\le d_1,\ p_1+p_2\le d_2\}. \tag{60}\] The first projective fibers have size \(d_1\); on their two endpoint sections the second sizes are \(d_2\) and \(d_2-d_1\). This gives the four vertices \[(0,0),\ (d_1,0),\ (d_1,d_2-d_1),\ (0,d_2).\] The invariant boundary curves have positive areas \(d_1,d_2-d_1,d_1,d_2\).

Over either marked point the fiber consists of the strict surface fiber and the exceptional plane. The former has the corner \((0,0)\) cut by \(p_1+p_2\ge s\); its five edge lengths are \[s,\quad d_1-s,\quad d_2-d_1,\quad d_1,\quad d_2-s.\] The exceptional plane has size \(s\). All these numbers are positive by (57). The torus acts torically on both components: its tangent characters at the blown-up point are \(0,(1,0),(0,1)\), which induce the standard toric action on the exceptional plane.

A horizontal invariant irreducible curve must be a strict vertex section. To see the exhaustiveness of this assertion, intersect it with the generic fiber over \(C\). This is a finite set preserved by the connected torus, hence fixed pointwise. The curve is therefore generically in the fixed locus, whose four horizontal components are precisely the vertex sections. Their degrees, in the order of the displayed vertices, are \[ 9b-2s,\qquad 9(b-d_1),\qquad 9(b-d_2),\qquad 9(b-d_2). \tag{61}\] Indeed \(U_2\) restricts to \(0\) and \(-3M\) on its two sections. On the first of them the upper first section has \(U_1=-3M\), whereas on the second \(\mathcal N\) is trivial and both first sections have \(U_1=0\). Only the bottom vertex section passes through the two blowup centers, losing \(2s\). All numbers in (61) are positive; the first is \(3+s\).

We explain why these curve tests prove ampleness. The invariant irreducible curves just enumerated have only finitely many numerical classes. The boundary curves in the ordinary fibers vary in algebraic families with constant numerical class, and there are only two special fibers. Fix an ample class \(B\) on \(W\). For sufficiently small \(\epsilon>0\), every listed curve has nonnegative degree against \(\mathcal A|_W-\epsilon B\). Every curve is rationally, hence numerically, equivalent to a torus-invariant effective cycle [12]. This result applies to a connected solvable group on a projective variety and does not require finitely many orbits. Its irreducible components are invariant, since a connected group cannot permute finitely many components nontrivially. Thus \(\mathcal A|_W-\epsilon B\) is nef. The sum of a nef class and a positive ample class is ample [15], proving the assertion on \(W\). Finally, ampleness on the reduced central fiber is equivalent to ampleness on its components, and relative ampleness is open near that fiber. These standard projective ampleness facts apply to a sufficiently divisible integral multiple of \(\mathcal A\); see [14, 15, 24]. ◻

Proof. Expand first in the projective coordinate associated to \(U_1\). A term of degree \(k_1\) leaves coefficient class \[(3bN-3k_1)M+(Nd_2-k_1)U_2.\] Expanding this in the second projective coordinate gives precisely the range and the coefficient line in (62). Sections on the point blowup are exactly sections before the blowup vanishing to total order at least \(sN\) at each center. In local coordinates \((y,w_1,w_2)\), the coefficient of \(w_1^{k_1}w_2^{k_2}\) must therefore vanish in \(y\) to order at least \(m_k\), with no other condition.

Writing \(t=(k_1+k_2)/N\), the degree of the line in (62), divided by \(N\), is \[9(b-t)-2(s-t)_+.\] This decreases on \(0\le t\le d_2\), and is at least \(9(b-d_2)>0\). Its integral degrees therefore tend to infinity uniformly over the weights as \(N\) increases through divisible values. On a genus-one curve every line of degree at least two is basepoint-free: for each point, the evaluation map is surjective by the vanishing of \(H^1\) after subtracting that point; see [24]. This proves the assertion. ◻

Proof. Moment coordinates and the horizontal form. In a local holomorphic frame of \(M|_C\), the projective potential on the unmarked affine chart is \[ B(y,\rho)=\frac1N\log\left(\sum_k g_k(y)e^{k\cdot\rho}\right), \qquad \rho_j=\log|w_j|^2. \tag{65}\] Here \(g_k\) is the sum of squared absolute values of a basis of the coefficient space in (62), in its local frame. It is positive away from the marked points. Near a mark, with local coordinate \(y=0\), it has exactly the form \[ g_k(y)=|y|^{2m_k}a_k(y),\qquad a_k>0 \tag{66}\] with \(a_k\) smooth up to zero, by basepoint-freeness after factoring off the required zeros.

The moments are \(p=\partial_\rho B\). For fixed \(y\), the Hessian \(B_{\rho\rho}\) is a positive multiple of the covariance matrix of the weight vectors with positive coefficients, and is positive definite since the weights span the plane. Its gradient is a diffeomorphism onto the interior of \(\mathcal Q\). Indeed, for \(p\) in that interior the function \(B-p\cdot\rho\) is proper and strictly convex: the maximum of the linear forms \((k/N-p)\cdot\rho\) grows at least linearly in \(|\rho|\). It has a unique minimum, smoothly depending on \(y,p\).

We check the axes explicitly. Put \(r_j=|w_j|^2\) and write \[F(y,r)=N^{-1}\log\sum_k g_k(y)r_1^{k_1}r_2^{k_2}.\] Then \(p_j=r_jF_{r_j}\). At \(r_j=0\), \(F_{r_j}>0\), because the complete system contains weights with \(k_j=1\). The derivative of \(p_j\) in another radial variable vanishes on this face. On its active variables, the logarithmic Hessian of the face system is positive definite. Thus the Jacobian of \(r\mapsto p\) is block triangular with invertible diagonal blocks at every lower face, including the origin. The inverse radii are smooth through those faces, and \[z_j=w_j\sqrt{F_{r_j}/\pi},\qquad \pi|z_j|^2=p_j,\] are smooth disk coordinates there. Compactness gives these coordinates on neighborhoods of all the prescribed truncated regions, with positive margins from the upper faces and the omitted base points.

For fixed \(p\) set \[ G(y,p)=\inf_\rho\bigl(B(y,\rho)-p\cdot\rho\bigr). \tag{67}\] This partial Legendre transform is the usual reduced Kähler potential; see [4]. On lower faces this is interpreted using the corresponding face system. The envelope identity gives \(\mathrm d_yG=\mathrm d_yB\) at the selected radii. Therefore the form in moment and phase coordinates is \[ \sum_{j=1}^2\mathrm dp_j\wedge\mathrm d\theta_j +\mathrm d\bigl(\mathrm d_y^cG\bigr). \tag{68}\] The one-form \(\mathrm d_y^cG\) extends smoothly through the axes: it is \(\mathrm d_y^cF\) evaluated at the smooth inverse radii. Possible terms \(p_j\log p_j\) in \(G\) are independent of the base and disappear upon this differentiation.

The base form \(\kappa(p)=\mathrm d_y\mathrm d_y^cG\) is positive. For instance, on the open orbit put \(\zeta_j=\log w_j\), so \(\rho_j=\zeta_j+\overline{\zeta_j}\). The positive Hermitian Hessian of \(B\) has blocks \[\begin{pmatrix} B_{y\bar y}&B_{y\rho}\\ B_{\rho\bar y}&B_{\rho\rho} \end{pmatrix}.\] Differentiating \(B_\rho=p\) at fixed \(p\) yields \[G_{y\bar y}= B_{y\bar y}-B_{y\rho}B_{\rho\rho}^{-1}B_{\rho\bar y}>0.\] At an axis, restrict the original Kähler form to the coordinate complex submanifold and use the same calculation on the active variables. At the origin it is the positive restriction to the zero section.

These base forms are global. If another frame of \(M\) is \(e'=he\), put \(\lambda=\log|h|^2\) and \(\phi=\arg(h)/(2\pi)\). The changes are \[\rho'_j=\rho_j-3\lambda,\quad \theta'_j=\theta_j-3\phi,\quad B'=B-3b\lambda,\quad G'=G-3(b-p_1-p_2)\lambda.\] Since \(\mathrm d^c\lambda=\mathrm d\phi\), the primitive in (68) changes by \(-3b\,\mathrm d\phi\), whose exterior derivative is zero. Thus the local potentials have the transition law of the real class \[ 3(b-p_1-p_2)c_1(M|_C). \tag{69}\] We will use this class after extending the curvature across the marked points.

Over a bordered \(\Sigma\), choose a smooth frame of \(M\), which is possible since \(H^2(\Sigma;\mathbb Z)=0\). In the resulting product disk coordinates, subtracting the standard fiber form leaves \[\eta=\kappa(y,p)+ \sum_j\mathrm dp_j\wedge\alpha_j(y,p),\] where the \(\alpha_j\) are smooth base one-forms. There are no fiber–fiber terms. Closure implies \(\partial_{p_j}\kappa=\mathrm d_C\alpha_j\) and \(\partial_{p_i}\alpha_j=\partial_{p_j}\alpha_i\). Choose \(\mathrm d_C\gamma_0=\kappa(y,0)\) and define \[\Gamma(y,p)=\gamma_0+ \int_0^1\sum_j p_j\alpha_j(y,tp)\,\mathrm dt.\] Radial homotopy in the downward domain gives \(\mathrm d\Gamma=\eta\). All choices can be made on a slightly larger bordered surface and a slightly larger truncated polytope, proving (63) with the asserted neighborhood extensions.

The two marked-point contributions to the base area. The form (63) now gives the required local model. To determine its limiting integrated area, we compute the curvature concentrated at the omitted marks. Write \(\xi=k/N\) and \(v(\xi)=(s-\xi_1-\xi_2)_+\). By (66), for \(\lambda=\log|y|^2<0\) the potential \(B\) differs by a uniformly bounded amount from \[A_\lambda(\rho)= \max_{\xi\in\mathcal Q\cap N^{-1}\mathbb Z^2} \bigl(\xi\cdot\rho+\lambda v(\xi)\bigr).\] The bound is uniform in \(\rho\) and \(y\) near the mark: there are finitely many \(a_k\), bounded above and below by positive constants. Take \(N\) divisible enough to include the vertices of the subdivision of \(\mathcal Q\) by \(\xi_1+\xi_2=s\). Barycentric interpolation in either cell, on which \(v\) is affine, shows that for every \(p\in\mathcal Q\) and every \(\rho\), \[A_\lambda(\rho)-p\cdot\rho\ge\lambda v(p).\] This bound is sharp. If \(p_1+p_2\le s\), take \(\rho=(\lambda,\lambda)\), for which \(A_\lambda=\lambda s\). If \(p_1+p_2\ge s\), take \(\rho=0\), for which \(A_\lambda=0\). Taking infima and using the bounded difference proves \[ G(y,p)=(s-p_1-p_2)_+\log|y|^2+O(1). \tag{70}\] This argument also applies on axes and at \(p_1+p_2=s\); indeed its bounded error is uniform in \(p\).

For each fixed \(p\), subtracting the logarithmic term in (70) leaves a bounded subharmonic function on the punctured coordinate disk. It extends subharmonically over the mark and has no atomic curvature there: an atom would give an unbounded logarithmic singularity. With our normalization \(\mathrm d\mathrm d^c\log|y|^2=\delta_0\), the extended current of \(G\) has an atom of mass \((s-p_1-p_2)_+\) at each mark. Its total mass, by (69), is \(9(b-p_1-p_2)\). Equivalently, compare its local potentials with smooth potentials in that real class; the difference is a global locally integrable function and its exact curvature has total integral zero. Removing the two atoms gives precisely (64).

The reduced form is smooth at every other base point, so deleting finitely many further points changes none of these integrals. Along an increasing bordered exhaustion the integrated positive forms give continuous functions on the compact set \(\Delta\), increasing pointwise to the continuous function \(H_0\). Dini’s theorem makes their convergence uniform. ◻

Proof. The region avoids the central double locus, which is the second upper face \(p_1+p_2=d_2\), and it avoids the two point blowups. It is therefore in the submersion locus. Parallel transport for the averaged total form exists for short time on a neighborhood of this compact set by Lemma 11. The connection is invariant under the first circle, so transport \(\Phi\) is both symplectic and equivariant. Hence \[\mathrm d(p_1^{\widehat V}\circ\Phi)=\mathrm dp_1^W\] there. The difference is constant on the connected region. The region includes points with first affine coordinate zero. They are fixed points on the first bottom section, away from the double locus and the blowups. The corresponding fixed component in the total space is the strict transform of the global zero section. Short equivariant transport of these points stays in that component, so both normalized moments are zero. The constant is consequently zero. Finally, the relevant inverse image is closed in the total space. A compact set disjoint from it stays disjoint under sufficiently short transport. ◻

Proof. If \(k=0\) the assertion is immediate. Otherwise the strict inequalities allow a rational \(a>R_0\), with \(1/2<a<1\), and auxiliary \(r_i>R_i\), \(1\le i\le k\), such that \[ r_i<s=1-a,\qquad \sum_{i=1}^k r_i^3<1-a^3. \tag{71}\] Use the notation (49). The limiting moment domain is \[\Delta_\infty= \{p_1,p_2\ge0:\ p_1\le s,\ p_1+p_2\le b\}.\] Since \(b>s\), direct integration of (64) gives \[\begin{align*} \int_{\Delta_\infty}H_0(p)\,\mathrm dp_1\mathrm dp_2 &=\int_0^s \left(\frac92(b-u)^2-(s-u)^2\right)\,\mathrm du \tag{72}\\ &=\frac32\bigl(b^3-(b-s)^3\bigr)-\frac{s^3}{3} =\frac{s}{2}-\frac{s^2}{2}+\frac{s^3}{6} =\frac{1-a^3}{6}. \end{align*}\] The full cap of size \(r_i<s\) is supported inside this domain and has integral \[\int_{p_1,p_2\ge0}(r_i-p_1-p_2)_+\, \mathrm dp_1\mathrm dp_2=\frac{r_i^3}{6}.\]

Fix \(d_1\) with \(s<d_1<b\). Choose rational \(h_1,c,h_2,d_2\) with \[ \max_i r_i<h_1<c<s<d_1<d_2<b,\qquad s<h_2<d_2, \tag{73}\] so close to \(h_1=s\) and \(h_2=b\) that on \[\Delta=\{p_1,p_2\ge0:\ p_1\le h_1,\ p_1+p_2\le h_2\}\] the function \[P(p)=H_0(p)-\sum_{i=1}^k(r_i-p_1-p_2)_+\] has positive integral, and hence positive mean. Figure 2 shows the truncated and limiting domains, the cut level, and the change of slope of the limiting density.

Such choices are possible by (71), (72), and convergence of these domains to \(\Delta_\infty\). The cap supports are already wholly inside every sufficiently close truncation.

The hypotheses of Lemma 9 hold. First, \(H_0\) is positive on \(\Delta\). As a function of \(t=p_1+p_2\), its slopes are \(-7\) for \(t<s\) and \(-9\) for \(t>s\), so it is concave. Each negative cap is concave as well, proving concavity of \(P\). The auxiliary simplices lie strictly away from the two outer faces. Choose \(\tau<1\) with \(\tau h_1>\max_i r_i\) and \(\tau h_2>\max_i r_i\). If \(p\notin\tau\Delta\), either \(p_1>\tau h_1\) or \(p_1+p_2>\tau h_2\); in either case all caps vanish. Thus \(P=H_0\) there, with a uniform positive lower bound. We have used the concave limiting density \(H_0\) throughout; only the model’s integrated areas are approximated by bordered pieces.

Construct the cubic family with the chosen \(d_1,d_2\). Remove from its base \(C\) the marked points and all intersections with \(\ell\cup Q\). The remaining curve has genus one and admits connected bordered exhaustions of genus one. Lemma 22 supplies the models (63), with uniform limiting area \(H_0\), over these exhaustions. Lemma 9 therefore embeds disjoint closed balls of capacities \(R_i<r_i\), \(1\le i\le k\), compactly in one such model, with neighborhood embeddings. The entire chosen model has \(p_1\le h_1<c\), is away from the double locus, and avoids the inverse images of \(\ell\cup Q\) under the map to the plane.

Transfer this compact region into a nearby fiber \(\widehat V\). Lemma 23 keeps it below \(p_1<c\) and preserves the stated avoidance. Apply the cut of Lemma 19. The balls now lie in \(Y\) off infinity and off the two forbidden vertical submanifolds over \(D\cap E\) and \(D\cap H_\infty\). The cut form is invariant and is in exactly the polarization class of Lemma 18.

Use Proposition 12 for the first normal-cone degeneration. The allowed relative Moser identification of Lemma 10 preserves infinity and the two disjoint forbidden vertical submanifolds. It follows that the compact packing transfers into \(T\setminus(E\cup H_\infty)\), with a Kähler form in class \(H-aE\). Lemma 14, using relative Moser for the disjoint divisors \(E,H_\infty\), identifies this complement symplectically with \[\left\{z\in\mathbb C^3: a<\pi\sum_{j=1}^3|z_j|^2<1\right\}.\] Add the central closed ball of capacity \(R_0<a\). It is disjoint from all these compact shell balls, and every source lies inside the open target. Each operation was defined on a neighborhood of the relevant compact set; restricting these neighborhoods if necessary gives the required neighborhood embeddings of the closed balls. ◻