Context

The fibration picture is related to the special Lagrangian formulation of mirror symmetry proposed by Strominger–Yau–Zaslow (Strominger et al. 1996). In the hyperkähler setting, rotation of complex structure relates special Lagrangian and holomorphic Lagrangian geometry; the original paper already discusses this relation for K3 surfaces (Strominger et al. 1996, sec. 4.1). Early explicit nef-isotropic formulations appear in Hassett–Tschinkel (Hassett and Tschinkel 2001, Remark 3.12) and Sawon (Sawon 2003, Conjecture 4.1). The former asks for semiampleness, while the latter relates the existence of a nef square-zero divisor to the existence of an abelian fibration. Matsushita established the half-dimensional and Lagrangian character of fibrations from projective irreducible symplectic manifolds (Matsushita 1999, 2001); his equidimensionality theorem also controls every component of every fiber (Matsushita 2000, Theorem 1). Thus, once semiampleness is known, the geometry of the resulting map fits the expected holomorphic symplectic fibration picture.

Analytic positivity gives another source of progress. Verbitsky proves that a nontrivial nef isotropic bundle with a smooth semipositive Hermitian metric has a nonzero section in some positive power (Verbitsky 2010, Theorem 1.9). Höring–Lazić–Lehn obtain nonvanishing or coisotropic-locus alternatives for nef isotropic classes. In complex dimension four, they prove semiampleness if the class contains a closed positive current whose Lelong numbers all vanish (Höring et al. 2025, Theorem D and Corollary E). These results show how additional analytic positivity or geometric information can turn the numerical condition into sections.

The deformation approach has been particularly effective. Soldatenkov and Verbitsky show that a fixed isotropic class is semiample at every nef point of its deformation component as soon as it is semiample at one point (Soldatenkov and Verbitsky 2025, Theorem 3.7(v)). For the known deformation types, earlier constructions and birational results were obtained by Bayer–Macrì (Bayer and Macrì 2014, Theorem 1.5) and Markman (Markman 2014, Theorem 1.3 and Remark 1.8) for \(\mathrm{K3}^{[n]}\)-type, by Yoshioka (Yoshioka 2016, Proposition 3.38) for generalized Kummer type, and by Mongardi–Rapagnetta (Mongardi and Rapagnetta 2021, Theorem 7.2) and Mongardi–Onorati (Mongardi and Onorati 2022, Theorem 2.2) for the two O’Grady types. These works combine moduli spaces of sheaves and stability conditions with monodromy and deformation methods. Their conclusions include fibrations on particular moduli spaces, results on the given manifold under nefness, and fibrations on birational models. Soldatenkov–Verbitsky’s deformation theorem implies the strong conjecture for all known hyperkähler deformation classes (Soldatenkov and Verbitsky 2025, sec. 1.3). This theorem treats compact Kähler manifolds and preserves the specified class; those two features are essential to our return from a nonprojective rotation to the original pair \((X,L)\).

The task left by this deformation theorem is to produce one semiample deformation of the prescribed class. We do so by constructing a torus and rotating the complex structure. The geometric input after rotation is the theorem of Greb–Lehn–Rollenske: a nonprojective hyperkähler manifold containing a holomorphic Lagrangian torus has a holomorphic Lagrangian fibration (Greb et al. 2013, Theorem 4.1).

The degeneration construction has a period-theoretic antecedent in Todorov’s work on maximally unipotent degeneration and vanishing torus cycles. His arithmetic-cusp construction relates isotropic classes to degenerations when \(b_2\ge5\) (Todorov 2003, sec. 4). Here the given class is retained as a labelled vector in the full second-cohomology lattice through family realization and semistable reduction. Starting with the supplied isotropic vector imposes no additional lower bound on the second Betti number.

On the analytic side, Zhang develops the persistence of special Lagrangian tori in Calabi–Yau structures close to a flat cylinder (Zhang 2017, sec. 4), using McLean’s deformation operator (McLean 1998). Li’s Fermat-family theorem gives special Lagrangian torus fibrations on generic regions whose relative volumes tend to one along a subsequence (Li 2022a, Theorem 1.1); (Li 2022b) surveys the metric SYZ framework. For polarized algebraic maximal degenerations of compact Calabi–Yau manifolds, Blum–Liu’s valuative independence theorem (Blum and Liu 2026, Theorem 1.1) and Li’s comparison theorems (Li 2026, Theorems 1.4 and 1.8) give special Lagrangian torus fibrations on open sets whose normalized Calabi–Yau volumes are arbitrarily close to one near the degeneration. The semiampleness argument needs a local torus together with its coordinate isotopy class, which connects its geometry to the prescribed monodromy. We obtain that torus by proving convergence to a full smooth flat cylinder in the actual logarithmic coordinates; the final perturbation uses the Zhang–McLean method.

The proof in four steps

Write \(c_1(L)=m_0e\), where \(m_0>0\) and \(e\) is primitive integral. The proof produces a semiample deformation of \(e\) and then returns to \(X\).

1. Retain the prescribed class in monodromy. Keep \(e\) as a labelled integral cohomology class; it need not remain of type \((1,1)\) during this construction. Choose a polarization on a deformation of \(X\) whose marked class \(h\) is perpendicular to \(e\). A rational three-dimensional period subspace containing \(e\) gives an arithmetic curve with a cusp labelled by \(\mathbb Qe\). We realize that curve by an algebraic family and track its full second-cohomology monodromy through semistable reduction (Proposition 12).

2. Construct a torus in a known isotopy class. For a Calabi–Yau degeneration of complex dimension \(d\), with \(s=-\log|t|\), multiply the polarized Ricci-flat metric by \(s\). Intersection numbers control both traces of this metric relative to logarithmic coordinates. Together with the curvature and projective-map energy estimates, they select noncollapsed centers and produce a metric limit with flat regular part. We identify the full limit with a smooth flat cylinder, including its apparent singularities. In the actual logarithmic coordinates, convergence is smooth on a fixed tube containing an entire angular torus (Lemma 11). A small special Lagrangian perturbation retains its coordinate isotopy class (Theorem 4). Here special Lagrangian means that the Kähler form vanishes on the torus and the holomorphic volume form has constant phase there.

3. Rotate using cohomology. For a suitable fiber \(Y\), write its hyperkähler triple as \((I,J,K)\), with \(I\) its original complex structure. The coordinate torus is parameterized consistently around a loop of the family. Its restriction map therefore kills the image of \(M-1\), where \(M\) is monodromy on \(H^2(Y,\mathbb R)\). The chosen periods put \([\omega_K]\) in that image. A cohomological rotation criterion (Lemma 15) turns the special Lagrangian torus into a holomorphic Lagrangian torus for \(J\). The rotated periods make \(Y'=(Y,J)\) nonprojective and force every rational isotropic \((1,1)\) class to lie on \(\mathbb Qe\). Thus the Greb–Lehn–Rollenske fibration makes \(e\) semiample on \(Y'\).

4. Return to the given manifold. The markings and the positive sign of \(e\) place \(X\) and \(Y'\) in the same fixed-class deformation locus. The Soldatenkov–Verbitsky theorem then gives semiampleness on \(X\) itself. Fujiki’s formula and Matsushita’s equidimensionality theorem give the stated fiber properties.

The cylinder argument extends harmonic functions across the possible singular set to obtain a global derivative bound. Path lifting identifies the metric completion, and bounded holomorphic coframes remove its apparent singularities. In the rotation criterion, cohomological vanishing of one Kähler form suffices: the special Lagrangian period and equality in Wirtinger’s inequality supply the pointwise conclusion.

Organization and conventions

Sections 2 and 3 prove the torus theorem independently of hyperkähler period theory. Section 4 constructs the prescribed degeneration; Lemma 13 separates the underlying polarized family and free period quotient from the cusp calculation. Section 5 rotates the torus and proves Theorem 1; Section 6 deduces Corollary 2.

Unspecified cohomology coefficients are real. We normalize \(q\) integrally; its signature is \((3,b_2-3)\), and the same symbol denotes its associated symmetric bilinear form. Fujiki’s relation is \[ \int_X a^{2n}=c_Xq(a)^n,\qquad c_X>0. \tag{1}\] The external inputs are stated where used: the Calabi–Yau theorem, hyperkähler period and Kähler-cone theory, arithmetic period-map algebraicity, semistable reduction, noncollapsed Einstein-limit and harmonic-function theory, and the fibration, deformation, and boundedness theorems cited above.

Coordinate volume and mixed intersections

Let \(g_t^{(0)}\) and \(\omega_t^{(0)}\) be the Ricci-flat metric and Kähler form supplied by the Calabi–Yau theorem (Yau 1978), with \([\omega_t^{(0)}]=a_t\). The useful scale is \[ g_t=s g_t^{(0)},\qquad \omega_t=s\omega_t^{(0)},\qquad \mu_t=\mathrm dV_t=\frac{\omega_t^d}{d!},\qquad \mu_t(Y_t)=c_a s^d, \quad c_a=\frac1{d!}\int_{Y_t}a_t^d>0. \tag{4}\] The last constant is independent of \(t\) by topological local triviality over the punctured disk. Scaling does not change either of the special Lagrangian conditions.

Take \(z_1,\ldots,z_d\) to be algebraic local equations of the corresponding components. They extend to rational functions on the algebraic total space. The remaining unit in the local equation of \(\pi\) can be absorbed into \(z_0\); no rationality of this last coordinate will be used. Shrink the analytic coordinate neighborhood so that \(|z_i|<r_*<1\) there. On its smooth fibers set \[x_i=\log|z_i|,\qquad y_i=\arg z_i\in\mathbb R/(2\pi\mathbb Z),\qquad \gamma=\sum_{i=1}^d(\mathrm dx_i^2+\mathrm dy_i^2),\qquad \omega_\gamma=\sum_{i=1}^d\mathrm dx_i\wedge\mathrm dy_i.\] Thus \(x_i+iy_i\) are local holomorphic logarithms, and \(\gamma\) is the standard flat metric on \(\mathbb R^d\times(\mathbb R/2\pi\mathbb Z)^d\). Its angular circles have fixed length; we will compare \(g_t\) with this metric. Let \[\Sigma=\left\{b\in\mathbb R^d:b_i<0\ (1\le i\le d),\quad \sum_i b_i>-1\right\}.\] For a relatively compact subset \(B\Subset\Sigma\), write \[\mathcal T_t(B)=\{x/s\in B,\quad y\in(\mathbb R/2\pi\mathbb Z)^d\}.\] For small \(t\) this tube is contained in the coordinate neighborhood. Indeed all \(|z_i|\), including \(|z_0|=\exp(-s-\sum_i x_i)\), are bounded by \(e^{-\delta_B s}\) for some \(\delta_B>0\). In particular, these tubes converge uniformly to \(o\) in the total space.

Use the convention \(|\eta_t|^2=2^{-d}i^{d^2}\eta_t\wedge\overline{\eta_t}\), so that (3) gives \(|\eta_t|^2=|f(z)|^2\mathrm dV_\gamma\) in the chart. Ricci-flatness implies that its volume form is a constant multiple of \(|\eta_t|^2\); integration determines that constant. Hence \[ \mathrm dV_t=\frac{c_a s^d}{I_t}|\eta_t|^2, \qquad \frac{\mathrm dV_t}{\mathrm dV_\gamma} =\frac{c_a s^d}{I_t}|f(z)|^2 \longrightarrow\vartheta>0 \quad\hbox{uniformly on }\mathcal T_t(B) \tag{5}\] after a subsequence. To obtain the convergence, first extract a limit of \(c_a s^d/I_t\) using (2), and then use \(f(z)\to f(o)\ne0\). In particular, the two volume measures are uniformly comparable on each fixed interior tube.

Global energies and many noncollapsed centers

The two trace bounds have different roles. The bound for \(\mathop{\mathrm{tr}}_{g_t}\gamma\) controls logarithmic functions on intrinsic balls; the bound for \(\mathop{\mathrm{tr}}_\gamma g_t\) supplies many noncollapsed centers. We also record two energies whose averages tend to zero.

Choose concentric open Euclidean balls \(B_0\Subset B_1\Subset B_2\Subset\Sigma\), and a smooth function \(0\le\rho\le1\) supported compactly in \(B_2\) and identically one on \(B_1\). We will select centers in \(\mathcal T_t(B_0)\). Define functions on the entire compact fiber by extending the following by zero: \[ U_0=s\rho(x/s),\qquad U_j=x_j\rho(x/s)\quad(1\le j\le d), \qquad H_t=\sum_{j=0}^d|\nabla U_j|_{g_t}^2. \tag{9}\] These extensions are smooth because their support is compactly contained in the chart for each fixed \(t\). On this support \(x_j/s\) is bounded, and \[\mathrm dU_0=\sum_i\rho_i(x/s)\,\mathrm dx_i, \qquad \mathrm dU_j=\rho(x/s)\,\mathrm dx_j +\frac{x_j}{s}\sum_i\rho_i(x/s)\,\mathrm dx_i.\] It follows that \(H_t\le C\mathop{\mathrm{tr}}_{g_t}\gamma\) on the support and that \[ \frac1{\mu_t(Y_t)}\int_{Y_t}H_t\,\mathrm dV_t\le C_H. \tag{10}\]

Choose a fixed projective embedding of the family over a neighborhood of the disk, written \((\Phi,\pi):\mathcal Y\hookrightarrow\mathbb P^M\times\Delta\). Give \(\mathbb P^M\) a fixed Fubini–Study metric and set \[K_t=|\mathop{\mathrm{Rm}}_{g_t}|^2+|\mathrm d\Phi_t|_{g_t}^2.\] The Ricci-flat Kähler Chern–Weil identity expresses \(\int_{Y_t}|\mathop{\mathrm{Rm}}_{g_t^{(0)}}|^2\mathrm dV_{g_t^{(0)}}\) as a universal dimension-dependent positive constant times \(\int_{Y_t}c_2(Y_t)\,a_t^{d-2}\). This is independent of \(t\). The energy of the holomorphic map \(\Phi_t\) in the unscaled metric is likewise a fixed multiple of \(\int_{Y_t}\Phi_t^*[\omega_{\rm FS}]\,a_t^{d-1}\). Under \(g_t=s g_t^{(0)}\), squared curvature scales by \(s^{-2}\), map-energy density by \(s^{-1}\), and volume by \(s^d\). Therefore \[ \begin{split} \int_{Y_t}|\mathop{\mathrm{Rm}}_{g_t}|^2\,\mathrm dV_t&\le C s^{d-2},\\ \int_{Y_t}|\mathrm d\Phi_t|_{g_t}^2\,\mathrm dV_t&\le C s^{d-1},\\ m_t:=\frac1{\mu_t(Y_t)}\int_{Y_t}K_t\,\mathrm dV_t &\le C(s^{-2}+s^{-1})\longrightarrow0. \end{split} \tag{11}\] This also covers \(d=2\), when the curvature integral in the unscaled metric is purely topological.

One center with estimates at every radius

We now choose one of the noncollapsed centers so that the average cutoff energy stays bounded, and the average curvature and embedding energies tend to zero, on every ball about it. The maximal function makes this simultaneous choice possible.

For a nonnegative integrable function \(h\) on \(Y_t\), let \[\mathcal M_t h(q)=\sup_{r>0} \frac1{\mu_t(B_{g_t}(q,r))} \int_{B_{g_t}(q,r)}h\,\mathrm dV_t\] be the centered maximal function. Bishop–Gromov gives doubling with constant \(2^N\). The usual disjoint-ball covering argument therefore has a constant \(D_N\) independent of \(t\) and gives \[ \mu_t\{\mathcal M_t h>\lambda\} \le\frac{D_N}{\lambda}\int_{Y_t}h\,\mathrm dV_t. \tag{14}\] For completeness, take at each point of the set on the left a ball whose average exceeds \(\lambda\), select disjoint balls whose fivefold enlargements cover a compact subset of that set, and use \(\mu_t(5B)\le5^N\mu_t(B)\). Sum the integrals on the disjoint balls and exhaust the set by compact subsets. Radii larger than the diameter simply give the whole manifold, so no diameter bound is needed.

Choose \(A\) so large that \(D_NC_H/A<\delta/3\). Set \(\epsilon_t=\sqrt{m_t}+s^{-1}\), which is positive and tends to zero. The estimate (14), first for \(H_t\) and then for \(K_t\), shows that \[\mu_t\{\mathcal M_tH_t>A\}<\frac\delta3\mu_t(Y_t),\qquad \mu_t\{\mathcal M_tK_t>\epsilon_t\} \le D_N\sqrt{m_t}\,\mu_t(Y_t).\] For small \(t\) the latter quantity is also less than \(\delta\mu_t(Y_t)/3\). These two exceptional sets cannot cover \(E_t\). We can therefore select \(p_t\in E_t\) with \[ \begin{gathered} \mu_t(B_{g_t}(p_t,1))\ge v,\\ \frac1{\mu_t(B_{g_t}(p_t,R))} \int_{B_{g_t}(p_t,R)}H_t\,\mathrm dV_t\le A,\\ \frac1{\mu_t(B_{g_t}(p_t,R))} \int_{B_{g_t}(p_t,R)}K_t\,\mathrm dV_t\le\epsilon_t \quad\hbox{for every }R>0. \end{gathered} \tag{15}\] After a further subsequence, \[b_t:=x(p_t)/s\longrightarrow b_*\in\overline{B_0}\subset B_1.\] In particular \(\rho(b_t)=1\), \(U_0(p_t)=s\), and \(U_j(p_t)=x_j(p_t)\).

We next turn average gradient control into an estimate centered at the actual value of the function at \(p_t\). The ball Poincaré inequality for nonnegative Ricci curvature, a consequence of the segment inequality (Cheeger and Colding 1996), states that \[\frac1{\mu_t(B(q,r))}\int_{B(q,r)}|U-U_{B(q,r)}|^2\,\mathrm dV_t \le C_Nr^2\frac1{\mu_t(B(q,2r))} \int_{B(q,2r)}|\nabla U|^2\,\mathrm dV_t.\] Here \(U_{B(q,r)}\) denotes the mean and doubling absorbs any equivalent choice of enlarged radius. Applied at \(p_t\) to \(U_j\), this gives an upper bound \(CA r^2\). Writing \(a_r=(U_j)_{B(p_t,r)}\) and comparing the smaller ball with the larger one by doubling, we also obtain \[\begin{split} |a_{r/2}-a_r|^2 &\le\frac1{\mu_t(B(p_t,r/2))} \int_{B(p_t,r/2)}|U_j-a_r|^2\,\mathrm dV_t\\ &\le C A r^2. \end{split}\] Since \(U_j\) is smooth, \(a_{2^{-k}R}\to U_j(p_t)\) as \(k\to\infty\). Summing the preceding bounds over dyadic radii yields \(|a_R-U_j(p_t)|\le C\sqrt A\,R\). Combining this with Poincaré proves \[ \frac1{\mu_t(B_{g_t}(p_t,R))} \int_{B_{g_t}(p_t,R)}|U_j-U_j(p_t)|^2\,\mathrm dV_t \le C R^2 \quad(0\le j\le d,\ R>0). \tag{16}\]

The cutoff functions now locate each fixed intrinsic ball in the crossing chart. Divide (16) by \(s^2\) and use \(b_t\to b_*\). On every fixed-radius ball, \(U_0/s\to1\) and \(U_j/s\to(b_*)_j\) in \(L^2\). Where \(U_0/s>1/2\), the point lies in the coordinate tube and \[\frac{x_j}{s}=\frac{U_j/s}{U_0/s}.\] Chebyshev’s inequality consequently gives, for every \(R,\eta>0\), \[ \mu_t\left(B_{g_t}(p_t,R)\setminus \{q\hbox{ in the chart}:|x(q)/s-b_*|<\eta\}\right) \longrightarrow0. \tag{17}\] The assertion concerns absolute volume, because Bishop–Gromov gives the uniform bound \(\mu_t(B_{g_t}(p_t,R))\le\omega_NR^N\), where \(\omega_N\) is the volume of the Euclidean unit ball.

The regular part of the pointed limit

Noncollapse permits us to pass to a limit. Vanishing curvature energy will make its regular set flat; vanishing embedding energy will place each regular compact set inside the chosen crossing chart.

The noncollapse in (15) and \(\mathop{\mathrm{Ric}}(g_t)=0\) give, after extraction, a pointed measured Gromov–Hausdorff limit \[(Y_t,g_t,p_t,\mu_t)\longrightarrow(Z,d_Z,p,\mu).\] We use the following precise consequences of noncollapsed Ricci limit theory and Einstein regularity (Cheeger and Colding 1997; Colding 1997; Cheeger and Naber 2015). The space \(Z\) is complete and proper; \(\mu\) is its \(N\)-dimensional Riemannian volume on its open dense regular set \(G\). Regular points are the points with a Euclidean tangent cone. Compact subsets of \(G\) admit smooth convergence identifications with regions in \(Y_t\), under which the metrics converge in \(C^\infty\). The closed singular set \(S=Z\setminus G\) satisfies \[ \dim_{\rm H} S\le N-4, \qquad \mu(B_Z(q,r))\le\omega_Nr^N. \tag{18}\] The codimension assertion is Theorem 1.4 of (Cheeger and Naber 2015); the smooth convergence follows from Einstein \(\epsilon\)-regularity (Cheeger and Naber 2015, Theorem 2.11) and elliptic bootstrapping. Complex structures also converge smoothly on \(G\): they are parallel orthogonal tensors for the converging metrics. Their limit \(I_\infty\) is parallel and integrable, so the limiting metric \(g\) is Kähler on \(G\).

Every compact smooth region lies in the image of a fixed-radius ball about \(p_t\) for all sufficiently small \(t\) along the sequence. By (15) and the volume upper bound, its curvature energy is at most \(\epsilon_t\omega_NR^N\), which tends to zero. Smooth metric convergence therefore implies \[ \mathop{\mathrm{Rm}}_g=0\quad\hbox{on }G. \tag{19}\] It remains to put these smooth regions in the coordinate chart about \(o\). We use the following small-energy argument for the maps \(\Phi_t\).

The maps \(\Phi_t\) are holomorphic between Kähler manifolds and hence harmonic. On a compact subdomain of \(G\), pull them back using the smooth convergence identifications. Their energies tend to zero by (15), so Lemma 7 applies. Every local constant limit is \(\Phi(o)\). Indeed a fixed smaller regular ball has positive limiting volume, whereas the exceptional set in (17) has volume tending to zero. That ball therefore contains points with \(x/s\) in a fixed compact subset of \(\Sigma\). Such points converge to \(o\) in the total space, and their images converge to \(\Phi(o)\). Local uniform convergence of the maps identifies the constant on each regular ball separately, regardless of whether \(G\) is connected. It follows, by a finite cover, that \[ \Phi_t\longrightarrow\Phi(o) \quad\hbox{uniformly on every compact subset of }G. \tag{21}\]

Since \((\Phi,\pi)\) is an embedding, there is a neighborhood of \((\Phi(o),0)\) whose inverse image lies in the coordinate neighborhood (3). Thus the corresponding regions of \(Y_t\) are eventually in that chart. On such a region all \(|z_i|<r_*<1\), and \(\sum_{i=0}^d\log|z_i|=-s\). For \(1\le j\le d\) this gives \[-s-d\log r_*\le x_j\le\log r_*.\] In particular \(x_j/s\) is uniformly bounded there. Each \(x_j\) is pluriharmonic, being locally the real part of \(\log z_j\), and is therefore harmonic for \(g_t\). Its boundedness and (17) give local \(L^2\) convergence of \(x_j/s\) to \((b_*)_j\) on the smooth regions. Interior harmonic estimates for the smoothly converging metrics improve this to \[ \frac{x_j}{s}\longrightarrow(b_*)_j \quad\hbox{in }C^\infty_{\rm loc}(G),\qquad 1\le j\le d. \tag{22}\] As \(b_*\in B_1\) and \(\rho=1\) there, every compact subset of \(G\) has a neighborhood on whose corresponding regions \(\rho(x/s)=1\) for all sufficiently small \(t\). Hence on these regions \(U_j=x_j\) for \(j\ge1\). The all-radius energy bound (15) and the centered oscillation estimate (16) therefore apply to the actual logarithmic coordinates. In the next section they give a global coordinate map on the limit, including its possible singular points.

The coordinate map and its derivative bounds

On every compact subset of the regular set \(G\) the actual logarithmic chart is defined for all sufficiently small \(t\), and \(\rho=1\) there. The functions \[x_j-U_j(p_t),\qquad j=1,\ldots,d,\] are harmonic on these subsets. Their local \(L^2\) bounds, from (16), and interior elliptic estimates give smooth subsequential limits \(u_j\). The angular coordinates have values in the compact circles \(\mathbb R/(2\pi\mathbb Z)\) and satisfy \(\mathrm d\arg z_j=-\mathrm dx_j\circ I_t\), where \(I_t\) is the fiber complex structure. A diagonal subsequence therefore also gives smooth circle-valued limits \(\vartheta_j\). Thus \[ F=(u_1+i\vartheta_1,\ldots,u_d+i\vartheta_d):G\longrightarrow H \tag{23}\] is holomorphic. Diagonal extraction here and below is over a countable exhaustion of the smooth manifold \(G\).

The coordinate density in (5) converges to a positive constant. Consequently the absolute real Jacobian of \(F\), computed using the limit metric and \(\gamma\), is the constant \(J_0=\vartheta^{-1}>0\). In particular \(F\) is a local biholomorphism. It is also injective. Indeed, if two distinct regular points had the same image, choose disjoint small regular neighborhoods of these points on which \(F\) is invertible. Smooth convergence and the inverse function theorem imply that their approximating logarithmic charts both contain one fixed smaller neighborhood of the common image. This contradicts the injectivity of the logarithmic chart on each nearby fiber.

For every \(R>0\), exhaustion of \(B(p,R)\cap G\) by regular compact subsets, smooth convergence, and the estimates at \(p_t\) give, with a fixed constant \(C\), \[\begin{align*} \int_{B(p,R)\cap G}|u_j|^2\,\mathrm d\mu &\le CR^2\mu(B(p,2R)),\tag{24}\\ \int_{B(p,R)\cap G}|\mathrm du_j|^2\,\mathrm d\mu &\le C\mu(B(p,2R)). \tag{25}\end{align*}\] Enlarging the radii slightly before taking limits avoids any issue with boundaries of balls.

To extend these functions across \(S\), we use the Sobolev structure on the whole noncollapsed Ricci limit. Pointed measured Gromov–Hausdorff convergence implies pointed measured Gromov convergence by (Gigli et al. 2015, Proposition 3.30). For finite \(N>2\), Lott–Villani’s (Lott and Villani 2009, Corollary E.44) preserves nonnegative \(N\)-Ricci curvature under pointed measured Gromov–Hausdorff convergence; in particular, the limit satisfies \(\mathrm{CD}(0,N)\). Here \(N=2d\ge4\). The finite-dimensional curvature-dimension condition in the pointed setting is also treated in (Gigli et al. 2015, sec. 4.2), while (Gigli et al. 2015, Theorem 7.2) preserves quadratic Cheeger energy. These statements allow measures finite on bounded sets, without a finite-total-volume assumption. The limit measure has full support: every open set contains a regular ball of positive Riemannian volume. Together with (Erbar et al. 2015, Theorem 3.17(i)), these stability results give the \(\mathrm{RCD}(0,N)\) structure used below. At zero curvature the reduced and ordinary curvature-dimension conditions coincide.

On a sufficiently small regular ball, short minimizing \(Z\)-segments stay in a larger regular ball. Ambient and Riemannian distances therefore agree locally. Restrictions of Lipschitz approximants and classical weak lower semicontinuity give the lower bound for the relaxed Sobolev energy; smooth compactly supported approximants, extended by zero, give the upper bound. Locality and exhaustion therefore identify this energy on \(G\) with ordinary Riemannian Sobolev energy.

The metric completion is the full cylinder

We have an injective local biholomorphism on \(G\) with uniformly bounded derivative and inverse derivative. To extend this identification over \(Z\), we first compare the intrinsic distance on \(G\) with the limit distance, and then continue the inverse along paths in the cylinder.

We use two elementary consequences of the small dimension of a removed set. First, in a Euclidean ball or flat cylinder, if \(\dim_{\mathrm H}E<N-1\), any two points outside \(E\) can be joined outside \(E\) by polygonal paths of length arbitrarily close to their distance. The set \(E\) need not be closed: intermediate vertices can be chosen arbitrarily near a shortest segment, outside the cones of rays through \(E\) from the endpoints. Each bad cone is a countable union of Lipschitz images of \(E\) times an interval, and has dimension at most \(\dim_{\mathrm H}E+1<N\). One works in Euclidean lifts for the cylinder.

Second, if \(E\) is closed and \(\dim_{\mathrm H}E<N-2\), the complement of \(E\) in a convex Euclidean ball is simply connected. Indeed a loop has positive distance from \(E\) and can be approximated, through a homotopy in the complement, by a polygonal loop. Cone its finitely many edges to a generic point of the ball. For each edge, the bad cone centers lie in countably many bounded Lipschitz images of \(E\) times the edge times a real interval: writing an intersection point as \((1-t)x+tc\) solves for the center \(c\), and one restricts to \(t\) bounded away from zero. This bad set has dimension at most \(\dim_{\mathrm H}E+2<N\). A center outside all the bad sets gives the required filling.

Removing the apparent singularities

The completion is now the whole cylinder. We next show that its metric is smooth across \(E=F(S)\), which will prove that \(S\) was empty and permit smooth convergence over every compact cylinder tube.

Transfer the metric on \(G\) to \(H\setminus E\). It is a flat Kähler metric \(g_\infty\), uniformly bounded above and below by positive multiples of \(\gamma\). Consider a small convex complex coordinate ball \(D\subset H\). Its complement of \(E\) is simply connected by the second avoidance fact. A unitary parallel holomorphic coframe therefore exists there: \[\theta_a=\sum_{j=1}^d A_{aj}(w)\,\mathrm dw_j,\qquad g_\infty=\sum_a|\theta_a|^2.\] The coefficient matrix \(A\) and its inverse are uniformly bounded, by the two-sided metric bounds.

Each bounded holomorphic coefficient extends holomorphically across \(E\). Here is a direct removal argument. Cover the compact exceptional set in a smaller ball by balls with \(\sum r_i^{N-1}\to0\), take finite subcovers, and use Lipschitz cutoffs \(\chi_k\) vanishing near \(E\). Then \(\|\mathrm d\chi_k\|_{L^1}\to0\). Testing the Cauchy–Riemann equation for the bounded coefficient against \(\chi_k\) times a smooth test form and passing to the limit proves the distributional Cauchy–Riemann equation on the whole smaller ball. The distributional equation implies holomorphicity. The inverse bound and continuity show that the extended matrix is nonsingular. The identities expressing that the coframe is closed, and that the resulting metric is flat and Kähler, extend from the dense complement. Thus \(g_\infty\) extends as a smooth flat Kähler metric across \(E\). Extensions on overlapping balls agree by density.

The distance of this extended metric is exactly the completion distance already obtained. In fact any piecewise smooth path in a coordinate ball can be approximated in length by paths avoiding \(E\), by polygonal approximation, the avoidance argument, and uniform continuity of the smooth metric. Subdivision gives the assertion for arbitrary paths. Conversely every path off \(E\) has its original length. Lemma 9 and completion now identify the two distances.

It follows that every point of \(Z\) has Euclidean tangent cones, so every point is regular in the original Einstein-limit sense. Thus \(S\) is empty. The smooth convergence theorem for Einstein metrics applies at every point of the cylinder, including those not initially known to be regular.

The extended metric is translation invariant. To see this, lift it to \(\mathbb C^d\), where flatness and simple connectivity give a global parallel unitary holomorphic coframe. Its coefficients in the coordinates \(w_j\) are bounded entire functions, again by the uniform metric bounds. Liouville’s theorem makes them constant.

The smooth limit must now be expressed in the original logarithmic coordinates. This will let us perturb a torus while retaining its angular parametrization. Write \[\mathcal L_t(q)=\bigl(x_j(q)-x_j(p_t)+i\arg z_j(q)\bigr)_{j=1}^d \in H, \qquad Q_r=\{w\in H:|\operatorname{Re}w|<r\}.\] Here \(r>0\) is fixed, and \(\mathcal L_t\) is defined on the part of \(Y_t\) in the crossing chart. The real shifts do not alter its angular coordinates. Normalize \(\eta_t\) by a positive scalar to obtain a holomorphic volume form \(\Omega_t\) with the Calabi–Yau normalization for \(g_t\).

A special Lagrangian torus in the nearby fibers

Fix a tube \(Q_{2r}\) and use the charts of Lemma 11 to regard its Calabi–Yau data as tensors on this one fixed domain. We suppress the pullback symbols. The polarizing line bundle extends over the total space and is topologically trivial on a sufficiently small contractible coordinate neighborhood of the maximal crossing. Thus \(\omega_t\) is exact on \(Q_{2r}\). Its constant limit \(\omega_\infty\) has zero period on every angular coordinate two-torus. Since its coefficients are constant, every angular coefficient vanishes. Thus \[T_0=\{\operatorname{Re}w=0\}\subset Q_{2r}\] is Lagrangian for \(\omega_\infty\). The constant nonzero form \(\Omega_\infty\) has constant phase on \(T_0\), making it special Lagrangian.

We perturb this torus using the persistence argument for Calabi–Yau torus tubes in (Zhang 2017, sec. 4, Lemmas 4.3–4.5). Its linearization is McLean’s special Lagrangian deformation operator (McLean 1998). Identify normal vector fields along \(T_0\) with one-forms \(\alpha\) by contraction with \(\omega_\infty\), and use a fixed normal graph chart \(\iota_\alpha:T_0\to Q_r\). Choose a sufficiently large integer \(k\) and \(0<\beta<1\). The source space is \[\mathcal V=\{\alpha\in C^{k+1,\beta}(T^*T_0): \alpha\perp\mathcal H^1(T_0)\},\] where orthogonality and harmonic forms use the flat metric on \(T_0\). All graphs with \(\alpha\) in a fixed sufficiently small ball of \(\mathcal V\) lie in \(Q_r\).

The periods \(\int_{T_0}\Omega_t\) converge to the nonzero flat period. Multiply each \(\Omega_t\), and its limit, by the phase making this period positive real. These phases converge. For each small graph set \[\mathcal P_t(\alpha)= \bigl(\iota_\alpha^*\omega_t, \iota_\alpha^*\operatorname{Im}\Omega_t\bigr).\] The first component is exact because \(\omega_t\) is exact. The second has zero integral because graph isotopy preserves \(\int_{T_0}\Omega_t\), whose imaginary part vanishes after the phase choice. Thus the target is the fixed Banach space \(\mathcal W\) consisting of exact \(C^{k,\beta}\) two-forms and exact \(C^{k,\beta}\) top forms on \(T_0\). These exact-form spaces are closed.

We have \(\mathcal P_\infty(0)=0\). Differentiation of pullbacks of closed forms gives \[ D\mathcal P_\infty(0)\alpha =\bigl(\mathrm d\alpha,c\,\mathrm d(*\alpha)\bigr),\qquad c\ne0, \tag{27}\] where the star is the Hodge star of \(T_0\). The second identity follows by contraction in an orthonormal Lagrangian frame. This operator is an isomorphism \(\mathcal V\to\mathcal W\). To see this, rewrite its second equation as \(\delta\alpha=f\), with \(f\) of mean zero. For an exact two-form \(\xi\), the Hodge Green operator \(\mathcal G\) gives \[\alpha=\delta\mathcal G\xi+\mathrm d\mathcal Gf.\] The Hodge identities give \(\mathrm d\alpha=\xi\) and \(\delta\alpha=f\). The solution is orthogonal to harmonic forms, and the only solution of the homogeneous equations in \(\mathcal V\) is zero. Elliptic estimates make the inverse bounded, with one derivative of gain.

Lemma 11 gives \(\mathcal P_t\to\mathcal P_\infty\) in \(C^1\) on a fixed small ball of \(\mathcal V\), with values in \(\mathcal W\). In particular, \(\mathcal P_t(0)\to0\), and their derivatives are uniformly close to the fixed invertible operator (27) after the ball is made smaller. The inverse function theorem therefore produces \(\alpha_t\to0\) in \(C^{k+1,\beta}\) with \(\mathcal P_t(\alpha_t)=0\) for every sufficiently late member of the subsequence. This argument requires convergence of the ambient tensors, not differentiability with respect to the degenerating parameter. On each connected Lagrangian graph, the real restriction of \(\Omega_t\) is nowhere zero. Its sign is therefore constant, and the positive period makes it positive. Thus the graph is calibrated by \(\operatorname{Re}\Omega_t\) and hence minimal. The minimal-graph equation is elliptic; smoothness of the ambient data and elliptic regularity therefore make each graph smooth.

The graphs are embedded, and \(\iota_{u\alpha_t}\), \(0\le u\le1\), is an isotopy to \(T_0\) inside \(Q_r\). The actual chart \(\Psi_t\) transports this isotopy to the smooth fiber, ending at an angular coordinate torus in the prescribed crossing chart. The graph is Lagrangian and has constant volume-form phase, so it is the torus required by Theorem 4.

The initial sequence was arbitrary. If the conclusion failed for arbitrarily small nonzero parameters, choose \(t_\nu\to0\) so that no \(Y_{t_\nu}\) contained a torus with the stated properties. The preceding construction gives such tori on a tail of a subsequence, a contradiction. This proves the assertion for every sufficiently small nonzero parameter and completes the proof of Theorem 4.

A positive polarization orthogonal to the null class

We use the following standard inputs from hyperkähler period theory: unobstructed deformations and local Torelli; surjectivity of the period map from the selected connected marked component; the projectivity criterion for a positive integral \((1,1)\) class; and the Kähler-cone description, whose walls are orthogonal to negative integral \((1,1)\) classes. See (Huybrechts 1999, 2003; Verbitsky 2013, 2019). The corrected projectivity criterion is (Huybrechts 2003, Theorem 2), and the Kähler-cone description in terms of negative MBM classes is due to Amerik–Verbitsky (Amerik and Verbitsky 2015, Theorems 1.19 and 6.2); see also (Soldatenkov and Verbitsky 2025, sec. 2.1).

The first two construction steps also have an independent use: they provide an algebraic family and a quotient in which equality of period lifts detects equality of monodromy on the full second-cohomology lattice. We isolate precisely this construction, before choosing a ternary subdomain or a cusp.

We now apply Lemma 13 to the primitive class fixed at the start of this section. Its open marked period image is the input for the next step.

A rational ternary domain containing the fixed vector

We now find a rational period subspace which meets \(U\) and has \(\mathbb Qe\) as a cusp. View \(D_h\) as the space of positive oriented real two-planes in \(V_{\mathbb R}\). Let \(W\in U\). Write \(e=e_W+e_-\) in the orthogonal decomposition \(V_{\mathbb R}=W\oplus W^\perp\). Since \(W^\perp\) is negative definite, \(e_W\ne0\); otherwise the nonzero vector \(e\) would have negative square. Moreover \[q(e_-)=-q(e_W)<0.\] It follows that \(W+\mathbb Re=W\oplus\mathbb Re_-\) has signature \((2,1)\).

For an explicit rational approximation, choose an oriented basis \(x,y\) of \(W\) and rational vectors \(x_j,y_j\in V\) tending to \(x,y\). For all sufficiently large \(j\) the plane \(W_j=\langle x_j,y_j\rangle_{\mathbb R}\) is positive, has the chosen orientation, and lies in \(U\). Nondegeneracy and signature are open conditions on the Gram matrix, so \[P_j=\langle e,x_j,y_j\rangle_{\mathbb Q}\] has signature \((2,1)\) for all sufficiently large \(j\). Fix such a \(P=P_j\). The vector \(e\) has remained fixed throughout this approximation. Let \(D_P\) be the component of its positive-plane domain that contains \(W_j\). Then \(D_P\subset D_h\) and \(D_P\cap U\ne\varnothing\).

An arithmetic subgroup of \(SO(P,q)\) preserving \(D_P\) can be made to act on \(\Lambda\) by extending it identically on \(P^\perp\). To see the integral issue explicitly, set \[\Lambda_0=(P\cap\Lambda)\oplus(P^\perp\cap\Lambda).\] This is a finite-index sublattice of \(\Lambda\). Choose \(N\) with \(N\Lambda\subset\Lambda_0\). If an isometry \(g\) of \(P\cap\Lambda\) is congruent to the identity modulo \(mN\), with \(m\in\mathbb Z_{>0}\), its extension satisfies \((g-1)\Lambda\subset m\Lambda_0\subset m\Lambda\). The same holds for its inverse, so the extension preserves \(\Lambda\); choosing \(m\) sufficiently divisible places it in \(\Gamma\). Take such a finite-index subgroup \(\Gamma_P\). It is torsion-free and its inclusion into \(\Gamma\) is an inclusion of actual lattice isometries.

The quotient \[C_P=\Gamma_P\backslash D_P\] is a smooth arithmetic curve. Its holomorphic map \(j:C_P\to Q_h\) is algebraic by Borel’s theorem (Borel 1972, Theorem 3.10). Since \(P\) contains the rational null line \(\mathbb Qe\), its smooth compactification \(\overline C_P\) has a cusp \(c_e\) represented by that line. We always retain this chosen lift of the cusp in the boundary of \(D_P\), even if other translates represent the same point of \(\overline C_P\).

The construction also covers the two smallest possible ranks. If \(r=4\), then \(V\) itself is a rational three-space of signature \((2,1)\) and we simply take \(P=V\). Its noncompact arithmetic curve has a cusp because the given \(e\) is rational and isotropic. If \(r=5\), the domain of signature \((2,2)\) is still a Hermitian symmetric domain and the same ternary subdomain works. General constructions that obtain a null vector from Meyer’s theorem need an additional rank hypothesis; here \(e\) is already supplied.

An algebraic curve reaching the specified cusp

It remains to reach this cusp in the actual polarized family and identify its monodromy. Form the algebraic fiber product \[Z=S\times_{Q_h}C_P.\] Choose a point \(z_0\) at which the marked period lift from \(S\) equals the chosen point of \(U\cap D_P\). The projection \(Z\to C_P\) is submersive near \(z_0\), because the local period map from \(S\) is submersive. Thus the component through \(z_0\) dominates \(C_P\). It contains an irreducible algebraic curve through a point in this selected local branch which dominates \(C_P\): intersect a quasi-projective embedding with general hyperplanes through a smooth point, choosing their tangent intersection transverse to the fiber of \(Z\to C_P\), and take the component through that point. Normalize this curve and call it \(B\). At the selected point the resulting map \(g:B\to C_P\) is nonconstant. Pullback from \(S\) gives a projective polarized family over \(B\) with the chosen marked local branch.

Take smooth projective completions. The nonconstant map of curves extends to a finite surjective morphism \[\bar g:\overline B\longrightarrow\overline C_P.\] There is a finite set \(F\subset\overline C_P\), containing all cusps, branch values, and images of the finitely many points omitted from \(B\), such that over \(C_P\setminus F\) the map is a finite unramified covering and its source lies in \(B\). If necessary move the chosen starting point a little within the same local branch so that it maps outside \(F\).

The following path choice fixes the marking at infinity. Start with its selected lift in \(D_P\). The inverse image in \(D_P\) of \(F\cap C_P\) is a locally finite discrete set. Its complement is path connected. Consequently we may take a path from the initial lift towards the boundary point represented by the specific vector \(e\), avoiding that discrete set; sufficiently far along the path it lies in a cusp neighborhood of \(e\). Project the path to \(C_P\setminus F\) and lift it through the covering from \(B\), starting at the chosen point. The lift approaches some point \(b_e\in\overline B\) over \(c_e\). In a punctured disk around \(b_e\) we therefore have the desired smooth algebraic polarized family.

Transport the original marking along this lifted path. The resulting family period lift to \(D_h\) and the inclusion of the chosen \(D_P\)-valued lift start at the same point and project to the same map into \(Q_h\). Uniqueness of lifts for the covering \(D_h\to Q_h\) makes them equal all along the path and on the universal cover of the resulting punctured disk.

Freeness now identifies the monodromy on all of \(\Lambda\). For a loop in \(B\), let \(M_B\in\Gamma\) be its second-cohomology monodromy and let \(\gamma_B\in\Gamma_P\subset \Gamma\) be the deck transformation of its modular-curve lift, using compatible conventions. Equivariance and the preceding equality of period lifts imply \[M_B\widetilde\Phi(\widetilde b) =\gamma_B\widetilde\Phi(\widetilde b).\] Hence \(\gamma_B^{-1}M_B\) fixes a point of \(D_h\). The action is free, so \(M_B=\gamma_B\) as isometries of the full lattice. In particular, near the selected cusp the actual family monodromy fixes our labelled \(e\) and is the identity on \(P^\perp\). Thus the free action and the equality of the initial marked lifts retain the precise class needed in the rotation argument.

The exact period and monodromy formulas

We have obtained a family whose marked periods lie in \(D_P\) and approach the cusp \(\mathbb Qe\). A rational hyperbolic basis makes both their growth and their monodromy explicit. Choose \(f_0\in P\) with \(q(e,f_0)=1\) and put \(f=f_0-\frac12q(f_0)e\). Then \(q(f)=0\) and \(q(e,f)=1\). The orthogonal complement of \(\langle e,f\rangle\) in \(P\) is a positive rational line; choose a nonzero rational vector \(v\) on it and put \(a=q(v)>0\). This proves (28).

For every positive period in \(P_{\mathbb C}\) the pairing with \(e\) is nonzero. Indeed, a positive real two-plane orthogonal to \(e\) would put \(e\) in its negative-definite orthogonal complement. Normalize the period by \(q(p,e)=1\). Writing \(p=f+\tau v+\beta e\), its nullity gives \(\beta=-a\tau^2/2\). Positivity becomes \[q(p,\bar p)=a|\tau|^2-\frac a2(\tau^2+\bar\tau^2) =2a(\operatorname{Im}\tau)^2>0.\] Change the sign of \(v\) so that the component \(D_P\) is parametrized by \(\operatorname{Im}\tau>0\). The point \(\tau=\infty\) represents the line \(\mathbb Qe\).

For a rational number \(u\), define an isometry \(U_u\) of \(P\) by \[U_ue=e,\qquad U_uv=v-au e,\qquad U_uf=f+u v-\frac{au^2}{2}e.\] The Gram matrix in (28) verifies directly that \(U_u\) preserves \(q\), and substitution gives \[U_up(\tau)=p(\tau+u).\] Taking \(u\) to be a sufficiently divisible positive integer makes this transformation belong to \(\Gamma_P\). The cusp stabilizer in the torsion-free orientation-preserving arithmetic group is generated by such a translation \(\tau\mapsto\tau+w_0\), with \(w_0>0\). Here the width \(w_0\) is rational because the basis \(e,f,v\) is rational and the generator preserves that rational structure. The standard cusp coordinate is \[u_c=\exp(2\pi i\tau/w_0).\] If a local parameter \(t\) upstairs has ramification degree \(b\) at the cusp, then \[ u_c=t^b\beta(t),\qquad \beta(0)\ne0, \qquad \operatorname{Im}\tau =\frac{bw_0}{2\pi}(-\log|t|)+O(1). \tag{35}\] The error is uniform near the puncture. A positive local loop corresponds to \(U_{bw_0}\). The already established equality of monodromy representations gives (31) with \(w=bw_0\). Computing \((U_w-1)\) on the displayed basis gives (32). Further finite base changes replace \(w\) by a positive integral multiple, so all the image and nonvanishing statements survive.

Because only finitely many exceptional values were removed from \(\overline C_P\), a sufficiently small punctured cusp neighborhood contains none of them. In the preceding path construction we may therefore choose its final lift in \(D_P\) to be the ray \(\tau=\lambda+iy\) for any specified irrational \(\lambda\) and all large \(y\). The ray lifts to the chosen finite cover after its initial branch is fixed. It continues to lift after any further finite base change. The resulting path in the final parameter \(t\) may be nonradial.

Semistable reduction and the logarithmic form

Take a projective algebraic model over a neighborhood of \(b_e\) in \(\overline B\). One-parameter semistable reduction in characteristic zero, followed by restriction to a small disk, gives, after finite base change and a projective modification which is an isomorphism over the punctured base, a smooth total space \(\mathcal Y\) with reduced simple normal crossings central fiber; see (Kempf et al. 1973, II, Semi-stable Reduction Theorem, pp. 53–54). The smooth fibers and their marked periods remain those already constructed. The polarization on the punctured family extends to a line bundle on \(\mathcal Y\): choose a rational section, close its divisor in the smooth total space, and use that every Weil divisor on a smooth variety is Cartier. Denote the extension by \(\mathcal A\). Its restriction to each smooth fiber is ample; relative ampleness on the semistable model is unnecessary.

The invariant vector \(e\) is now a single-valued flat section of \(R^2\pi_*\mathbb Z\) on \(\Delta^*\). The pairing \(q(-,e)\) is a single-valued holomorphic functional on its associated flat vector bundle. Its restriction to the Hodge line is nowhere zero, by (30). Therefore it trivializes that Hodge line. Holomorphic base change identifies the Hodge line with the bundle of relative holomorphic two-forms, so there is a unique holomorphic relative form \(\sigma\) with \(q([\sigma],e)=1\). On the universal cover its class is exactly \(p(\tau)\), including its scalar. Thus \(\eta=\sigma^n\) is a single-valued holomorphic relative volume form on \(\Delta^*\).

Polarizing the Fujiki identity (Fujiki 1987), or simply comparing the coefficient of \(z^nw^n\) in its application to \(z[\sigma]+w[\bar\sigma]\), gives \[\binom{2n}{n}\int_{Y_t}\sigma_t^n\wedge\bar\sigma_t^n =c_X2^n q(p(\tau),\overline{p(\tau)})^n.\] For the positive volume convention in \(|\eta_t|^2\), any fixed normalization factor can be absorbed in a constant. Equations (30) and (35) consequently give \[\int_{Y_t}|\eta_t|^2 =c' (\operatorname{Im}\tau)^{2n}\asymp s^{2n}=s^d, \qquad c'>0,\] proving (33) uniformly near the puncture.

Let \(\mathcal K_{\log}\) denote the invertible sheaf \[\det\!\left( \Omega^1_{\mathcal Y}(\log Y_0) \big/\pi^*\Omega^1_{\Delta}(\log\{0\})\right).\] The quotient is locally free of rank \(d\) in the semistable coordinates. On \(\Delta^*\) its determinant is the ordinary relative canonical line, and \(\eta\) is a section of it there. We prove that \(\eta\) extends holomorphically as a section of \(\mathcal K_{\log}\).

First consider a point of \(Y_0\) lying on only one component. There the family has product coordinates \((z_1,\ldots,z_d,t)\). Write \[\eta_t=F(z,t)\,\mathrm dz_1\wedge\cdots\wedge\mathrm dz_d.\] On a smaller product chart the submean inequality in the fiber variables, followed by the global upper volume bound, gives \[|F(z,t)|^2\le C\int_{Y_t}|\eta_t|^2\le C' s^d.\] For \(m\ge1\) the negative Laurent coefficient in the \(t\) variable satisfies, uniformly on a smaller \(z\)-polydisk, \[\left|\frac1{2\pi i}\int_{|t|=\rho} F(z,t)t^{m-1}\,\mathrm dt\right| \le C\rho^m|\log\rho|^{d/2}\longrightarrow0.\] All negative Laurent coefficients vanish. Hence \(F\) extends holomorphically through \(t=0\). This extends \(\eta\) over the smooth locus of the reduced central divisor. The crossing locus has complex codimension at least two in the smooth total space. Local trivializations of \(\mathcal K_{\log}\) and Hartogs’ extension theorem then extend \(\eta\) uniquely over that locus. Thus the polynomial bound in \(s\) suffices for logarithmic extension.

Maximal growth forces a nonzero maximal residue

We have the semistable family and the extended logarithmic form. To apply Theorem 4, we must still find a maximal crossing with nonzero coefficient. The lower growth bound forces one, as follows. At a point where precisely \(k\) components of \(Y_0\) meet, take coordinates with \[t=z_0\cdots z_{k-1},\qquad 1\le k\le d+1.\] A local generator of \(\mathcal K_{\log}\) restricts to the fiber as \[\frac{\mathrm dz_1}{z_1}\wedge\cdots\wedge \frac{\mathrm dz_{k-1}}{z_{k-1}} \wedge\mathrm dz_k\wedge\cdots\wedge\mathrm dz_d.\] Its extended holomorphic coefficient is bounded on a smaller chart. In the variables \(x_i=-\log|z_i|\), \(1\le i\le k-1\), the fiber part of a fixed coordinate polydisk has an expanding simplex as its radial domain: each \(x_i\) is bounded below, and \(\sum_i x_i\le s+O(1)\). Its dimension is \(k-1\). Integrating the angular variables and the remaining ordinary coordinates therefore yields \[ \int_{Y_t\cap\text{chart}}|\eta_t|^2 \le C(1+s^{k-1}). \tag{36}\]

For \(k=d+1\), the crossing is an isolated point \(o\). Suppose its coefficient \(F\) vanishes at \(o\). To improve (36), choose a polydisk \(|z_i|<\delta<1\), put \(A=-\log\delta\), and write \[x_i=A+(s-(d+1)A)u_i,\qquad u_i>0,\quad \sum_{i=1}^d u_i<1.\] For almost every \(u\) in this fixed simplex, all \(d+1\) coordinates \(z_0,\ldots,z_d\) tend to zero as \(s\to\infty\), uniformly in their arguments. The rescaled integrand \(|F|^2\) thus tends to \(|F(o)|^2=0\) almost everywhere. It is uniformly bounded and the simplex and angle torus have finite measure. Dominated convergence gives \[ \int_{Y_t\cap\text{chart}}|\eta_t|^2=o(s^d) \quad\text{if }F(o)=0. \tag{37}\]

Properness gives finitely many such charts covering the central fiber and, after shrinking the base, all nearby fibers. If there were no maximal crossings, every chart would contribute \(O(s^{d-1})\). If maximal crossings existed but all their coefficients vanished, their contributions would be \(o(s^d)\) by (37), with all other contributions again \(O(s^{d-1})\). Either conclusion contradicts the lower bound in (33). At least one maximal crossing therefore has nonzero coefficient. This proves (34). At that point the functions \(z_1,\ldots,z_d\) may be chosen to be algebraic local defining functions of the corresponding components; the unit in the product equation is absorbed in \(z_0\). They are thus restrictions of rational functions on the algebraic total space, as required for the intersection estimates in Theorem 4. This completes the proof of Proposition 12.

The primitive class and the chosen fiber

Recall the primitive class \(e\) and line bundle \(E\) constructed at the start of Section 4. Write \(E_X=E\), so that \(L\simeq E_X^{\otimes m_0}\). The class \(e\) is nonzero, nef, and \(q(e)=0\); its sign was fixed so that, for every Kähler class \(\kappa\) on \(X\), \[ q(e,\kappa)>0. \tag{38}\]

Fix \(\lambda\in\mathbb R\setminus\mathbb Q\) and use the family of Proposition 12. Its rational classes \(e,f,v,h\) satisfy \[q(e)=q(f)=0,\quad q(e,f)=1,\quad q(e,v)=q(f,v)=0,\quad a:=q(v)>0,\] and \(h\perp P:=\langle e,f,v\rangle_{\mathbb Q}\). The normalized period and the local monodromy \(M\) on \(H^2\) are \[ p(\tau)=f+\tau v-\frac a2\tau^2e, \qquad\tau=\lambda+iy,\quad y>0, \tag{39}\] \[ Me=e,\qquad Mv=v-aw e,\qquad Mf=f+w v-\frac{aw^2}{2}e,\quad w\ne0. \tag{40}\] Consequently, \[ (M-1)P=\langle e,v\rangle_{\mathbb Q}. \tag{41}\] Theorem 4, applied along the prescribed vertical ray, provides a smooth fiber \(Y=Y_t\), its polarized Ricci-flat metric \(g\), and an embedded special Lagrangian torus \(T\subset Y\). Here \(\dim_{\mathbb C}Y=2n\), \(\dim_{\mathbb R}T=2n\), and \(T\) is isotopic to a coordinate torus in a semistable chart \(z_0z_1\cdots z_{2n}=t\). A constant rescaling of \(g\), as used in the torus construction, does not affect these properties.

Write the present complex structure as \(I\). Choose a hyperkähler triple \((I,J,K)\) for \(g\), with \(IJ=K\), so that \[ \omega_J+i\omega_K=c\sigma_t,\qquad c>0, \qquad [\omega_I]\in\mathbb R_{>0}h. \tag{42}\] The positive scalar normalizes the length of \(\sigma_t\); its given phase fixes the ordered pair \((J,K)\). Since \([\sigma_t]=p(\tau)\), \[ [\omega_K]=cy(v-a\lambda e),\qquad q(e,[\omega_J])=c>0. \tag{43}\] The second equality follows from \(q(e,p(\tau))=1\) and will retain the required sign of \(e\) after rotation.

Restriction, calibration, and complex rotation

The following criterion isolates the rotation argument. Cohomological vanishing is enough because the special Lagrangian condition turns volume into a period.

We verify the criterion for the torus constructed above, using its coordinate isotopy class. This is where the monodromy information is needed.

For a coordinate torus \(T_0=(\mathbb R/2\pi\mathbb Z)^{2n}\) choose its fixed radii \(r_1,\ldots,r_{2n}\). Around the circle \(t(\theta)=t\exp(i\theta)\), the formula \[z_j=r_j\exp(iy_j)\quad(1\le j\le2n),\qquad z_0=\frac{t\exp(i\theta)}{z_1\cdots z_{2n}}\] defines a family of embeddings of this same parameterized \(T_0\). All absolute values remain fixed, so the family stays inside the chart, and the embeddings at \(\theta=0\) and \(2\pi\) agree as parameterized maps. The induced restriction maps form a morphism from the fiber cohomology local system to the constant local system \(H^k(T_0,\mathbb R)\). Therefore, in every degree \(k\), \[ i_0^*M_k=i_0^*:H^k(Y,\mathbb R)\longrightarrow H^k(T_0,\mathbb R), \tag{45}\] where \(M_k\) is monodromy in degree \(k\). This statement controls the restriction of classes in degree two, as well as the torus’s top-degree period. Isotopy identifies the restriction maps for \(T_0\) and \(T\). Thus, for \(i:T\hookrightarrow Y\), \(i^*(M-1)=0\) on \(H^2(Y,\mathbb R)\). Equations (41) and (43) give \[ i^*[\omega_K]=0. \tag{46}\] Replacing monodromy by its inverse or a positive power preserves the image in (41), so the conclusion is independent of this convention and of the finite base changes.

Lemma 15 now makes \(T\) a holomorphic Lagrangian complex torus on the rotated manifold \(Y'=(Y,J)\).

The rational isotropic ray after rotation

The period plane of \(Y'\) is \[ \langle[\omega_K],[\omega_I]\rangle_{\mathbb R} =\langle v-a\lambda e,h\rangle_{\mathbb R}. \tag{47}\] It is perpendicular to \(e\), so \(e\) is an integral \((1,1)\) class on \(Y'\); see Figure 1.

Let \(x\in H^{1,1}(Y')\cap H^2(Y',\mathbb Q)\). Orthogonality to the period plane says \[q(x,h)=0,\qquad q(x,v)=a\lambda q(x,e).\] The three numbers \(q(x,v)\), \(a\), and \(q(x,e)\) are rational, whereas \(a\ne0\) and \(\lambda\notin\mathbb Q\). Therefore \[ q(x,e)=q(x,v)=0. \tag{48}\]

The restriction of \(q\) to \(H^{1,1}(Y',\mathbb R)\) has signature \((1,b_2-3)\). In a Lorentzian space, the orthogonal complement of a nonzero null vector is negative semidefinite with radical its own line. Applying this to \(e\) and (48) proves \[ \begin{gathered} q(x)\le0\quad\text{for }x\in H^{1,1}(Y')\cap H^2(Y',\mathbb Q),\\ q(x)=0\quad\Longrightarrow\quad x\in\mathbb Qe. \end{gathered} \tag{49}\] In particular, \(Y'\) is nonprojective: a projective hyperkähler manifold has an integral ample class of positive BBF square. This is the necessary direction of the projectivity criterion (Huybrechts 1999, 2003). The rational Picard space may contain negative-square classes; its isotropic classes are exactly those on \(\mathbb Qe\). Equation (43) places \(e\) in the closure of the positive cone containing the Kähler class \([\omega_J]\).

A semiample point and deformation back to \(X\)

We state explicitly the two external results used in this step.

The projectivity assertion follows from the line-bundle construction of the algebraic reduction in (Greb et al. 2013, sec. 4.2) and Stein factorization.

The fibration defined by a generated power

We derive the fibration description from semiampleness, using the Fujiki argument for Lagrangian fibers of Matsushita (Matsushita 1999, 2001). Let \(m>0\) make \(L^{\otimes m}\) globally generated, let \[g:X\longrightarrow Z\subset\mathbb P H^0(X,L^{\otimes m})^*\] be its morphism to the image, and take its Stein factorization \(g=\nu\circ f\): \[X\xrightarrow{\ f\ }B\xrightarrow{\ \nu\ }Z.\] The space \(B\) is normal, \(f\) has connected fibers, and \(\nu\) is finite. Since \(Z\) is projective, \(B\) is projective and \(H:=\nu^*\mathcal O_Z(1)\) is ample. By construction \[ L^{\otimes m}\simeq f^*H. \tag{51}\]

Put \(\ell=c_1(L)\) and choose a Kähler class \(\kappa\) on \(X\). The positive sign gives \(q(\ell,\kappa)>0\). Fujiki’s formula and \(q(\ell)=0\) give the polynomial identity \[ \int_X(\kappa+t\ell)^{2n} =c_X\bigl(q(\kappa)+2tq(\kappa,\ell)\bigr)^n, \qquad c_X>0. \tag{52}\] Comparison of coefficients shows \[\int_X\ell^k\kappa^{2n-k} \begin{cases} >0,&0\le k\le n,\\ =0,&n<k\le2n. \end{cases}\] On the other hand, a pullback of the Fubini–Study form by \(g\) represents \(m\ell\), is semipositive, and has generic rank \(\dim Z=\dim B\). Its \(k\)th power wedged with a Kähler form has positive integral precisely for \(k\le\dim B\). Thus \[ \dim_{\mathbb C}B=n. \tag{53}\]

Let \(F\) be a general smooth fiber. Its Poincaré dual class is a positive multiple of \(\ell^n\). Indeed, choose a very ample power \(H^{\otimes a_0}\) and intersect \(n\) general members of its linear system on \(B\). Their intersection is a nonempty finite set of transverse points in the smooth regular-value locus of \(f\). Pulling back gives the union of the corresponding smooth fibers and represents \((a_0m)^n\ell^n\). These fibers have the same homology class, by smooth proper transport along the connected regular-value locus. This proves the claim.

For \(n\ge2\), polarization of Fujiki’s relation gives \[ \int_X\ell^n\kappa^{n-2}\sigma\overline\sigma=0. \tag{54}\] To see the vanishing directly, each summand in the polarized formula is a product of pairings of the \(2n\) degree-two factors. Every copy of \(\ell\) is orthogonal to \(\ell\), \(\sigma\), and \(\overline\sigma\), so it would have to pair with a copy of \(\kappa\). There are \(n\) copies of \(\ell\) and only \(n-2\) copies of \(\kappa\), and therefore every summand is zero. Using the fiber class, (54) gives \[\int_F\sigma|_F\wedge\overline{\sigma|_F} \wedge\kappa|_F^{\,n-2}=0.\] In unitary coordinates this integrand is a fixed positive constant times the squared pointwise norm of \(\sigma|_F\) times the Kähler volume form. Hence \(\sigma|_F=0\). For \(n=1\), the restriction of a holomorphic two-form to the curve \(F\) vanishes automatically. The fibers in question have complex dimension \(n\) by (53), so they are Lagrangian.

The same assertion holds for every irreducible component of every fiber by Matsushita’s equidimensionality theorem (Matsushita 2000, Theorem 1). Its hypotheses are exactly those now established: \(X\) is Kähler, \(f\) is proper and surjective with normal base, and a general smooth fiber is Lagrangian. The holomorphic symplectic form is closed, as every holomorphic form on a compact Kähler manifold is. Thus every fiber component has dimension \(n\) and is isotropic on its smooth locus. This also gives the Lagrangian fibration used in Section 6.

Conversely, an ample bundle has a globally generated positive power, so (51) implies semiampleness of \(L\). This proves the equivalence between semiampleness and the fibration description in Theorem 1.

Projective-space bases

We finish with the additional conclusion for projective \(X\) in Corollary 3.