Exercise: The main theorem. What does the strong hyperkähler SYZ conjecture assert, and what is proven here?
For a compact IHS Kähler manifold X of complex dimension 2n with n ≥ 1, the strong hyperkähler SYZ conjecture says: if a holomorphic line bundle L has first Chern class c_1(L) that is nonzero, nef (lies in the closure of the Kähler cone of X), and isotropic (q(c_1(L)) = 0), then L is semiample, meaning some positive tensor power L^{⊗m} is globally generated. The paper proves it in full (Theorem (Main)). More precisely, there exist a positive integer m, a normal projective variety B of dimension n, an ample line bundle A on B, and a surjective holomorphic map f: X → B with connected fibers such that L^m ≅ f^*A; every irreducible component of every fiber is n-dimensional, and the holomorphic 2-form σ vanishes on its smooth locus. This turns a class on the boundary of the Kähler cone into a genuine fibration half-dimensional in the fibers. Citation: Verbitsky 2010, Conjecture 1.7 (Hyperkähler SYZ conjecture and semipositive line bundles), https://arxiv.org/abs/0811.0639v2.

Exercise: IHS Kähler manifold. What is one, and why the name?
An IHS Kähler manifold is a simply connected compact Kähler manifold X with H^0(X, Ω^2_X) = ℂσ for a holomorphic 2-form σ that is nondegenerate everywhere. "Irreducible holomorphic symplectic" is the standard name. Complex dimension is even, write 2n. Examples: K3 surfaces (n = 1); higher-dimensional analogues from moduli spaces of stable sheaves on K3 surfaces (K3^[n]-type) and abelian surfaces (generalized Kummer type); and two exotic examples by O'Grady (n = 3 and n = 5). "Irreducible" means σ generates the whole cohomology in a specific algebraic sense, ruling out products of lower-dimensional pieces. Big theorem behind much of the subject: every deformation of such an X is again IHS (Huybrechts 1999, https://arxiv.org/abs/alg-geom/9705025; erratum 2003, https://arxiv.org/abs/math/0106014).

Exercise: The BBF form q. What is the Beauville–Bogomolov–Fujiki form, and why is it important?
For any IHS manifold X there is a canonical quadratic form q: H^2(X, ℝ) → ℝ, called the Beauville–Bogomolov–Fujiki form (BBF). Reference: Beauville 1983 (Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom.); Bogomolov 1996 for the K3 case. q is integral on H^2(X, ℤ) up to a fixed positive rescaling; nondegenerate; and has signature (3, b_2(X) − 3). The +3 directions are essentially the underlying σ + σ̄ + Kähler plane; the −(b_2 − 3) directions drive negativity of the Kähler cone picture. We also write q(u, v) for the associated symmetric bilinear form, and q(u) = q(u, u). We say u is isotropic (or "square-zero") when q(u) = 0, positive when q(u) > 0, and negative when q(u) < 0. The Kähler cone of X is described in terms of q: the classes with q(u) > 0 form a cone with one positive sheet, and inside it, the Kähler cone is cut out by hyperplanes orthogonal to "MBM classes" (Amerik–Verbitsky 2015, Theorems 1.19 and 6.2, https://arxiv.org/abs/1401.0479).

Exercise: Nef, isotropic, semiample. Define each in one line.
Nef: c_1(L) lies in the closure of the Kähler cone of X; equivalently, L carries a smooth Hermitian metric whose curvature form is semipositive. Isotropic: q(c_1(L)) = 0. Semiample: some positive tensor power L^{⊗m} is globally generated, i.e. the evaluation map H^0(X, L^{⊗m}) → L^{⊗m}_x is surjective at every x ∈ X.

Exercise: The three corollaries. What extra conclusions does the paper draw?
Corollary (Finiteness): for each fixed n ≥ 1, there are only finitely many smooth deformation types of compact IHS Kähler manifolds of complex dimension 2n with b_2 ≥ 5. (Two manifolds are "the same smooth deformation type" if they live in a connected smooth proper family of IHS manifolds.) Corollary (Projective Case): if in Theorem (Main) the manifold X is additionally projective, then the base B is ℙ^n_ℂ, and there do not exist a prime divisor D on B, an effective divisor E on X, and an integer r > 1 with f^*D = rE -- i.e. no multiple fibers over codimension one. The second corollary is proved in Section "Projective-space bases"; it uses a companion paper (OpenAI 2026, see the link in the header). The finiteness corollary uses Theorem (Main) combined with a boundedness theorem from the minimal model program (Engel–Filipazzi–Greer–Mauri–Svaldi 2025, https://arxiv.org/abs/2507.00973v1).

Exercise: The Fujiki relation. Why does a nonzero nef isotropic class have numerical dimension exactly n?
Fujiki's relation (Fujiki 1987, https://doi.org/10.2969/aspm/01010105) says: for a compact IHS X of complex dimension 2n there is a positive constant c_X such that for every cohomology class a in H^2(X, ℝ),
    ∫_X a^{2n} = c_X · q(a)^n.
Equivalently, in the cohomology ring of X, "top self-intersections are q to the n". Take a nef L with q(ℓ) = 0, ℓ = c_1(L), and a Kähler class κ. Expand (κ + tℓ)^{2n} using Fujiki: the polynomial in t is c_X · (q(κ) + 2 t q(κ, ℓ))^n. Comparing the coefficient of t^k gives ∫_X ℓ^k κ^{2n − k} = 0 for k > n and a positive multiple of q(κ, ℓ)^k q(κ)^{n − k} for k ≤ n. Since q(κ, ℓ) > 0 and q(κ) > 0 (both standard for a Kähler class κ and a nonzero nef isotropic ℓ), those coefficients are positive precisely for 0 ≤ k ≤ n, and zero for n < k ≤ 2n. That is exactly the definition of "numerical dimension n". So a nonzero nef isotropic class on a 2n-fold has numerical dimension n; this number is used everywhere below.

Exercise: Historical context. Who proved what before, and what was still open at the start of this paper?
The fibration picture is tied to mirror symmetry via the Strominger–Yau–Zaslow conjecture (Strominger–Yau–Zaslow 1996, https://arxiv.org/abs/hep-th/9606040v2): a Calabi–Yau manifold should admit a special Lagrangian torus fibration with half-dimensional fibers. The hyperkähler analogue says: any nonzero nef isotropic class on an IHS manifold should be semiample, and the resulting map should be a Lagrangian fibration. Key prior pieces. Matsushita (Topology 1999, https://arxiv.org/abs/alg-geom/9709033; addendum 2001, https://arxiv.org/abs/math/9903045; Math. Res. Lett. 2000, https://doi.org/10.4310/MRL.2000.v7.n4.a4) proved that any fibration of a projective IHS manifold is Lagrangian with half-dimensional fibers and that every fiber component is n-dimensional. Hassett–Tschinkel (2001, Remark 3.12, https://arxiv.org/abs/math/9910021v2) and Sawon (2003, Conjecture 4.1, https://arxiv.org/abs/math/0404362v1) formulated the semiampleness question explicitly. Verbitsky (2010, Theorem 1.9, https://arxiv.org/abs/0811.0639v2) proved: a nontrivial nef isotropic bundle with a smooth semipositive Hermitian metric already has a section in some power. Horing–Lazic–Lehn (2025, https://arxiv.org/abs/2508.14634v2) got semiampleness in dimension four under an extra Lelong-number condition. Bayer–Macri (2014, https://arxiv.org/abs/1301.6968v3), Markman (2014, https://arxiv.org/abs/1301.6584v3), Yoshioka (2016, https://arxiv.org/abs/1206.4838v2), Mongardi–Rapagnetta (2021, https://doi.org/10.1016/j.matpur.2020.12.006), Mongardi–Onorati (2022, https://doi.org/10.1007/s00209-021-02966-6) handled the known deformation types one by one. Two deep results make the deformation-proof work. Soldatenkov–Verbitsky (2025, Theorem 3.7(v), https://arxiv.org/html/2409.09142v3) prove that a fixed isotropic class is semiample at every nef point of its deformation component as soon as it is semiample at one point. Greb–Lehn–Rollenske (2013, Theorem 4.1, https://arxiv.org/abs/1105.3410v3) prove that a *nonprojective* IHS manifold containing a holomorphic Lagrangian torus has a Lagrangian fibration. So the whole problem reduces to constructing one semiample deformation of the *given* class on a *possibly nonprojective* manifold. That is what this paper does.

Exercise: The four-step proof plan. Give the whole proof in four sentences before diving into details.
Write c_1(L) = m_0 · e with m_0 > 0 and e a primitive integral class, still isotropic and nef. The proof shows e is semiample on X and then upgrades to L = E^{⊗m_0}. Step 1 (Section 3, "Family construction"): construct a projective semistable degeneration π: Y → Δ whose generic fiber Y_t is an IHS manifold with a polarization h perpendicular to e, and whose marked period function p(τ) = f + τ v − (a/2) τ^2 e has a cusp at e. Along the way track the full second-cohomology monodromy M through semistable reduction, so that near the cusp M is the identity on the Q-span of e and f, and M v = v − aw e. Step 2 (Sections 4–5, "Torus theorem"): for a very general smooth fiber Y_t, construct a special Lagrangian torus T ⊂ Y_t that is isotopic to a coordinate angle torus in the crossing chart. The proof is a sequence of metric compactness estimates at the degeneration scale, culminating in the identification of a flat cylinder almost-everywhere in the pointed limit, and then in a Zhang–McLean perturbation. Step 3 (Section 6, "Rotation"): rotate the complex structure on Y_t from I to a second hyperkähler complex structure J, using i^*[ω_K] = 0 from the coordinate isotopy class of T; the calibration/Wirtinger inequality shows T becomes complex Lagrangian for J. Then Y' = (Y_t, J) is nonprojective and its only rational isotropic (1,1) class is Q e; Greb–Lehn–Rollenske then give a Lagrangian fibration on Y', making e semiample there. Step 4 (Section 6, "Return"): Soldatenkov–Verbitsky transfer e-semiampleness back to the original X, and Matsushita plus Fujiki clean up the fiber description.

Exercise: Why the deformation route? Why not try to construct sections directly on X?
Because the numerical hypothesis alone is too weak: c_1(L) is on the boundary of the Kähler cone, and there is no obvious semipositive metric, so Kodaira vanishing, Hörmander L^2 estimates, and other "analytic positivity" tools do not apply as-is. The good strategy is to *change the complex structure* while holding e fixed as a labeled lattice class, until we land at a point where e becomes an actual algebraic fibration class. Soldatenkov–Verbitsky say that this is enough: semiampleness spreads through the whole fixed-class deformation locus. What remains is finding a *single* point Y' in the locus where e is semiample, and the paper finds that point by first finding a special Lagrangian torus on some fiber Y_t of a degeneration, then rotating to make Y_t a nonprojective hyperkähler on which a Lagrangian fibration is forced. Key subtlety: the abstract complexity of finding a torus is dwarfed by the *bookkeeping* -- we must retain e as a labeled class through every base change, compactification, and rotation, so the class that is semiample on Y' literally equals the one we started with on X. That bookkeeping is the role of the family construction.

Exercise: The Yau conjecture / Strominger–Yau–Zaslow version. How does the paper relate to it?
The metric SYZ conjecture says: near a degeneration of a Calabi–Yau manifold to a maximally unipotent boundary point, the rescaled Ricci-flat metrics collapse to a half-dimensional space, and the generic fiber admits a special Lagrangian torus fibration. Li's Fermat family theorem (Li 2022, Theorem 1.1, https://arxiv.org/abs/1912.02360v1; survey https://arxiv.org/abs/2204.11363v1) gives such fibrations on generic regions. Blum–Liu (2026, https://arxiv.org/abs/2604.27890v1) and Li (2026, https://arxiv.org/abs/2605.00516v1) handle valuative independence and metric SYZ for maximum degenerations of polarized CY. Zhang (2017, Section 4, https://arxiv.org/abs/0911.1028v2) develops persistence of Calabi–Yau torus tubes near a flat cylinder. The present paper does not prove the full metric SYZ conjecture; it proves the *algebraic* strong hyperkähler SYZ conjecture. What it *borrows* from metric SYZ is the technical machinery: the compactness estimates, the flat-cylinder limit, the McLean/Zhang perturbation. The output of that machinery, for this paper's purposes, is a single special Lagrangian torus T ⊂ Y_t with a controlled isotopy class; the torus is then used algebraically via rotation.

Exercise: Setup for the family construction. What are e, E, L, m_0, and the sign convention on e?
Write c_1(L) = m_0 · e with m_0 > 0 an integer and e ∈ H^2(X, ℤ) a primitive integral class, still nef and isotropic. By Hodge theory H^1(X, O_X) = 0, so c_1: Pic(X) → H^2(X, ℤ) is injective on a compact Kähler manifold; hence e = c_1(E) for a unique line bundle E, and L ≅ E^{⊗m_0}. Throughout the deformation part e is retained as the same *labeled lattice vector* in the integral cohomology of a fixed differentiable manifold; it need not remain of type (1,1) — it may leave the Kähler cone — but its integrality and its identity as a lattice vector are preserved by every step. Sign: e is fixed so that q(e, κ_X) > 0 for every Kähler class κ_X on X. In a Lorentzian 3-dimensional space (the H^{1,1} slice of H^2), a nonzero null vector in the closure of the positive cone pairs positively with every vector in that cone — so this sign is intrinsic, not a choice. Because we later want semiampleness of e on a rotated complex structure, we must retain both e and this sign through every step.

Exercise: Positive polarization orthogonal to e. Why does an h exist with q(h) > 0 and q(h, e) = 0?
Lemma (Polarized Family Construction, part 1). Let q have signature (3, r − 3) with r = rank H^2(X, ℤ), and e ≠ 0 primitive with q(e) = 0. Then there is a primitive h ∈ H^2(X, ℤ) with q(h) > 0 and q(h, e) = 0. Proof: the radical of e^⊥ is ℝ e, and after modding it out, signature(e^⊥ / ℝ e) = (2, r − 4). In particular e^⊥ contains positive vectors. Because e is rational, e^⊥ is Q-defined, so rational approximation gives a rational positive vector; scale to a primitive integral one and call it h. Role: h will be the polarization we impose on the deformation family. Since q(h, e) = 0, moving in the h-polarized deformation component never mixes h with e, so e stays exactly where we put it. This is the first example of *bookkeeping by construction*: the perpendicularity q(h, e) = 0 is not a technical convenience, it is what allows us to keep track of e while we deform.

Exercise: The algebraic family and the arithmetic quotient Q_h. Why does this setup give an honest algebraic family?
Lemma (Polarized Family Construction, part 2). Take a manifold X_h in the marked deformation component of X on which h is ample (by period surjectivity and generic vanishing of extra NS classes: for almost every period with q(p, h) = 0, the NS lattice is ℤ h). Fix a positive integer k with k h very ample, so H^1(X_h, k h) = 0; bases of sections embed X_h into ℙ^M, giving a Hilbert point. Deformations of the polarized pair plus frames of the section space are smooth by local Torelli; the Hilbert scheme component has finite automorphism group acting freely on frames, so it has a smooth point with submersive period map. Restrict the universal family to a smooth connected Zariski-open base S and keep the polarization k h. Let Γ ⊂ G_h(ℤ) = {g ∈ O(Λ, q): gh = h, g D_h = D_h} be a torsion-free finite-index congruence subgroup acting on the full lattice Λ = H^2(X, ℤ); pass to a finite étale cover S → S with monodromy in Γ (this uses SGA 1, Exposé XII, Theorem 5.1). The arithmetic quotient Q_h = Γ \ D_h (where D_h is the h-polarized period domain component containing [X_h]) is smooth quasi-projective by Baily–Borel 1966 (https://annals.math.princeton.edu/1966/84-3/p11); the descended period map Φ: S → Q_h is algebraic by Borel 1972 (https://doi.org/10.4310/jdg/1214430642). The Γ-action on D_h is *free*: the stabilizer of a positive oriented 2-plane is compact in O(V_ℝ, q), hence finite, hence trivial in the torsion-free Γ.

Exercise: Rational ternary subspace P and arithmetic curve C_P with cusp.
Lemma (Polarized Family Construction, part 3). In V_ℝ = h^⊥_ℝ, decompose e = e_W + e_− for W a positive 2-plane period in the open set U ⊂ D_h that S actually realizes. Since W^⊥ is negative-definite and e is not zero, e_W ≠ 0 and q(e_−) = −q(e_W) < 0; hence W + ℝ e = W ⊕ ℝ e_− has signature (2,1). This shows that *some* positive 3-space containing e has the right signature. Now approximate W by rational 2-planes W_j = <x_j, y_j>_ℝ, keeping {e, x_j, y_j} rational and of signature (2,1); set P = <e, x_j, y_j>_ℚ for large j and let D_P ⊂ D_h be the positive-plane domain of P containing W_j. Because S is algebraic and open in Q_h, D_P ∩ U ≠ ∅ for large j. To make the arithmetic structure compatible with the full lattice Λ = H^2(X, ℤ), set Λ_0 = (P ∩ Λ) ⊕ (P^⊥ ∩ Λ) ⊂ Λ; Λ_0 has finite index, and an isometry of P ∩ Λ_P congruent to the identity modulo m · N (N large enough that N Λ ⊂ Λ_0) extends identically on P^⊥ ∩ Λ to an isometry of Λ. Take such a congruence subgroup Γ_P ⊂ SO(P, q) acting on D_P; then Γ_P ⊂ Γ. The quotient C_P := Γ_P \ D_P is a smooth arithmetic curve, and because P contains the rational null line ℚ e, its smooth compactification C̄_P has a cusp c_e represented by ℚ e. The inclusion j: C_P → Q_h is algebraic (Borel 1972).

Exercise: The algebraic curve B reaching the cusp, plus the transport of the marking.
Form the fiber product Z = S ×_{Q_h} C_P. Choose a point z_0 of Z at which the marked period lift from S equals the chosen point of U ∩ D_P; there the map Z → C_P is submersive (since the local period map of S is submersive). Intersect a quasi-projective embedding of Z with general hyperplanes through a smooth point near z_0 whose tangent intersection is transverse to the fiber of Z → C_P; take the component through that point and normalize: this is an irreducible smooth algebraic curve B with a nonconstant map b: B → C_P, dominating C_P and carrying the pulled-back family from S. Take smooth projective completions; b̄: B̄ → C̄_P is a finite surjective morphism of curves. Remove a finite set F ⊂ C̄_P containing every cusp, every branch value, and every image of a point of B̄ \ B. Then over C_P \ F the morphism b is a finite unramified covering and its source lies in B. (If necessary slide the starting point slightly within the same local branch of S so it maps outside F.) This is the family we will use; from here on, the base is a punctured disk around some b_e ∈ B̄ over c_e.

Exercise: The path lift and the equality of period lifts.
In the D_P cover, the inverse image of F ∩ C_P is a locally finite discrete set; its complement is path connected. Take a path from the initial marked lift of c_e-part in D_P towards the boundary point represented by e, avoiding that discrete set; eventually it lies in a cusp neighborhood of e. Project this path to C_P \ F, lift it through the finite unramified covering B → C_P \ F, and note that the lift tends to a point b_e ∈ B̄ over c_e. So a small punctured disk around b_e carries our desired smooth algebraic polarized family approaching the cusp. *Transport of the marking:* as the family is transported over the path, its period lift to D_h is a continuous lift of the period map S → Q_h; by uniqueness of lifts for the covering D_h → Q_h, this D_h-valued lift begins at the same point as the constructed D_P-valued lift, and both descend to the same map on the base. Therefore they are equal on the universal cover of the punctured disk. That equality of lifts — not just equality of images in Q_h — is exactly what lets us identify the family monodromy with the arithmetic monodromy next.

Exercise: Identifying monodromy M on the full lattice.
Lemma (Monodromy Identification). For a loop in B, let M_B ∈ Γ ⊂ O(Λ, q) be its second-cohomology monodromy, and γ_B ∈ Γ_P ⊂ Γ the deck transformation acting on the D_P-lift. Equivariance and the equality M_B Φ̃ = γ_B Φ̃ at every lift point imply γ_B^{−1} M_B fixes Φ̃(b̃) ∈ D_h. Since Γ acts *freely* on D_h (proved above by compactness of the stabilizer of a positive oriented 2-plane), γ_B^{−1} M_B = 1, and M_B = γ_B as isometries of the *full* lattice Λ. Two consequences. (i) The actual family monodromy fixes our labeled e, is the identity on P^⊥, and on P it acts by the explicit Γ_P element defined below. (ii) Because the equality of lifts is exact — not just up to a finite index in the arithmetic subgroup — the identification is not only up to base change; it pins down the labeling of e on the nose. That is the technical core of "retain the prescribed class in monodromy" from the four-step plan.

Exercise: The exact period and monodromy formulas on P.
Choose f_0 ∈ P with q(e, f_0) = 1; set f = f_0 − ½ q(f_0) e; then q(f) = 0 and q(e, f) = 1. Let <e, f>^⊥ ∩ P be the positive rational line; pick v ≠ 0 there and put a = q(v) > 0. This is a "hyperbolic basis" (e, f, v) of P with q(e) = q(f) = 0, q(e, f) = 1, q(e, v) = q(f, v) = 0, q(v) = a > 0. Every positive period p ∈ P_ℂ pairs nontrivially with e: a positive 2-plane orthogonal to e would put e into its negative-definite orthogonal complement, impossible. So normalize to q(p, e) = 1. Write p = f + τ v + β e; nullity gives β = −(a/2) τ^2. Positivity q(p, p̄) = 2 a (Im τ)^2 gives the upper-half-plane model {Im τ > 0} for D_P, with the cusp at τ = ∞. Flip v if necessary so the chosen component is Im τ > 0. Define U_w for rational w > 0 by U_w e = e, U_w v = v − a w e, U_w f = f + w v − (a w^2 / 2) e. Direct computation with the Gram matrix confirms q(U_w x) = q(x) on the basis; and U_w p(τ) = p(τ + w). Reduction mod a sufficiently divisible integer puts U_w in Γ_P; the cusp stabilizer of the torsion-free orientation-preserving Γ_P is generated by such a translation τ ↦ τ + w_0, w_0 > 0 rational (rational width because e, f, v are rational). Let u_c = exp(2π i τ / w_0) be the cusp coordinate; if t has ramification b at the cusp, u_c = t^b β(t) with β(0) ≠ 0, so Im τ = (b w_0 / 2π) (−log|t|) + O(1). Now the monodromy around a small positive loop in t is U_w with w = b w_0, so
    Me = e,   Mv = v − a w e,   Mf = f + w v − (a w^2 / 2) e,
    M is identity on P^⊥ (in Λ_ℚ),
    (M−1) P = <e, v>_ℚ,   (M−1)^2 f = −a w^2 e ≠ 0,   (M−1)^3 = 0.
The "middle nonvanishing" (M−1)^2 f ≠ 0 says M is not a single Jordan block; it is important in the restriction computation below, because after rotation the class [ω_K] will sit precisely in the image of M−1 with its nonzero component on the e-direction.

Exercise: Semistable reduction and logarithmic extension of η = σ^n.
Take a projective algebraic model of the family over a neighborhood of b_e in B̄. One-parameter semistable reduction in characteristic zero (Kempf–Knudsen–Mumford–Saint-Donat, Toroidal Embeddings I, 1973, Chapter II, pages 53–54, https://link.springer.com/book/10.1007/BFb0070318) followed by restriction to a small disk yields, after finite base change and a projective modification which is an isomorphism over the punctured base, a smooth total space Y with reduced simple-normal-crossing central fiber Y_0. The smooth fibers are still the same IHS manifolds and their marked periods are unchanged. The polarization extends to a line bundle A on 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. A is ample on each smooth fiber; relative ampleness on the semistable model is unnecessary. The flat integral class e is a single-valued section of R^2π_*ℤ on Δ*, and the pairing q(−, e) is a single-valued holomorphic functional on the flat bundle. Its restriction to the Hodge line of relative 2-forms is nowhere zero because q(p(τ), e) = 1; so q(−, e) trivializes the Hodge line, and there is a unique relative 2-form σ with q([σ], e) = 1 and [σ] = p(τ) on the universal cover. Put η = σ^n; it is a single-valued holomorphic relative volume form on Δ*. Now for the extension statement. Let K_log = det(Ω^1_Y(log Y_0) / π^*Ω^1_Δ(log 0)) ⊂ the ordinary relative canonical line; this is the relative canonical taken in the log sense along Y_0. Claim: η extends holomorphically as a section of K_log. First consider a point of Y_0 lying on only one component; there Y is product coordinates (z_1, ..., z_d, t), and locally η_t = F(z, t) dz_1 ∧ ... ∧ dz_d. On a smaller product chart the sub-mean inequality together with the global upper bound gives |F(z, t)|^2 ≤ C ∫_{Y_t} |η_t|^2 ≤ C' s^d. The negative Laurent coefficient in t satisfies |(1/2πi) ∫_{|t|=ρ} F(z, t) t^{m−1} dt| ≤ C ρ^m |log ρ|^{d/2} → 0 for m ≥ 1, so all negative coefficients vanish and F extends holomorphically through t = 0. This covers the smooth locus of the central divisor; the crossing locus has complex codimension ≥ 2 in the smooth Y, so Hartogs extends η uniquely over it. The argument uses only the polynomial upper bound from the growth hypothesis, not the sharp constant.

Exercise: Volume growth s^d exactly (this is the growth hypothesis of Theorem (Torus)).
Polarize Fujiki's relation. For σ any holomorphic 2-form on Y_t, Fujiki says ∫_{Y_t} (z σ + w σ̄)^{2n} = c_X q(z [σ] + w [σ̄])^n for all z, w; comparing the z^n w^n coefficient gives binom(2n, n) ∫ σ^n ∧ σ̄^n = c_X 2^n q(p, p)^n, with p = [σ]. For p(τ) = f + τ v − (a/2) τ^2 e, an explicit computation gives q(p, p) = 2 a (Im τ)^2. And Im τ = (b w_0 / 2π)(−log |t|) + O(1). So q(p, p)^{2n/(2)} = q(p, p)^n ≈ (const) · (log 1/|t|)^{2n} = s^d. Therefore c s^d ≤ I_t := ∫_{Y_t} |η_t|^2 ≤ C s^d for positive constants c, C independent of t near 0. Same Fujiki relation as numerical dimension n, same growth rate at the cusp. Good: growth is not an extra assumption here, it is a consequence of the way we normalized σ.

Exercise: Maximal growth forces a maximal crossing with nonzero residue.
Fix a point of Y_0 where exactly k components meet, 1 ≤ k ≤ d + 1. Take coordinates with t = z_0 ... z_{k−1} and consider the local generator g = (dz_1 / z_1) ∧ ... ∧ (dz_{k−1} / z_{k−1}) ∧ dz_k ∧ ... ∧ dz_d of K_log. Its extended holomorphic coefficient (call it h(z)) is bounded on a smaller coordinate polydisk. Fix such a chart; in the variables x_i = −log |z_i| for 1 ≤ i ≤ k−1, the fiber part of the chart is an expanding simplex: each x_i is bounded below, and Σ x_i ≤ s + O(1), of dimension k−1. Integrating the angular variables and the remaining ordinary coordinates yields ∫_{Y_t ∩ chart} |η_t|^2 ≤ C (1 + s^{k−1}). If k = d + 1 (maximal crossing), the crossing is an isolated point o. See what happens if the coefficient h(o) = 0. Dilation trick: put A = −log δ for a polydisk radius δ, and rescale x_i = A + (s − (d+1)A) u_i over the standard simplex {u_i > 0, Σ u_i < 1}. For almost every u, all d+1 coordinates z_0, ..., z_d tend to zero as s → ∞, uniformly in their arguments; the rescaled integrand |h|^2 tends to |h(o)|^2 = 0 pointwise on the simplex and is uniformly bounded. Dominated convergence gives ∫_{Y_t ∩ chart} |η_t|^2 = o(s^d) if h(o) = 0. Now cover the compact central fiber by finitely many such charts (properness, then shrink the disk so nearby fibers are covered). If there were no maximal crossings (all k ≤ d), the total ∫ would be O(s^{d−1}). If there were maximal crossings but all coefficients vanished, the total would be o(s^d). Either contradicts the lower bound c s^d ≤ ∫ |η_t|^2 = I_t. So at least one maximal crossing has nonzero residue h(o) ≠ 0. At that point the functions z_1, ..., z_d may be chosen to be restrictions of rational functions on the algebraic total space (algebraic local defining functions), which the intersection estimates of the next section need.

Exercise: Proposition (Family). Assemble the ingredients.
The prescribed-cusp family π: Y → Δ constructed above has: (1) Y smooth, Y_0 reduced SNC, every Y_t IHS in the marked deformation component of X; a line bundle A on Y restricts on each Y_t to an ample bundle with c_1 = k h; the family is the analytic restriction of an algebraic family. (2) A rational subspace P = <e, f, v>_ℚ ⊂ h^⊥ of signature (2, 1) with q(e) = q(f) = 0, q(e, f) = 1, q(e, v) = q(f, v) = 0, q(v) = a > 0; periods p(τ) = f + τ v − (a/2) τ^2 e; q(p, p) = 2 a (Im τ)^2; monodromy M as in the previous exercise; M is identity on P^⊥; (M−1) P = <e, v>_ℚ, (M−1)^2 f = −a w^2 e ≠ 0, (M−1)^3 = 0. (3) A single-valued holomorphic relative σ on Y|Δ* with q([σ], e) = 1, η = σ^n extends to K_log, c s^d ≤ ∫ |η|^2 ≤ C s^d, and there is a maximal crossing o ∈ Y_0 with residue h(o) ≠ 0. (4) For any prescribed irrational λ ∈ ℝ \ ℚ, there is a path in Δ* along which τ = λ + i y and y → +∞. Properties (1)–(3) supply all hypotheses of Theorem (Torus) on any sufficiently small fiber along this path. Property (4) is the input to the rotation step: choosing λ irrational forces the rotated period plane to contain no rational isotropic class except the original Q e.

Exercise: Theorem (Torus). State the special Lagrangian torus theorem.
Theorem (Torus). Let π: Y → Δ be the restriction to a disk of a projective algebraic family over a smooth curve. Assume Y is smooth, Y_0 is a reduced simple-normal-crossings divisor, and the fibers Y_t (t ≠ 0) are smooth connected projective manifolds of complex dimension d ≥ 2 with trivial canonical bundle. Let a_t be polarizing classes given by a line bundle on Y; the line bundle need not be relatively ample at 0. Let η_t be nowhere-vanishing holomorphic d-forms that extend as a section of the relative logarithmic canonical line. Put s = −log |t|, and suppose for constants c, C > 0 that c s^d ≤ I_t := ∫_{Y_t} |η_t|^2 ≤ C s^d for all 0 < |t| << 1. Suppose that at some point o ∈ Y_0 exactly d+1 components meet and that the logarithmic coefficient of η at o is nonzero: so in some coordinate neighborhood of o, z_0 z_1 ... z_d = t and η_t = f(z) dz_1/z_1 ∧ ... ∧ dz_d/z_d with f(o) ≠ 0. Equip Y_t with the Ricci-flat Kähler metric in the polarization class a_t. Then for every sufficiently small nonzero t, Y_t contains a smooth embedded special Lagrangian real d-torus isotopic in Y_t to an angular coordinate torus in the chart z_0 z_1 ... z_d = t. Here "special Lagrangian" means the Kähler form restricts to zero on T and the holomorphic volume form η_t has constant phase on the oriented torus T. In the application d = 2n. The theorem is independent of hyperkähler theory; it is a general Calabi–Yau degeneration result.

Exercise: Scaling, tubes, and coordinate volume density.
Let g^{(0)}_t and ω^{(0)}_t be the Ricci-flat Calabi–Yau metric and Kähler form on Y_t with [ω^{(0)}_t] = a_t (Calabi–Yau theorem; Yau 1978, https://doi.org/10.1002/cpa.3160310304). The useful scale is g_t = s g^{(0)}_t, ω_t = s ω^{(0)}_t, μ_t = ω_t^d / d!, giving μ_t(Y_t) = c_a s^d with c_a = (1/d!) ∫_{Y_t} a_t^d independent of t. Take z_1, ..., z_d to be algebraic local equations of the corresponding components in the crossing chart; they extend to rational functions on the algebraic total space, but we only need z_0 absorbed into the unit. Set x_i = log |z_i| and y_i = arg z_i ∈ ℝ / 2πℤ, γ = Σ (dx_i^2 + dy_i^2), ω_γ = Σ dx_i ∧ dy_i: this is the standard flat metric on ℝ^d × (ℝ/2πℤ)^d. Define the open simplex Σ = {b ∈ ℝ^d: b_i < 0, Σ b_i > −1}; for a relatively compact subset B << Σ, the tube T_t(B) = {x/s ∈ B, y ∈ (ℝ/2πℤ)^d}. For small t all |z_i| including |z_0| = exp(−s − Σ x_i) are bounded by e^{−δ_B s}, so the tubes are compactly inside the chart. Now the coordinate density: by Ricci-flatness dV_t is a constant multiple of |η_t|^2 = |f(z)|^2 dV_γ, and integration determines the constant. Hence on T_t(B), dV_t / dV_γ = (c_a s^d / I_t) |f(z)|^2 → θ > 0 uniformly after a sub, using c s^d ≤ I_t ≤ C s^d and f(z) → f(o) ≠ 0. So the two measures are comparable on every fixed interior tube.

Exercise: Lemma (Mixed Intersections). The integral bounds that drive everything.
Statement: for every fixed B << Σ and 0 ≤ k ≤ d, ∫_{T_t(B)} ω_γ^k ∧ ω_t^{d−k} ≤ C_B s^d. Consequently, with traces over real Riemannian metrics, ∫_{T_t(B)} tr_γ g_t dV_γ ≤ C_B s^d and ∫_{T_t(B)} tr_{g_t} γ dV_t ≤ C_B s^d. Proof: resolve the finitely many rational maps z_i: Y ⇢ ℙ^1 simultaneously to get a projective modification ϖ: Y_hat → Y and morphisms f_i: Y_hat → ℙ^1 with ϖ an isomorphism over the chart being used. Fix a Fubini–Study form ω_FS on ℙ^1; for A_b([u:v]) = [e^b u : v], define the smooth closed semipositive form α_{s,t} = (1/s) ∫_0^s Σ_i f_i^* A_b^* ω_FS db on Y_hat_t. Its class equals Σ_i f_i^* [ω_FS] and is independent of s and t; so every mixed intersection ∫ α_{s,t}^k ∧ (ϖ^* ω^{(0)}_t)^{d−k} is a fixed finite number. In a logarithmic coordinate w_i = x_i + i y_i, the i-th summand before averaging is a positive constant times e^{2(b + x_i)} / (1 + e^{2(b + x_i)})^2 dx_i ∧ dy_i. For x/s ∈ B both −x_i and s + x_i are at least δ_B s (some δ_B > 0). Hence the interval [−x_i − 1, −x_i + 1] lies in [0, s] for large s, and on that interval the scalar factor is bounded below. Summing over i gives α_{s,t} ≥ (c_B/s) ω_γ on the tube. Positivity of wedge products now gives ∫_{T_t(B)} ω_γ^k ∧ ω_t^{d−k} ≤ C_B s^k s^{d−k} ∫ α_{s,t}^k ∧ (ϖ^* ω^{(0)}_t)^{d−k} ≤ C_B' s^d. The real trace identity (tr_h k) dV_h = (2/(d−1)!) ω_k ∧ ω_h^{d−1} with k = 1 and k = d−1 converts the mixed bounds into the two trace bounds.

Exercise: Global energies, vanishing average curvature, and many noncollapsed centers.
The two trace bounds have different jobs. tr_{g_t} γ controls logarithmic functions on intrinsic balls; tr_γ g_t supplies many noncollapsed centers. Choose concentric open Euclidean balls B_0 << B_1 << B_2 << Σ and a smooth cutoff ρ ∈ [0, 1] supported in B_2, identically 1 on B_1. Define functions on the whole compact fiber, extended by zero: U_0 = s ρ(x/s), U_j = x_j ρ(x/s) for 1 ≤ j ≤ d, and H_t = Σ_{j=0}^d |∇ U_j|_{g_t}^2. These extensions are smooth. On the support, x_j/s is bounded, dU_0 = Σ ρ_i (x/s) dx_i and dU_j = ρ(x/s) dx_j + (x_j/s) Σ ρ_i (x/s) dx_i, so H_t ≤ C tr_{g_t} γ and (1/μ_t(Y_t)) ∫ H_t dV_t ≤ C_H. For the energies of curvature and projective embedding: fix (Φ, π): Y ↪ ℙ^M × Δ; give ℙ^M a Fubini–Study metric; set K_t = |Rm_{g_t}|^2 + |dΦ_t|_{g_t}^2. The Ricci-flat Kähler Chern–Weil identity makes ∫ |Rm_{g^{(0)}}|^2 dV_{g^{(0)}} = (universal constant) · ∫ c_2(Y_t) a_t^{d−2}, independent of t. Likewise the holomorphic map Φ_t has fixed energy. Under g_t = s g^{(0)}_t, those scale as ∫ |Rm_{g_t}|^2 dV_t ≤ C s^{d−2} and ∫ |dΦ_t|_{g_t}^2 dV_t ≤ C s^{d−1}, so m_t := (1/μ_t(Y_t)) ∫ K_t dV_t ≤ C(s^{−2} + s^{−1}) → 0.

Exercise: Lemma (Many Centers) + Lemma (Choose One). Getting one good center with all-radius estimates.
Lemma (Many Centers). There are δ, v > 0 and measurable sets E_t ⊂ T_t(B_0) with μ_t(E_t) ≥ δ μ_t(Y_t) and μ_t(B_{g_t}(q, 1)) ≥ v for all q ∈ E_t. Proof sketch: the region sB_0 contains ≥ c_0 s^d disjoint translates of a fixed small cube; multiply by a fixed angle cube to get disjoint coordinate boxes Q in T_t(B_0). By the first trace bound at least half satisfy ∫_Q tr_γ g_t dV_γ ≤ C_0. By coordinate density, each such box has μ_t(Q) between fixed m_0, m_1. For a, b ∈ Q, a convexity/Jacobian change-of-variables estimate bounds the double integral of d_{g_t}(a, b) by a constant. Then Markov's inequality picks E_Q ⊂ Q of measure ≥ μ_t(Q)/2 on which the average distance is small; Bishop–Gromov (Ric(g_t) = 0) converts "μ_t(B(a, R_0)) large" into "μ_t(B(a, 1)) ≥ v". Union over good boxes gives a fixed positive fraction of total volume that is noncollapsed. Lemma (Choose One). For a nonnegative integrable h on Y_t, let M_t h(q) = sup_{r>0} (1/μ_t(B_{g_t}(q, r))) ∫_{B_{g_t}(q, r)} h dV_t be the centered maximal function. Doubling gives weak-(1,1): μ_t{M_t h > λ} ≤ (D_N/λ) ∫ h dV_t. Choose A so large that D_N C_H / A < δ/3; put ε_t = √m_t + s^{−1} → 0. The two exceptional sets {M_t H_t > A} and {M_t K_t > ε_t} each have measure < δ μ_t(Y_t)/3 for small t, so E_t is not covered; there is p_t ∈ E_t with μ_t(B(p_t, 1)) ≥ v, and for every R > 0, the average of H_t and of K_t over B(p_t, R) are bounded by A and ε_t respectively. Pass to a sub with b_t := x(p_t)/s → b_* ∈ closure(B_0) ⊂ B_1. In particular ρ(b_t) = 1, U_0(p_t) = s, U_j(p_t) = x_j(p_t).

Exercise: Poincaré + Chebyshev locate balls in the chart, and the flat regular set of the pointed limit.
The ball Poincaré inequality for Ric ≥ 0 (Cheeger–Colding segment inequality, https://annals.math.princeton.edu/articles/13331) at p_t for U_j gives an upper bound CA r^2 on the "oscillation vs mean" of U_j in B(p_t, r). Combine with dyadic radii and doubling to get the centered L^2 estimate: (1/μ_t(B(p_t, R))) ∫_{B(p_t, R)} |U_j − U_j(p_t)|^2 dV_t ≤ C R^2 for all 0 ≤ j ≤ d and all R > 0. Divide by s^2 and use b_t → b_*: on each fixed-radius ball, U_0/s → 1 and U_j/s → (b_*)_j in L^2. Where U_0/s > 1/2, the point is in the coordinate tube with x_j/s = (U_j/s)/(U_0/s). Chebyshev's inequality gives, for every R, η > 0, volume-tending-to-zero of B_{g_t}(p_t, R) minus the set {q in chart : |x(q)/s − b_*| < η}. Bishop–Gromov bounds μ_t(B(p_t, R)) ≤ ω_N R^N. Now pass to the pointed measured Gromov–Hausdorff limit (Y_t, g_t, p_t, μ_t) → (Z, d_Z, p, μ). The compactness, curvature, and singularity theory of Cheeger–Colding (Cheeger–Colding I, https://doi.org/10.4310/jdg/1214459974; Colding, https://annals.math.princeton.edu/articles/13099) and Cheeger–Naber (https://doi.org/10.4007/annals.2015.182.3.5) gives: Z complete proper; μ is the N-dimensional Riemannian volume on the dense regular set G; S = Z \ G is closed with dim_H S ≤ N − 4 and μ(B_Z(q, r)) ≤ ω_N r^N; compact subsets of G admit smooth convergence identities with regions of Y_t; complex structures converge smoothly on G to a parallel integrable limit I_∞. Because every compact smooth region lies in a fixed-radius ball about p_t with curvature energy ≤ ε_t ω_N R^N → 0, the limit metric has Rm_g = 0 on G. Flat regular set. Then the embedding maps Φ_t are holomorphic between Kähler manifolds, hence harmonic; with their energy on regular compact sets tending to zero by the K_t bound, Lemma (Vanishing Energy, next section) forces dΦ_t → 0; every local constant is Φ(o), so Φ_t → Φ(o) uniformly on compact subsets of G. From the embedding and the log equation Σ_{i=0}^d log |z_i| = −s, the corresponding regions of Y_t lie in the coordinate chart with each x_j pluriharmonic and bounded; interior harmonic estimates improve L^2 convergence to C^∞_loc: x_j/s → (b_*)_j. Since b_* ∈ closure(B_1) has ρ = 1 there, every compact subset of G has a neighborhood where U_j = x_j for all small t; the all-radius bounds then apply to the actual logarithmic coordinates, ready for the cylinder identification.

Exercise: Lemma (Vanishing Energy). Why do zero-energy harmonic maps on a smooth ball collapse to constants?
Lemma (Vanishing Energy). Let h_ν be metrics converging smoothly on a fixed smooth domain D of real dimension N, and let ψ_ν: (D, h_ν) → P be smooth harmonic maps to a fixed compact Riemannian manifold P. If their energies tend to zero on every relatively compact subdomain, then sup_K |dψ_ν| → 0 for every K << D, and after extraction they converge smoothly to a constant on each connected relatively compact domain. Proof sketch. Smooth harmonic maps are stationary, so the stress tensor |dψ|^2 h − 2 ψ^* g_P has zero divergence. The energy functional Θ_ν(q, r) = r^{2−N} ∫_{B_{h_ν}(q, r)} |dψ_ν|^2 dV is nondecreasing in r up to a bounded factor (test the stress tensor against radial vector fields, use ∇^2(r^2/2) = h + O(r^2) in geodesic balls, discard a nonnegative term, integrate). If derivatives were unbounded, maximize dist_{h_ν}(q, ∂D') |dψ_ν(q)| over a fixed intermediate domain D'; at a maximizer q_ν, put L_ν = |dψ_ν(q_ν)| → ∞. On the ball B_{h_ν}(q_ν, L_ν^{−1}) we have |dψ_ν| ≤ 2 L_ν. Rescale h_ν by L_ν^2; the rescaled maps v_ν on unit balls have |dv_ν(0)| = 1 and |dv_ν| ≤ 2, and the rescaled metrics converge smoothly to a Euclidean metric. Harmonic-map Bochner inequality then gives Δ|dv_ν|^2 ≥ −C |dv_ν|^2; the mean-value inequality with uniform constants gives 1 ≤ C ∫_{B(0,1)} |dv_ν|^2 = C L_ν^{N−2} ∫_{B_{h_ν}(q_ν, L_ν^{−1})} |dψ_ν|^2 dV = C Θ_ν(q_ν, L_ν^{−1}) → 0, contradiction. Therefore the derivatives are uniformly bounded on interior domains, and the same Bochner + mean-value argument on fixed smaller balls gives sup_K |dψ_ν|^2 ≤ C E_ν → 0. Elliptic estimates give C^∞ convergence to a constant on connected domains. Reference for stationary harmonic maps: Lin 1999, https://annals.math.princeton.edu/articles/12464.

Exercise: Sobolev extension of the harmonic coordinates across the singular set S.
Continue with F = (u_1 + i ϑ_1, ..., u_d + i ϑ_d) defined on the regular set G by: u_j = smooth subsequential limit of x_j − x_j(p_t), ϑ_j = smooth subsequential limit of arg z_j. The u_j are harmonic on compact subsets of G (as limits of pluriharmonic x_j under smooth metric convergence); u_j has local L^2 bounds from the centered Poincaré; the ϑ_j have compact-circle values and satisfy d ϑ_j = −d u_j ∘ I_t, giving holomorphic limits. F is a local biholomorphism by the coordinate-density Jacobian J_0 = θ^{−1} > 0, injective, with L^2 and finite-energy bounds ∫_{B(p, R) ∩ G} |u_j|^2 dμ ≤ C R^2 μ(B(p, 2R)) and ∫ |d u_j|^2 dμ ≤ C μ(B(p, 2R)). To extend u_j across S: use RCD(0, N) structure on Z (Gigli–Mondino–Savaré, https://doi.org/10.1112/plms/pdv047; Lott–Villani, https://math.berkeley.edu/~lott/LottVillani.pdf; Erbar–Kuwada–Sturm, https://doi.org/10.1007/s00222-014-0563-7). Lemma (Capacity Extension). Each u_j extends to a weakly harmonic function in W^{1,2}_loc(Z). Proof: cover the compact singular portion of a bounded subdomain by balls of radii r_i with Σ r_i^{N−2} small; a Lipschitz cutoff on a doubled covering ball has squared gradient integral ≤ C r_i^{N−2}; take minima of complementary cutoffs to get χ_k that vanish near S, → 1 off S, 0 ≤ χ_k ≤ 1, and ∫ |dχ_k|^2 → 0. T_A(u_j) = max(−A, min(A, u_j)) is Sobolev; χ_k T_A(u_j) is compactly supported inside G and locally ambient Lipschitz (local agreement of distances + boundedness); weak compactness and lower semicontinuity give a Sobolev extension; let A → ∞. Weak harmonicity: test ∫ ⟨d u_j, d(χ_k φ)⟩ over G; the extra ⟨d u_j, d χ_k⟩ term goes to zero by Cauchy–Schwarz and ∫ |d u_j|^2 bounded; the d u_j term converges to the integral over Z because S has measure zero.

Exercise: Global gradient bounds and the derivative + inverse derivative bounds on F.
Lemma (Capacity Extension continued). We use the local harmonic gradient estimate on noncollapsed Ricci limits, or the RCD(0, N) formulation of Hua–Kell–Xia 2016 (Theorem 1.2 and 4.1(a), https://arxiv.org/abs/1308.3607v2): sup_{B(q, R/2)} |d u| ≤ C(N)/R · (1/μ(B(q, 2R)) ∫_{B(q, 2R)} |u|^2 dμ)^{1/2}. For signed u, apply the mean-value estimate to u^2 first: sup_{B(q, R)} |u| ≤ C · normalized L^2 on B(q, 2R); add twice the supremum and apply the positive-harmonic gradient estimate on B(q, R/2); if sup = 0, trivial. Fix q; take R ≥ 2 d_Z(p, q). Then B(q, 2R) ⊂ B(p, 3R) and B(p, 6R) ⊂ B(q, 7R); doubling compares μ(B(q, 2R)) with μ(B(p, 6R)). The centered L^2 bound limits the right side to a constant independent of q and R. Hence |d u_j| ≤ C on G. Similarly, holomorphicity bounds the angular derivatives, so |dF| ≤ A for some fixed A. Since the product of the N real singular values of dF is J_0 > 0, the smallest singular value is at least J_0 / A^{N−1}, giving a uniform bound on the inverse derivative |(dF)^{−1}| ≤ A^{N−1} / J_0. So F is a bi-Lipschitz-equivalence between the flat cylinder H = ℂ^d / (2π i ℤ^d) (with its standard γ) on the tangent-metric side and (G, g) on the Z side, but only at the level of infinitesimal metrics; we must still show that globally G is the whole cylinder.

Exercise: Path distance, F extends to Z, and the completion is a bi-Lipschitz homeomorphism to H.
Lemma (Regular Paths). The intrinsic path distance on G equals the restriction of d_Z. In particular F extends to a Lipschitz map Z → H. Proof: fix a, b ∈ G and small balls A, B about them with closures in G. Minimizing segments between A, B stay in a compact ball K (segment inequality). Cover K's singular portion by small balls B(q_i, r_i), disjoint from A ∪ B after doubling, with Σ r_i^{N−1} small. Put f = Σ r_i^{−1} 1_{B(q_i, 2 r_i)}. The volume upper bound μ(B(q_i, 2 r_i)) ≤ ω_N (2 r_i)^N gives ∫ f dμ ≤ C Σ r_i^{N−1} small. The segment inequality for nonnegative integrable f on Ric ≥ 0 limits (Cheeger–Colding III, Theorem 2.15, https://doi.org/10.4310/jdg/1214342146) says the average over endpoints in A, B, of inf_γ ∫_γ f over minimizing segments γ, is bounded by C √(∫ f dμ) √(diam(A) + diam(B)) → can be made small. So some minimizing segment has ∫_γ f < 1. A segment that meets B(q_i, r_i) with endpoints outside B(q_i, 2 r_i) spends length ≥ r_i in the doubled ball, giving ∫_γ f ≥ 1; so the chosen segment misses S. Short paths inside A, B join it to a, b. Letting radii → 0 gives dist_G(a, b) ≤ d_Z(a, b); the reverse holds for every path. Lipschitz from the inverse derivative bound; density of G in Z and completeness of H extend F to F: Z → H.

Lemma (Full Cylinder). The extension F: Z → H is a bi-Lipschitz homeomorphism. Proof: set E = F(S); dim_H E ≤ N − 4; we will prove closedness only after constructing the inverse on all of H. Local openness of F on G gives a starting point of F(G) \ E; lift any finite polygonal path in H \ E through the local inverse on G. A finite-parameter end cannot appear: the inverse-derivative bound makes the lift Cauchy, and Z is complete; its image is the prescribed path end, outside F(S), so in G where the local inverse continues. Injectivity on G gives uniqueness. The dimension-avoidance facts below guarantee H \ E ⊂ F(G) and give a Lipschitz bound d_Z(K(a), K(b)) ≤ C d_H(a, b) for a, b ∈ H \ E. Both the domain H \ E and the image of K are dense: domain by dim_H E < N, image by local openness at each regular neighborhood and density of G. Completing K gives Lipschitz K: H → Z. Compositions F K and K F equal the identity on dense subsets, hence everywhere. So F is a bi-Lipschitz homeomorphism. Also E is closed, since S is closed. So the metric completion of the regular set is the whole flat cylinder H.

Exercise: Avoidance facts in dimension codimension ≥ 1 and ≥ 2.
Two elementary consequences of small dimension of a removed set E, used repeatedly. Fact 1. In a Euclidean ball or flat cylinder, if dim_H E < N − 1, any two points outside E can be joined outside E by polygonal paths of length arbitrarily close to their distance. Proof: put intermediate vertices arbitrarily near a shortest segment, on the correct side of the cones of rays through E from the endpoints. Each bad cone is a countable union of Lipschitz images of E × an interval, dimension ≤ dim_H E + 1 < N. Work in Euclidean lifts for the cylinder. Fact 2. If E is closed and dim_H E < N − 2, then the complement of E in a convex Euclidean ball is simply connected. Proof: a loop has positive distance from E and can be approximated, through a homotopy in the complement, by a polygonal loop. Cone the 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 × edge × a real interval (writing intersection as (1 − t) x + t c solves for t, c; one restricts to t bounded away from 0). Dimension ≤ dim_H E + 2 < N; a center outside all bad sets gives the required filling. Applied to E = F(S) ⊂ H with dim_H E ≤ N − 4: both facts apply. They supply the two structural facts used in proving that F is bi-Lipschitz and that its metric extends across E.

Exercise: Removing the apparent singularities E = F(S) by bounded holomorphic coframe extension.
Transfer the metric on G to H \ E; it is a flat Kähler metric g_∞, uniformly bounded above and below by positive multiples of the standard γ on H (from the previous |dF| and |(dF)^{−1}| bounds). Consider a small convex complex coordinate ball D ⊂ H. By Fact 2 (with dim_H E ≤ N − 4), D \ E is simply connected. Hence a unitary parallel holomorphic coframe exists there: θ_a = Σ_j A_{a j}(w) dw_j, with g_∞ = Σ_a |θ_a|^2 on the complement of E; the coefficient matrix A and its inverse are uniformly bounded, by the two-sided metric bounds. Each coefficient extends holomorphically across E: direct removal. Cover the compact exceptional set E ∩ D' (for D' << D) by balls with Σ r_i^{N−1} → 0, take finite subcovers, use Lipschitz cutoffs χ_k vanishing near E. Then ||dχ_k||_{L^1} → 0. Testing the CR equation for the coefficient against χ_k times a smooth test form and passing to the limit gives the distributional CR equation on the smaller ball D', hence holomorphicity by Weyl's lemma. Nonsingularity of the extended matrix at E follows from continuity + the uniform lower bound on the inverse. The identities expressing that the coframe is closed, and that the resulting metric is flat and Kähler, extend from the dense complement by continuity. So g_∞ extends as a smooth flat Kähler metric across E, and extensions on overlapping balls agree by density. Consequence: the metric-distance completion already obtained has the smooth flat-Kähler-metric distance of the extension; the apparent singularities are removable.

Exercise: Distance equality, S = ∅, and translation invariance.
Distance equality. Any piecewise smooth path in a coordinate ball can be approximated in length by paths avoiding E: polygonal approximation + Fact 1 + uniform continuity of the smooth metric. Subdivision gives the assertion for arbitrary paths. Conversely any path off E has its original length. Lemma (Regular Paths) plus completion identifies the two distances. S = ∅. The extended metric is smooth flat at every point of Z, so every point has Euclidean tangent cones, so every point is regular in the original Einstein-limit sense. Hence the singular set S = Z \ G is empty. The smooth convergence theorem for Einstein metrics then applies at every point of the cylinder, including those not initially regular. Translation invariance. Lift the extended metric to ℂ^d, the universal cover of H; flatness + simple connectivity give a global parallel unitary holomorphic coframe. Its coefficients in the w_j coordinates are bounded entire functions, by the uniform metric bounds; Liouville makes them constant. So g_∞ is translation invariant. Together with the constant limits of ω_∞ and Ω_∞ (from the next exercise), the limit is exactly the standard flat Calabi–Yau structure on the cylinder H.

Exercise: Lemma (Actual Logarithmic Convergence). Convergence in the *original* logarithmic coordinates.
We have Z ≅ H with a smooth flat metric, but we need convergence in the original coordinates (the actual arg z_j), so we can perturb a torus in the labeled angular coordinate torus. Set L_t(q) = (x_j(q) − x_j(p_t) + i arg z_j(q))_{j=1}^d ∈ H, defined on the part of Y_t in the crossing chart; Q_r = {w ∈ H : |Re w| < r}. The real shift does not change the angular coordinates. Normalize η_t by a positive scalar to get a holomorphic volume form Ω_t with the Calabi–Yau normalization for g_t. Lemma (Actual Logarithmic Convergence). For each fixed r > 0, the chart L_t contains Q_r in its image for all small t along the selected sub; the inverse Ψ_t: Q_r → Y_t satisfies Ψ_t^* g_t → g_∞, Ψ_t^* ω_t → ω_∞, Ψ_t^* Ω_t → Ω_∞ in C^∞_loc(Q_r), with constant tensors and Ω_∞ ≠ 0; and the angular coordinates of Ψ_t are the original arg z_j. Proof sketch. The inclusion Q_r ⊂ L_t(chart) is immediate from x(p_t)/s ∈ B_0 << Σ: a bounded real shift leaves all z_0, ..., z_d exponentially small. To identify inverses with the abstract GH maps: take smooth local identifications compatible with pointed GH convergence, available at every point of Z since S = ∅. The embedding-map and harmonic-coordinate arguments of Section 4 apply on each neighborhood, giving smooth local subsequential limits of L_t. On the intersection with the old regular set G, these limits equal F, because old and new identifications send each point to g_t-distance-tending-to-zero points, and interior harmonic estimates bound derivatives of the shifted logarithms and angular maps. G is dense in Z, so every local limit equals F throughout its neighborhood. The derivative of F is nonsingular; the inverse function theorem gives smooth convergence of local inverses on smaller image neighborhoods. Those inverses agree with Ψ_t because the actual logarithmic chart is injective. A finite cover of each compact subset of Q_r proves the asserted convergence of metrics and Kähler forms. This argument uses the actual arg z_j at every stage; no change of angular marking. Finally, Ω_t = (positive scalar) × f(z) dw_1 ∧ ... ∧ dw_d in these coordinates; the scalar converges to a positive limit by the coordinate-density; z → o exponentially; f(z) → f(o) with all derivatives in the w coordinates. This gives nonzero constant Ω_∞. Same for ω_t; g_∞ is already translation invariant by the preceding exercise.

Exercise: The angular torus T_0 in the flat cylinder is special Lagrangian for the limit.
Fix a tube Q_{2r} ⊂ Q_r' (with r' > r), and use Lemma (Actual Logarithmic Convergence) to regard the CY data (g_t, ω_t, Ω_t) as tensors on one fixed domain Q_{2r}. The polarizing line bundle on Y_t is topologically trivial on a sufficiently small contractible coordinate neighborhood of the maximal crossing o (its restriction to that chart has a smooth trivialization of the underlying line bundle, regardless of amplification). Therefore ω_t is exact as a 2-form on Q_{2r}: ω_t = d β_t for a 1-form β_t. The constant-form limit ω_∞ has zero period on every angular coordinate 2-torus: in the coordinates w_j = x_j + i y_j, ω_∞ = Σ_j dx_j ∧ dy_j, so ω_∞(∂_{y_i}, ∂_{y_j}) = 0, and every coefficient of ω_∞ has to vanish when evaluated on angular tangent vectors. Constant coefficients then force ω_∞|_{\{Re w = 0\}} = 0 identically. Hence T_0 := {Re w = 0} ⊂ Q_{2r} is Lagrangian for ω_∞. Moreover the constant nonzero form Ω_∞ has constant phase on T_0, because T_0 = {Re w = 0} has tangent vectors all of the form Σ c_j ∂_{y_j}, on which Ω_∞ evaluates to a single fixed complex constant independent of the point. So T_0 is special Lagrangian for the limit structure. We now perturb T_0 to a torus that is special Lagrangian for the actual (t-dependent) structure.

Exercise: Zhang–McLean persistence. From the flat torus T_0 to a smooth SL torus in Y_t.
Persistence of Calabi–Yau torus tubes: Zhang 2017, Section 4, Lemmas 4.3–4.5 (https://arxiv.org/abs/0911.1028v2). Its linearization is McLean's special Lagrangian deformation operator (McLean 1998, https://doi.org/10.4310/CAG.1998.v6.n4.a4). Setup. Identify normal vector fields along T_0 with one-forms α via contraction with ω_∞; use a fixed normal graph chart ι_α: T_0 → Q_r. Pick a large integer k and 0 < β < 1. Source space:
    V = {α ∈ C^{k+1, β}(T^* T_0) : α ⟂ H^1(T_0)},
the orthogonal complement of harmonic 1-forms; all α in a small ball give graphs in Q_r. Target space: W = exact C^{k, β} two-forms on T_0 × exact C^{k, β} top forms on T_0; both factors closed subspaces. Now calibrate the phases: the periods ∫_{T_0} Ω_t converge to the nonzero flat period (nonzero because Ω_∞ is a constant positive multiple of dw_1 ∧ ... ∧ dw_d on T_0, which has nonzero integral). Multiply each Ω_t, and its limit, by the phase making this period positive real; these phases converge. For a small graph set
    P_t(α) = (ι_α^* ω_t, ι_α^* Im Ω_t) ∈ W.
The first component lies in exact 2-forms because ω_t is exact on Q_{2r}. The second has zero integral because graph isotopy preserves ∫_{T_0} Ω_t (the form is closed), and after the phase choice Im of that integral is zero. P_∞(0) = 0. Differentiation of pullbacks of closed forms gives the linearization DP_∞(0) α = (dα, c d(*α)) with c ≠ 0; the second identity follows by contracting in an orthonormal Lagrangian frame. Invertible. Rewrite the second equation as δα = f (f mean zero). With ξ the prescribed exact two-form and Hodge Green operator G, α = δ G ξ + d G f. Hodge identities give dα = ξ and δα = f; the solution is orthogonal to harmonic forms; the only homogeneous solution in V is zero; elliptic estimates give bounded inverse with one derivative of gain. Assemble. Lemma (Actual Logarithmic Convergence) gives P_t → P_∞ in C^1 on a fixed small ball of V, with values in W; derivatives are uniformly close to the fixed invertible operator DP_∞(0) after the ball is made smaller. IFT: α_t → 0 in C^{k+1, β} with P_t(α_t) = 0 for every sufficiently late subsequence. Each resulting graph is embedded, Lagrangian, and calibrated by Re Ω_t (restriction of Ω_t to a connected Lagrangian graph has constant sign; the positive period makes it positive). Ellipticity of the minimal-graph equation plus smoothness of the ambient tensors (and Ω_t ≠ 0) gives C^∞ regularity. The isotopy ι_{u α_t}, 0 ≤ u ≤ 1, from the graph to T_0 lies in Q_r. The chart Ψ_t (Lemma Actual Logarithmic Convergence) transports to the smooth fiber Y_t, ending at a smooth embedded angular coordinate torus in the prescribed crossing chart. That is the torus required by Theorem (Torus).

Exercise: Full torus theorem for every sufficiently small nonzero t, not just along a subsequence.
The hidden quantifier. The compactness construction chose an arbitrary sequence of nonzero parameters t_ν → 0, extracted a subsequence and produced centers p_{t_ν} with noncollapsed unit balls and small curvature energy on every ball. The subsequence ambiguity came from compactness (Bishop–Gromov, Cheeger–Colding, singularity theory). Lemma (Actual Logarithmic Convergence) then gave smooth convergence of Ψ_t^* (g_t, ω_t, Ω_t) to constant tensors along that sub. The 3-Jordan-block structure of M and the nonzero residue made the resulting limiting structure the standard flat Calabi–Yau cylinder. All the intermediate arguments were uniform in t. So the construction produced SL tori on a tail of the subsequence, starting from any t_ν → 0. Hence no failure is possible at any point of a sufficiently small punctured disk. Formal version: Suppose for contradiction that the conclusion failed at some sequence t_ν → 0: choose t_ν so that no Y_{t_ν} contains a torus with the stated properties. The compactness construction gives such tori on a tail of a subsequence, a contradiction. So the assertion holds for every sufficiently small nonzero t, closing Theorem (Torus). This is the pattern for "universal quantifier from compactness": the proof iteratively extracts so many subsequences that the limit is forced; therefore a counterexample cannot exist.

Exercise: Rotation setup. What are I, J, K, and the class formulas on the chosen fiber?
Fix λ ∈ ℝ \ ℚ. Use the family of Proposition (Family) and go to a fiber Y = Y_t along the vertical ray τ = λ + i y, y >> 0, which contains the SL torus T of Theorem (Torus) (isotopic to a coordinate angle torus in the crossing chart z_0 z_1 ... z_{2n} = t). Write the present complex structure as I. Choose a hyperkähler triple (I, J, K) for the polarized Ricci-flat metric g, with IJ = K, so that (rotation–normalization):
    ω_J + i ω_K = c σ_t,     c > 0,     [ω_I] ∈ ℝ_{>0} h.
The positive scalar normalizes the length of σ_t; its phase fixes the ordered pair (J, K). Since [σ_t] = p(τ) = f + τ v − (a/2) τ^2 e:
    [ω_K] = c y (v − a λ e),      q(e, [ω_J]) = c > 0.
The second identity follows from q(e, p(τ)) = 1 and it will retain the positive sign of e after the rotation. So the triple does not live in an abstract vacuum: the classes are pinned to the period formulas obtained from the family construction. Throughout, x labels a cohomology class and we use parallel transport along the family to compare classes on different fibers.

Exercise: Lemma (Rotation Criterion). The Wirtinger-equality argument that turns SL torus into a J-complex Lagrangian torus.
Lemma (Rotation Criterion). Let (Y, g, I, J, K) be a hyperkähler manifold of real dimension 4n, n ≥ 1, IJ = K, and i: T ↪ Y a compact, connected, oriented embedded special Lagrangian submanifold for I. If i^* [ω_K] = 0 in H^2(T, ℝ), then T is a complex n-dimensional submanifold for J and is Lagrangian for σ_J = ω_K + i ω_I. If T is diffeomorphic to a real 2n-torus, then T is a complex torus for J. Proof. The normalized parallel holomorphic volume form for (Y, I, g) is Ω_I = (ω_J + i ω_K)^n / n!. In a symplectic unitary frame ω_J + i ω_K = Σ_{j=1}^n dz_{2j−1} ∧ dz_{2j}; its n-th power divided by n! is the unit volume form. On an orthonormal frame of any I-Lagrangian real 2n-plane, |Ω_I| = 1. So the SL calibration identity (Harvey–Lawson 1982, III.1 Theorem 1.10 and Corollary 1.11, https://doi.org/10.1007/BF02392726) gives vol_g(T) = |∫_T Ω_I|. Now open the wedge product: every term in the expansion carrying ω_K vanishes because i^*ω_K is exact (hypothesis) and all factors in the expansion are closed, so Stokes kills those terms; hence vol_g(T) = |∫_T ω_J^n / n!|. Wirtinger inequality (Harvey–Lawson 1982, II.6 Theorem 6.11): |ω_J^n / n!| ≤ vol_{g|_T} pointwise on real 2n-planes, with equality exactly on J-invariant 2n-planes. Thus vol_g(T) = |∫_T ω_J^n / n!| ≤ ∫_T |i^*ω_J^n / n!| ≤ vol_g(T), so all inequalities are equalities. The continuous nonnegative Wirtinger deficit has integral zero, hence is pointwise zero; so every tangent plane T_x T is J-invariant. Conclusion: T is a complex n-submanifold of Y' = (Y, J). For the Lagrangian claim: for J the holomorphic symplectic form is σ_J = ω_K + i ω_I; on T we already have ω_I|_T = 0 pointwise. For u, v ∈ T_x T, quaternion identities give ω_K(u, v) = g(IJ u, v) = ω_I(J u, v) = 0 (using J T_x T = T_x T and ω_I|_T ≡ 0). So σ_J|_T = 0. Thus T is a holomorphic Lagrangian submanifold of Y'. Now the torus case. Suppose T ≃ (S^1)^{2n}. The restriction of ω_J makes T Kähler (it is a complex n-submanifold of Y', and ω_J is Kähler on Y', hence on T). So b_1(T) = 2n and h^{1,0}(T) = n. The Albanese map α: T → Alb(T) has complex n-torus target and induces iso on integral first cohomology. Both integral cohomology rings are exterior algebras on H^1, so α^* is iso in every degree. Hence α has nonzero degree; holomorphicity makes it positive; equals one. Choose Kähler class ξ on T. Surjectivity on H^2(−, ℝ) writes ξ = α^* β for some real class β on Alb(T). If a fiber contained a positive-dimensional irreducible compact analytic V of dimension k, then ∫_V ξ^k = 0 (α|_V constant), contradicting Kähler positivity. So α is proper with finite fibers, hence finite. A finite degree-one holomorphic map to a normal complex space is an isomorphism. So T is a holomorphic Lagrangian complex torus in Y'.

Exercise: i^* [ω_K] = 0 from the coordinate isotopy class of T.
The class [ω_K] lies in the image of M − 1. To use Lemma (Rotation Criterion) we need i^* [ω_K] = 0 for our particular torus T. Here is where the isotopy class of T matters. For the coordinate torus T_0 = (ℝ/2πℤ)^{2n} with fixed radii r_1, ..., r_{2n}, go around the loop t(θ) = t exp(iθ) and define i_θ: T_0 ↪ Y_{t exp(iθ)} by z_j = r_j exp(i y_j) for 1 ≤ j ≤ 2n and z_0 = t exp(iθ) / (z_1 ... z_{2n}). All |z_j| remain fixed, so this family of embeddings stays inside the crossing chart, and the embeddings at θ = 0 and at θ = 2π are literally equal as parameterized maps of T_0. Therefore the induced restriction maps i_θ^* form a morphism from the fiber cohomology local system to the constant local system H^k(T_0, ℝ), giving i_0^* M_k = i_0^* in every degree k. In particular in degree two, i^* (M − 1) = 0 on H^2(Y, ℝ), where i = i_0 and the identification i^* ≅ (isotopy-independent) lets us use i = i_0. Now: equations from Proposition (Family) give (M − 1) P = <e, v>_ℚ, and [ω_K] = c y (v − a λ e) ∈ <e, v>_ℝ = image of M − 1 on P (extended by 0 on P^⊥, since M is identity there). So i^* [ω_K] = 0. Convenient: replacing M by its inverse or a positive power preserves the image <e, v>_ℚ, so the conclusion is independent of the monodromy convention and of the finite base changes made in constructing the family. This is what the paper calls "coordinate isotopy class": restriction to a coordinate torus in the crossing chart kills the image of monodromy.

Exercise: The rational isotropic ray after rotation: nonprojective, only Q e.
The period plane of Y' = (Y, J) is <[ω_K], [ω_I]>_ℝ = <v − a λ e, h>_ℝ. It is perpendicular to e, so e is of type (1, 1) on Y' (more correctly, e lies in H^{1,1}(Y') and in the positive cone containing the Kähler class, see below). Now let x ∈ H^{1,1}(Y') ∩ H^2(Y', ℚ). Orthogonality to the period plane says q(x, h) = 0 and q(x, v) = a λ q(x, e). The three numbers q(x, v), a, q(x, e) are rational; a ≠ 0; λ ∉ ℚ. Therefore q(x, e) = q(x, v) = 0. 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 to e and the two relations above: q(x) ≤ 0 for every rational (1,1) class x on Y', with q(x) = 0 iff x ∈ ℚ e. In particular, Y' is nonprojective: a projective hyperkähler manifold has an integral ample class of positive BBF square (Huybrechts projectivity criterion, https://arxiv.org/abs/math/0106014). The rational Picard space may contain negative-square classes; its isotropic classes are exactly on ℚ e. Equation [ω_K] = c y (v − a λ e) together with q(e, [ω_J]) = c > 0 places e in the closure of the positive cone containing the Kähler class [ω_J]. So the rotated Y' has: (i) a holomorphic Lagrangian complex torus T by Lemma (Rotation Criterion); (ii) nonprojectivity; (iii) exactly one rational isotropic (1,1) class, namely Q e.

Exercise: Theorem (GLR). Nonprojective hyperkähler with a holomorphic Lagrangian torus fibration.
Theorem (Greb–Lehn–Rollenske 2013, Theorem 4.1, https://arxiv.org/abs/1105.3410v3). If Z is a nonprojective compact irreducible hyperkähler manifold of complex dimension 2n containing a holomorphic Lagrangian complex torus S, then its algebraic dimension a(Z) equals n and it has a holomorphic algebraic reduction f: Z → B which is a Lagrangian fibration with S as a fiber. In particular the fibration has normal projective base B and provides a nontrivial semiample pullback line bundle on Z. Note the projectivity of B is not asserted verbatim in the main GLR statement alone; it follows from the explicit line-bundle construction of the algebraic reduction in (GLR Section 4.2) combined with Stein factorization. In the present paper it is used exactly in that form: "the fibration has normal projective base and provides a nontrivial semiample pullback bundle". This converts the existence of a single holomorphic Lagrangian torus on the nonprojective hyperkähler Y' into a genuine Lagrangian fibration over a projective base and hence into a nontrivial semiample line bundle on Y'.

Exercise: Theorem (SV fixed class). Semiampleness spreads through the fixed-class nef locus.
Theorem (Soldatenkov–Verbitsky 2025, Theorem 3.7(v), https://arxiv.org/html/2409.09142v3). Let N be a compact simply connected differentiable manifold of hyperkähler type with maximal holonomy. Let T°(N) be one connected component of its Teichmüller space (space of marked complex structures modulo smooth isotopy), and fix a nonzero primitive integral class e ∈ H^2(N, ℤ) with q(e) = 0. Let C_I ⊂ H^{1,1}(N, I) be the component of the positive cone containing the Kähler cone of (N, I); set
    T°_e = {[I] ∈ T°(N) : e ∈ H^{1,1}(N, I), e ∈ closure(C_I)}.
This is the "positive-sign" fixed-class locus. By (SV Proposition 2.7), T°_e is connected. For [I] ∈ T°_e, let E_I denote the unique holomorphic line bundle with c_1 = e (unique because H^1(N, O) = 0 and c_1 is injective on Pic when e is of type (1,1); on Y' we can use this as well). Define T°_{e, nef} ⊆ T°_e by requiring E_I nef, and T°_{e, sa} ⊆ T°_{e, nef} by requiring E_I semiample. Then T°_{e, sa} ≠ ∅ ⟹ T°_{e, sa} = T°_{e, nef}. Equivalently: if e is semiample at one point of its nef positive-sign locus, it is semiample everywhere in that locus. Let me emphasize the exact form used. The family formulation (SV Theorem 3.8) says: a nef isotropic bundle is semiample when its pair deforms, in a smooth connected family carrying a line bundle, to a semiample pair. The "positive-sign" hypothesis on the nef locus is automatically satisfied in our setting because e is placed in the closure of the positive cone by the period formulas (both on X and on Y'). So the locus form is enough: any single semiample point forces semiampleness everywhere on T°_{e, nef}.

Exercise: Semiampleness of e on Y' via GLR + Fujiki.
Apply Theorem (GLR) to Y' and its holomorphic Lagrangian torus T (from Lemma Rotation Criterion). Let H be an ample line bundle on the fibration base B, and A = f^*H the pullback line bundle on Y'. We compute the numerical invariants of b := c_1(A).
- A is semiample and nef.
- b ≠ 0. A semipositive representative of a globally generated power has positive trace on an open set where the map has positive rank; if b were 0 as a class, its wedge product with ω_J^{2n−1} would have positive integral yet be zero as class, contradiction.
- ∫_{Y'} b^{2n} = 0. The class b^{2n} = f^*[H]^{2n} = f^*(0) because dim B = n < 2n and [H]^{2n} = 0 for a line bundle on an n-dimensional base. Hence ∫ b^{2n} = 0.
- Fujiki: c_1(A)^{2n} = c_X q(b)^n. LHS is ∫ b^{2n} = 0; c_X ≠ 0; hence q(b) = 0.
So b is a nonzero rational (1,1) isotropic class on Y'. By the previous exercise, the only such class is on ℚ e; primitivity of e fixes b = k e for an integer k. Sign: b pairs positively with [ω_J] because A is nef with c_1(A) ≠ 0 and q(b) = 0; on the other hand e pairs positively with [ω_J] since q(e, [ω_J]) = c > 0. So k > 0. Let E_{Y'} be the unique bundle on Y' with class e; injectivity of c_1 gives A ≅ E_{Y'}^{⊗k}. A globally generated positive power of A is then a globally generated positive power of E_{Y'}. So the *specified primitive class* e is semiample on Y'.

Exercise: Return to X. Where do Y' and X sit in the same positive fixed-class locus?
The marking constructed in Proposition (Family) identifies Y = Y_t with a point of the same fixed Teichmüller component T°(N) as the original X. The hyperkähler twistor sphere through the triple (I, J, K) connects the complex structure I of Y to the complex structure J of Y' on the same marked differentiable manifold; hence Y' = (Y, J) also lies in T°(N). Now compare X and Y' in the fixed-class locus. On X: e is of type (1,1) by hypothesis (it is the class of the nef line bundle E, so sits in H^{1,1}(X, ℝ)) and has the positive sign q(e, κ_X) > 0 for every Kähler class κ_X (this is the initial sign choice). On Y': e is of type (1,1) because the rotated period plane <[ω_K], [ω_I]> = <v − aλ e, h> is perpendicular to e; and e pairs positively with [ω_J] by q(e, [ω_J]) = c > 0. So both endpoints X and Y' lie in T°_e for the fixed primitive e. Because e is semiample on Y', T°_{e, sa} is nonempty. Because e is nef on X (hypothesis), X lies in T°_{e, nef}. By Theorem (SV), T°_{e, sa} = T°_{e, nef}, so X ∈ T°_{e, sa}. Meaning: E_X is semiample on X. Choose s > 0 such that E_X^{⊗s} is globally generated. Then L^{⊗s} ≅ (E_X^{⊗s})^{⊗m_0} is globally generated. In particular the evaluation morphism H^0(X, L^{⊗s}) ⊗_ℂ O_X → L^{⊗s} is surjective at every point of X. This proves the semiampleness assertion of Theorem (Main).

Exercise: Fibration description. Which holomorphic map realizes L and how are its fibers described?
Let m > 0 make L^{⊗m} globally generated (from the last exercise, take m = m_0 s). Let g: X → Z ⊂ ℙ H^0(X, L^{⊗m})^* be the morphism to the image, and take its Stein factorization g = ν ∘ f, with f: X → B surjective with connected fibers and ν: B → Z finite. B is normal, hence B is projective because Z is, and H := ν^* O_Z(1) is ample on B. By construction L^{⊗m} ≅ f^* H.

Dimension of B. Put ℓ = c_1(L), κ a Kähler class on X. Positive sign gives q(ℓ, κ) > 0, and Fujiki with q(ℓ) = 0 gives ∫_X (κ + t ℓ)^{2n} = c_X (q(κ) + 2t q(κ, ℓ))^n. Comparing coefficients: ∫_X ℓ^k κ^{2n−k} is positive for 0 ≤ k ≤ n, zero for n < k ≤ 2n. On the other hand, g^* ω_FS represents m ℓ, is semipositive, and has generic rank dim Z = dim B. So (g^* ω_FS)^k ∧ ω_Kähler^{2n−k} has positive integral precisely for k ≤ dim B. Hence dim_ℂ B = n. So the base has dimension n.

Fiber is Lagrangian. Let F be a general smooth fiber. Its Poincaré dual class is a positive multiple of ℓ^n: choose a very ample power H^{⊗a_0} on B; its linear system has intersection that is a nonempty finite set of transverse points in the regular-value locus of f; pulling back gives the union of the corresponding smooth fibers and represents (a_0 m)^n ℓ^n; smooth proper transport gives the claim. For n ≥ 2, polarized Fujiki gives ∫_X ℓ^n κ^{n−2} σ σ̄ = 0 (every copy of ℓ is orthogonal to ℓ, σ, σ̄, so it must pair with a copy of κ; there are n copies of ℓ and n − 2 copies of κ, so every summand in the polarized Fujiki expansion vanishes). Using the fiber class: ∫_F σ|_F ∧ σ̄|_F ∧ κ|_F^{n−2} = 0; in unitary coordinates the integrand is a fixed positive constant times the squared norm of σ|_F times the Kähler volume form; so σ|_F = 0. For n = 1, a holomorphic 2-form restricted to a curve is automatically zero. So every general smooth fiber is Lagrangian of complex dim n. Extend to every irreducible fiber component by Matsushita's equidimensionality theorem (Matsushita, Math. Res. Lett. 2000, Theorem 1, https://doi.org/10.4310/MRL.2000.v7.n4.a4): its hypotheses are X Kähler, f proper surjective with normal base, general smooth fiber Lagrangian; conclusion: every irreducible fiber component has dim n and is isotropic on its smooth locus. Conversely, an ample bundle has a globally generated positive power, so L^{⊗m} ≅ f^* H implies L is semiample. So the fibration description and the semiampleness statement are equivalent.

Exercise: Corollary (Finiteness). Bounding smooth deformation types by Theorem (Main) plus boundedness.
Let X have complex dimension 2n and r = b_2(X) ≥ 5. The rational BBF form q is nondegenerate, indefinite, of signature (3, r − 3). Meyer's theorem: an indefinite rational quadratic form in at least five variables represents 0 nontrivially (Serre, A Course in Arithmetic, Chapter IV §3.2 Corollary 2, https://doi.org/10.1007/978-1-4684-9884-4). Clearing denominators and dividing by the positive gcd gives a primitive integral vector 0 ≠ e ∈ H^2(X, ℤ) with q(e) = 0. Consider periods perpendicular to e in the marked deformation component of X. Modding out the radical ℝe, e^⊥ / ℝe has signature (2, r − 4), so this period locus is nonempty. A generic period in it is perpendicular to no integral vector outside ℚ e: each additional orthogonality condition is a proper hyperplane section, and there are countably many. Period surjectivity therefore gives a deformation X_0 of X with NS(X_0) = ℤ e. There are no negative integral (1,1) classes on X_0 (because e generates the (1,1) lattice and q(e) = 0). The Kähler cone, described in terms of MBM classes, then equals the positive cone. One of e, −e lies in the closure of that cone; the corresponding unique line bundle is nonzero, nef, isotropic. Theorem (Main) makes it semiample, giving a holomorphic Lagrangian fibration on X_0. So *every* smooth deformation type as above contains a manifold with a Lagrangian fibration. Boundedness of Lagrangian-fibered irreducible holomorphic symplectic manifolds (Engel–Filipazzi–Greer–Mauri–Svaldi 2025, Theorem B and Corollary A.3, https://arxiv.org/abs/2507.00973v1) leaves only finitely many possibilities in each dimension. More precisely: their Corollary A.3 deforms the fibration to a projective one; Theorem B gives finiteness for the projective representatives; the deformations are locally trivial so preserve smoothness and give ordinary smooth deformations. Hence finitely many smooth deformation types. Note the role of b_2 ≥ 5: it enters only via Meyer's theorem (indefinite q of signature (3, r-3) needs dimension ≥ 5 for Meyer). Theorem (Main) itself has no such restriction, since its isotropic class e is part of the hypothesis.

Exercise: Corollary (Projective Case). How is ℙ^n_ℂ identified as the base and why no multiple fibers over codim-1?
Statement (companion paper: Projective-space bases of Lagrangian fibrations, OpenAI, September 2026, Theorem 1.1 plus the following paragraph on codimension-one multiple fibers; link in header). Suppose in Theorem (Main) X is additionally projective, and let f: X → B be its connected-fiber map. Then B ≃ ℙ^n_ℂ, and there do *not* exist a prime divisor D ⊂ B, an effective divisor E ⊂ X, and an integer r > 1 with f^* D = r E. Proof: choose a very ample line bundle M on X (unrelated to L); embed X into ℙ^N via M; the closed-graph embedding (f, i): X ↪ B × ℙ^N exhibits M as relatively very ample for f, so f is projective. All other hypotheses of the companion paper's Theorem 1.1 are supplied by Theorem (Main). Apply the companion: B is ℙ^n_ℂ, and the following no-multiple-fiber paragraph there gives the divisor assertion. Note: this is the one step in the paper that is genuinely outsourced. The rest of the paper proves the main theorem on its own.

Exercise: What wasn't assumed. Which hypotheses does the proof never use, and where does b_2 ≥ 5 enter?
The proof needs no projectivity of X: it works for compact Kähler X entirely. The rotated Y' is even nonprojective, and the argument runs through it explicitly using GLR's machinery. The boundedness step (finiteness corollary) uses b_2 ≥ 5, but only via Meyer's theorem to manufacture some isotropic integral vector; Theorem (Main) itself has no such restriction. No deformation type is fixed, no finiteness of n, no extra structure on π_1(X), no Kähler metric on X beyond existence. Where Kähler-ness is used: Hodge theory H^1(X, O) = 0 and c_1 injective on Pic; the Calabi–Yau theorem for the metric; the harmonic-coordinate theory; the GLR hypothesis of Kähler compactness on Y'. Where b_2 ≥ 5: Meyer's theorem for the finiteness corollary. Where n appears: Theorem (Main) for n = 1 essentially reduces to classical K3 semiampleness; the paper's proof still applies directly.

Exercise: Correctness bookkeeping. Which steps are most load-bearing and why?
(a) The labeled class preservation. e must survive every base change and modification. The congruence subgroup on Λ (integrality) plus freeness of Γ on D_h (so lifted period equality determines M on the nose) is what pins e down. Without it, we might only get e up to finite index, which would defeat the return step. (b) Coordinate isotopy of T. The torus T must be SL and must also be *isotopic to a coordinate torus in the crossing chart*; otherwise the restriction i^*[ω_K] = 0 argument fails because the local monodromy identities do not apply. The whole "smooth convergence in the actual logarithmic coordinates" technical block exists to supply exactly this. (c) λ ∉ ℚ. Choosing λ irrational forces the rotated period plane to contain no rational isotropic class except Q e. If λ were rational we could not claim Y' has only Q e as an isotropic (1,1) class. (d) Nonzero maximal residue. h(o) ≠ 0 is what makes the pointed GH limit a full cylinder with definite volume density; otherwise the torus story degenerates. (e) The 3-Jordan-block structure of M. (M − 1)^2 f = −aw^2 e ≠ 0 is needed so that [ω_K] ∈ image(M − 1) = <e, v>_ℚ; a 2-step nilpotent M would leave [ω_K] outside that image, and then i^*[ω_K] might not vanish. (f) Singular set removal. The bi-Lipschitz + coframe-extension + translation-invariance sequence is what upgrades abstract GH convergence to smooth convergence in the original log coordinates with the original angular marking. (g) Matsushita equidimensionality. Needed for *all* fiber components, not just the general ones. (h) SV fixed-class theorem. Needed for the return to X. Each of these steps is a plausible point of failure; each is explicit in the paper.

Exercise: The whole proof in one paragraph. Condense.
Let c_1(L) = m_0 e primitive integral, nef, isotropic. Choose h positive in e^⊥. Find a polarized family X_h over a smooth quasi-projective S; take the arithmetic quotient Q_h and a rational ternary subspace P = <e, f, v>_ℚ of signature (2, 1) containing e. Realize the cusp c_e via an algebraic curve, semistable-reduce, and retain e as a labeled lattice vector through the full monodromy M. Fujiki computes the volume growth c s^d ≤ ∫ |η|^2 ≤ C s^d; the maximal crossing cusp ends up with nonzero log residue. Theorem (Torus) then produces a special Lagrangian torus T ≅ {Re w = 0} in a generic smooth fiber Y_t, in a controlled coordinate isotopy class. The metric compactness machinery (Cheeger–Colding/Lott–Villani/RCD, Hua–Kell–Xia, path-lifting, coframe-removal) is what makes T exist. Choose a hyperkähler triple (I, J, K) on Y_t. Because T is SL for I and i^*[ω_K] = 0 (isotopy class), the calibration/Wirtinger equality upgrades T to a complex Lagrangian torus for J on Y' = (Y_t, J). Rotated periods give Y' nonprojective; the only rational isotropic (1,1) class on Y' lies on Q e. Greb–Lehn–Rollenske then produces a Lagrangian fibration on Y'; pulling back an ample, we find a nef isotropic b ≠ 0 with q(b) = 0, hence b = k e with k > 0; so e is semiample on Y'. The markings put X and Y' in the same positive fixed-class locus, and Soldatenkov–Verbitsky then give e semiample on X. Finally, Matsushita + Fujiki produce the half-dimensional Lagrangian fibration X → B realizing a positive power of L. The finiteness corollary follows from Meyer's theorem plus Engel–Filipazzi–Greer–Mauri–Svaldi boundedness. The projective case (base is ℙ^n_ℂ, no multiple fibers) is a corollary of a companion paper.