The strong hyperkähler SYZ conjecture

1. Introduction

A compact irreducible holomorphic symplectic Kähler manifold is a simply connected compact Kähler manifold X such that H^0(X,\Omega_X^2)=\C\sigma for a holomorphic two-form \sigma which is nondegenerate everywhere. Its complex dimension is even, say 2n. We write q=q_X for the Beauville–Bogomolov–Fujiki quadratic form, and also for its associated symmetric bilinear form. We denote the Kähler cone by \mathcal K_X. A line bundle L is nef if c_1(L) lies in the closure of the Kähler cone, isotropic if q(c_1(L))=0, and semiample if some positive tensor power is generated by its global holomorphic sections.

A nonzero nef isotropic class has numerical dimension n, by Fujiki’s relation: its powers have positive intersection with a Kähler class through degree n, and zero intersection in higher degrees. The strong hyperkähler SYZ conjecture asks whether every line bundle with such a class is semiample [43, Conjecture 1.7]. This would turn a class on the boundary of the Kähler cone into a holomorphic map with half-dimensional fibers. The numerical condition by itself supplies neither sections nor control of their base locus.

Theorem 1.1 resolves the strong hyperkähler SYZ conjecture positively. The conclusion holds on the given manifold X: a positive tensor power of L has no base points anywhere. No restriction on the dimension, deformation type, or second Betti number is imposed.

There is also a consequence for the number of possible smooth manifolds. Here a deformation type means an equivalence class under smooth proper deformations through compact irreducible holomorphic symplectic Kähler manifolds.

This follows by combining Theorem 1.1 with the boundedness of Lagrangian-fibered symplectic varieties proved by Engel–Filipazzi–Greer–Mauri–Svaldi [11]. Section 6 gives the deduction in the smooth category, including the role of the hypothesis b_2\ge5.

This is a consequence of the projective-space base theorem in the companion [36, Theorem 1.1 and the following paragraph on codimension-one multiple fibers]. Its application is given after the proof of Theorem 1.1, in Section 5.6.

1.1. Context

The fibration picture is related to the special Lagrangian formulation of mirror symmetry proposed by Strominger–Yau–Zaslow [41]. 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 [41, Section 4.1]. Early explicit nef-isotropic formulations appear in Hassett–Tschinkel [18, Remark 3.12] and Sawon [37, 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 [30, 32]; his equidimensionality theorem also controls every component of every fiber [31, 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 [43, 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 [19, 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 [40, Theorem 3.7(v)]. For the known deformation types, earlier constructions and birational results were obtained by Bayer–Macrì [3, Theorem 1.5] and Markman [29, Theorem 1.3 and Remark 1.8] for \mathrm{K3}^{[n]}-type, by Yoshioka [47, Proposition 3.38] for generalized Kummer type, and by Mongardi–Rapagnetta [35, Theorem 7.2] and Mongardi–Onorati [34, 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 [40, Section 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 [15, 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 [42, Section 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 [48, Section 4], using McLean’s deformation operator [33]. Li’s Fermat-family theorem gives special Lagrangian torus fibrations on generic regions whose relative volumes tend to one along a subsequence [24, Theorem 1.1]; [25] surveys the metric SYZ framework. For polarized algebraic maximal degenerations of compact Calabi–Yau manifolds, Blum–Liu’s valuative independence theorem [4, Theorem 1.1] and Li’s comparison theorems [26, 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.

1.2. 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. (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 \Q e. We realize that curve by an algebraic family and track its full second-cohomology monodromy through semistable reduction (Proposition 4.1).

2. (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 3.4). A small special Lagrangian perturbation retains its coordinate isotopy class (Theorem 2.1). Here special Lagrangian means that the Kähler form vanishes on the torus and the holomorphic volume form has constant phase there.

3. (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,\R). The chosen periods put [\omega_K] in that image. A cohomological rotation criterion (Lemma 5.1) 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 \Q e. Thus the Greb–Lehn–Rollenske fibration makes e semiample on Y'.

4. (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.

1.3. Organization and conventions

Sections 2 and 3 prove the torus theorem independently of hyperkähler period theory. Section 4 constructs the prescribed degeneration; Lemma 4.2 separates the underlying polarized family and free period quotient from the cusp calculation. Section 5 rotates the torus and proves Theorem 1.1; Section 6 deduces Corollary 1.2.

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

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.

2. Metric estimates at the torus scale

This section and the next construct a special Lagrangian torus near a maximal crossing of a polarized semistable degeneration. The coordinate isotopy class is essential: an angular coordinate torus is parameterized consistently around the puncture, so restriction to it is invariant under monodromy. Section 5 uses this identity in degree two. The construction applies to Calabi–Yau manifolds independently of the hyperkähler setting. We use d for the complex dimension and N=2d for the real dimension.

We begin with an arbitrary sequence of nonzero parameters tending to zero. The compactness argument will produce a subsequence on which every sufficiently late fiber contains the required torus. At the end of Section 3, a sequence of counterexamples will rule out failure anywhere in a sufficiently small punctured disk.

This section chooses centers with noncollapsed unit balls and controls the metric and logarithmic coordinates on every ball about them. Section 3 identifies the pointed limit as a smooth flat cylinder and perturbs its angular torus. All constants below are independent of t in a sufficiently small punctured disk, unless stated otherwise. Every extraction is from the arbitrary initial sequence.

2.1. 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 [46], with [\omega_t^{(0)}]=a_t. The useful scale is

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

Thus x_i+iy_i are local holomorphic logarithms, and \gamma is the standard flat metric on \R^d\times(\R/2\pi\Z)^d. Its angular circles have fixed length; we will compare g_t with this metric. Let

For a relatively compact subset B\Subset\Sigma, write

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 (2.2) gives |\eta_t|^2=|f(z)|^2\dd V_\gamma in the chart. Ricci-flatness implies that its volume form is a constant multiple of |\eta_t|^2; integration determines that constant. Hence

after a subsequence. To obtain the convergence, first extract a limit of c_a s^d/I_t using (2.1), and then use f(z)\to f(o)\ne0. In particular, the two volume measures are uniformly comparable on each fixed interior tube.

2.2. Global energies and many noncollapsed centers

The two trace bounds have different roles. The bound for \tr_{g_t}\gamma controls logarithmic functions on intrinsic balls; the bound for \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:

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

It follows that H_t\le C\tr_{g_t}\gamma on the support and that

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

The Ricci-flat Kähler Chern–Weil identity expresses \int_{Y_t}|\Rm_{g_t^{(0)}}|^2\dd V_{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

This also covers d=2, when the curvature integral in the unscaled metric is purely topological.

2.3. 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

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

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 (2.13), first for H_t and then for K_t, shows that

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

After a further subsequence,

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 [6], states that

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

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

The cutoff functions now locate each fixed intrinsic ball in the crossing chart. Divide (2.15) 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

Chebyshev’s inequality consequently gives, for every R,\eta>0,

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.

2.4. 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 (2.14) and \Ric(g_t)=0 give, after extraction, a pointed measured Gromov–Hausdorff limit

We use the following precise consequences of noncollapsed Ricci limit theory and Einstein regularity [7, 10, 9]. 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

The codimension assertion is Theorem 1.4 of [9]; the smooth convergence follows from Einstein \epsilon-regularity [9, 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 (2.14) 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

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 (2.14), so Lemma 2.4 applies. Every local constant limit is \Phi(o). Indeed a fixed smaller regular ball has positive limiting volume, whereas the exceptional set in (2.16) 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

Since (\Phi,\pi) is an embedding, there is a neighborhood of (\Phi(o),0) whose inverse image lies in the coordinate neighborhood (2.2). 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

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 (2.16) 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

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 (2.14) and the centered oscillation estimate (2.15) 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.

3. A flat cylinder and a special Lagrangian torus

We continue the proof of Theorem 2.1. All limits are taken along the subsequence selected in the preceding section. We write \mu for the limit Riemannian volume and

The standard flat metric on H is again denoted by \gamma. Our goal is to identify Z with this cylinder, obtain smooth convergence in the actual logarithmic coordinates, and perturb an angular torus. The possible singular set S=Z\setminus G must be handled first: flatness on G alone does not give convergence over an entire torus.

3.1. 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

are harmonic on these subsets. Their local L^2 bounds, from (2.15), and interior elliptic estimates give smooth subsequential limits u_j. The angular coordinates have values in the compact circles \R/(2\pi\Z) and satisfy \dd\arg z_j=-\dd x_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

is holomorphic. Diagonal extraction here and below is over a countable exhaustion of the smooth manifold G.

The coordinate density in (2.4) 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,

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 [14, Proposition 3.30]. For finite N>2, Lott–Villani’s [28, 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 [14, Section 4.2], while [14, 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 [12, 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.

3.2. 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.

3.3. 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:

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 \|\dd\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 3.2 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 \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

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.

3.4. A special Lagrangian torus in the nearby fibers

Fix a tube Q_{2r} and use the charts of Lemma 3.4 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

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 [48, Section 4, Lemmas 4.3–4.5]. Its linearization is McLean’s special Lagrangian deformation operator [33]. 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

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

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

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

The Hodge identities give \dd\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 3.4 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 (3.5) 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 2.1.

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 2.1.

4. A polarized degeneration with the prescribed isotropic class

The degeneration must supply two kinds of data. Its volume growth and a nonzero residue at a maximal crossing will allow us to apply Theorem 2.1. Its marked periods and second-cohomology monodromy will retain the prescribed isotropic class for rotation. We construct both in the same polarized family. All markings take values in

with its integral Beauville–Bogomolov–Fujiki form q. We write q(x,y) for the associated symmetric bilinear form, so that q(x)=q(x,x). Markings are obtained by deformation and parallel transport from the differentiable manifold underlying X. Fix the connected Teichmüller component containing X. Along deformation paths use Ehresmann trivializations anchored at X; the transported complex structures on its differentiable manifold then stay in this component.

The universal coefficient theorem and H_1(X,\Z)=0 show that \Lambda is torsion-free. Write c_1(L)=m_0e with m_0\in\Z_{>0} and e\in\Lambda primitive. Since H^1(X,\mathcal O_X)=0 by Hodge theory, the exponential sequence identifies \Pic(X) with the integral (1,1) classes. Thus e=c_1(E) for a unique line bundle E, and L\simeq E^{\otimes m_0}. The class e is nonzero, nef, and isotropic. It will remain the same labelled lattice vector, although it need not remain of type (1,1) during the construction. Its sign is fixed by

for every Kähler class \kappa_X on X: in the Lorentzian space H^{1,1}(X,\R), a nonzero null vector in the closure of the positive cone pairs positively with every vector in that cone.

The prescribed line \Q e and the normalization q(p,e)=1 are the data needed for rotation. We obtain them in three stages. First, choose a positive polarization perpendicular to e and realize an open set of its periods by an algebraic family. Next, find a rational three-dimensional period subspace containing e and follow its cusp inside that family, retaining the original marking. Finally, compute the periods at the cusp and use their growth to produce the nonzero logarithmic residue required by Theorem 2.1.

4.1. 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 [21, 22, 44, 45]. The corrected projectivity criterion is [22, Theorem 2], and the Kähler-cone description in terms of negative MBM classes is due to Amerik–Verbitsky [1, Theorems 1.19 and 6.2]; see also [40, Section 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 4.2 to the primitive class fixed at the start of this section. Its open marked period image is the input for the next step.

4.2. A rational ternary domain containing the fixed vector

We now find a rational period subspace which meets U and has \Q e as a cusp. View D_h as the space of positive oriented real two-planes in V_{\R}. Let W\in U. Write e=e_W+e_- in the orthogonal decomposition V_{\R}=W\oplus W^\perp. Since W^\perp is negative definite, e_W\ne0; otherwise the nonzero vector e would have negative square. Moreover

It follows that W+\R e=W\oplus\R e_- 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_{\R} is positive, has the chosen orientation, and lies in U. Nondegeneracy and signature are open conditions on the Gram matrix, so

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

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\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

is a smooth arithmetic curve. Its holomorphic map j:C_P\to Q_h is algebraic by Borel’s theorem [5, Theorem 3.10]. Since P contains the rational null line \Q e, 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.

4.3. 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

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

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

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.

4.4. The exact period and monodromy formulas

We have obtained a family whose marked periods lie in D_P and approach the cusp \Q e. 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 (4.1).

For every positive period in P_{\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

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 \Q e.

For a rational number u, define an isometry U_u of P by

The Gram matrix in (4.1) verifies directly that U_u preserves q, and substitution gives

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

If a local parameter t upstairs has ramification degree b at the cusp, then

The error is uniform near the puncture. A positive local loop corresponds to U_{bw_0}. The already established equality of monodromy representations gives (4.4) with w=bw_0. Computing (U_w-1) on the displayed basis gives (4.5). 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.

4.5. 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 [23, Chapter 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_*\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 (4.3). 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 [13], or simply comparing the coefficient of z^nw^n in its application to z[\sigma]+w[\bar\sigma], gives

For the positive volume convention in |\eta_t|^2, any fixed normalization factor can be absorbed in a constant. Equations (4.3) and (4.8) consequently give

proving (4.6) uniformly near the puncture.

Let \mathcal K_{\log} denote the invertible sheaf

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

On a smaller product chart the submean inequality in the fiber variables, followed by the global upper volume bound, gives

For m\ge1 the negative Laurent coefficient in the t variable satisfies, uniformly on a smaller z-polydisk,

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.

4.6. Maximal growth forces a nonzero maximal residue

We have the semistable family and the extended logarithmic form. To apply Theorem 2.1, 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

A local generator of \mathcal K_{\log} restricts to the fiber as

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

For k=d+1, the crossing is an isolated point o. Suppose its coefficient F vanishes at o. To improve (4.9), choose a polydisk |z_i|<\delta<1, put A=-\log\delta, and write

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

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 (4.10), with all other contributions again O(s^{d-1}). Either conclusion contradicts the lower bound in (4.6). At least one maximal crossing therefore has nonzero coefficient. This proves (4.7). 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 2.1. This completes the proof of Proposition 4.1.

5. Rotation and return to the original manifold

We now complete the proof of Theorem 1.1. The marking in Proposition 4.1 will be used throughout: all second cohomology groups are identified with the lattice of the differentiable manifold underlying the given X, and q denotes both the quadratic form and its associated symmetric bilinear form.

5.1. 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,

Fix \lambda\in\R\setminus\Q and use the family of Proposition 4.1. Its rational classes e,f,v,h satisfy

and h\perp P:=\langle e,f,v\rangle_{\Q}. The normalized period and the local monodromy M on H^2 are

Consequently,

Theorem 2.1, 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_{\C}Y=2n, \dim_{\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

The positive scalar normalizes the length of \sigma_t; its given phase fixes the ordered pair (J,K). Since [\sigma_t]=p(\tau),

The second equality follows from q(e,p(\tau))=1 and will retain the required sign of e after rotation.

5.2. 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=(\R/2\pi\Z)^{2n} choose its fixed radii r_1,\ldots,r_{2n}. Around the circle t(\theta)=t\exp(i\theta), the formula

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,\R). Therefore, in every degree k,

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,\R). Equations (5.4) and (5.6) give

Replacing monodromy by its inverse or a positive power preserves the image in (5.4), so the conclusion is independent of this convention and of the finite base changes.

Lemma 5.1 now makes T a holomorphic Lagrangian complex torus on the rotated manifold Y'=(Y,J).

5.3. The rational isotropic ray after rotation

The period plane of Y' is

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',\Q). Orthogonality to the period plane says

The three numbers q(x,v), a, and q(x,e) are rational, whereas a\ne0 and \lambda\notin\Q. Therefore

The restriction of q to H^{1,1}(Y',\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 (5.11) proves

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 [21, 22]. The rational Picard space may contain negative-square classes; its isotropic classes are exactly those on \Q e. Equation (5.6) places e in the closure of the positive cone containing the Kähler class [\omega_J].

5.4. 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 [15, Section 4.2] and Stein factorization.

The fixed-class deformation theorem.. Let N be a compact simply connected differentiable manifold of hyperkähler type with maximal holonomy, let \mathcal T^\circ(N) be one connected component of its Teichmüller space, and fix a nonzero primitive integral class e with q(e)=0. Write \mathcal C_I for the component of the positive cone containing the Kähler cone of (N,I), and set

This is the positive-sign locus of [40, Definition 2.6]. By [40, Proposition 2.7] it is connected. Let E_I denote the unique holomorphic bundle with class e. Define \mathcal T^\circ_{e,\mathrm{nef}} and \mathcal T^\circ_{e,\mathrm{sa}} by requiring E_I to be nef and semiample, respectively.

Its family formulation, [40, Theorem 3.8], says that a nef isotropic bundle is semiample when its pair deforms, in a smooth connected family carrying a line bundle, to a semiample pair. We use (5.13) directly.

Apply Theorem 5.2 to (Y',T). Choose an ample bundle H on its fibration base and let A be its pullback to Y'. The bundle A is semiample and nef, and its first Chern class b is nonzero: a semipositive representative of a generated power has positive trace on an open set where the map has positive rank. Since the base has dimension n, \int_{Y'}b^{2n}=0. Fujiki’s relation [13] gives q(b)=0. Equation (5.12) and the primitivity of e now give

Both b and e pair positively with [\omega_J]: for b this is the nonzero nef null-vector property, and for e it is (5.6). Hence k>0.

Let E_{Y'} be the unique bundle on Y' with first Chern class e. Injectivity of the first Chern class map gives

A globally generated positive power of A is therefore a globally generated positive power of E_{Y'}. Thus the specified primitive class e is semiample on Y'.

The marking furnished by Proposition 4.1 identifies Y with a point of the same ambient component \mathcal T^\circ(N) as the original X. The hyperkähler twistor sphere connects I to J on that same marked differentiable manifold, so Y' also belongs to this component. On X, e is of type (1,1) and has the positive sign by (5.1). On Y' the corresponding statements follow from (5.10) and (5.6). Consequently both endpoints lie in the same positive fixed-class locus \mathcal T^\circ_e. The semiample point Y' makes its semiample locus nonempty; the original point X belongs to its nef locus by hypothesis. Equation (5.13) now proves that E_X is semiample on X.

Choose s>0 such that E_X^{\otimes s} is globally generated. Then L^{\otimes s}\simeq(E_X^{\otimes s})^{\otimes m_0} is globally generated as well. In particular, on the original manifold the evaluation morphism

is surjective everywhere. This proves the semiampleness assertion of Theorem 1.1.

5.5. The fibration defined by a generated power

We derive the fibration description from semiampleness, using the Fujiki argument for Lagrangian fibers of Matsushita [30, 32]. Let m>0 make L^{\otimes m} globally generated, let

be its morphism to the image, and take its Stein factorization g=\nu\circ f:

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

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

Comparison of coefficients shows

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 kth power wedged with a Kähler form has positive integral precisely for k\le\dim B. Thus

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

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, (5.17) gives

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 (5.16), so they are Lagrangian.

The same assertion holds for every irreducible component of every fiber by Matsushita’s equidimensionality theorem [31, 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 (5.14) implies semiampleness of L. This proves the equivalence between semiampleness and the fibration description in Theorem 1.1.

5.6. Projective-space bases

We finish with the additional conclusion for projective X in Corollary 1.3.

6. Finiteness of smooth deformation types

We now deduce Corollary 1.2. The reduction from an arbitrary manifold with b_2\ge5 to one with a nef isotropic class is the argument of Sawon [37, Proposition 4.3]; we give it here to show precisely where the second-Betti-number hypothesis enters. Here deformation equivalence is generated by smooth proper families of compact irreducible holomorphic symplectic Kähler manifolds over connected complex bases. The dimension and second Betti number are constant in such a family.

We use the following smooth consequence of Engel–Filipazzi–Greer–Mauri–Svaldi [11, Theorem B and Corollary A.3]: in fixed dimension, compact irreducible holomorphic symplectic Kähler manifolds admitting a Lagrangian fibration have only finitely many smooth deformation types. Indeed, their Corollary A.3 deforms such a fibration to a projective one, and Theorem B gives finiteness for the projective representatives. The deformations in these results are locally trivial, meaning that each point has an analytic neighborhood which deforms as a product. They therefore preserve smoothness and give ordinary smooth deformations for the manifolds considered here.

The restriction b_2\ge5 enters only through Meyer’s theorem. Theorem 1.1 itself has no such restriction, since its isotropic class is part of the hypothesis.