==== TABLE OF CONTENTS ====
The-Strong-Hyperkahler-SYZ-Conjecture//figures/rotation.tex
The-Strong-Hyperkahler-SYZ-Conjecture//main.tex
The-Strong-Hyperkahler-SYZ-Conjecture//preamble.tex
The-Strong-Hyperkahler-SYZ-Conjecture//sections/completion.tex
The-Strong-Hyperkahler-SYZ-Conjecture//sections/finiteness.tex
The-Strong-Hyperkahler-SYZ-Conjecture//sections/introduction.tex
The-Strong-Hyperkahler-SYZ-Conjecture//sections/metric_estimates.tex
The-Strong-Hyperkahler-SYZ-Conjecture//sections/polarized_family.tex
The-Strong-Hyperkahler-SYZ-Conjecture//sections/rotation.tex
The-Strong-Hyperkahler-SYZ-Conjecture//references.bib
===========================

==== BEGIN The-Strong-Hyperkahler-SYZ-Conjecture//figures/rotation.tex ====
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=.92,>=Stealth,font=\small]
  \coordinate (O) at (0,0);
  \coordinate (I) at (3.5,-.65);
  \coordinate (J) at (-3,-.9);
  \coordinate (K) at (0,2.65);
  \fill[blue!9] (O)--(-2.35,-.705)--(-2.35,1.35)--(0,2.055)--cycle;
  \fill[orange!14] (O)--(2.65,-.492)--(2.65,1.563)--(0,2.055)--cycle;
  \draw[->,semithick] (O)--(I) node[right,align=left] {$[\omega_I]$\\$\in\R_{>0}h$};
  \draw[->,semithick] (O)--(J) node[left] {$[\omega_J]$};
  \draw[->,semithick] (O)--(K) node[above] {$[\omega_K]\in\R_{>0}(v-a\lambda e)$};
  \node[align=center] at (-1.27,.78) {period plane\\for $I$};
  \node[align=center] at (1.42,.90) {period plane\\for $J$};
  \fill (O) circle (1.2pt);
\end{tikzpicture}
\caption{The period planes before and after rotation, inside the
positive three-space spanned by the hyperk\"ahler forms.
The class $e$ is perpendicular to $[\omega_I]$ and $[\omega_K]$,
so it becomes a $(1,1)$ class for $J$; its pairing with
$[\omega_J]$ fixes the positive sign.}
\label{fig:rotation}
\end{figure}
==== END The-Strong-Hyperkahler-SYZ-Conjecture//figures/rotation.tex ====

==== BEGIN The-Strong-Hyperkahler-SYZ-Conjecture//main.tex ====
\documentclass[11pt]{amsart}
\input{preamble}
\title[The strong hyperk\"ahler SYZ conjecture]{The strong hyperk\"ahler SYZ conjecture}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\begin{abstract}
We prove the strong hyperk\"ahler SYZ conjecture: every holomorphic line
bundle with nonzero nef isotropic first Chern class on a compact
irreducible holomorphic symplectic K\"ahler manifold is semiample.
\end{abstract}
\maketitle
\tableofcontents
\input{sections/introduction}
\input{sections/metric_estimates}
\input{sections/completion}
\input{sections/polarized_family}
\input{sections/rotation}
\input{sections/finiteness}
\enlargethispage{2\baselineskip}
\bibliographystyle{astra-plain}
\bibliography{references}
\end{document}
==== END The-Strong-Hyperkahler-SYZ-Conjecture//main.tex ====

==== BEGIN The-Strong-Hyperkahler-SYZ-Conjecture//preamble.tex ====
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\hypersetup{pdftitle={The strong hyperkahler SYZ conjecture},
 pdfauthor={OpenAI},
 pdfsubject={The strong hyperkahler SYZ conjecture}}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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==== END The-Strong-Hyperkahler-SYZ-Conjecture//preamble.tex ====

==== BEGIN The-Strong-Hyperkahler-SYZ-Conjecture//sections/completion.tex ====
\section{A flat cylinder and a special Lagrangian torus}
\label{sec:completion}

We continue the proof of Theorem~\ref{thm:torus}. All limits are taken along
the subsequence selected in the preceding section. We write $\mu$ for the
limit Riemannian volume and
\[
        H=\C^d/(2\pi i\Z^d),\qquad N=2d.
\]
The standard flat metric on $H$ is again denoted by $\gamma$.
Our goal is to identify $Z$ with this cylinder, obtain smooth convergence
in the actual logarithmic coordinates, and perturb an angular torus.
The possible singular set $S=Z\setminus G$ must be handled first:
flatness on $G$ alone does not give convergence over an entire torus.

\subsection{The coordinate map and its derivative bounds}

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

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

For every $R>0$, exhaustion of $B(p,R)\cap G$ by regular compact subsets,
smooth convergence, and the estimates at $p_t$ give, with a fixed
constant $C$,
\begin{align}
 \int_{B(p,R)\cap G}|u_j|^2\,\dd\mu
       &\le CR^2\mu(B(p,2R)),\label{eq:linear-L2}\\
 \int_{B(p,R)\cap G}|\dd u_j|^2\,\dd\mu
       &\le C\mu(B(p,2R)).\label{eq:finite-energy}
\end{align}
Enlarging the radii slightly before taking limits avoids any issue
with boundaries of balls.

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

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

\begin{lemma}\label{lem:capacity-extension}
Each $u_j$ extends to a weakly harmonic function in
$W^{1,2}_{\mathrm{loc}}(Z)$. Moreover $\lvert\dd u_j\rvert\le C$ on $G$,
with a constant independent of the point. Both $\dd F$ and its local
inverse derivatives are uniformly bounded.
\end{lemma}
\begin{proof}
\emph{Sobolev extension.}
Recall that $\dim_{\mathrm H}S\le N-4$ and
$\mu(B(q,r))\le\omega_Nr^N$. On a fixed bounded subdomain, cover its
compact singular portion by balls of radii $r_i$ with
$\sum_i r_i^{N-2}$ arbitrarily small, and take a finite subcover.
A Lipschitz cutoff on a doubled
covering ball has squared gradient integral at most $C r_i^{N-2}$.
Taking minima of the complementary cutoffs, and exhausting bounded
subdomains, gives cutoffs $\chi_k$ which vanish near $S$, converge
to one off $S$, satisfy $0\le\chi_k\le1$, and have
$\int|\dd\chi_k|^2\to0$. Their exceptional supports can also be
chosen to have measure tending to zero. All these assertions are local;
one inserts a fixed exterior cutoff when necessary.

Let $T_A(u_j)=\max(-A,\min(A,u_j))$. For fixed $A$, the products
$\chi_kT_A(u_j)$ are Sobolev functions across $S$, after extension
by zero there. To justify this before any global path-distance
assertion, insert a fixed compactly supported exterior cutoff.
For each fixed $k$ the product has compact support inside $G$.
The local agreement of distances just noted, applied on a finite
regular cover of the support, shows that the product is locally
ambient Lipschitz;
boundedness controls pairs separated by a fixed positive distance.
It is therefore an ambient Sobolev function, with its ordinary
Riemannian energy on $G$. The extra derivative term has norm at most
$A\|\dd\chi_k\|_2$. Weak compactness and lower semicontinuity give a
Sobolev extension of $T_A(u_j)$ with the energy bound inherited from
$G$. Letting $A\to\infty$ and using
\eqref{eq:linear-L2}--\eqref{eq:finite-energy} gives the claimed
Sobolev extension of $u_j$.
For a compactly supported Lipschitz function $\varphi$, harmonicity
on $G$ gives
\[
 0=\int_G \chi_k\langle\dd u_j,\dd\varphi\rangle\,\dd\mu
       +\int_G\varphi\langle\dd u_j,\dd\chi_k\rangle\,\dd\mu .
\]
The second integral tends to zero by Cauchy--Schwarz, using
\eqref{eq:finite-energy}. The first converges to the integral over
$Z$, since $S$ has zero measure. This proves weak harmonicity.

\emph{Global derivative bounds.}
We use the local harmonic gradient estimate on noncollapsed
Ricci limits, or equivalently its $\mathrm{RCD}(0,N)$ formulation
\cite[Theorem~1.2 and Theorem~4.1(a)]{HuaKellXia}:
\begin{equation}\label{eq:weak-gradient}
 \sup_{B(q,R/2)}|\dd u|
 \le \frac{C(N)}{R}
       \left(\frac{1}{\mu(B(q,2R))}
                    \int_{B(q,2R)}|u|^2\,\dd\mu\right)^{1/2}.
\end{equation}
For this signed-function version, the mean-value estimate applied
to $u^2$ bounds $\sup_{B(q,R)}|u|$ by a constant times the
normalized $L^2$ norm on $B(q,2R)$. Add twice this supremum and
apply the positive-harmonic gradient estimate on the smaller ball;
if the supremum vanishes, the conclusion is immediate. Fix $q$ and
take $R\ge 2d_Z(p,q)$. Then
$B(q,2R)\subset B(p,3R)$ and $B(p,6R)\subset B(q,7R)$,
so doubling compares $\mu(B(q,2R))$ with $\mu(B(p,6R))$.
Equation~\eqref{eq:linear-L2} bounds the right side of
\eqref{eq:weak-gradient} by a constant independent of $q$ and $R$.
Thus $|\dd u_j|\le C$ on $G$.

Holomorphicity bounds the angular derivatives as well, so that
$|\dd F|\le A$ for a fixed $A$. Since the product of its $N$ real
singular values is $J_0>0$, its smallest singular value is at least
$J_0/A^{N-1}$. This gives the inverse derivative bound.
\end{proof}

\subsection{The metric completion is the full cylinder}

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

\begin{lemma}\label{lem:regular-paths}
The intrinsic path distance on $G$ equals the restriction of $d_Z$.
In particular $F$ extends to a Lipschitz map $F:Z\to H$.
\end{lemma}
\begin{proof}
Fix $a,b\in G$ and small balls $A,B$ about them whose closures are
contained in $G$. All minimizing segments between these balls lie
in one compact ball. Cover its singular portion by sufficiently
small balls $B(q_i,r_i)$, disjoint from $A\cup B$ even after doubling,
such that $\sum_i r_i^{N-1}$ is as small as desired. Put
\[
             f=\sum_i r_i^{-1}{\bf1}_{B(q_i,2r_i)}.
\]
The volume upper bound gives
$\int f\,\dd\mu\le C\sum_i r_i^{N-1}$.
The segment inequality for nonnegative integrable functions on Ricci
limits \cite[Theorem~2.15]{CheegerColdingIII} bounds the average, over
endpoints in $A,B$, of the infimum of $\int_\gamma f$ over minimizing
segments $\gamma$ between them. Since $\int f$ can be made arbitrarily
small, some endpoints have this infimum less than one. Choose a
minimizing segment for which $\int_\gamma f<1$; the infimum need not
be attained.

A segment meeting $B(q_i,r_i)$, with its endpoints outside
$B(q_i,2r_i)$, spends length at least $r_i$ in the doubled ball;
it would have $\int f\ge1$. The chosen segment therefore misses
$S$. Short paths inside $A,B$ join it to $a,b$. Letting their radii
tend to zero proves that the path-distance infimum in $G$ is at
most $d_Z(a,b)$; the reverse inequality holds for every path.
The derivative bound of Lemma~\ref{lem:capacity-extension} now
makes $F$ Lipschitz for $d_Z$. The regular set is dense and $H$
is complete, so $F$ extends to $Z$.
\end{proof}

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

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

\begin{lemma}\label{lem:full-cylinder}
The extension $F:Z\to H$ is a bi-Lipschitz homeomorphism.
\end{lemma}
\begin{proof}
Set $E=F(S)$. Its Hausdorff dimension is at most $N-4$; we will prove
closedness only after constructing the inverse on all of $H$.
Local openness of $F$ on $G$ supplies a point of $F(G)\setminus E$.
Start there and lift any finite polygonal path in $H\setminus E$
by the local inverse on $G$. The lift cannot end at a finite
parameter: the inverse derivative bound makes it Cauchy, and
completeness supplies a limit $z\in Z$. Its image is the prescribed
point of the path. This point is outside $F(S)$, hence $z\in G$,
where the local inverse continues the lift. Injectivity on $G$
gives uniqueness. The first avoidance fact shows that
$H\setminus E\subset F(G)$ and that its inverse $K$ obeys
\[
       d_Z(K(a),K(b))\le C\,d_H(a,b)
           \quad(a,b\in H\setminus E).
\]

Both the domain and the image of $K$ are dense. The domain is
dense by the dimension bound. To see density of the image in
$Z$, use local openness of $F$ on each regular neighborhood and
the density of $G$. Completing $K$ thus gives a Lipschitz map
$K:H\to Z$. The compositions $FK$ and $KF$ equal the identity on
dense subsets and hence everywhere. This proves the assertion.
It also makes $E$ closed, since $S$ is closed.
\end{proof}

\subsection{Removing the apparent singularities}

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

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

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

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

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

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

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

\begin{lemma}[Convergence in the original logarithmic coordinates]
\label{lem:actual-log-convergence}
For every fixed $r>0$, the chart $\mathcal L_t$ contains $Q_r$ in its
image for all sufficiently small $t$ along the selected subsequence.
Its inverse $\Psi_t:Q_r\to Y_t$ satisfies
\[
 \Psi_t^*g_t\longrightarrow g_\infty,\qquad
 \Psi_t^*\omega_t\longrightarrow\omega_\infty,\qquad
 \Psi_t^*\Omega_t\longrightarrow\Omega_\infty
       \quad\hbox{in }C^\infty_{\mathrm{loc}}(Q_r),
\]
where the limits are constant tensors and $\Omega_\infty\ne0$.
The angular coordinates of $\Psi_t$ are the original $\arg z_j$.
\end{lemma}
\begin{proof}
The inclusion $Q_r\subset\mathcal L_t(Y_t\cap\text{chart})$ follows
already from $x(p_t)/s\in B_0\Subset\Sigma$: a bounded real shift leaves
all $d+1$ coordinates $z_0,\ldots,z_d$ exponentially small. The issue
is to identify these coordinate inverses with the smooth convergence
maps.

Take smooth local identifications compatible with the pointed
Gromov--Hausdorff convergence, now available at every point of $Z$.
The embedding-map and harmonic-coordinate arguments of
Section~\ref{sec:metric-estimates} apply on each such neighborhood.
They give smooth local subsequential limits of $\mathcal L_t$.
On its intersection with the old regular set, these limits equal $F$.
Indeed, the old and new identifications send each point to points
whose $g_t$-distance tends to zero; on a slightly larger smooth
neighborhood, the interior harmonic estimates bound the derivatives
of the shifted logarithms and their angular maps. Thus the two
limits coincide there. The old regular set is dense, so every local
limit equals $F$ throughout its neighborhood.

The derivative of $F$ is nonsingular. The inverse function theorem
therefore gives smooth convergence of the local inverse maps on
smaller image neighborhoods. Those inverses agree with $\Psi_t$,
because the actual logarithmic chart is injective. A finite cover
of each compact subset of $Q_r$ now proves the asserted convergence
of the metrics and K\"ahler forms. This argument uses the actual
circle-valued functions $\arg z_j$ at every stage; no change of their
angular marking occurs.

Finally, in these coordinates $\Omega_t$ is a positive normalizing
scalar times $f(z)\,\dd w_1\wedge\cdots\wedge\dd w_d$.
The scalar converges to a positive limit by
\eqref{eq:coordinate-density}. On every fixed tube, $z\to o$
exponentially and $f(z)\to f(o)$ with all derivatives in the $w$
coordinates. This gives the asserted nonzero constant limit of
$\Omega_t$. The metric limit is translation invariant by the
preceding argument, so all three limiting tensors are constant.
\end{proof}

\subsection{A special Lagrangian torus in the nearby fibers}

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

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

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

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

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

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

The initial sequence was arbitrary. If the conclusion failed for
arbitrarily small nonzero parameters, choose $t_\nu\to0$ so that no
$Y_{t_\nu}$ contained a torus with the stated properties. The preceding
construction gives such tori on a tail of a subsequence, a contradiction.
This proves the assertion for every sufficiently small nonzero
parameter and completes the proof of Theorem~\ref{thm:torus}.
==== END The-Strong-Hyperkahler-SYZ-Conjecture//sections/completion.tex ====

==== BEGIN The-Strong-Hyperkahler-SYZ-Conjecture//sections/finiteness.tex ====
\section{Finiteness of smooth deformation types}
\label{sec:finiteness}

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

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

\begin{proof}[Proof of Corollary~\ref{cor:finiteness}]
Let $X$ have complex dimension $2n$ and $r=b_2(X)\ge5$.
Its rational BBF form is nondegenerate and indefinite, of signature
$(3,r-3)$. Meyer's theorem says that an indefinite rational quadratic
form in at least five variables represents zero nontrivially
\cite[Chapter~IV, \S3.2, Corollary~2]{SerreArithmetic}. Clearing denominators and dividing by the positive
divisibility supplies a primitive integral vector
\[
             0\ne e\in H^2(X,\Z),\qquad q(e)=0.
\]

Consider periods perpendicular to $e$ in the marked deformation
component of $X$. The quotient $e^\perp/\R e$ 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 such additional orthogonality condition is a proper
hyperplane section, and there are countably many of them.
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 $X_0$.
The K\"ahler-cone description recalled in
Section~\ref{sec:family} consequently identifies its K\"ahler cone
with its positive cone. One of $e,-e$ lies in the closure of
that cone; the corresponding unique line bundle is nonzero,
nef, and isotropic.

Theorem~\ref{thm:main} makes this bundle semiample and gives a
holomorphic Lagrangian fibration on $X_0$. Thus every smooth
deformation type under consideration contains a manifold with
such a fibration. The smooth consequence of
\cite[Theorem~B and Corollary~A.3]{EFGMS} stated above leaves
only finitely many possibilities in dimension $2n$.
\end{proof}

The restriction $b_2\ge5$ enters only through Meyer's theorem.
Theorem~\ref{thm:main} itself has no such restriction, since its
isotropic class is part of the hypothesis.
==== END The-Strong-Hyperkahler-SYZ-Conjecture//sections/finiteness.tex ====

==== BEGIN The-Strong-Hyperkahler-SYZ-Conjecture//sections/introduction.tex ====
\section{Introduction}

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

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

\begin{theorem}\label{thm:main}
Let $X$ be a compact irreducible holomorphic symplectic K\"ahler
manifold of complex dimension $2n$, where $n\ge1$. If a holomorphic
line bundle $L$ satisfies
\[
 c_1(L)\ne0,\qquad c_1(L)\in\overline{\mathcal K_X},
                  \qquad q_X(c_1(L))=0,
\]
then $L$ is semiample.

More precisely, there exist a positive integer $m$, a normal
projective variety $B$, an ample line bundle $A$ on $B$, and a
surjective holomorphic map with connected fibers
\[
                 f:X\longrightarrow B,\qquad L^m\simeq f^*A,
\]
such that $\dim_{\C}B=n$. Every irreducible component of every
fiber has complex dimension $n$, and $\sigma$ vanishes on its
smooth locus.
\end{theorem}

Theorem~\ref{thm:main} resolves the strong hyperk\"ahler SYZ
conjecture positively. The conclusion holds on the given manifold
$X$: a positive tensor power of $L$ has no base points anywhere.
No restriction on the dimension, deformation type, or second Betti
number is imposed.

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

\begin{corollary}\label{cor:finiteness}
For each integer $n\ge1$, there are only finitely many smooth
deformation types of compact irreducible holomorphic symplectic
K\"ahler manifolds of complex dimension $2n$ with $b_2\ge5$.
\end{corollary}

This follows by combining Theorem~\ref{thm:main} with the boundedness
of Lagrangian-fibered symplectic varieties proved by
Engel--Filipazzi--Greer--Mauri--Svaldi~\cite{EFGMS}.
Section~\ref{sec:finiteness} gives the deduction in the smooth
category, including the role of the hypothesis $b_2\ge5$.

\begin{corollary}[Projective case]\label{cor:projective-base}
In Theorem~\ref{thm:main}, assume additionally that $X$ is projective,
and let $f:X\to B$ be the associated connected-fiber map furnished there.
Then $B\simeq\mathbb P^n_{\C}$. Moreover, there do not exist a prime
divisor $D\subset B$, an effective divisor $E$ on $X$, and an integer
$r>1$ such that $f^*D=rE$.
\end{corollary}

This is a consequence of the projective-space base theorem in the
companion \cite[Theorem~1.1 and the following paragraph on
codimension-one multiple fibers]{ProjectiveSpaceBases2026}.
Its application is given after the proof of Theorem~\ref{thm:main},
in Section~\ref{subsec:projective-base}.

\subsection{Context}

The fibration picture is related to the special Lagrangian formulation
of mirror symmetry proposed by Strominger--Yau--Zaslow~\cite{SYZ}.
In the hyperk\"ahler setting, rotation of complex structure relates
special Lagrangian and holomorphic Lagrangian geometry; the original
paper already discusses this relation for K3 surfaces
\cite[Section~4.1]{SYZ}.
Early explicit nef-isotropic formulations appear in
Hassett--Tschinkel~\cite[Remark~3.12]{HassettTschinkel} and
Sawon~\cite[Conjecture~4.1]{Sawon}. The former asks for
semiampleness, while the latter relates the existence of a nef
square-zero divisor to the existence of an abelian fibration.
Matsushita established the
half-dimensional and Lagrangian character of fibrations from
projective irreducible symplectic manifolds
\cite{Matsushita1999,MatsushitaAddendum}; his equidimensionality theorem
also controls every component of every fiber
\cite[Theorem~1]{MatsushitaEquidimensional}.
Thus, once semiampleness is known, the geometry of the resulting map
fits the expected holomorphic symplectic fibration picture.

Analytic positivity gives another source of progress. Verbitsky proves
that a nontrivial nef isotropic bundle with a smooth semipositive
Hermitian metric has a nonzero section in some positive power
\cite[Theorem~1.9]{VerbitskySYZ}.
H\"oring--Lazi\'c--Lehn obtain nonvanishing or coisotropic-locus
alternatives for nef isotropic classes. In complex dimension four,
they prove semiampleness if the class contains a closed positive
current whose Lelong numbers all vanish
\cite[Theorem~D and Corollary~E]{HoringLazicLehn}.
These results show how additional analytic positivity or geometric
information can turn the numerical condition into sections.

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

The task left by this deformation theorem is to produce one
semiample deformation of the \emph{prescribed} class. We do so by
constructing a torus and rotating the complex structure. The
geometric input after rotation is the theorem of Greb--Lehn--Rollenske:
a nonprojective hyperk\"ahler manifold containing a holomorphic
Lagrangian torus has a holomorphic Lagrangian fibration
\cite[Theorem~4.1]{GLR}.

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

On the analytic side, Zhang develops the persistence of special
Lagrangian tori in Calabi--Yau structures close to a flat cylinder
\cite[Section~4]{Zhang}, using McLean's deformation
operator~\cite{McLean}. Li's Fermat-family theorem gives special
Lagrangian torus fibrations on generic regions whose relative volumes
tend to one along a subsequence~\cite[Theorem~1.1]{LiFermat};
\cite{LiSurvey} surveys the metric SYZ framework.
For polarized algebraic maximal degenerations of compact Calabi--Yau
manifolds, Blum--Liu's valuative
independence theorem~\cite[Theorem~1.1]{BlumLiu2026} and Li's
comparison theorems~\cite[Theorems~1.4 and~1.8]{Li2026}
give special Lagrangian torus fibrations on open sets whose normalized
Calabi--Yau volumes are arbitrarily close to one near the degeneration.
The semiampleness argument needs a local torus together with its coordinate
isotopy class, which connects its geometry to the prescribed monodromy.
We obtain that torus by proving convergence to a full smooth flat
cylinder in the actual logarithmic coordinates; the final perturbation
uses the Zhang--McLean method.

\subsection{The proof in four steps}

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

\begin{enumerate}[leftmargin=*,label=\textup{(\arabic*)}]
\item \emph{Retain the prescribed class in monodromy.}
Keep $e$ as a labelled integral cohomology class; it need not remain
of type $(1,1)$ during this construction.
Choose a polarization on a deformation of $X$ whose marked class
$h$ is perpendicular to $e$.
A rational three-dimensional period subspace containing $e$
gives an arithmetic curve with a cusp labelled by $\Q e$.
We realize that curve by an algebraic family and track its full
second-cohomology monodromy through semistable reduction
(Proposition~\ref{prop:family}).

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

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

\item \emph{Return to the given manifold.}
The markings and the positive sign of $e$ place $X$ and $Y'$ in
the same fixed-class deformation locus. The
Soldatenkov--Verbitsky theorem then gives semiampleness on $X$
itself. Fujiki's formula and Matsushita's equidimensionality theorem
give the stated fiber properties.
\end{enumerate}

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

\subsection{Organization and conventions}

Sections~\ref{sec:metric-estimates} and~\ref{sec:completion} prove
the torus theorem independently of hyperk\"ahler period theory.
Section~\ref{sec:family} constructs the prescribed degeneration;
Lemma~\ref{lem:polarized-family-construction} separates the underlying
polarized family and free period quotient from the cusp calculation.
Section~\ref{sec:rotation} rotates the torus and proves
Theorem~\ref{thm:main}; Section~\ref{sec:finiteness} deduces
Corollary~\ref{cor:finiteness}.

Unspecified cohomology coefficients are real. We normalize $q$
integrally; its signature is $(3,b_2-3)$, and the same symbol
denotes its associated symmetric bilinear form. Fujiki's relation is
\begin{equation}\label{eq:fujiki}
        \int_X a^{2n}=c_Xq(a)^n,\qquad c_X>0.
\end{equation}
The external inputs are stated where used: the Calabi--Yau theorem,
hyperk\"ahler period and K\"ahler-cone theory, arithmetic period-map
algebraicity, semistable reduction, noncollapsed Einstein-limit
and harmonic-function theory, and the fibration, deformation, and
boundedness theorems cited above.
==== END The-Strong-Hyperkahler-SYZ-Conjecture//sections/introduction.tex ====

==== BEGIN The-Strong-Hyperkahler-SYZ-Conjecture//sections/metric_estimates.tex ====
\section{Metric estimates at the torus scale}
\label{sec:metric-estimates}

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

\begin{theorem}[A special Lagrangian torus near a maximal crossing]
\label{thm:torus}
Let $\pi:\mathcal Y\to\Delta$ be the restriction to a disk of a projective
algebraic family over a smooth curve.  Assume that $\mathcal Y$ is smooth,
$\mathcal Y_0$ is a reduced simple normal crossings divisor, and the fibers
$Y_t$ for $t\ne0$ are smooth connected projective manifolds of complex
dimension $d\ge2$ with trivial canonical bundle.  Let $a_t$ be polarizing
classes given by a line bundle on $\mathcal Y$; this line bundle need not
be relatively ample at $0$.  Let $\eta_t$ be nowhere vanishing holomorphic
$d$-forms which extend as a section of the relative logarithmic canonical
line bundle.  Put $s=-\log|t|$, and suppose that, for constants $c,C>0$,
\begin{equation}
 c s^d\le I_t:=\int_{Y_t}|\eta_t|^2\le C s^d,
 \qquad 0<|t|\ll1.
 \label{eq:log-volume-growth}
\end{equation}
Suppose, moreover, that at some point $o\in\mathcal Y_0$ exactly $d+1$
components meet and that the logarithmic coefficient of $\eta$ at $o$ is
nonzero.  Thus in a coordinate neighborhood of $o$,
\begin{equation}
 z_0z_1\cdots z_d=t,
 \qquad
 \eta_t=f(z)\,\frac{\dd z_1}{z_1}\wedge\cdots\wedge
                         \frac{\dd z_d}{z_d},
 \qquad f(o)\ne0.
 \label{eq:crossing-chart}
\end{equation}
Equip $Y_t$ with the Ricci-flat K\"ahler metric in $a_t$.
For every sufficiently small nonzero $t$, the fiber $Y_t$ contains a
smooth embedded special Lagrangian real $d$-torus isotopic in $Y_t$ to an
angular coordinate torus in \eqref{eq:crossing-chart}.
Here special Lagrangian means that the K\"ahler form restricts to zero and
that $\eta_t$ has constant phase on the oriented torus.
\end{theorem}

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

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

\subsection{Coordinate volume and mixed intersections}

Let $g_t^{(0)}$ and $\omega_t^{(0)}$ be the Ricci-flat metric and K\"ahler
form supplied by the Calabi--Yau theorem \cite{Yau}, with
$[\omega_t^{(0)}]=a_t$.  The useful scale is
\begin{equation}
 g_t=s g_t^{(0)},\qquad \omega_t=s\omega_t^{(0)},\qquad
 \mu_t=\dd V_t=\frac{\omega_t^d}{d!},\qquad
 \mu_t(Y_t)=c_a s^d,
 \quad c_a=\frac1{d!}\int_{Y_t}a_t^d>0.
 \label{eq:torus-scaling}
\end{equation}
The last constant is independent of $t$ by topological local triviality
over the punctured disk.  Scaling does not change either of the special
Lagrangian conditions.

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

Use the convention
$|\eta_t|^2=2^{-d}i^{d^2}\eta_t\wedge\overline{\eta_t}$, so that
\eqref{eq:crossing-chart} gives
$|\eta_t|^2=|f(z)|^2\dd V_\gamma$ in the chart.  Ricci-flatness implies
that its volume form is a constant multiple of $|\eta_t|^2$; integration
determines that constant.  Hence
\begin{equation}
 \dd V_t=\frac{c_a s^d}{I_t}|\eta_t|^2,
 \qquad
 \frac{\dd V_t}{\dd V_\gamma}
       =\frac{c_a s^d}{I_t}|f(z)|^2
       \longrightarrow\vartheta>0
 \quad\hbox{uniformly on }\mathcal T_t(B)
 \label{eq:coordinate-density}
\end{equation}
after a subsequence.  To obtain the convergence, first extract a limit
of $c_a s^d/I_t$ using \eqref{eq:log-volume-growth}, and then use
$f(z)\to f(o)\ne0$.  In particular, the two volume measures are uniformly
comparable on each fixed interior tube.

\begin{lemma}[Mixed-intersection estimates]
\label{lem:mixed-estimates}
For every fixed $B\Subset\Sigma$ and $0\le k\le d$,
\begin{equation}
 \int_{\mathcal T_t(B)}\omega_\gamma^k\wedge\omega_t^{d-k}
       \le C_B s^d.
 \label{eq:mixed-estimates}
\end{equation}
Consequently, with traces taken for real Riemannian metrics,
\begin{equation}
 \int_{\mathcal T_t(B)}\tr_\gamma g_t\,\dd V_\gamma\le C_Bs^d,
 \qquad
 \int_{\mathcal T_t(B)}\tr_{g_t}\gamma\,\dd V_t\le C_Bs^d.
 \label{eq:trace-estimates}
\end{equation}
\end{lemma}

\begin{proof}
Resolve the finitely many rational maps $z_i:\mathcal Y\dashrightarrow
\mathbb P^1$ simultaneously.  We obtain a projective modification
$\varpi:\widehat{\mathcal Y}\to\mathcal Y$ and morphisms
$f_i:\widehat{\mathcal Y}\to\mathbb P^1$, with $\varpi$ an isomorphism
over their common regular locus, in particular over the coordinate
neighborhood being used.  After shrinking the punctured disk, its
nearby fibers $\widehat Y_t$ are smooth.  This follows from generic
smoothness for the fixed algebraic model; no resolution depending on
$s$ is required.

Fix a Fubini--Study form $\omega_{\rm FS}$ on $\mathbb P^1$.  For
$A_b([u:v])=[e^bu:v]$, define the smooth closed semipositive form
\[
 \alpha_{s,t}=\frac1s\int_0^s\sum_{i=1}^d
                f_i^* A_b^*\omega_{\rm FS}\,\dd b
 \quad\hbox{on }\widehat Y_t.
\]
Its cohomology class is the restriction of
$\sum_i f_i^*[\omega_{\rm FS}]$, independently of $s$, since each $A_b$
is homotopic to the identity.  Thus every intersection
\begin{equation}
 \int_{\widehat Y_t}\alpha_{s,t}^k\wedge
                  (\varpi^*\omega_t^{(0)})^{d-k}
 \label{eq:fixed-mixed-intersection}
\end{equation}
is a fixed finite number.  Both classes involved extend over the fixed
model, so this number is independent of $t$ as well.

In a logarithmic coordinate $w_i=x_i+iy_i$, the $i$th summand before
averaging is a fixed positive constant times
\[
 \frac{e^{2(b+x_i)}}{(1+e^{2(b+x_i)})^2}
                   \dd x_i\wedge\dd y_i.
\]
For $x/s\in B$, both $-x_i$ and $s+x_i$ are at least
$\delta_Bs$.  Consequently the interval
$[-x_i-1,-x_i+1]$ lies in $[0,s]$ for large $s$.  On this interval the
displayed scalar factor is bounded below by a positive constant.
Summing over $i$ proves, as Hermitian forms on the tube,
\[
 \alpha_{s,t}\ge\frac{c_B}{s}\omega_\gamma.
\]
The pullback $\varpi^*\omega_t^{(0)}$ is semipositive, including on the
exceptional divisors.  Positivity of wedge products therefore gives
\[
 \begin{split}
 \int_{\mathcal T_t(B)}\omega_\gamma^k\wedge\omega_t^{d-k}
 &\le C_Bs^k s^{d-k}
       \int_{\widehat Y_t}\alpha_{s,t}^k\wedge
                        (\varpi^*\omega_t^{(0)})^{d-k}\\
 &\le C_B' s^d.
 \end{split}
\]
This proves \eqref{eq:mixed-estimates}.  Finally, for Hermitian metrics
$h,k$ with K\"ahler forms $\omega_h,\omega_k$, the real trace identity is
\[
 (\tr_h k)\,\dd V_h
       =\frac{2}{(d-1)!}\omega_k\wedge\omega_h^{d-1}.
\]
Use $k=1$ and $k=d-1$ in \eqref{eq:mixed-estimates} to obtain the two
assertions in \eqref{eq:trace-estimates}.
\end{proof}

\subsection{Global energies and many noncollapsed centers}

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

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

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

\begin{lemma}[A positive fraction of noncollapsed centers]
\label{lem:many-noncollapsed-centers}
There exist constants $\delta,v>0$ and measurable sets
$E_t\subset\mathcal T_t(B_0)$ such that
\begin{equation}
 \mu_t(E_t)\ge\delta\mu_t(Y_t),\qquad
 \mu_t(B_{g_t}(q,1))\ge v\quad(q\in E_t).
 \label{eq:many-noncollapsed-centers}
\end{equation}
All balls here and below use the intrinsic distance of the complete
manifold $(Y_t,g_t)$.
\end{lemma}

\begin{proof}
Fix a rectangular angle chart of positive size in $(\R/2\pi\Z)^d$.
The region $sB_0$ contains at least $c_0s^d$ disjoint translates of a
fixed small $d$-dimensional cube.  Taking their products with one fixed
angle cube gives at least $c_0s^d$ disjoint convex coordinate boxes $Q$,
all of the same Euclidean size, inside $\mathcal T_t(B_0)$.
By the first estimate of \eqref{eq:trace-estimates}, at least half of
these boxes satisfy
\begin{equation}
 \int_Q\tr_\gamma g_t\,\dd V_\gamma\le C_0
 \label{eq:good-box-trace}
\end{equation}
for a constant $C_0$ independent of $t$.  By
\eqref{eq:coordinate-density}, every such box has
$m_0\le\mu_t(Q)\le m_1$ for fixed positive $m_0,m_1$.

For $a,b\in Q$, the straight coordinate segment lies in $Q$, and its
length bounds the intrinsic distance from above.  Consequently
\[
 d_{g_t}(a,b)\le |a-b|_\gamma
   \int_0^1\sqrt{\tr_\gamma g_t((1-r)a+rb)}\,\dd r.
\]
Write $h=\sqrt{\tr_\gamma g_t}$.  For $0\le r\le1/2$, first integrate
in $a$ and change variables $z=(1-r)a+rb$.  Its Jacobian is
$(1-r)^N\ge2^{-N}$ and $z$ stays in $Q$.  Thus
\[
 \int_Q\int_Q h((1-r)a+rb)\,\dd a\,\dd b
       \le2^N\vol_\gamma(Q)\int_Qh\,\dd V_\gamma.
\]
For $1/2\le r\le1$, use $b$ first and the Jacobian $r^N$ instead.
The Euclidean diameter and volume of $Q$ are fixed.  Cauchy--Schwarz and
\eqref{eq:good-box-trace} now bound the double integral of
$d_{g_t}(a,b)$ uniformly.  Comparing measures gives a uniform constant
$C_1$ such that
\[
 \frac1{\mu_t(Q)^2}\int_Q\int_Qd_{g_t}(a,b)\,
                       \dd V_t(a)\,\dd V_t(b)\le C_1.
\]

The set $E_Q$ of $a\in Q$ for which
$\mu_t(Q)^{-1}\int_Qd_{g_t}(a,b)\dd V_t(b)\le2C_1$
has measure at least $\mu_t(Q)/2$.  For each such $a$, a second application
of Markov's inequality gives
\[
 \mu_t\bigl(B_{g_t}(a,R_0)\bigr)\ge\mu_t(Q)/2\ge m_0/2,
 \qquad R_0=\max\{1,4C_1\}.
\]
Since $\Ric(g_t)=0$, Bishop--Gromov comparison yields
$\mu_t(B_{g_t}(a,1))\ge R_0^{-N}m_0/2$.
Finally take the union of $E_Q$ over the disjoint good boxes.  There are
at least $(c_0/2)s^d$ of them and each contributes measure at least
$m_0/2$.  In view of \eqref{eq:torus-scaling}, this union has a fixed
positive fraction of the total volume, as required.
\end{proof}

\subsection{One center with estimates at every radius}

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

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

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

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

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

\subsection{The regular part of the pointed limit}

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

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

Every compact smooth region lies in the image of a fixed-radius ball
about $p_t$ for all sufficiently small $t$ along the sequence.  By
\eqref{eq:good-center} and the volume upper bound, its curvature energy
is at most $\epsilon_t\omega_NR^N$, which tends to zero.  Smooth metric
convergence therefore implies
\begin{equation}
 \Rm_g=0\quad\hbox{on }G.
 \label{eq:flat-regular-set}
\end{equation}
It remains to put these smooth regions in the coordinate chart about
$o$.  We use the following small-energy argument for the maps $\Phi_t$.

\begin{lemma}[Vanishing energy on a smooth domain]
\label{lem:small-energy}
Let $h_\nu$ be metrics converging smoothly on a fixed smooth domain $D$
of real dimension $N$,
and let $\psi_\nu:(D,h_\nu)\to P$ be smooth harmonic maps to a fixed
compact Riemannian manifold.  If their energies tend to zero on every
relatively compact subdomain, then
$\sup_K|\dd\psi_\nu|\to0$ for every $K\Subset D$.
After extraction they converge smoothly to a constant on each connected
relatively compact domain.
\end{lemma}

\begin{proof}
Smooth harmonic maps are stationary, which is the property used in the
energy monotonicity argument below.  Fix nested relatively
compact domains around $K$.  Their metrics have uniform smooth bounds,
and geodesic balls with centers in an intermediate domain have a
uniformly positive admissible radius $r_0$.
For a smooth harmonic map the stress tensor
$|\dd\psi|^2h-2\psi^*g_P$ has zero divergence.  Testing this identity
against radial vector fields and using
$\nabla^2(r^2/2)=h+O(r^2)$ in these geodesic balls gives
\[
 \Theta_\nu'(q,r)\ge-Cr\Theta_\nu(q,r),\qquad
 \Theta_\nu(q,r)=r^{2-N}
       \int_{B_{h_\nu}(q,r)}|\dd\psi_\nu|^2\,\dd V_{h_\nu}.
\]
The nonnegative radial derivative term has been discarded here.
Integration gives the uniform monotonicity consequence
\begin{equation}
 \Theta_\nu(q,r)\le C\Theta_\nu(q,r_0)
     \le Cr_0^{2-N}E_\nu\longrightarrow0
       \quad(0<r\le r_0),
 \label{eq:harmonic-map-monotonicity}
\end{equation}
where $E_\nu$ is the energy on a fixed larger domain.  This is the
stationary harmonic-map monotonicity argument; see also \cite{Lin}.

If derivatives were unbounded on $K$, maximize
$\operatorname{dist}_{h_\nu}(q,\partial D')
 |\dd\psi_\nu(q)|$ on a fixed intermediate domain $D'$.
At maximizing points $q_\nu$, put
$L_\nu=|\dd\psi_\nu(q_\nu)|$.  Then $L_\nu\to\infty$ and the product
just maximized tends to infinity.  On the ball of radius
$L_\nu^{-1}$ about $q_\nu$, point selection gives
$|\dd\psi_\nu|\le2L_\nu$ for large $\nu$.  Rescale the domain metric by
$L_\nu^2$.  The rescaled maps $v_\nu$ on unit balls satisfy
$|\dd v_\nu(0)|=1$ and $|\dd v_\nu|\le2$, while the rescaled domain
metrics converge smoothly to a Euclidean metric.  The harmonic-map
Bochner inequality and these derivative bounds imply
\[
 \Delta|\dd v_\nu|^2\ge-C|\dd v_\nu|^2.
\]
The local mean-value inequality, with uniform constants on these
smooth balls, now gives
\[
 1\le C\int_{B(0,1)}|\dd v_\nu|^2
   =C L_\nu^{N-2}
          \int_{B_{h_\nu}(q_\nu,L_\nu^{-1})}
                         |\dd\psi_\nu|^2\,\dd V_{h_\nu}
   =C\Theta_\nu(q_\nu,L_\nu^{-1})\longrightarrow0,
\]
a contradiction.  Thus the derivatives are uniformly bounded on
interior domains.  Apply the same Bochner and mean-value argument on
fixed smaller balls: it gives
$\sup_K|\dd\psi_\nu|^2\le C E_\nu\to0$.
Compactness of $P$ and interior elliptic estimates for the smooth
harmonic-map equation then give smooth subsequential convergence; the
limit has zero derivative and is constant on connected domains.
\end{proof}

The maps $\Phi_t$ are holomorphic between K\"ahler manifolds and hence
harmonic.  On a compact subdomain of $G$, pull them back using the
smooth convergence identifications.  Their energies tend to zero by
\eqref{eq:good-center}, so Lemma~\ref{lem:small-energy} applies.
Every local constant limit is $\Phi(o)$.  Indeed a fixed smaller regular
ball has positive limiting volume, whereas the exceptional set in
\eqref{eq:coordinate-concentration} has volume tending to zero.
That ball therefore contains points with $x/s$ in a fixed compact
subset of $\Sigma$.  Such points converge to $o$ in the total space,
and their images converge to $\Phi(o)$.  Local uniform convergence of
the maps identifies the constant on each regular ball separately,
regardless of whether $G$ is connected.
It follows, by a finite cover, that
\begin{equation}
 \Phi_t\longrightarrow\Phi(o)
       \quad\hbox{uniformly on every compact subset of }G.
 \label{eq:embedding-map-limit}
\end{equation}

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

==== BEGIN The-Strong-Hyperkahler-SYZ-Conjecture//sections/polarized_family.tex ====
\section{A polarized degeneration with the prescribed isotropic class}
\label{sec:family}

The degeneration must supply two kinds of data. Its volume growth and
a nonzero residue at a maximal crossing will allow us to apply
Theorem~\ref{thm:torus}. Its marked periods and second-cohomology
monodromy will retain the prescribed isotropic class for rotation.
We construct both in the same polarized family.
All markings take values in
\[
 \Lambda=H^2(X,\Z),\qquad \Lambda_{\Q}=\Lambda\otimes_{\Z}\Q,
 \qquad r=\operatorname{rank}\Lambda,
\]
with its integral Beauville--Bogomolov--Fujiki form $q$.  We write
$q(x,y)$ for the associated symmetric bilinear form, so that
$q(x)=q(x,x)$.  Markings are obtained by deformation and parallel
transport from the differentiable manifold underlying $X$.
Fix the connected Teichm\"uller component containing $X$. Along
deformation paths use Ehresmann trivializations anchored at $X$;
the transported complex structures on its differentiable manifold
then stay in this component.

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

\begin{samepage}
\begin{proposition}[The prescribed-cusp family]
\label{prop:family}\label{prop:polarized-family}
Put $d=2n$.  There exist a primitive vector $h\in\Lambda$ with
$q(h)>0$ and $q(h,e)=0$, a rational subspace
\[
 P=\langle e,f,v\rangle_{\Q}\subset h^\perp,
 \qquad \operatorname{sign}(q|_P)=(2,1),
\]
and a projective semistable family $\pi:\mathcal Y\to\Delta$ with
the following properties.  Here $\Delta$ is a sufficiently small disk
in a smooth algebraic curve, and $\Delta^*=\Delta\setminus\{0\}$.
\begin{enumerate}
\item The total space $\mathcal Y$ is smooth, the central fiber is a
reduced simple normal crossings divisor, and every $Y_t$, $t\ne0$,
is an irreducible holomorphic symplectic manifold in the marked
deformation component of $X$.  A line bundle $\mathcal A$ on
$\mathcal Y$ restricts to an ample bundle on every $Y_t$ with
$c_1(\mathcal A|_{Y_t})=kh$ for a fixed positive integer $k$ under
parallel transport.  The family is the analytic restriction of an
algebraic family.
\item The rational vectors can be chosen so that
\begin{equation}
 \label{eq:family-hyperbolic-basis}
 q(e)=q(f)=0,\quad q(e,f)=1,\quad
 q(e,v)=q(f,v)=0,\quad a:=q(v)>0.
\end{equation}
On the universal cover of $\Delta^*$ the marked Hodge line is spanned
by the normalized vector
\begin{equation}
 \label{eq:family-period}
 p(\tau)=f+\tau v-\frac a2\tau^2e,
 \qquad \operatorname{Im}\tau>0.
\end{equation}
The function $\tau$ tends to the cusp represented by the original
line $\Q e$, and
\begin{equation}
 \label{eq:family-period-identities}
 q(p(\tau))=0,\qquad q(p(\tau),e)=1,\qquad
 q(p(\tau),\overline{p(\tau)})
       =2a(\operatorname{Im}\tau)^2.
\end{equation}
For a consistent convention for second-cohomology monodromy there
is a rational $w>0$ such that
\begin{equation}
 \label{eq:family-monodromy}
 Me=e,\qquad Mv=v-aw e,\qquad
 Mf=f+w v-\frac{aw^2}{2}e,
 \qquad M|_{P^\perp}=1.
\end{equation}
The orthogonal complement in the last formula is taken in
$\Lambda_{\Q}$.  In particular,
\begin{equation}
 \label{eq:family-monodromy-image}
 (M-1)P=\langle e,v\rangle_{\Q},\quad
 (M-1)^2f=-aw^2e\ne0,\quad (M-1)^3=0.
\end{equation}
The same image statement holds if the monodromy convention uses
$M^{-1}$.
\item There is a single-valued holomorphic relative symplectic form
$\sigma_t$ on the smooth family, uniquely normalized by
$q([\sigma_t],e)=1$.  The forms $\eta_t=\sigma_t^n$ extend to a
holomorphic section of the relative logarithmic top-form line
bundle of the semistable family.  With $s=-\log|t|$,
\begin{equation}
 \label{eq:family-volume-growth}
 c s^d\le \int_{Y_t}|\eta_t|^2\le C s^d
\end{equation}
for positive constants $c,C$.  There is a point $o\in Y_0$ where
$d+1$ components meet and coordinates satisfying
\begin{equation}
 \label{eq:family-maximal-residue}
 t=z_0\cdots z_d,\qquad
 \eta=F(z)\,\frac{\dd z_1}{z_1}\wedge\cdots\wedge
                   \frac{\dd z_d}{z_d},\qquad F(o)\ne0.
\end{equation}
\item For any prescribed irrational number $\lambda$, there is a
path tending to $0$ in $\Delta^*$, with a continuous choice of lift,
along which $\tau=\lambda+iy$ and $y\to+\infty$.
\end{enumerate}
\end{proposition}
\end{samepage}

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

\subsection{A positive polarization orthogonal to the null class}

We use the following standard inputs from hyperkähler period theory:
unobstructed deformations and local Torelli; surjectivity of the
period map from the selected connected marked component; the
projectivity criterion for a positive integral $(1,1)$ class; and
the Kähler-cone description, whose walls are orthogonal to negative
integral $(1,1)$ classes.  See
\cite{Huybrechts,HuybrechtsErratum,Verbitsky,VerbitskyErratum}.
The corrected projectivity criterion is
\cite[Theorem~2]{HuybrechtsErratum}, and the K\"ahler-cone description
in terms of negative MBM classes is due to Amerik--Verbitsky
\cite[Theorems~1.19 and~6.2]{AmerikVerbitsky}; see also
\cite[Section~2.1]{SV}.

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

\begin{lemma}[A positive polarization and a free period quotient]
\label{lem:polarized-family-construction}
Let $X$ be a compact irreducible holomorphic symplectic K\"ahler
manifold, fix its connected marked deformation component, and put
$\Lambda=H^2(X,\Z)$ with its integral BBF form $q$.
For each nonzero primitive $e\in\Lambda$ with $q(e)=0$, there are
\begin{enumerate}
\item a primitive $h\in\Lambda$ with $q(h)>0$ and $q(h,e)=0$,
and a manifold in the chosen marked component on which $h$ is ample;
\item a smooth projective algebraic family over a smooth connected
quasi-projective base $S$, whose fibers are irreducible holomorphic
symplectic manifolds in that component, with a relative polarization
of marked class $kh$ for one $k\in\Z_{>0}$;
\item a torsion-free finite-index subgroup
\[
 \Gamma\subset G_h(\Z)
 :=\{g\in O(\Lambda,q):gh=h,\ gD_h=D_h\},
\]
where $D_h$ is the component of the polarized period domain containing
the chosen ample period, such that the family monodromy is contained
in $\Gamma$ and the descended period map
\[
       \Phi:S\longrightarrow Q_h:=\Gamma\backslash D_h
\]
is algebraic. The quotient $Q_h$ is smooth and quasi-projective,
$D_h\to Q_h$ is a covering map with free $\Gamma$-action, and a local
marked lift of $\Phi$ is submersive at some point of $S$.
\end{enumerate}
More generally, conclusions \textup{(2)} and \textup{(3)} hold for any
prescribed primitive positive $h\in\Lambda$ which is ample on a
manifold in the chosen marked component.
The markings are obtained by deformation and parallel transport from
$X$. No finite-index assertion about the image of the family monodromy
inside $\Gamma$ is part of this construction.
\end{lemma}

\begin{proof}
\emph{The positive polarization.}
The signature of $q$ is $(3,r-3)$.  The existence of the nonzero
null vector $e$ implies $r\ge4$.  The radical of $e^\perp$ is $\R e$,
and
\[
 \operatorname{sign}(e^\perp/\R e)=(2,r-4).
\]
Thus $e^\perp$ contains positive real vectors.  Since $e$ is rational,
$e^\perp$ is defined over $\Q$; rational approximation inside this
space supplies a rational positive vector. Taking the primitive
integral generator of its ray gives $h$ with
$q(h)>0$ and $q(h,e)=0$.

Set $V=h^\perp\subset\Lambda_{\Q}$; its signature is $(2,r-3)$.
The polarized period domain is
\[
 \{[p]\in\mathbb P(V_{\C}):q(p)=0,\ q(p,\bar p)>0\}.
\]
A generic point of this domain is orthogonal to no integral class
outside $\Q h$: the excluded sets are countably many proper
hyperplane sections.  Period surjectivity realizes such a point
in the selected marked component.  Its Néron--Severi lattice is
$\Z h$.  One of $h,-h$ is in the component of the positive cone
containing the Kähler cone.  There are no negative integral
$(1,1)$ classes on this manifold, so the Kähler-cone description
makes that choice a Kähler class.  After replacing $h$ by $-h$ if
necessary, it is the class of an ample bundle.  This sign change
does not change $V$ or $e$.  Fix the resulting polarized manifold
and let $D_h$ denote the component of the polarized period domain
containing its period.  Locally, the periods of its polarized
deformations fill an open subset of $D_h$.

\smallskip\noindent\emph{The algebraic family and the free arithmetic quotient.}

We need an algebraic family with an open marked period image.
Choose $k\in\Z_{>0}$ so that the bundle with class $kh$ on the preceding
manifold is very ample and has vanishing higher cohomology.  The
bundle extends locally over polarized deformations; bases of its
sections give nearby embeddings with the same Hilbert polynomial.
Locally, deformations of a complete-linear-system embedding are
deformations of the polarized pair together with projective frames
of these section spaces.  The pair deformation space is smooth by
local Torelli, and cohomology vanishing lets the frames vary freely.
There are no infinitesimal automorphisms, and finite automorphisms
act freely on frames.  A reduced irreducible Hilbert-scheme component
therefore has a smooth point with submersive polarized period map.

Restrict its universal family to a smooth connected Zariski-open
base $S$ of smooth irreducible holomorphic symplectic fibers,
retaining a submersive point.  This restriction is algebraic:
Hodge numbers are locally constant in a smooth projective family,
and the degeneracy locus of the relative two-form has closed image
in the base by properness.  Simple connectivity and differentiable
type are preserved in the connected smooth family.  The universal
hyperplane bundle has class $kh$.

Let
\[
 G_h(\Z)=\{g\in O(\Lambda,q):gh=h,\ gD_h=D_h\}.
\]
Choose a torsion-free finite-index congruence subgroup
$\Gamma\subset G_h(\Z)$ whose elements have determinant one.
Both groups act on the full lattice
$\Lambda$.  Restriction embeds $G_h(\Z)$ in $O(V,q)$,
because $\Lambda_{\Q}=\Q h\oplus V$; its image is arithmetic.
For example, a congruence subgroup acting trivially on the finite
gluing quotient between $\Z h\oplus(\Lambda\cap V)$ and $\Lambda$
extends across this quotient, which shows the required finite-index
property in the integral orthogonal group of $V$.

The family monodromy is contained in $G_h(\Z)$: it preserves the
BBF form and the class $kh$, hence also $h$, and preserves the
component of its periods.  Pass to the finite cover of $S$
corresponding to the inverse image of $\Gamma$ under this
monodromy representation.  A finite topological cover of a smooth
complex quasi-projective variety is algebraic and finite étale
\cite[Expos\'e~XII, Theorem~5.1]{SGA1}.
We keep one connected component and continue to call the base $S$.
This step requires only that $\Gamma$ has finite index in
$G_h(\Z)$; no hypothesis on the index of the family monodromy
image is used.

The action of $\Gamma$ on $D_h$ is free.  Indeed, the stabilizer
of a positive oriented two-plane is a compact subgroup of
$O(V_{\R},q)$.  Its intersection with a discrete integral subgroup
is finite and is therefore trivial in the torsion-free group
$\Gamma$.  This argument also removes any kernel of the projective
action.  Consequently
\[
 Q_h:=\Gamma\backslash D_h
\]
is a smooth arithmetic quotient and $D_h\to Q_h$ is a covering
map.  The Baily--Borel theorem \cite{BailyBorel} makes $Q_h$
quasi-projective.  Borel's algebraicity theorem
\cite[Theorem~3.10]{Borel} then makes the descended period map
\[
 \Phi:S\longrightarrow Q_h
\]
algebraic.  We have given the Hilbert-family construction directly;
compare \cite[Lemma~4.5]{Soldatenkov}.  At a chosen point $s_0\in S$, fix
the marking obtained by transport from the starting polarized
manifold and a local lift of $\Phi$ to $D_h$.  After shrinking an
analytic neighborhood, this lift is submersive and its image
contains an open set $U\subset D_h$ with the actual marking.
Freeness will later identify monodromy on the full second-cohomology
lattice from equality of these lifted periods.
\end{proof}

We now apply Lemma~\ref{lem:polarized-family-construction} to the
primitive class fixed at the start of this section. Its open marked
period image is the input for the next step.

\subsection{A rational ternary domain containing the fixed vector}

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

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

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

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

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

\subsection{An algebraic curve reaching the specified cusp}

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

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

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

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

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

\subsection{The exact period and monodromy formulas}

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

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

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

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

\subsection{Semistable reduction and the logarithmic form}

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

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

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

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

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

\subsection{Maximal growth forces a nonzero maximal residue}

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

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

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

\begin{corollary}
\label{cor:family-torus}
Fix an irrational $\lambda$ and choose the lifted path in
Proposition~\ref{prop:family} with $\tau=\lambda+iy$.
For every sufficiently large $y$, the corresponding smooth fiber
contains an embedded special Lagrangian real $d$-torus for its
Ricci-flat metric in the polarization class.  This torus is
isotopic to a coordinate angle torus in the chart
\eqref{eq:family-maximal-residue}.  The period and monodromy on
that fiber are the exact marked data
\eqref{eq:family-period}--\eqref{eq:family-monodromy-image}.
\end{corollary}

\begin{proof}
The algebraicity, projectivity, extension of the polarization,
volume growth, logarithmic extension, and nonzero coefficient
required by Theorem~\ref{thm:torus} are all supplied by
Proposition~\ref{prop:family}.  The theorem applies to every
sufficiently small parameter, in particular along the chosen path,
where $\operatorname{Re}\tau=\lambda$ and the marking is already fixed.
\end{proof}
==== END The-Strong-Hyperkahler-SYZ-Conjecture//sections/polarized_family.tex ====

==== BEGIN The-Strong-Hyperkahler-SYZ-Conjecture//sections/rotation.tex ====
\section{Rotation and return to the original manifold}
\label{sec:rotation}

We now complete the proof of Theorem~\ref{thm:main}.  The marking in
Proposition~\ref{prop:family} will be used throughout: all second
cohomology groups are identified with the lattice of the differentiable
manifold underlying the given $X$, and $q$ denotes both the quadratic
form and its associated symmetric bilinear form.

\subsection{The primitive class and the chosen fiber}

Recall the primitive class $e$ and line bundle $E$ constructed at the
start of Section~\ref{sec:family}. Write $E_X=E$, so that
$L\simeq E_X^{\otimes m_0}$. The class $e$ is nonzero, nef, and
$q(e)=0$; its sign was fixed so that, for every K\"ahler class
$\kappa$ on $X$,
\begin{equation}
                    q(e,\kappa)>0.
\label{eq:rotation-original-sign}
\end{equation}

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

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

\subsection{Restriction, calibration, and complex rotation}

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

\begin{lemma}[A cohomological rotation criterion]
\label{lem:rotation-criterion}
Let $(Y,g,I,J,K)$ be a hyperk\"ahler manifold of real dimension $4n$,
where $n\ge1$ and $IJ=K$, and let $i:T\hookrightarrow Y$ be a compact,
connected, oriented embedded
special Lagrangian submanifold for the complex structure $I$.
If $i^*[\omega_K]=0$ in $H^2(T,\R)$, then $T$ is a complex
$n$-dimensional submanifold for $J$ and is Lagrangian for
$\omega_K+i\omega_I$. If $T$ is diffeomorphic to a real $2n$-torus,
then it is a complex torus for $J$.
\end{lemma}
\begin{proof}
The normalized parallel holomorphic volume form for $(Y,I,g)$ is
\[
                 \Omega_I=\frac{(\omega_J+i\omega_K)^n}{n!}.
\]
To check the normalization, in a symplectic unitary frame one has
$\omega_J+i\omega_K=\sum_{j=1}^n dz_{2j-1}\wedge dz_{2j}$;
its $n$th power divided by $n!$ is the unit holomorphic volume
form.  It has absolute value one on an orthonormal frame of any
$I$-Lagrangian real $2n$-plane. The special Lagrangian calibration identity
\cite[III.1, Theorem~1.10 and Corollary~1.11]{HarveyLawson}
therefore gives
\[
                    \vol_g(T)=\left|\int_T\Omega_I\right|.
\]
All terms containing $\omega_K$ in this integral vanish:
the hypothesis makes $i^*\omega_K$ exact,
while all the factors in the expansion are closed.  Stokes' theorem
then yields
\begin{equation}
 \vol_g(T)
 =\left|\int_T\frac{\omega_J^n}{n!}\right|
 \le\int_T\left|\frac{i^*\omega_J^n}{n!}\right|
 \le\vol_g(T).
\label{eq:rotation-wirtinger}
\end{equation}
The last inequality is the pointwise Wirtinger inequality
\cite[II.6, Theorem~6.11]{HarveyLawson}.
Its continuous, nonnegative deficit has integral zero, so equality
holds at every point of $T$.  Equality planes are exactly the
$J$-invariant real $2n$-planes.  Thus $T$ is a complex
$n$-dimensional submanifold of
\[
                              Y'=(Y,J).
\]
Its orientation may now be chosen to be the complex orientation;
the absolute period in \eqref{eq:rotation-wirtinger} made the
initial choice immaterial.

For $J$ the holomorphic symplectic form is
$\sigma_J=\omega_K+i\omega_I$.  We already have
$\omega_I|_T=0$ pointwise.  Since $JT_xT=T_xT$, for
$u,v\in T_xT$ the quaternion identities give
\[
              \omega_K(u,v)=g(IJu,v)=\omega_I(Ju,v)=0.
\]
Hence $\sigma_J|_T=0$: the rotated submanifold is holomorphically
Lagrangian.

Suppose now that $T$ is diffeomorphic to a real $2n$-torus. The
restriction of $\omega_J$ makes $T$ K\"ahler.
Consequently $b_1(T)=2n$ and $h^{1,0}(T)=n$.  Its Albanese map
\[
                 \alpha:T\longrightarrow\operatorname{Alb}(T)
\]
has a complex $n$-torus as target and induces an isomorphism on
integral first cohomology.  Both integral cohomology rings are
exterior algebras on first cohomology, so $\alpha^*$ is an
isomorphism in every degree.  In particular, $\alpha$ has
nonzero degree; holomorphicity makes that degree positive, hence
equal to one, and $\alpha$ is surjective.

Choose a K\"ahler class $\xi$ on $T$.  Surjectivity on $H^2(-,\R)$
writes $\xi=\alpha^*\beta$ for some real class $\beta$ on
$\operatorname{Alb}(T)$.  If a fiber contained a positive-dimensional
irreducible compact analytic subvariety $V$ of dimension $k$, then
\[
                        \int_V\xi^k=0,
\]
since $\alpha|_V$ is constant.  This contradicts K\"ahler
positivity.  Thus $\alpha$ is proper with finite fibers, hence
finite.  A finite degree-one holomorphic map to a normal complex
space is an isomorphism.  This proves that $T$ is a holomorphic
Lagrangian complex torus in $Y'$.
\end{proof}

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

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

Lemma~\ref{lem:rotation-criterion} now makes $T$ a holomorphic
Lagrangian complex torus on the rotated manifold $Y'=(Y,J)$.

\subsection{The rational isotropic ray after rotation}

The period plane of $Y'$ is
\begin{equation}
       \langle[\omega_K],[\omega_I]\rangle_{\R}
       =\langle v-a\lambda e,h\rangle_{\R}.
\label{eq:rotation-j-period}
\end{equation}
It is perpendicular to $e$, so $e$ is an integral $(1,1)$ class
on $Y'$; see Figure~\ref{fig:rotation}.

\input{figures/rotation}

Let
$x\in H^{1,1}(Y')\cap H^2(Y',\Q)$.  Orthogonality to the period
plane says
\[
                q(x,h)=0,\qquad q(x,v)=a\lambda q(x,e).
\]
The three numbers $q(x,v)$, $a$, and $q(x,e)$ are rational,
whereas $a\ne0$ and $\lambda\notin\Q$.  Therefore
\begin{equation}
             q(x,e)=q(x,v)=0.
\label{eq:rotation-rational-orthogonality}
\end{equation}

The restriction of $q$ to $H^{1,1}(Y',\R)$ has signature
$(1,b_2-3)$.  In a Lorentzian space, the orthogonal complement of
a nonzero null vector is negative semidefinite with radical its
own line.  Applying this to $e$ and
\eqref{eq:rotation-rational-orthogonality} proves
\begin{equation}
 \begin{gathered}
 q(x)\le0\quad\text{for }x\in H^{1,1}(Y')\cap H^2(Y',\Q),\\
 q(x)=0\quad\Longrightarrow\quad x\in\Q e.
 \end{gathered}
\label{eq:rotation-unique-ray}
\end{equation}
In particular, $Y'$ is nonprojective: a projective hyperk\"ahler
manifold has an integral ample class of positive BBF square.
This is the necessary direction of the projectivity criterion
\cite{Huybrechts,HuybrechtsErratum}.  The rational Picard space
may contain negative-square classes; its isotropic classes are
exactly those on $\Q e$.  Equation~\eqref{eq:rotation-k-class}
places $e$ in the closure of the positive cone containing the
K\"ahler class $[\omega_J]$.

\subsection{A semiample point and deformation back to \texorpdfstring{$X$}{X}}

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

\begin{theorem}[Greb--Lehn--Rollenske {\cite[Theorem~4.1]{GLR}}]
\label{thm:nonprojective-torus}
If a nonprojective compact irreducible hyperk\"ahler manifold $Z$ of
complex dimension $2n$ contains a holomorphic Lagrangian complex
torus $S$, then its algebraic dimension is $n$ and it has a
holomorphic algebraic reduction $Z\to B$ which is a Lagrangian
fibration with $S$ as a fiber.  In particular, this fibration has
normal projective base and provides a nontrivial semiample
pullback bundle on $Z$.
\end{theorem}
The projectivity assertion follows from the line-bundle construction
of the algebraic reduction in \cite[Section~4.2]{GLR} and Stein
factorization.

\paragraph{The fixed-class deformation theorem.}
Let $N$ be a compact simply connected differentiable manifold
of hyperk\"ahler type with maximal holonomy, let
$\mathcal T^\circ(N)$ be one connected component of its
Teichm\"uller space, and fix a nonzero primitive integral class
$e$ with $q(e)=0$.  Write $\mathcal C_I$ for the component of
the positive cone containing the K\"ahler cone of $(N,I)$, and set
\[
 \mathcal T^\circ_e
 =\{[I]\in\mathcal T^\circ(N):
       e\in H^{1,1}(N,I),\ e\in\overline{\mathcal C_I}\}.
\]
This is the positive-sign locus of
\cite[Definition~2.6]{SV}.  By \cite[Proposition~2.7]{SV} it is
connected.  Let $E_I$ denote the unique holomorphic bundle with
class $e$.  Define $\mathcal T^\circ_{e,\mathrm{nef}}$ and
$\mathcal T^\circ_{e,\mathrm{sa}}$ by requiring $E_I$ to be nef
and semiample, respectively.

\begin{theorem}[Soldatenkov--Verbitsky {\cite[Theorem~3.7(v)]{SV}}]
\label{thm:fixed-class}
With this notation,
\begin{equation}
 \mathcal T^\circ_{e,\mathrm{sa}}\ne\varnothing
 \quad\Longrightarrow\quad
 \mathcal T^\circ_{e,\mathrm{nef}}
      =\mathcal T^\circ_{e,\mathrm{sa}}.
\label{eq:rotation-sv}
\end{equation}
\end{theorem}
Its family formulation, \cite[Theorem~3.8]{SV}, says that a nef
isotropic bundle is semiample when its pair deforms, in a smooth
connected family carrying a line bundle, to a semiample pair.
We use \eqref{eq:rotation-sv} directly.

Apply Theorem~\ref{thm:nonprojective-torus} to $(Y',T)$.  Choose an ample bundle $H$
on its fibration base and let $A$ be its pullback to $Y'$.
The bundle $A$ is semiample and nef, and its first Chern class
$b$ is nonzero: a semipositive representative of a generated
power has positive trace on an open set where the map has
positive rank.  Since the base has dimension $n$,
$\int_{Y'}b^{2n}=0$.  Fujiki's relation \cite{Fujiki} gives
$q(b)=0$.  Equation~\eqref{eq:rotation-unique-ray} and the
primitivity of $e$ now give
\[
                             b=k e,\qquad k\in\Z.
\]
Both $b$ and $e$ pair positively with $[\omega_J]$: for $b$ this
is the nonzero nef null-vector property, and for $e$ it is
\eqref{eq:rotation-k-class}.  Hence $k>0$.

Let $E_{Y'}$ be the unique bundle on $Y'$ with first Chern class
$e$.  Injectivity of the first Chern class map gives
\[
                          A\simeq E_{Y'}^{\otimes k}.
\]
A globally generated positive power of $A$ is therefore a
globally generated positive power of $E_{Y'}$.  Thus the
\emph{specified primitive class} $e$ is semiample on $Y'$.

The marking furnished by Proposition~\ref{prop:family} identifies
$Y$ with a point of the same ambient component
$\mathcal T^\circ(N)$ as the original $X$.  The hyperk\"ahler
twistor sphere connects $I$ to $J$ on that same marked
differentiable manifold, so $Y'$ also belongs to this component.
On $X$, $e$ is of type $(1,1)$ and has the positive sign by
\eqref{eq:rotation-original-sign}.  On $Y'$ the corresponding
statements follow from \eqref{eq:rotation-j-period} and
\eqref{eq:rotation-k-class}.  Consequently both endpoints lie
in the same positive fixed-class locus $\mathcal T^\circ_e$.
The semiample point $Y'$ makes its semiample locus nonempty;
the original point $X$ belongs to its nef locus by hypothesis.
Equation~\eqref{eq:rotation-sv} now proves that $E_X$ is
semiample on $X$.

Choose $s>0$ such that $E_X^{\otimes s}$ is globally generated.
Then $L^{\otimes s}\simeq(E_X^{\otimes s})^{\otimes m_0}$ is
globally generated as well.  In particular, on the original
manifold the evaluation morphism
\[
 H^0(X,L^{\otimes s})\otimes_{\C}\mathcal O_X
                    \longrightarrow L^{\otimes s}
\]
is surjective everywhere.  This proves the semiampleness assertion
of Theorem~\ref{thm:main}.

\subsection{The fibration defined by a generated power}

We derive the fibration description from semiampleness, using the
Fujiki argument for Lagrangian fibers of Matsushita
\cite{Matsushita1999,MatsushitaAddendum}.  Let $m>0$ make $L^{\otimes m}$ globally
generated, let
\[
 g:X\longrightarrow Z\subset\mathbb P H^0(X,L^{\otimes m})^*
\]
be its morphism to the image, and take its Stein factorization
$g=\nu\circ f$:
\[
                   X\xrightarrow{\ f\ }B\xrightarrow{\ \nu\ }Z.
\]
The space $B$ is normal, $f$ has connected fibers, and $\nu$ is
finite.  Since $Z$ is projective, $B$ is projective and
$H:=\nu^*\mathcal O_Z(1)$ is ample.  By construction
\begin{equation}
                         L^{\otimes m}\simeq f^*H.
\label{eq:rotation-ample-pullback}
\end{equation}

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

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

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

The same assertion holds for every irreducible component of every
fiber by Matsushita's equidimensionality theorem
\cite[Theorem~1]{MatsushitaEquidimensional}. Its hypotheses are
exactly those now established: $X$ is K\"ahler, $f$ is proper and
surjective with normal base, and a general smooth fiber is
Lagrangian. The holomorphic symplectic form is closed, as every
holomorphic form on a compact K\"ahler manifold is. Thus every
fiber component has dimension $n$ and is isotropic on its smooth
locus. This also gives the Lagrangian fibration used in
Section~\ref{sec:finiteness}.

Conversely, an ample bundle has a globally generated positive
power, so \eqref{eq:rotation-ample-pullback} implies semiampleness
of $L$. This proves the equivalence between semiampleness and the
fibration description in Theorem~\ref{thm:main}.

\subsection{Projective-space bases}
\label{subsec:projective-base}

We finish with the additional conclusion for projective $X$ in
Corollary~\ref{cor:projective-base}.

\begin{proof}
Choose a very ample line bundle $H$ on $X$, independently of $L$, with
embedding $i:X\hookrightarrow\mathbb P^N$; the closed graph embedding
$(f,i):X\hookrightarrow B\times\mathbb P^N$ exhibits $H$ as relatively
very ample for $f$, so $f$ is projective. The remaining hypotheses are
supplied by Theorem~\ref{thm:main}; hence the companion's
Theorem~1.1~\cite{ProjectiveSpaceBases2026} identifies $B$ with
$\mathbb P^n_{\C}$, and the following no-multiple-fiber paragraph there
gives the divisor assertion.
\end{proof}
==== END The-Strong-Hyperkahler-SYZ-Conjecture//sections/rotation.tex ====

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@misc{ProjectiveSpaceBases2026,
 author = {{OpenAI}},
 title = {{Projective-space bases of Lagrangian fibrations}},
 howpublished = {OpenAI Math Release preprint
                 \href{https://github.com/openai/math/blob/main/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf}{OAI:Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026}},
 year = {2026}
}
==== END The-Strong-Hyperkahler-SYZ-Conjecture//references.bib ====