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@@ -1,19 +1,19 @@
 
 
 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.
+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), ./pdf/the-fibration-picture-is-tied-to-1996-0811.0639v2-https-arxiv-org-abs-hep-th-9606040v2.pdf.
 
 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).
+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, ./pdf/an-ihs-kahler-manifold-is-a-1999.pdf; erratum 2003, ./pdf/an-ihs-kahler-manifold-is-a-1999-math0106014-https-arxiv-org-abs-alg-geom-9705025.pdf).
 
 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).
+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, ./pdf/for-any-ihs-manifold-x-there-1983-1401.0479-varietes-kahleriennes-dont-la-premiere-classe-de.pdf).
 
 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).
+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, ./pdf/corollary-finiteness-for-each-fixed-n-2026-2507.00973v1-see-the-link-in-the-header-the.pdf).
 
 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),
@@ -21,16 +21,16 @@ Fujiki's relation (Fujiki 1987, https://doi.org/10.2969/aspm/01010105) says: for
 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.
+The fibration picture is tied to mirror symmetry via the Strominger–Yau–Zaslow conjecture (Strominger–Yau–Zaslow 1996, ./pdf/the-fibration-picture-is-tied-to-1996-hep-th9606040.pdf): 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, ./pdf/the-fibration-picture-is-tied-to-1996-https-arxiv-org-abs-hep-th-9606040v2.pdf; addendum 2001, ./pdf/the-fibration-picture-is-tied-to-1996-math9903045-https-arxiv-org-abs-hep-th-9606040v2.pdf; 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, ./pdf/the-fibration-picture-is-tied-to-1996-math9910021-https-arxiv-org-abs-hep-th-9606040v2.pdf) and Sawon (2003, Conjecture 4.1, ./pdf/the-fibration-picture-is-tied-to-1996-math0404362-https-arxiv-org-abs-hep-th-9606040v2.pdf) formulated the semiampleness question explicitly. Verbitsky (2010, Theorem 1.9, ./pdf/the-fibration-picture-is-tied-to-1996-0811.0639v2-https-arxiv-org-abs-hep-th-9606040v2.pdf) proved: a nontrivial nef isotropic bundle with a smooth semipositive Hermitian metric already has a section in some power. Horing–Lazic–Lehn (2025, ./pdf/the-fibration-picture-is-tied-to-1996-2508.14634v2-https-arxiv-org-abs-hep-th-9606040v2.pdf) got semiampleness in dimension four under an extra Lelong-number condition. Bayer–Macri (2014, ./pdf/the-fibration-picture-is-tied-to-1996-1301.6968v3-https-arxiv-org-abs-hep-th-9606040v2.pdf), Markman (2014, ./pdf/the-fibration-picture-is-tied-to-1996-1301.6584v3-https-arxiv-org-abs-hep-th-9606040v2.pdf), Yoshioka (2016, ./pdf/the-fibration-picture-is-tied-to-1996-1206.4838v2-https-arxiv-org-abs-hep-th-9606040v2.pdf), 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, ./pdf/the-fibration-picture-is-tied-to-1996-1105.3410v3-https-arxiv-org-abs-hep-th-9606040v2.pdf) 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.
+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 ineuality 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.
+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, ./pdf/the-metric-syz-conjecture-says-near-2022-1912.02360v1-theorem-1-1.pdf; survey ./pdf/the-metric-syz-conjecture-says-near-2022-2204.11363v1-theorem-1-1-https-arxiv-org-abs.pdf) gives such fibrations on generic regions. Blum–Liu (2026, ./pdf/the-metric-syz-conjecture-says-near-2022-2604.27890v1-theorem-1-1-https-arxiv-org-abs.pdf) and Li (2026, ./pdf/the-metric-syz-conjecture-says-near-2022-2605.00516v1-theorem-1-1-https-arxiv-org-abs.pdf) handle valuative independence and metric SYZ for maximum degenerations of polarized CY. Zhang (2017, Section 4, ./pdf/the-metric-syz-conjecture-says-near-2022-0911.1028v2-theorem-1-1-https-arxiv-org-abs.pdf) 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.
@@ -61,7 +61,7 @@ Choose f_0 ∈ P with q(e, f_0) = 1; set f = f_0 − ½ q(f_0) e; then q(f) = 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.
+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 ineuality 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.
@@ -85,22 +85,22 @@ Exercise: Global energies, vanishing average curvature, and many noncollapsed ce
 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).
+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 ineuality 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.
+The ball Poincaré ineuality for Ric ≥ 0 (Cheeger–Colding segment ineuality, 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 ineuality 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.
+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 ineuality then gives Δ|dv_ν|^2 ≥ −C |dv_ν|^2; the mean-value ineuality 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.
+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, ./pdf/unknown-nd-continue-with-f-u-1-i-1.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.
+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), ./pdf/lemma-capacity-extension-continued-we-use-2016-1308.3607v2-theorem-1-2-and-4-1-a.pdf): 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 (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 ineuality). 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 ineuality 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.
 
@@ -120,7 +120,7 @@ Exercise: The angular torus T_0 in the flat cylinder is special Lagrangian for t
 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:
+Persistence of Calabi–Yau torus tubes: Zhang 2017, Section 4, Lemmas 4.3–4.5 (./pdf/the-metric-syz-conjecture-says-near-2022-0911.1028v2-theorem-1-1-https-arxiv-org-abs.pdf). 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.
@@ -137,16 +137,16 @@ The positive scalar normalizes the length of σ_t; its phase fixes the ordered p
 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'.
+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 ineuality (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 ineualities 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.
+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, ./pdf/an-ihs-kahler-manifold-is-a-1999-math0106014-https-arxiv-org-abs-alg-geom-9705025.pdf). 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'.
+Theorem (Greb–Lehn–Rollenske 2013, Theorem 4.1, ./pdf/the-fibration-picture-is-tied-to-1996-1105.3410v3-https-arxiv-org-abs-hep-th-9606040v2.pdf). 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
@@ -172,7 +172,7 @@ Dimension of B. Put ℓ = c_1(L), κ a Kähler class on X. Positive sign gives q
 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.
+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, ./pdf/corollary-finiteness-for-each-fixed-n-2026-2507.00973v1-see-the-link-in-the-header-the.pdf) 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.