# Metric SYZ conjecture for certain toric Fano hypersurfaces

Yang Li

###### Abstract

We prove the metric version of the SYZ conjecture for a class of Calabi-Yau
hypersurfaces inside toric Fano manifolds, by solving a variational problem
whose minimizer may be interpreted as a global solution of the real Monge-
Ampère equation on certain polytopes. This does not rely on discrete symmetry.

##  1 Introduction

The metric aspect of the Strominger-Yau-Zaslow (SYZ) conjecture [28] asks for
the following:

######  Conjecture 1.1.

(_cf._ [18][21][28]) Let $X_{t}$ be a 1-parameter maximally degenerate family
of polarized $d$-dimensional Calabi-Yau manifolds over the punctured disc
$\mathbb{D}_{t}^{*}$, then for any given $0<\delta\ll 1$, and all $t$ small
enough depending on $\delta$, there exists a special Lagrangian
$T^{d}$-fibration on an open subset of $X_{t}$ for $0<|t|\ll 1$, whose
normalized Calabi-Yau measure is at least $1-\delta$.

The case of Abelian varieties is classical, while the case of K3 surfaces is
known through the trick of hyperkähler rotation [12][26]. The first nontrivial
example in all dimensions is the Fermat family of Calabi-Yau hypersurfaces
[17]

$X_{s}=\\{Z_{0}\ldots
Z_{d+1}+e^{-s}\sum_{0}^{d+1}Z_{i}^{d+2}=0\\}\subset\mathbb{CP}^{d+1},\quad
s\gg 1.$

The proof of the SYZ conjecture in this case exploits the large discrete
symmetry group, to simplify some rather delicate combinatorial problems.

A turn of thinking came about in [18], where we reduced the SYZ conjecture to
a problem of algebraic geometry. In brief, there is a non-archimedean (NA)
analogue of the complex Monge-Ampère (MA) equation, defined in terms of
intersection theory, and the deep work of Boucksom et al. [4][3][2][5] proved
the existence and uniqueness of the solution. To deduce the SYZ conjecture,
what is left to prove is a ‘comparison property’ [18], which morally asserts
the NA MA solution is not too wild.

For this new strategy to be convincing, it is requisite to verify the
comparison property in examples. The classical case of Abelian varieties was
checked by Goto-Odaka [10]. The recent work of Pille-Schneider [27] and
Hultgren et al. [14] independently verified the comparison property for some
generalisation of the Fermat family, but both works still rely heavily on the
large discrete symmetry group, and thus still only apply to certain
hypersurfaces inside $\mathbb{CP}^{d+1}$.

A notable contribution in [14] is to introduce a global variational problem.
The existence and uniqueness of the minimizer is quite transparent, but the
local PDE nature of the minimizer is not manifest. It is essentially known in
[14] that if one can deduce a local real Monge-Ampère equation from the
minimizer, then it would naturally produce the unique solution of the NA MA
solution, and the ‘comparison property’ would be a corollary. The present
paper aims to make progress on this strategy, by proving the expected
properties of the minimizer for more examples without explicit reliance on the
discrete symmetry.

Let $\Delta\subset M_{\mathbb{R}}$ be an integral reflexive Delzant polytope,
and $X_{\Delta}$ be the associated smooth toric Fano manifold of dimension
$d+1$, with an ample polarization $L\to X_{\Delta}$, which needs not be the
anticanonical polarization. The origin $0\in\Delta$ corresponds to a
distinguished section $X_{can}\in H^{0}(X_{\Delta},-K_{X_{\Delta}})$, which
defines the toric boundary of $X_{\Delta}$. Let $F\in
H^{0}(X_{\Delta},-K_{X_{\Delta}})$ be a generic section, and by assumption the
divisor $\\{F=0\\}\subset X_{\Delta}$ is smooth, and intersects all the toric
boundary strata transversely, and in particular does not pass through the
finite number of toric fixed points on $X_{\Delta}$. We will consider the
family of Calabi-Yau hypersurfaces as $t\to 0$:

$X_{t}=\\{X_{can}+tF=0\\}\subset X_{\Delta}\times\mathbb{C}^{*}_{t}.$ (1)

Algebro-geometrically $X_{t}$ degenerate to the toric boundary.

We will assume one extra property on the reflexive polytope $\Delta$, whose
dual polytope is denoted as $\Delta^{\vee}$:

$\langle m,n\rangle\neq 0,\quad\forall m\in\text{vertex}(\Delta),\quad\forall
n\in\text{vertex}(\Delta^{\vee}).$ (2)

The main outcome is

######  Theorem 1.2.

The SYZ conjecture 1.1 holds for the family $X_{t}$ as $t\to 0$.

The main new content of this paper is contained in section 2, which
establishes a structure theory for the minimizer of the variational problem,
and has the flavour of convex analysis, variational calculus, and real MA
equation. This part does not require prior knowledge of Kähler geometry, and
may be of independent interest to analysts. In section 3 we briefly sketch the
complex geometric aspect towards the application to the SYZ conjecture,
relying on the aforementioned results in [17][19][18][14].

######  Remark 1.3.

J. Hultgren informs the author that together with R. Andreasson they have
independently found an equivalent condition in terms of optimal transport
theory, for the minimizer of the variational problem to admit the expected PDE
interpretation [1]. Their upcoming paper will also contain counterexamples,
demonstrating in particular that some nontrivial condition on the reflexive
polytope is necessary in this strategy.

######  Acknowledgement.

The author is a current Clay Research Fellow based at MIT. He would like to
especially thank J. Hultgren for pointing out his upcoming work, and for
helpful comments on this manuscript. He also thanks S. Boucksom, L. Pille-
Schneider, and S. Sun for discussions in the past.

##  2 The variational problem

###  2.1 Motivation: A global real Monge-Ampère equation?

Let $M_{\mathbb{R}}\simeq\mathbb{Z}^{d+1}\otimes_{\mathbb{Z}}\mathbb{R}$ and
$N_{\mathbb{R}}=M_{\mathbb{R}}^{\vee}$ be a pair of dual vector spaces, and
let $\Delta\subset M_{\mathbb{R}}$ be an integral reflexive polytope, with
dual polytope $\Delta^{\vee}\subset N_{\mathbb{R}}$, so that

$\Delta^{\vee}=\\{x\in N_{\mathbb{R}}|\langle m,x\rangle\leq 1,\forall
m\in\text{vertex}(\Delta)\\},$ $\Delta=\\{p\in M_{\mathbb{R}}|\langle
n,p\rangle\leq 1,\forall n\in\text{vertex}(\Delta^{\vee})\\},$

both $\Delta,\Delta^{\vee}$ contain the origin, and all the vertices are
integral points. Given negative real numbers $\mu(n)$ for all
$n\in\text{vertex}(\Delta^{\vee})$, and $\lambda(m)$ for all
$m\in\text{vertex}(\Delta)$, we can define the convex functions on
$N_{\mathbb{R}}$ and $M_{\mathbb{R}}$,

$L_{\lambda}(x)=\max_{m}\langle x,m\rangle+\lambda(m),\quad
L_{\mu}(p)=\max_{n}\langle p,n\rangle+\mu(n).$

These in turn define the convex polytopes containing the origin in the
interior,

$\Delta_{\lambda}^{\vee}=\\{L_{\lambda}(x)\leq 0\\}\subset
N_{\mathbb{R}},\quad\Delta_{\mu}=\\{L_{\mu}(p)\leq 0\\}\subset
M_{\mathbb{R}}.$

For instance, the choice $\lambda=-1$ recovers $\Delta^{\vee}$, and the choice
$\mu=-1$ recovers $\Delta$.

The normal vectors to the top dimensional faces are labelled by the vertices
$m,n$, which are primitive integral vectors. This integral structure on the
faces induce a canonical Lebesgue measure $dx$ and $dp$ on
$\partial\Delta_{\lambda}^{\vee}$ and $\partial\Delta_{\mu}$ respectively,
which we normalize to probability measures

$d\mathcal{L}^{\vee}=\frac{1}{\int_{\partial\Delta_{\lambda}^{\vee}}dx}dx,\quad
d\mathcal{L}=\frac{1}{\int_{\partial\Delta_{\mu}}dp}dp.$ (3)

We shall use the notation $|E|$ to denote the measure of subsets $E$ with
respect to $d\mathcal{L}$ or $d\mathcal{L}^{\vee}$.

Our secret goal is to formulate and solve a version of the real MA equation on
$\partial\Delta_{\lambda}^{\vee}$ and $\partial\Delta_{\mu}$. A direct attempt
faces the following difficulties:

  * •

The polytopes do not have natural global affine structures. On the overlap of
the candidate charts, the transition functions will only be piecewise affine.

  * •

Consequently, the condition of being a convex function depends on local
charts.

  * •

The definition of the real MA equation via local charts will not be compatible
with piecewise affine transition functions.

An insight of Hultgren et al. [14], carried out in a very symmetric special
case, is to adopt instead a _global variational formulation_. The price is
that the PDE nature of the problem is not manifest.

###  2.2 Legendre transform on polytopes

We define the Legendre transforms (referred to as ‘c-transforms’ in [14],
which comes from optimal transport nomenclature),

$L^{\infty}(\partial\Delta_{\mu})\to
L^{\infty}(\partial\Delta_{\lambda}^{\vee}),\quad
L^{\infty}(\partial\Delta_{\lambda}^{\vee})\to
L^{\infty}(\partial\Delta_{\mu}),$

such that for $\phi\in L^{\infty}(\partial\Delta_{\lambda}^{\vee})$ and
$\psi\in L^{\infty}(\partial\Delta_{\mu})$,

$\psi^{*}(x)=\sup_{p\in\partial\Delta_{\mu}}\langle
x,p\rangle-\psi(p),\quad\phi^{*}(p)=\sup_{x\in\partial\Delta_{\lambda}^{\vee}}\langle
x,p\rangle-\phi(x).$ (4)

The theory works symmetrically for $\phi,\psi$.

Some of the immediate formal properties are listed below (_cf._ [14, section
3]):

######  Lemma 2.1.

(Equicontinuity) The Legendre transforms $\phi^{*},\psi^{*}$ have uniformly
bounded Lipschitz constant.

######  Lemma 2.2.

(Involution) For arbitrary $\phi\in
L^{\infty}(\partial\Delta_{\lambda}^{\vee})$, the double Legendre transform
$\phi^{**}\leq\phi$. If $\phi=\psi^{*}$ for some $\psi$, then
$\phi^{**}=\phi$.

######  Lemma 2.3.

(Monotonicity) If $\phi_{1}\geq\phi_{2}$, then $\phi_{1}^{*}\leq\phi_{2}^{*}$.
For any constant $c\in\mathbb{R}$, we have $(\phi+c)^{*}=\phi^{*}-c$. Morever
$\left\lVert\phi_{1}^{*}-\phi_{2}^{*}\right\rVert_{C^{0}}\leq\left\lVert\phi_{1}-\phi_{2}\right\rVert_{C^{0}}$.

We then introduce a class of functions $\mathcal{P}\subset
C^{0}(\partial\Delta_{\lambda}^{\vee})$ and $\mathcal{P}^{\vee}\subset
C^{0}(\partial\Delta_{\mu})$, defined as the images of $L^{\infty}$ under the
Legendre transform. By the involution property, the Legendre transform sets up
a canonical isomorphism $\mathcal{P}\simeq\mathcal{P}^{\vee}$, which is
isometric with respect to the $C^{0}$-norms by Lemma 2.3. The formula

$\phi(x)=\max_{p\in\partial\Delta_{\mu}}\langle
x,p\rangle-\phi^{*}(p),\quad\phi^{*}(p)=\max_{x\in\partial\Delta_{\lambda}^{\vee}}\langle
x,p\rangle-\phi(x)$

provides a canonical _extension_ of functions in
$\mathcal{P},\mathcal{P}^{\vee}$ to convex functions on the ambient Euclidean
spaces $N_{\mathbb{R}}$ and $M_{\mathbb{R}}$. We can thus think of
$\mathcal{P},\mathcal{P}^{\vee}$ as ‘ _global convex functions_ ’, which in
particular have gradient contained in $\Delta_{\mu},\Delta_{\lambda}^{\vee}$
respectively.

###  2.3 Conjugate sets and gradients

We begin with a classical analogy. In the classical Legendre transform theory
for convex functions on $\mathbb{R}^{n}$, for a given position variable $x$,
the gradient $\nabla\phi(x)$ is recovered by the conjugate momentum $p$
achieving $\phi(x)=\langle x,p\rangle-\phi^{*}(p)$. In the global context of
polytopes, finding a reasonable analogue of the conjugate $p$ to $x$ is a
rather delicate question, and there are at least two natural notions.

######  Definition 2.4.

Given a function $\phi\in\mathcal{P}$, let
$x\in\partial\Delta_{\lambda}^{\vee}$, then the _conjugate set_ is

$\bar{\nabla}\phi(x):=\\{p\in\partial\Delta_{\mu}|\phi^{*}(p)+\phi(x)=\langle
x,p\rangle\\}.$

For $E\subset\partial\Delta_{\lambda}^{\vee}$, the conjugate set is

$\bar{\nabla}\phi(E):=\\{p\in\partial\Delta_{\mu}|\phi^{*}(p)+\phi(x)=\langle
x,p\rangle,\text{ for some }x\in E\\}.$

We leave the reader to define the Legendre dual version
$\bar{\nabla}\phi^{*}$.

In the polytope setting, there is a more subtle new notion, which better
captures the classical intuition of gradient. To motivate this, we first delve
a little into polyhedral geometry. The top dimensional faces
$\Delta_{m}^{\vee}$ of $\partial\Delta_{\lambda}^{\vee}$ are labelled by the
vertices $m$ of $\Delta$, and concretely

$\Delta_{m}^{\vee}=\\{\langle
x,m\rangle=-\lambda(m)\\}\subset\partial\Delta_{\lambda}^{\vee}.$

Likewise the vertices $n$ of $\Delta^{\vee}$ label the faces
$\Delta_{n}\subset\partial\Delta_{\mu}$.

Given $\phi\in\mathcal{P}$, we can restrict $\phi$ to a convex function on
$\text{Int}(\Delta_{m}^{\vee})$, which is just a convex domain in some
Euclidean hyperplane. The classical gradient of this restricted function
naturally lies in the quotient space $M_{\mathbb{R}}/\mathbb{R}m$, and a
moment of thought reveals the gradient is contained in the image of
$\Delta_{\mu}$ under the quotient map $M_{\mathbb{R}}\to
M_{\mathbb{R}}/\mathbb{R}m$.

######  Lemma 2.5.

Each fibre under the quotient map

$\Delta_{\mu}\subset M_{\mathbb{R}}\to\text{Image}(\Delta_{\mu})\subset
M_{\mathbb{R}}/\mathbb{R}m$

intersects each of the following two unions of faces
$H_{m}^{+},H_{m}^{-}\subset\partial\Delta_{\mu}$ uniquely at one point:

$H_{m}^{+}=\bigcup\\{\Delta_{n}:\text{vertex $n\in\partial\Delta^{\vee}$ with
$\langle n,m\rangle=1$}\\},$ $H_{m}^{-}=\bigcup\\{\Delta_{n}:\text{vertex
$n\in\partial\Delta^{\vee}$ with $\langle n,m\rangle\leq-1$}\\},$

In particular $H_{m}^{+},H_{m}^{-}$ each projects homeomorphically onto
$\text{Image}(\Delta_{\mu})\subset M_{\mathbb{R}}/\mathbb{R}m$.

######  Remark 2.6.

We leave the reader to write down the Legendre dual statement involving the
self explanatory notations $H_{n}^{+},H_{n}^{-}$.

###### Proof.

For every point $p\in\Delta_{\mu}$, the line $p+\mathbb{R}m$ intersects the
convex polytope $\Delta_{\mu}$ in a line segment, with two endpoints (unless
we are in the degenerate setting where the line segment becomes a point). Now
the vertices $n$ prescribe the normal vectors to the faces. In order for the
line segment to exit $\Delta_{\mu}$ in the positive (resp. negative) $m$
direction, we need $\langle n,m\rangle>0$ (resp. $<0$). By assumption, all
$n,m$ are integral vectors, so $\langle m,n\rangle$ is an integer. Furthermore
by assumption $\langle m,n\rangle\leq 1$ for any choice of vertices $m,n$. The
result follows. ∎

It is desirable that the gradient $p$ at $x$ is contained in $H_{m}^{+}$ so
that it has a unique natural identification with a point in
$M_{\mathbb{R}}/\mathbb{R}m$.

######  Definition 2.7.

Let $x\in\partial\Delta_{\lambda}^{\vee}$, which may lie on possibly several
faces. Then the _gradient set_ of $x$ is the intersection

$\nabla\phi(x)=\bar{\nabla}\phi(x)\cap\bigcap_{m}\\{H_{m}^{+}:x\in\Delta_{m}^{\vee}\\}.$

We say $x$ has _anomalous conjugate points_ if
$\bar{\nabla}\phi(x)\setminus\nabla\phi(x)$ is nonempty. Likewise we define
$\nabla\phi(E)$ for subsets $E$, and the Legendre dual versions etc.

######  Remark 2.8.

If $p\in\bar{\nabla}\phi(x)$ is a conjugate point, then tautologically
$x\in\bar{\nabla}\phi^{*}(p)$. However $p\in\nabla\phi(x)$ does not
automatically imply $x\in\nabla\phi^{*}(p)$; one sufficient condition is that
$p$ lies on only one face.

While conjugate points exist quite obviously by the definition of the Legendre
transform, the existence of at least one gradient $p$ for any given $x$
requires a little more thought:

######  Lemma 2.9.

If $x\in\partial\Delta_{\lambda}^{\vee}$ has an anomalous conjugate point
$p^{\prime}$, then it also admits a gradient $p\in\nabla\phi(x)$ (so in
particular gradients always exist).

###### Proof.

Suppose $x$ lies on the intersection of some faces $\Delta_{m_{i}}^{\vee}$. By
the definition of conjugate point $\phi^{*}(p^{\prime})+\phi(x)=\langle
x,p\rangle.$ Consider the polyhedral set $(p^{\prime}+\sum_{i}\mathbb{R}_{\geq
0}m_{i})\cap\Delta_{\mu}$, which must have some extremal point $p$ such that
$(p+\sum_{i}\mathbb{R}_{\geq 0}m_{i})\cap\Delta_{\mu}$ consists of only one
point $p$. Then for each $m_{i}$, there must be some $n_{i}$ with $\langle
m_{i},n_{i}\rangle>0$ such that $p\in\Delta_{n_{i}}$, hence $p\in\bigcap
H_{m_{i}}^{+}$.

We claim $p$ is also a conjugate point of $x$. We write $p=p^{\prime}+\sum
s_{i}m_{i}$ with $s_{i}\geq 0$. Now since $\phi^{*}$ as a function on
$M_{\mathbb{R}}$ has gradient contained inside $\Delta_{\lambda}^{\vee}$, we
have

$\phi^{*}(p)-\phi^{*}(p^{\prime})\leq\max_{y\in\Delta_{\lambda}^{\vee}}\langle
p-p^{\prime},y\rangle\leq\sum s_{i}\max_{y\in\Delta_{\lambda}^{\vee}}\langle
m_{i},y\rangle\leq-\sum s_{i}\lambda(m_{i}).$

Hence

$\phi^{*}(p)+\phi(x)\leq\phi^{*}(p^{\prime})+\phi(x)-\sum
s_{i}\lambda(m_{i})=\langle x,p^{\prime}\rangle-\sum
s_{i}\lambda(m_{i})=\langle x,p^{\prime}+\sum s_{i}m_{i}\rangle=\langle
x,p\rangle.$

This combined with the tautological inequality $\phi^{*}(p)+\phi(x)\geq\langle
x,p\rangle$ shows that the equalities hold, so $p$ is a conjugate point, hence
a gradient. ∎

######  Remark 2.10.

The arguments above will not work if we attempt to define the gradient to lie
in $H_{m}^{-}$ instead of $H_{m}^{+}$. In fact, very often there is no
conjugate point in $H_{m}^{-}$.

######  Corollary 2.11.

Let $p\in\bar{\nabla}\phi(x)$, such that there is an anomalous conjugate point
$p^{\prime}$ with $p=p^{\prime}+\sum s_{i}m_{i}$ and $s_{i}\geq 0$ as above.
Then for each $m_{i}$ with $x\in\Delta_{m_{i}}^{\vee}$, either $s_{i}=0$ or
$\bar{\nabla}\phi^{*}(p)\subset\Delta_{m_{i}}^{\vee}$.

###### Proof.

Since the equalities are achieved in the proof of Lemma 2.9, we have
$\phi^{*}(p)-\phi^{*}(p^{\prime})=-\sum s_{i}\lambda(m_{i})$. For any
$y\in\partial\Delta_{\lambda}^{\vee}$, we have

$\begin{split}\phi(y)\geq\langle
y,p^{\prime}\rangle-\phi^{*}(p^{\prime})=\langle
y,p^{\prime}\rangle-\phi^{*}(p)-\sum s_{i}\lambda(m_{i})\\\ =\langle
y,p\rangle-\phi^{*}(p)-\sum s_{i}(\lambda(m_{i})+\langle
y,m_{i}\rangle).\end{split}$

Now suppose $y$ is any conjugate point of $p$, then $\phi(y)=\langle
y,p\rangle-\phi^{*}(p)$, whence

$\sum s_{i}(\lambda(m_{i})+\langle y,m_{i}\rangle)\geq 0.$

By construction $\lambda(m_{i})+\langle y,m_{i}\rangle\leq 0$ for any
$y\in\Delta_{\lambda}^{\vee}$. For any $i$, as long as $s_{i}>0$, then
$\langle y,m_{i}\rangle=-\lambda(m_{i})$, namely $y\in\Delta_{m_{i}}^{\vee}$.
∎

######  Remark 2.12.

In the special case that $x$ lies on the interior of one face, then there is
only one $m=m_{i}$ involved, with $p=p^{\prime}+sm$, and since $p\neq
p^{\prime}$, we are forced to have $s>0$, and
$\bar{\nabla}\phi^{*}(p)\subset\Delta_{m}^{\vee}$.

######  Proposition 2.13.

(Comparison of gradient notions) Let $x\in\text{Int}(\Delta_{m}^{\vee})$, then
$p\in H_{m}^{+}$ is a gradient in $\nabla\phi(x)$, if and only if $p\in
M_{\mathbb{R}}/\mathbb{R}m$ under the identification
$H_{m}^{+}\simeq\text{Image}(\Delta_{\mu})\subset M_{\mathbb{R}}/\mathbb{R}m$
is a classical gradient of the convex function $\phi$ on
$\text{Int}(\Delta_{m}^{\vee})$.

###### Proof.

If $p\in\nabla\phi(x)$, then restricted to $\text{Int}(\Delta_{m}^{\vee})$ we
have

$\phi(y)\geq\langle y,p\rangle-\phi^{*}(p)=\phi(x)+\langle
y-x,p\rangle,\quad\forall y\in\text{Int}(\Delta_{m}^{\vee}),$

which means $p$ is a classical gradient.

For the converse, we suppose $p\in M_{\mathbb{R}}/\mathbb{R}m$ is a classical
gradient at $x$, which is identified uniquely with $p\in\Delta_{n}\subset
H_{m}^{+}$. Since $\Delta_{\lambda}^{\vee}$ lies in the half space $\langle
m,x\rangle+\lambda(m)\leq 0$, and $\langle n,m\rangle=1$, we can write any
$y\in\partial\Delta_{\lambda}^{\vee}$ as

$y=y^{\prime}-sn,\quad s\geq 0,\quad\langle m,y^{\prime}\rangle+\lambda(m)=0.$

Recall $\phi$ has a canonical extension to a convex function on
$N_{\mathbb{R}}$. Restricted to the hyperplane $\langle
m,\cdot\rangle+\lambda(m)=0$, the classical gradient property implies

$\phi(y^{\prime})\geq\phi(x)+\langle y^{\prime}-x,p\rangle.$

But since the convex function $\phi$ on $N_{\mathbb{R}}$ has gradient
contained in $\Delta_{\mu}$,

$\phi(y^{\prime})\leq\phi(y)+\max_{\Delta_{\mu}}\langle
y^{\prime}-y,\cdot\rangle\leq\phi(y)-s\mu(n).$

Combining the above,

$\phi(y)\geq\phi(x)+\langle y^{\prime}-x,p\rangle+s\mu(n)=\phi(x)+\langle
y^{\prime}-x-sn,p\rangle=\phi(x)+\langle y-x,p\rangle.$

Since $y$ is arbitrary, we deduce $\phi(x)+\phi^{*}(p)=\langle x,p\rangle$,
namely $p\in\bar{\nabla}\phi(x)$. Since $x$ lies on only one face, and $p\in
H_{m}^{+}$, we conclude $p\in\nabla\phi(x)$. ∎

###  2.4 Variational problem

We now set up a global functional on $\mathcal{P}\simeq\mathcal{P}^{\vee}$:

$\mathcal{F}(\phi)=\int_{\partial\Delta_{\lambda}^{\vee}}\phi
d\mathcal{L}^{\vee}+\int_{\partial\Delta_{\mu}}\phi^{*}d\mathcal{L}.$ (5)

This functional is manifestly symmetric in terms of the Legendre transform;
this ‘mirror symmetry’ is fundamental to our approach. In this variational
problem, the _existence of a minimizer_ is quite transparent, but the local
PDE nature of the minimizer is less obvious, since the definition of the
Legendre transform is not local, and the differentiability of the functional
at the critical point is not a priori clear.

######  Theorem 2.14.

The functional admits a minimizer in $\mathcal{P}$, henceforth denoted as
$\phi$.

###### Proof.

(_cf._ [14, Thm 5.2]) By Lemma 2.3 and the fact that
$d\mathcal{L},d\mathcal{L}^{\vee}$ are both probability measures, for any
$c\in\mathbb{R}$ we have $\mathcal{F}(\phi+c)=\mathcal{F}(\phi)$. This allows
us to impose without loss that $\max_{\partial\Delta_{\lambda}^{\vee}}\phi=0$.
Then by the uniform Lipschitz estimate in $\mathcal{P}$, we can envoke Arzela-
Ascoli to take a $C^{0}$ limit of a minimizing sequence of the functional. By
Lemma 2.3, the Legendre transformed functions also converge in $C^{0}$, so the
limit attains the minimum. ∎

######  Theorem 2.15.

The minimizer is unique up to an additive constant.

###### Proof.

(compare [14, Thm 5.2]) Suppose $\tilde{\phi}$ is another minimizer, and we
consider $\phi^{\prime}=\frac{1}{2}(\phi+\tilde{\phi})$, which still lies in
$\mathcal{P}$ by the convexity of the function class $\mathcal{P}$. Now for
any $p\in\partial\Delta_{\mu}$,

$\begin{split}&\phi^{\prime*}(p)=\max\langle x,p\rangle-\phi^{\prime}(x)\\\
\leq&\frac{1}{2}\max(\langle x,p\rangle-\phi(x))+\frac{1}{2}\max(\langle
x,p\rangle-\tilde{\phi}(x))=\frac{1}{2}(\phi^{*}(p)+\tilde{\phi}^{*}(p)),\end{split}$

whence
$\mathcal{F}(\phi^{\prime})\leq\frac{1}{2}(\mathcal{F}(\phi)+\mathcal{F}(\tilde{\phi}))$.
This forces $\phi^{\prime}$ to be also a minimizer, and the equality is
achieved. If $p\in\nabla\phi^{\prime}(x)$, then the equality
$\phi^{\prime*}(p)=\frac{1}{2}(\phi^{*}(p)+\tilde{\phi}^{*}(p))$ implies

$(\phi(x)+\phi^{*}(p)-\langle
x,p\rangle)+(\tilde{\phi}(x)+\tilde{\phi}^{*}(p)-\langle x,p\rangle)=0,$

which forces $p\in\nabla\phi(x)$ and $p\in\nabla\tilde{\phi}(x)$.

When we restrict to any open face $\text{Int}(\Delta_{m}^{\vee})$ of
$\partial\Delta_{\lambda}^{\vee}$, the functions $\phi,\tilde{\phi}$ are
convex functions on convex domains, and $\nabla\phi(x),\nabla\tilde{\phi}(x)$
are identified with the classical gradients of convex functions, which
naturally lie inside the quotient space $M_{\mathbb{R}}/\mathbb{R}m$ (_cf._
section 2.3). Since $\nabla\phi=\nabla\tilde{\phi}$ Lebesgue-a.e, the
difference $\phi-\tilde{\phi}$ is a constant on the face (_cf._ [14, Lem
5.3]). By the continuity of the functions, and matching the functions on the
intersection of different faces, we see that the constant must be independent
of the face. ∎

###  2.5 Variational inequality

We now derive a fundamental _variational inequality for the minimizer_.

######  Proposition 2.16.

For any measurable $E\subset\partial\Delta_{\lambda}^{\vee}$, we have
$|\bar{\nabla}\phi(E)|\geq|E|.$ Completely analogously
$|\bar{\nabla}\phi^{*}(E)|\geq|E|$ for any measurable subset
$E\subset\partial\Delta_{\mu}$.

###### Proof.

We consider the function $\phi-t1_{E}$ for $0<t\ll 1$, where $1_{E}$ denotes
the characteristic function of $E$. By Lemma 2.3,

$\phi^{*}\leq(\phi-t1_{E})^{*}\leq\phi^{*}+t.$

Morever $(\phi-t1_{E})^{*}>\phi^{*}$ at $p\in\partial\Delta_{\mu}$, only when

$\sup_{x\in E}(\langle p,x\rangle-\phi(x))\geq\phi^{*}(p)-t.$

Hence

$\begin{split}t|\\{p:\sup_{x\in E}(\langle
p,x\rangle-\phi(x))\geq\phi^{*}(p)-t\\}|\geq\int_{\partial\Delta_{\mu}}((\phi-t1_{E})^{*}-\phi^{*})d\mathcal{L}.\end{split}$

Since $\phi$ is a minimizer, plugging in $(\phi-t1_{E})^{**}$ as a competitor
(the caveat being that $(\phi-t1_{E})$ may not lie in $\mathcal{P}$), we get

$\int_{\partial\Delta_{\mu}}(\phi-t1_{E})^{*}-\phi^{*}\geq-\int_{\partial\Delta_{\lambda}^{\vee}}(\phi-t1_{E})^{**}-\phi\geq-\int_{\partial\Delta_{\lambda}^{\vee}}(\phi-t1_{E})-\phi=t|E|.$

Here the second inequality uses Lemma 2.2. Combining the above and cancelling
the $t$ factor,

$|\\{p:\sup_{x\in E}(\langle
p,x\rangle-\phi(x))\geq\phi^{*}(p)-t\\}|\geq|E|,\quad\forall 0<t\ll 1.$

Taking the $t\to 0$ limit, and recalling $\phi^{*}(p)\geq\langle
x,p\rangle-\phi(x)$, we get

$|\\{p:\sup_{x\in E}(\langle p,x\rangle-\phi(x))=\phi^{*}(p)\\}|\geq|E|.$

For closed subsets $E$, the sup can be replaced by max by the continuity of
$\phi$, so the LHS subset is the conjugate set $\bar{\nabla}\phi(E)$. For
general Borel subsets $E$, we take a compact exhaustion of $E$. For any
compact $K\subset E$, we observe

$|\bar{\nabla}\phi(E)|\geq|\bar{\nabla}\phi(K)|\geq|K|.$

Since the Lebesgue measure is inner regular, we obtain
$|\bar{\nabla}\phi(E)|\geq|E|$ by taking the limit $K\uparrow E$. ∎

######  Corollary 2.17.

If no point in the measurable subset $E\subset\partial\Delta_{\lambda}^{\vee}$
has any anomalous conjugate point, then $|\nabla\phi(E)|\geq|E|$.

###### Proof.

If there is no anomalous conjugate point, then
$\nabla\phi(E)=\bar{\nabla}\phi(E)$. We then apply the variational inequality.
∎

######  Remark 2.18.

We can now explain the heuristic why the variational problem should be related
to the _real MA equation_. In the ideal situation, there is no anomalous
conjugate point, and $\nabla\phi,\nabla\phi^{*}$ are both bijective on points,
and define mutually inverse maps. Then

$|\nabla\phi(E)|\geq|E|=|\nabla\phi^{*}(\nabla\phi(E))|\geq|\nabla\phi(E)|.$

This forces equality everywhere. On each open face of
$\partial\Delta_{\lambda}^{\vee}$ the gradient agrees with the classical
notion in convex function theory, and $|\nabla\phi(E)|=|E|$ is just the weak
formulation of the real MA equation. To make this argument work more
rigorously, one needs to control the anomalous conjugate points, and the
multivalued nature of gradient.

###  2.6 Anomalous conjugate points

The remaining technical difficulty mainly comes from the possible existence of
anomalous conjugate points.

We introduce the subset
$\mathcal{S}^{\vee}\subset\partial\Delta_{\lambda}^{\vee}$ as
$\mathcal{S}^{\vee}=\mathcal{S}^{\vee}_{1}\cup\mathcal{S}^{\vee}_{2}$, where

$\mathcal{S}^{\vee}_{1}=\bigcup\\{\text{lower dim faces of
$\partial\Delta_{\lambda}^{\vee}$}\\},$
$\mathcal{S}^{\vee}_{2}=\bigcup_{m}\\{x|\bar{\nabla}\phi(x)\cap
H_{m}^{+}\text{ contains at least two points}\\}.$

In particular $\nabla\phi(x)$ is single valued outside $\mathcal{S}^{\vee}$.
Similarly the reader may write down the Legendre dual analogue $\mathcal{S}$.
We now recall a basic fact in classical convex function theory.

######  Lemma 2.19.

The subset $\mathcal{S}^{\vee}\subset\partial\Delta_{\lambda}^{\vee}$ has
Lebesgue measure zero. A similar statement holds for
$\mathcal{S}\subset\partial\Delta_{\mu}$.

###### Proof.

(_cf._ [13, section 1.1.1]) We restrict $\phi$ to the the open face
$\text{Int}(\Delta_{m}^{\vee})$, to obtain a classical convex function on a
convex domain. As explained in section 2.3, the notion of gradient inside
$H_{m}^{+}$ is naturally identified with the classical notion which takes
value in $M_{\mathbb{R}}/\mathbb{R}m$. Since our convex functions are
Lipschitz, they are differentiable Lebesgue-a.e. The multi-valued gradient
problem happens on the non-differentiability locus, which has measure zero.
Now the global statement follows by taking the union of all the faces. ∎

The following lemmas say that the anomalous conjugate has controlled effect on
the gradient set. In this section, let $E$ be a measurable subset of
$\Delta_{m}^{\vee}$, such that any $x\in E$ admits some anomalous conjugate
point in $H_{m}^{-}$. Let $F^{\prime}=\bar{\nabla}\phi(E)\cap H_{m}^{-}$, and
let $F\subset H_{m}^{+}$ be the image of $F^{\prime}$ under the projection
$H_{m}^{-}\to H_{m}^{+}$ (_cf._ section 2.3). By the proof of Lemma 2.9, we
have $F\subset\bar{\nabla}\phi(E)\cap H_{m}^{+}$.

######  Lemma 2.20.

$|F|\leq|E|$.

###### Proof.

By construction every $p\in F\setminus\mathcal{S}$ is of the form
$p=p^{\prime}+sm$ with $s>0$ and $p^{\prime}\in F^{\prime}$. By Cor. 2.11
concerning the anomalous conjugate points, we must have
$\bar{\nabla}\phi^{*}(p)\subset\Delta_{m}^{\vee}$. Since $p\in
H_{m}^{+}\setminus\mathcal{S}$, it lies on only one face $\Delta_{n}\subset
H_{m}^{+}$ with $\langle n,m\rangle=1$, whence $\Delta_{m}^{\vee}\subset
H_{n}^{+}$. We then have
$\bar{\nabla}\phi^{*}(p)\subset\Delta_{m}^{\vee}\subset H_{n}^{+},$ whence

$\bar{\nabla}\phi^{*}(p)=\bar{\nabla}\phi^{*}(p)\cap
H_{n}^{+}=\nabla\phi^{*}(p).$

Since $p\notin\mathcal{S}$, we know that $\nabla\phi^{*}(p)$ consists of only
one point, so in fact $\nabla\phi^{*}(p)=\\{x\\}$. But $p\in
F\subset\bar{\nabla}\phi(E)$, so $x\in E$ and $\bar{\nabla}\phi^{*}(p)\subset
E$.

In summary we get the inclusion
$\bar{\nabla}\phi^{*}(F\setminus\mathcal{S})\subset E.$ Now applying the
variational inequality Prop. 2.16,

$|E|\geq|\bar{\nabla}\phi^{*}(F\setminus\mathcal{S})|\geq|F\setminus\mathcal{S}|=|F|.$

The last equality uses that $\mathcal{S}$ has zero measure, by Lemma 2.19. ∎

######  Lemma 2.21.

$|F^{\prime}|\leq|F|$. When the equality is achieved, then $F^{\prime}$ has
zero measure except perhaps on the faces $\Delta_{n}$ with $\langle
n,m\rangle=-1$.

###### Proof.

By Lemma 2.5, under the quotient map $M_{\mathbb{R}}\to
M_{\mathbb{R}}/\mathbb{R}m$, the sets $H_{m}^{+}$ and $H_{m}^{-}$ each project
homeomorphically to the image of $\Delta_{\mu}$. The projection map is
piecewise affine, and we focus on some face $\Delta_{n}$. With respect to the
natural Lebesgue measures $dp$ induced by the integral structure, the quotient
map $\Delta_{n}\to M_{\mathbb{R}}/\mathbb{R}m$ has Jacobian factor $|\langle
n,m\rangle|$. For $\Delta_{n}\subset H_{m}^{+}$ this factor is equal to one,
and for $\Delta_{n}\subset H_{m}^{-}$ this factor is a positive integer (_cf._
Lemma 2.5). Thus under the projection $H_{m}^{-}\to H_{m}^{+}$, the Jacobian
factor is $\geq 1$. Since the normalized measure $d\mathcal{L}$ is a constant
multiple of $dp$, the Jacobian factor is still $\geq 1$ with respect to
$d\mathcal{L}$. This shows $|F^{\prime}|\leq|F|$. The equality case forces the
Jacobian factor to be equal to one, which means $|\langle n,m\rangle|=1$. ∎

######  Proposition 2.22.

$|E|=|F|=|F^{\prime}|$. In particular the equality forces that $F^{\prime}$
has zero measure except perhaps on the faces $\Delta_{n}$ with $\langle
n,m\rangle=-1$.

###### Proof.

By Lemma 2.20, 2.21, we already know $|E|\geq|F|\geq|F^{\prime}|$, so it
suffices to prove $|F^{\prime}|\geq|E|$. We write
$H_{m}^{-}=\bigcup\Delta_{n_{i}}$. For each $i$, we have the set inclusion
$\Delta_{m}^{\vee}\subset H_{n_{i}}^{-}$, and since any point of $E$ is
conjugate to some point in $H_{m}^{-}\cap\bar{\nabla}\phi(E)=F^{\prime}$, we
conclude

$E\subset\bar{\nabla}\phi^{*}(F^{\prime})\cap\Delta_{m}^{\vee}\subset\bigcup_{i}\bar{\nabla}\phi^{*}(F^{\prime}\cap\Delta_{n_{i}})\cap
H_{n_{i}}^{-},$

whence $|E|\leq\sum_{i}|\bar{\nabla}\phi^{*}(F^{\prime}\cap\Delta_{n_{i}})\cap
H_{n_{i}}^{-}|.$

Recall that our setup has the Legendre duality symmetry. For any fixed $i$, we
now let $F^{\prime}\cap\Delta_{n_{i}}$ play the role of $E$ in the Legendre
dual version of Lemma 2.20, 2.21. The role of $F^{\prime}$ is then played by
$\bar{\nabla}\phi^{*}(F^{\prime}\cap\Delta_{n_{i}})\cap H_{n_{i}}^{-}$, and we
conclude

$|\bar{\nabla}\phi^{*}(F^{\prime}\cap\Delta_{n_{i}})\cap
H_{n_{i}}^{-}|\leq|F^{\prime}\cap\Delta_{n_{i}}|.$

Summing over $i$ and combining the above,

$|E|\leq\sum_{i}|\bar{\nabla}\phi^{*}(F^{\prime}\cap\Delta_{n_{i}})\cap
H_{n_{i}}^{-}|\leq\sum_{i}|F^{\prime}\cap\Delta_{n_{i}}|=|F^{\prime}|.$

This is the promised reverse inequality. ∎

######  Remark 2.23.

The equality forces a very strong rigidity, and it is an interesting question
whether it forces $|E|=0$ in fact.

######  Corollary 2.24.

$|E|=|\nabla\phi(E)\cap F|\leq|\nabla\phi(E)|$.

###### Proof.

Since $|E|=|F|$ by Prop. 2.22, it suffices to prove
$F\setminus\nabla\phi(E)\cup\mathcal{S}$ has measure zero. Now every point in
$p\in F\setminus\mathcal{S}$ has a unique gradient $x=\nabla\phi^{*}(p)\in E$,
and if $x\notin\mathcal{S}^{\vee}$ then $p=\nabla\phi(x)$. This shows that

$F\setminus\nabla\phi(E)\cup\mathcal{S}\subset\text{Image of
}(\bar{\nabla}\phi(\mathcal{S}^{\vee}\cap E)\cap H_{m}^{-})\text{ under
}H_{m}^{-}\to H_{m}^{+}.$

By applying Prop. 2.22 again to $E\cap\mathcal{S}^{\vee}$, we see

$|F\setminus\nabla\phi(E)\cup\mathcal{S}^{\vee}|\leq|\text{Image}(\bar{\nabla}\phi(\mathcal{S}^{\vee}\cap
E)\cap H_{m}^{-})|=|E\cap\mathcal{S}^{\vee}|=0,$

so the result follows. ∎

###  2.7 Anomalous conjugate points II

In this section we aim to prove

######  Proposition 2.25.

For any $E\subset\partial\Delta_{\lambda}^{\vee}$, we have
$|\nabla\phi(E)|\leq|E|$.

######  Lemma 2.26.

Assume the subset $E\subset\Delta_{m}^{\vee}$ has the property that any $x\in
E$ admits some conjugate in a face $\Delta_{n}$ with $\langle m,n\rangle=0$.
Then $|E\cap\nabla\phi^{*}(\partial\Delta_{\mu})|=0$.

###### Proof.

Consider a point
$x\in(E\cap\nabla\phi^{*}(\partial\Delta_{\mu}))\setminus\mathcal{S}^{\vee}$.
Since $x\in\nabla\phi^{*}(\partial\Delta_{\mu})$, we write
$x=\nabla\phi^{*}(p)$. Since $x$ does not lie on $\mathcal{S}^{\vee}$, it has
a unique gradient, so $p=\nabla\phi(x)$.

Now by assumption $x$ has some anomalous conjugate $p^{\prime}$ in a face
$\Delta_{n}$ with $\langle m,n\rangle=0$. By Lemma 2.14, there exists a
gradient of the form $p^{\prime}+sm$ for $s>0$. The uniqueness of the gradient
then forces $p=p^{\prime}+sm$. Since $p^{\prime}\in\Delta_{n}$, we have
$\langle p^{\prime},n\rangle+\mu(n)=0$, hence

$\langle p,n\rangle+\mu(n)=\langle p-p^{\prime},n\rangle=s\langle
m,n\rangle=0.$

This means $p\in\Delta_{n}$. Since $x$ lies on $\Delta_{m}^{\vee}$ but not on
$\mathcal{S}^{\vee}$, it must be in the interior of $\Delta_{m}^{\vee}$, so by
the definition of gradient $x=\nabla\phi^{*}(p)$ forces $\langle
m,n\rangle=1$. This contradicts $\langle m,n\rangle=0$. This contradiction
shows that $x$ cannot exist, which means
$(E\cap\nabla\phi^{*}(\partial\Delta_{\mu}^{\vee}))\setminus\mathcal{S}^{\vee}$
is empty. Since $\mathcal{S}^{\vee}$ has measure zero by Lemma 2.19, we see
$|E\cap\nabla\phi^{*}(\partial\Delta_{\mu})|=0$. ∎

We also record the Legendre dual statement.

######  Lemma 2.27.

Suppose the subset $G\subset\Delta_{n}$ has the property that any $p\in G$
admits a conjugate point in $\Delta_{m}^{\vee}$ with $\langle m,n\rangle=0$.
Then $|G\cap\nabla\phi(\partial\Delta_{\lambda}^{\vee})|=0$.

We can now prove Prop. 2.25.

###### Proof.

(Prop. 2.25) First, we notice it suffices to prove Prop. 2.25 only for
$E\subset\Delta_{m}^{\vee}$. This is because we can decompose $E$ into the
union of $E_{m}\subset\Delta_{m}^{\vee}$ for all the possible $m$, and once we
know $|\nabla\phi(E_{m})|\leq|E_{m}|$ for every $m$, it would follow that

$|\nabla\phi(E)|\leq\sum|\nabla\phi(E_{m})|\leq\sum|E_{m}|=|E|.$

The same argument shows more generally that if we can partition $E$ into a
union of subsets, and the intersections have measure zero, then it suffices to
prove the statement for the subsets.

We will partition $\nabla\phi(E)\setminus\mathcal{S}$ into several subsets.

  1. 1.

Let $G_{1}$ be the union over $n$ of all the
$p\in\nabla\phi(E)\cap\Delta_{n}\setminus\mathcal{S}$, which admit an
anomalous conjugate point in some $\Delta_{m^{\prime}}^{\vee}$ with $\langle
m^{\prime},n\rangle=0$. By Lemma 2.27, its size $|G_{1}|=0$.

  2. 2.

Let $G_{2}$ consist of all the $p\in\nabla\phi(E)\setminus\mathcal{S}$ which
admit an anomalous conjugate point, but not covered by the first case. We
decompose $G_{2}\subset\nabla\phi(E)\subset H_{m}^{+}$ into the union of
$G_{2,n}=G_{2}\cap\Delta_{n}$ for all $\Delta_{n}\subset H_{m}^{+}$. By the
Legendre dual version of Cor. 2.24 applied to $G_{2,n}$, we have
$|G_{2,n}|\leq|\nabla\phi^{*}(G_{2,n})|,$ whence

$|G_{2}|=\sum|G_{2,n}|\leq\sum|\nabla\phi^{*}(G_{2,n})|.$

Next, for any $p\in\nabla\phi(E)\setminus\mathcal{S}$, we can write
$p=\nabla\phi(x)$ for $x\in E$, and since $p$ is not in $\mathcal{S}$, it has
a unique gradient point which is $x\in E$. Thus
$\nabla\phi(\nabla\phi^{*}(p))$ contains $p$, and $\nabla\phi^{*}(p)\subset
E\subset\Delta_{m}^{\vee}$. Since the subset of points in $\Delta_{m}^{\vee}$
with multiple gradients has zero measure (_cf._ Lemma 2.19), we see that the
intersections between the sets $\nabla\phi^{*}(G_{2,n})$ have zero measure,
whence

$|G_{2}|\leq\sum|\nabla\phi^{*}(G_{2,n})|=|\nabla\phi^{*}(G_{2})|.$

  3. 3.

Let $G_{3}$ consist of all the $p\in\nabla\phi(E)\setminus\mathcal{S}$ with no
anomalous conjugate point. Any such $p$ lies outside of $\mathcal{S}$, hence
has a unique gradient, which must then be its unique conjugate point. The set
$\nabla\phi^{*}(G_{3})\subset E$ is the union of these conjugate points. By
the Legendre dual version of Cor. 2.17 we have
$|\nabla\phi^{*}(G_{3})|\geq|G_{3}|$.

By construction $\nabla\phi(E)\setminus\mathcal{S}=G_{1}\cup G_{2}\cup G_{3}$,
hence by the above discussions

$\begin{split}|\nabla\phi(E)\setminus\mathcal{S}|\leq|G_{1}|+|G_{2}|+|G_{3}|\leq|\nabla\phi^{*}(G_{2})|+|\nabla\phi^{*}(G_{3})|.\end{split}$

The same argument in item 2 above shows that
$|\nabla\phi^{*}(G_{2})\cap\nabla\phi^{*}(G_{3})|=0$, so

$|\nabla\phi(E)\setminus\mathcal{S}|\leq|\nabla\phi^{*}(G_{2})|+|\nabla\phi^{*}(G_{3})|=|\nabla\phi^{*}(G_{2}\cup
G_{3})|\leq|E|.$

The last inequality here is because $\nabla\phi^{*}(G_{2}\cup G_{3})\subset
E$. Now since $\mathcal{S}$ has measure zero, we get

$|\nabla\phi(E)|=|\nabla\phi(E)\setminus\mathcal{S}|\leq|E|,$

as required. ∎

###  2.8 Structure theorems for the minimizer

In this section we will list certain structural properties of the minimizer
$\phi$, which in the best situation lead to the real MA equation.

######  Proposition 2.28.

The set function $E\mapsto|\nabla\phi(E)|$ defines a _Borel measure_ on
$\partial\Delta_{\lambda}^{\vee}$, which is absolutely continuous with respect
to the Lebesgue measure. Likewise with the Legendre dual analogue.

###### Proof.

Since $|\nabla\phi(E)|\leq|E|$ by Prop. 2.25, we see that $|E|=0$ implies
$|\nabla\phi(E)|=0$. Next we justify the countable additivity axiom. Let
$E=\bigcup E_{i}$ be a disjoint partition, then clearly
$|\nabla\phi(E)|\leq\sum|\nabla\phi(E_{i})|$. To see that the equality holds,
without loss we may assume $E$ is disjoint from the measure zero set
$\mathcal{S}^{\vee}$, so that the gradient is a single valued map. Morever,
the mutual intersections of $\nabla\phi(E_{i})$ is contained in $\mathcal{S}$,
which again has measure zero, so

$\sum|\nabla\phi(E_{i})|=|\bigcup_{i}\nabla\phi(E)|=|\nabla\phi(E)|$

as required. ∎

We now introduce a _good-bad decomposition_ on
$\partial\Delta_{\lambda}^{\vee}$. The _good set_
$\mathcal{G}^{\vee}\subset\partial\Delta_{\lambda}^{\vee}$ consists of the
following two types of points $x\in\partial\Delta_{\lambda}^{\vee}$:

  * •

Type I good point: $x$ has no anomalous conjugate point, so that all conjugate
points are gradients of $x$;

  * •

Type II good point: $x$ is contained in some face $\Delta_{m}^{\vee}$ and $x$
admits an anomalous conjugate point $p\in\bar{\nabla}\phi(x)\cap H_{m}^{-}$.

The _bad set_ $\mathcal{B}^{\vee}\subset\Delta_{\lambda}^{\vee}$ consists of
all $x$ contained in some face $\Delta_{m}^{\vee}$, and which admits some
anomalous conjugate point $p\in\Delta_{n}$ with $\langle m,n\rangle=0$.
Clearly
$\partial\Delta_{\lambda}^{\vee}=\mathcal{G}^{\vee}\cup\mathcal{B}^{\vee}$. We
leave the reader to write down the Legendre dual version
$\partial\Delta_{\mu}=\mathcal{G}\cup\mathcal{B}$.

######  Theorem 2.29.

(Property of _good-bad decomposition_) If $E\subset\mathcal{G}^{\vee}$, then
$|\nabla\phi(E)|=|E|$. If $E\subset\mathcal{B}^{\vee}$, then
$|\nabla\phi(E)|=0$. In particular
$|\mathcal{G}^{\vee}\cap\mathcal{B}^{\vee}|=0$.

###### Proof.

Since the measure $E\mapsto|\nabla\phi(E)|$ has finite additivity, it suffices
to consider $E$ of single types. For type I good points, we use
$|\nabla\phi(E)|\leq|E|$ from Prop. 2.25 and $|\nabla\phi(E)|\geq|E|$ from
Cor. 2.17. For type II good points, we use Cor. 2.24, Prop. 2.25 and finite
additivity.

Now we consider the bad points. By finite additivity, without loss
$E\subset\Delta_{m}^{\vee}$. Since
$|\mathcal{S}^{\vee}|=|\nabla\phi(\mathcal{S}^{\vee})|=0$, without loss we can
suppose $E$ is disjoint from $\mathcal{S}^{\vee}$, so every $x\in E$ has a
unique gradient $p=\nabla\phi(x)$. By assumption $E$ consists of bad points,
so there is an anomalous conjugate $p^{\prime}\in\Delta_{n}$ with $\langle
m,n\rangle=0$. By Lemma 2.9, the unique gradient is of the form
$p=p^{\prime}+sm$ for $s>0$. Thus

$\langle p,n\rangle=\langle p^{\prime},n\rangle+s\langle n,m\rangle=-\mu(n),$

whence $p$ lies on $\Delta_{n}$ as well. Since $p$ is a gradient of $x$, it
must also lie on $H_{m}^{+}$, so it falls into $H_{m}^{+}\cap\Delta_{n}$,
which has measure zero. ∎

One lesson of Prop. 2.29 is that $\nabla\phi$ only sees the good set, and an
analogous version holds for $\nabla\phi^{*}$. In fact modulo measure zero
sets, these two maps are in some sense inverse to each other, when we restrict
to the good set.

######  Theorem 2.30.

(_Legendre duality_) The maps $\nabla\phi$ and $\nabla\phi^{*}$ have the
following properties:

  1. 1.

For any $E\subset\mathcal{G}^{\vee}$, we have
$|\nabla\phi(E)\cap\mathcal{B}|=0$, and $|\nabla\phi(E)\cap\mathcal{G}|=|E|$.

  2. 2.

We consider the maps between subsets of $\mathcal{G}^{\vee}$ and $\mathcal{G}$
defined by

$\begin{cases}E\mapsto\nabla\phi(E)\cap\mathcal{G},\quad\forall
E\subset\mathcal{G}^{\vee},\\\
F\mapsto\nabla\phi^{*}(F)\cap\mathcal{G}^{\vee},\quad\forall
F\subset\mathcal{G}.\end{cases}$

The mutual compositions agree with $E,F$ up to a measure zero subset.

  3. 3.

In particular $|\mathcal{G}|=|\mathcal{G}^{\vee}|$ and
$|\mathcal{B}|=|\mathcal{B}^{\vee}|$.

###### Proof.

The claim that $|\nabla\phi(E)\cap\mathcal{B}|=0$ follows from the Legendre
dual version of Lemma 2.26. By Prop. 2.28 and Thm. 2.29, we have
$|\mathcal{G}\cap\mathcal{B}|=0$, and

$|\nabla\phi(E)\cap\mathcal{G}|=|\nabla\phi(E)|-|\nabla\phi(E)\cap\mathcal{B}|=|\nabla\phi(E)|=|E|.$

This proves item 1.

Consequently, the two maps in item 2 _preserve the measures_. To prove the
composition statement, without loss $E$ is disjoint from the measure zero set
$\mathcal{S}^{\vee}$. For any
$p\in\nabla\phi(E)\cap\mathcal{G}\setminus\mathcal{S}$, the
$\nabla\phi^{*}(p)$ uniquely recovers the point $x\in E$ with
$p=\nabla\phi(x)$. Thus the composition of $\nabla\phi$ and $\nabla\phi^{*}$
recovers a full measure subset of $E$. This proves item 2.

Now by item 1, 2 applied to $\mathcal{G}^{\vee}$, we get
$|\mathcal{G}|=|\mathcal{G}^{\vee}|$. Since
$\partial\Delta_{\lambda}^{\vee}=\mathcal{G}^{\vee}\cup\mathcal{B}^{\vee}$ and
$|\mathcal{G}^{\vee}\cap\mathcal{B}^{\vee}|=0$, we have

$|\mathcal{B}|=1-|\mathcal{G}|=1-|\mathcal{G}^{\vee}|=|\mathcal{B}^{\vee}|,$

which proves item 3. ∎

The most important special case for us is the following:

######  Theorem 2.31.

The following conditions are equivalent:

  1. 1.

The size of the bad set $|\mathcal{B}|=0$, or equivalently
$|\mathcal{B}^{\vee}|=0$. (For instance, this works if $\langle n,m\rangle=0$
never holds for any vertices of $\Delta,\Delta^{\vee}$.)

  2. 2.

The images of $\nabla\phi$ and $\nabla\phi^{*}$ have full measure.

When this holds, then $|\nabla\phi(E)|=|E|$ for any
$E\subset\partial\Delta_{\lambda}^{\vee}$, and likewise with $\nabla\phi^{*}$.
The two maps $\nabla\phi$ and $\nabla\phi^{*}$ compose to the identity modulo
measure zero sets.

###### Proof.

By Thm. 2.30, the vanishing of $|\mathcal{B}|$ would force
$|\mathcal{B}^{\vee}|=0$. Thus the good sets have full measure, and we know
from Thm. 2.30 that $\nabla\phi$ and $\nabla\phi^{*}$ on the good sets are
measure preserving, mutually inverse maps modulo zero measure. In particular
the images of $\nabla\phi$ and $\nabla\phi^{*}$ have full measure.

Conversely, suppose the images of $\nabla\phi$ and $\nabla\phi^{*}$ have full
measure. By Thm. 2.29,

$|\nabla\phi(\partial\Delta_{\lambda}^{\vee})|=|\nabla\phi(\mathcal{G}^{\vee})|+|\nabla\phi(\mathcal{B}^{\vee})|=|\mathcal{G}^{\vee}|,$

which then forces $|\mathcal{G}^{\vee}|=1$, namely $|\mathcal{B}^{\vee}|=0$.
This shows that item 2 implies item 1, and the rest follow from Thm. 2.30. ∎

When the equivalence conditions in Thm. 2.31 is satisfied, then the
conclusions admit classical interpretations. When we restrict $\phi$ to an
open face $\text{Int}(\Delta_{m}^{\vee})$ to obtain a convex function still
denoted as $\phi$, then $\nabla\phi$ is identified with the classical gradient
(_cf._ section 2.3). Recall from (3) how the normalized measure is related to
the Lebesgue measure induced by the integral structure. Morever, the
projection map $H_{m}^{+}\to M_{\mathbb{R}}/\mathbb{R}m$ has Jacobian factor
one with respect to the integral Lebesgue measure (_cf._ the proof of Lemma
2.21). In summary, the fact that $|\nabla\phi(E)|=|E|$ with respect to the
normalized measures, translates into the weak Alexandrov formulation of the
following real MA equation over $\text{Int}(\Delta_{m}^{\vee})$:

$\det(D^{2}\phi)=C_{0}=\frac{\int_{\partial\Delta_{\mu}}dp}{\int_{\partial\Delta_{\lambda}^{\vee}}dx}.$
(6)

Completely analogously, we have the weak Alexandrov formulation of the
following real MA equation over the open faces $\text{Int}(\Delta_{n})$:

$\det(D^{2}\phi^{*})=C_{0}^{-1}.$ (7)

This expresses the local PDE nature of the variational problem.

###  2.9 Some examples

We first give some simple examples to which Theorem 2.31 applies.

######  Example 2.32.

We revisit the standard simplex example [14, section 1.1][17]. We take the
standard basis $e_{0},e_{1},\ldots e_{d+1}$ inside $\mathbb{Z}^{d+2}$, and
define

$M=\\{p\in\mathbb{Z}^{d+2}|\sum p_{i}=0\\},\quad
N=M^{\vee}=\mathbb{Z}^{d+2}/\mathbb{Z}(1,\ldots 1),\quad$

and $M_{\mathbb{R}}=M\otimes\mathbb{R},N_{\mathbb{R}}=N\otimes\mathbb{R}.$ The
polytope $\Delta\subset M_{\mathbb{R}}$ is the convex hull of

$m_{i}=(d+2)e_{i}-\sum_{0}^{d+1}e_{j},\quad i=0,\ldots d+1.$

Its dual polytope $\Delta^{\vee}$ is also a simplex, with vertices given by

$n_{0}=(-1,0,\ldots,0),\ldots n_{d+1}=(0,\ldots,0,-1).$

Thus

$\langle m_{i},n_{j}\rangle=\begin{cases}-(d+1),\quad&i=j,\\\ 1,\quad&i\neq
j.\end{cases}$

In particular $\langle n,m\rangle=0$ never occurs, and Thm 2.31 applies. This
example corresponds to hypersurfaces inside $\mathbb{CP}^{d+1}$.

######  Example 2.33.

Suppose we have $k$ integral reflexive polytopes $\Delta_{i}\subset
M_{\mathbb{R},i}$, with dual polytopes $\Delta_{i}^{\vee}\subset
N_{\mathbb{R},i}$, such that the pairing $\langle m,n\rangle\neq 0$ for all
possible vertices $m,n$ of $\Delta_{i}$ and $\Delta^{\vee}_{i}$. Then we can
take

$N_{\mathbb{R}}=\oplus_{i}N_{\mathbb{R},i},\quad
M_{\mathbb{R}}=\oplus_{i}M_{\mathbb{R},i},\quad\Delta\subset
M_{\mathbb{R}},\quad\Delta^{\vee}\subset N_{\mathbb{R}}.$

The vertices of $\Delta^{\vee}$ are of the form $(0,\ldots,n_{i},0,\ldots)$
with $n_{i}\in\text{vertex}(\Delta_{i}^{\vee})$, while the vertices of
$\Delta$ are of the form $(m_{1},\ldots m_{k})$ for
$m_{i}\in\text{vertex}(\Delta_{i})$. It is immediate to check that the pairing
$\langle m,n\rangle\neq 0$ still holds for $\Delta,\Delta^{\vee}$. Algebro-
geometrically, this is just taking the product of several toric Fano
varieties.

We now give a simple example which shows that $|\mathcal{B}|=0$ might fail if
$\langle m,n\rangle=0$ is allowed.

######  Example 2.34.

This example lives inside $\mathbb{R}^{2}$. Let $\Delta$ be the convex full of

$(1,0),(0,1),(-1,1),(-1,0),(0,-1),(1,-1).$

Then $\Delta^{\vee}$ is the convex hull of $\pm(1,0),\pm(1,1),\pm(0,1).$
Notice for instance that $m_{0}=(1,-1)$ is orthogonal to $n_{0}=(1,1)$. We can
take $\Delta_{\lambda}^{\vee}=\Delta^{\vee}$, and let

$\Delta_{\mu}=\\{(x,y)\in\mathbb{R}^{2}:|x|\leq 1,|y|\leq
1,|x+y|\leq\epsilon\\},$

for some given constant $0<\epsilon<1$. When $\epsilon\ll 1$, then
$\Delta_{\mu}$ is concentrated along a line segment.

We claim that $|\mathcal{B}|\neq 0$. Otherwise by Thm 2.31, we have
$|\nabla\phi(E)|=|E|$ for any $E$. We take $E=\Delta_{m}^{\vee}$, so that by
the definition of gradient $\nabla\phi(E)\subset H_{m}^{+}$. Thus

$|\Delta_{m}^{\vee}|=|\nabla\phi(\Delta_{m}^{\vee})|\leq|H_{m}^{+}|.$

For the choice $m=m_{0}=(1,-1)$, this leads to a contradition for small enough
$\epsilon$.

Algebro-geometrically, the corresponding Fano manifold $X_{\Delta}$ is the
blow up of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ at two toric fixed points, so
there are two disjoint exceptional divisors $E_{1},E_{2}$. Adjusting
$\epsilon$ amounts to changing the Kähler class on $X_{\Delta}$ in a
1-parameter family, with $\epsilon=1$ corresponding to
$c_{1}(\pi_{1}^{*}[\mathcal{O}_{\mathbb{P}^{1}}]+\pi_{2}^{*}[\mathcal{O}_{\mathbb{P}^{1}}])$,
and decreasing $\epsilon$ corresponding to subtracting multiplies of
$E_{1}+E_{2}$. The family $X_{t}$ of Calabi-Yau hypersurfaces (_cf._ the
introduction) is here just a pencil of elliptic curves. The second cohomology
of an elliptic curve has rank one, so the restriction of all these Kähler
classes to the elliptic curve will all be proportional. A plausible
interpretation of the above example, is that once an ample polarization $L\to
X$ on the degenerating Calabi-Yau hypersurfaces is fixed, there is still some
flexibility to extend this polarization to $L\to X_{\Delta}$, and for an
inappropriate choice of $L\to X_{\Delta}$, the variational problem may not be
helpful for the SYZ conjecture.

###  2.10 Open questions

Our work suggests a number of natural questions, which are relevant for
further applications to the SYZ conjecture, and which may be of some
independent interest to PDE theorists.

  1. 1.

Thm. 2.31 makes it clear that the failure for the minimizer to satisfy
$|\nabla\phi(E)|=|E|$ is measured by the _size of the bad set_
$|\mathcal{B}|$. Is it possible to compute or estimate $|\mathcal{B}|$ in
general, when we allow $\langle m,n\rangle=0$?

  2. 2.

Assume $|\mathcal{B}|=0$. Is it possible for the anomalous conjugate point to
exist?

  3. 3.

Assume $|\mathcal{B}|=0$. When can we prove that _the gradient is unique
everywhere_ (instead of Lebesgue-a.e), namely $\nabla\phi(x)$ consists of only
one point for every $x\in\partial\Delta_{\lambda}^{\vee}$?

  4. 4.

The Legendre dual version of the above question asks if $\nabla\phi^{*}(p)$
consists of only one point for every $p\in\partial\Delta_{\mu}$.

This question is closely related to the _strict convexity_ of the restriction
$\phi$ to a convex function on the open faces. This has implication on the
regularity question of $\phi$, since a strictly convex solution of the real MA
equation is known to be smooth.

If both $\nabla\phi$ and $\nabla\phi^{*}$ are single valued, they would define
mutually inverse maps setting up a homeomorphism between
$\partial\Delta_{\lambda}^{\vee}$ and $\partial\Delta_{\mu}$. This global
version of Legendre duality is desirable from the viewpoint of potential
applications to mirror symmetry.

  5. 5.

The polytopes $\partial\Delta_{\lambda}^{\vee}$ and $\partial\Delta_{\mu}$
have no natural global smooth structure; the only a priori information is that
the open faces $\text{Int}(\Delta_{m}^{\vee})$ and $\text{Int}(\Delta_{n})$
have natural affine structures, and in particular smooth structures.

Now suppose the previous questions have positive answers, so that $\phi$ and
$\phi^{*}$ are smooth on the respective open faces, and the gradient maps set
up a global homeomorphism. Then we can transfer $\text{Int}(\Delta_{n})$ to
the open subset
$\nabla\phi^{*}(\text{Int}(\Delta_{n}))\subset\partial\Delta_{\lambda}^{\vee}$,
and on the overlap with the open faces of $\partial\Delta_{\lambda}^{\vee}$
the smooth structures are compatible. This suggests that there is a _hidden
smooth structure with singularity_ on $\partial\Delta_{\lambda}^{\vee}$, where
the singular locus lies on the subset of the lower dimensional faces of
$\partial\Delta_{\lambda}^{\vee}$, which maps under $\nabla\phi$ to the lower
dimensional faces of $\partial\Delta_{\mu}$. Morever, this smooth structure
with singularity is compatible with Legendre duality.

It would be desirable for mirror symmetry (_cf._ the Kontsevich-Soibelman
conjecture [15]) that the singular locus is of _Hausdorff codimension two_.

######  Remark 2.35.

J. Hultgren informs the author that their upcoming work [1] will contain more
discussions on the affine structure with singularity.

##  3 Application to the SYZ conjecture

For backgrounds on the non-archimedean (NA) geometry, and the previous
literature on its relation to Kähler geometry, the reader may refer to [5]
[27, section 1][18]. A recent survey on the progress in the metric aspects of
the SYZ conjecture can be found in [21].

We now sketch how the result of section 2 implies Theorem 1.2. This argument
is essentially known, and consists of assembling ingredients from the
literature.

###  3.1 Complex geometry setup

We work in the context of the introduction, so $X_{\Delta}$ is a smooth toric
Fano manifold associated to the reflexive Delzant polytope $\Delta$, and
$X=\cup X_{t}$ (_cf._ (1)) is a union of Calabi-Yau hypersurfaces degenerating
to the toric boundary. This degeneration family admits a natural model

$\mathcal{X}=\\{X_{can}+tF=0\\}\subset X_{\Delta}\times\mathbb{C}.$

Upon base change, $X$ can be regarded as a family over
$\text{Spec}\mathbb{C}(\\!(t)\\!)$, and $\mathcal{X}$ defines a _model_ over
$\text{Spec}\mathbb{C}[\\![t]\\!]$, which is divisorial log terminal (_dlt_)
since by assumption the singularity structure on the central fibre is étale
locally of the form (_cf._ [14, Page 25])

$\mathcal{X}\simeq V(x_{1}\ldots
x_{k}-ty)\subset\mathbb{A}^{k+2}_{x_{i},t,y}\times\mathbb{A}^{d-k},\quad
d=\dim_{\mathbb{C}}X_{t}.$

The central fibres is identified with the toric boundary of $X_{\Delta}$, and
consists of many divisors labelled by the vertices $n\in\Delta^{\vee}$, each
with multiplicity one. The _dual complex_ $\Delta_{\mathcal{X}}$ of the dlt
model $\Delta_{\mathcal{X}}$ has vertices corresponding to these divisors, and
the tropicalization map gives a natural identification

$\text{trop}:\Delta_{\mathcal{X}}\simeq\Delta^{\vee}\subset N_{\mathbb{R}}.$

The model $\mathcal{X}$ is _minimal_ , meaning that the logarithmic relative
canonical bundle is trivial. As a caveat, our dlt model $\mathcal{X}$ is not
$\mathbb{Q}$-factorial.

Any ample line bundle $L\to X_{\Delta}$ corresponds to a moment polytope
$\Delta_{\mu}$ for some choice of $\mu(n)<0$ for
$n\in\text{vertex}(\Delta^{\vee})$, such that the piecewise linear function
$\mu:N_{\mathbb{R}}\to\mathbb{R}$ extending the values $\mu(n)$ is a ‘strictly
concave’ function. This induces a polarization line bundle $L\to X$, which in
particular prescribes the Kähler class $c_{1}(L)$ on $X_{t}$. We denote
$(L^{d})=\int_{X_{t}}c_{1}(L)^{d}$.

######  Remark 3.1.

The concavity condition on $\mu$ is related to ampleness, and if we drop it,
the line bundle $L\to X_{\Delta}$ will only be big, even though the induced
line bundle $L\to X$ might still be ample. On the other hand, in section 2
this condition is not needed. Once we drop it, then for some
$n\in\text{vertex}(\Delta^{\vee})$ the face $\Delta_{n}$ may be empty, but
this does not affect the arguments.

By the adjunction formula, $X$ over some small punctured disc admits a nowhere
vanishing holomorphic volume form $\Omega$, which induces holomorphic volume
forms $\Omega_{t}$ on the hypersurfaces $X_{t}$, hence normalized Calabi-Yau
measures on $X_{t}$

$\mu_{t}=\frac{\Omega_{t}\wedge\overline{\Omega}_{t}}{\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}}.$
(8)

The metric SYZ conjecture concerns the Calabi-Yau metrics $(X_{t},\omega_{t})$
in the class $c_{1}(L)$, satisfying the complex MA equation

$\omega_{t}^{d}=(L^{d})\mu_{t}.$ (9)

The SYZ conjecture predicts that given any small $0<\delta\ll 1$, for all $t$
small enough depending on $\delta$, then $(X_{t},\omega_{t})$ contains an open
subset with $\mu_{t}$-measure at least $1-\delta$, which admits a special
Lagrangian torus fibration.

###  3.2 Non-archimedean geometry

The work of Boucksom-Jonsson [4] established the following picture. The family
$X$ viewed as a smooth projective variety over
$\text{Spec}\mathbb{C}(\\!(t)\\!)$ gives rise to the _Berkovich space_
$X^{an}$, which can be regarded as the limit of $X_{t}$ as $t\to 0$ under the
hybrid topology. The Berkovich space contains the _essential skeleton_
$Sk(X)$, and under the hybrid topology convergence, the family of probability
measures $\mu_{t}$ converge to a Lebesgue type probability measure $\mu_{0}$
supported on $Sk(X)\subset X^{an}$. This has a concrete description. Given a
dlt model $\mathcal{X}$, there is a canonical embedding of its dual complex
$\Delta_{\mathcal{X}}$ into $X^{an}$, and if $\mathcal{X}$ is minimal, then

$\text{emb}:\Delta_{\mathcal{X}}\simeq Sk(X)\subset X^{an}.$

In our case of Calabi-Yau hypersurfaces, under the canonical isomorphisms

$\Delta_{\mathcal{X}}\xrightarrow{emb}Sk(X)\xrightarrow{trop}\partial\Delta^{\vee}\subset
N_{\mathbb{R}},$

the probability measure $\mu_{0}$ is up to a global constant the Lebesgue
measure $dx$ induced by the integral structure on the open faces of
$\partial\Delta^{\vee}$. When the dust settles, $\mu_{0}=d\mathcal{L}^{\vee}$
(_cf._ (3)), where we choose $\lambda=-1$ so that
$\Delta_{\lambda}^{\vee}=\Delta^{\vee}$.

Given the model $\mathcal{X}$, every point on $X^{an}$ has a centre
$c_{\mathcal{X}}(x)$ which is a scheme theoretic point in $\mathcal{X}_{0}$.
The map $c_{\mathcal{X}}:X^{an}\to\mathcal{X}_{0}$ is anticontinuous, which
means the preimage of closed subsets of $\mathcal{X}_{0}$ are open subsets of
$X^{an}$. In particular, the preimage of the finitely many toric fixed points
in $\mathcal{X}_{0}$ is an open subset $U\subset X^{an}$, and the evaluation
of the logarithmic coordinates provides a natural _retraction map_ from $U$ to
the open faces of $\Delta_{\mathcal{X}}$, which is identified with the
_tropicalization map_ $U\to\partial\Delta^{\vee}\subset N_{\mathbb{R}}$.

Boucksom-Favre-Jonsson [3][2][5] developed a NA pluripotential theory, which
concerns the analogue of semipositive metrics and Monge-Ampère measures on
$X^{an}$. Its central result concerns the solution of the NA Calabi
conjecture. The special case relevant to us is the following:

######  Theorem 3.2.

Let $L\to X$ be an ample line bundle, then there is a unique up to constant
semipositive metric $\left\lVert\cdot\right\rVert_{CY}$ on $L\to X^{an}$,
whose NA MA measure is equal to the Lebesgue measure $(L^{d})\mu_{0}$
supported on $Sk(X)\subset X^{an}$.

It is convenient to think about a semipositive metric
$\left\lVert\cdot\right\rVert$ in terms of a potential function $\varphi$.
Concretely in our case, we can choose any local trivialising section $s$ of
the line bundle $L\to X_{\Delta}$, and evaluate $\varphi=-\log\left\lVert
s\right\rVert$. We say $\left\lVert\cdot\right\rVert$ satisfies the ‘weak
comparison property’, if under the retraction map from $U$ to the open faces
of $\Delta_{\mathcal{X}}$, the potential $\varphi$ is constant on fibres.
Notice that the pair $(\mathcal{X},\mathcal{X}_{0})$ is a semistable SNC pair
near the toric fixed points. Vilsmeier [29] implies that

######  Theorem 3.3.

[29][14, Lem. 6.1][27, Thm 3.17] Under the weak comparison property, then the
NA MA measure $\text{MA}_{NA}$ on $U\subset X^{an}$ is related to the real MA
measure $\text{MA}_{\mathbb{R}}$ on the open faces of
$\Delta_{\mathcal{X}}\simeq Sk(X)\simeq\partial\Delta^{\vee}$, by the equation

$1_{Sk(X)}\text{MA}_{NA}(\left\lVert\cdot\right\rVert)=d!\text{MA}_{\mathbb{R}}(\varphi).$

A key link to the SYZ conjecture is provided by [18].

######  Theorem 3.4.

The weak comparison property for the metric
$\left\lVert\cdot\right\rVert_{CY}$ implies the SYZ conjecture 1.1 for the
given family $X_{t}$.

######  Remark 3.5.

The semistable SNC setting (instead of the dlt setting) was assumed in [18],
but the proof only uses the semistable SNC condition on those divisor
intersection strata contributing to $Sk(X)$. If the reader prefers to work
with SNC models, one can apply the discussion to any SNC resolution of
$\mathcal{X}$ isomorphic to $\mathcal{X}$ on its smooth locus. The choice of
the resolution does not matter because the blow up along the singular locus of
$\mathcal{X}$ has no effect on $U\subset X^{an}$.

Our current strategy, which follows [14], is to produce
$\left\lVert\cdot\right\rVert_{CY}$ via solving the variational problem. We
are given the polytopes $\Delta,\Delta^{\vee}$,
$\Delta_{\lambda}^{\vee}=\Delta^{\vee}$ and $\Delta_{\mu}$, and we assume that
the equivalent conditions in Thm. 2.31 applies (eg. when (2) holds). Then
section 2 produces the minimizer $\phi$ on $\partial\Delta^{\vee}$, and by
Thm. 2.31, this implies that $\phi$ restricted to the open faces
$\text{Int}(\Delta_{m}^{\vee})$ satisfies the _real MA equation_ in the weak
formulation.

This minimizer $\phi$ extends canonically to a convex function on
$N_{\mathbb{R}}$ by

$\phi(x)=\max_{p\in\partial\Delta_{\mu}}\langle x,p\rangle-\phi^{*}(p).$

By [27, Thm. C], this convex function $\phi$ corresponds canonically to a
semipositive toric metric on the Berkovich analytification of the toric Fano
manifold $X_{\Delta}$, hence induces by restriction a semipositive metric
$\Phi$ on $L\to X^{an}$. This concretely works as follows. The integral points
$m\in M_{\mathbb{R}}$ correspond to monomial sections $s^{m}$ which induce
logarithms $\log|s^{m}|$ on $X^{an}$, and by linear interpolation we can make
sense of $\log|s^{p}|=\sum p_{i}\log|s^{i}|$, where $s^{i}$ corresponds to a
$\mathbb{Z}$-basis of $M_{\mathbb{R}}$. Then the semipositive metric is
defined by

$\Phi=\max_{p\in\partial\Delta_{\mu}}\log|s^{p}|-\phi^{*}(p).$ (10)

######  Lemma 3.6.

The semipositive metric $\Phi$ on $L\to X^{an}$ satisfies the weak comparison
property.

###### Proof.

Over the open face $\text{Int}(\Delta_{m}^{\vee})$, upon choosing a local
trivialising section as the monomial $s=s^{m}$, then $\log|s^{p}|$ is
identified with the local function on $X^{an}$

$\log|\frac{s^{p}}{s}|=\langle x,p\rangle-\langle x,m\rangle=\langle
x,p\rangle+\lambda(m)=\langle x,p\rangle-1.$

Here the point $x\in N_{\mathbb{R}}$ is the image under the tropicalization
map $U\to\partial\Delta^{\vee}\subset N_{\mathbb{R}}$. Thus the local
potential of $\Phi$ over the open face factorizes through the tropicalization
map, and is identified with the function $\phi-1$. ∎

######  Corollary 3.7.

The NA MA measure of $\Phi$ is supported on $Sk(X)$, and agrees with the
Lebesgue type measure $(L^{d})\mu_{0}$.

###### Proof.

Over any open face $\text{Int}(\Delta_{m}^{\vee})$, the convex function $\phi$
satisfies the weak formulation of the real MA equation (6),

$\det(D^{2}\phi)=C_{0}=\frac{\int_{\partial\Delta_{\mu}}dp}{\int_{\partial\Delta_{\lambda}^{\vee}}dx}.$

Applying Theorem 3.3, the NA MA measure over these open faces is supported
inside $Sk(X)\subset X^{an}$, and can be identified as $d!$ times the real MA
measure.

We need to figure out the normalisation constants. Tracing through the
conventions (_cf._ (3)), the NA MA measure is equal to

$d!\text{MA}_{\mathbb{R}}(\phi)=d!C_{0}|dx|=(d!C_{0}\int_{\partial\Delta_{\lambda}^{\vee}}dx)d\mathcal{L}^{\vee}=(d!\int_{\partial\Delta_{\mu}}dp)d\mathcal{L}^{\vee}=(d!\int_{\partial\Delta_{\mu}}dp)\mu_{0}.$
(11)

######  Claim 3.8.

The factor $d!\int_{\partial\Delta_{\mu}}dp=(L^{d})$.

To see the claim, we start with the asymptotic Riemann-Roch formula

$h^{0}(X_{t},kL)\sim\frac{(L^{d})}{d!}k^{d},\quad k\gg 1.$

On the other hand, by the short exact sequence on the toric Fano manifold
$X_{\Delta}$

$0\to kL\otimes\mathcal{O}(-X_{t})\to kL\to kL|_{X_{t}}\to 0$

together with the Serre vanishing of
$h^{1}(X_{t},kL\otimes\mathcal{O}(-X_{t}))$ for $k\gg 1$ (since $L$ is ample),

$h^{0}(X_{t},kL)=h^{0}(X_{\Delta},kL)-h^{0}(X_{\Delta},kL\otimes\mathcal{O}(-X_{t}))=h^{0}(X_{\Delta},kL)-h^{0}(X_{\Delta},kL\otimes\mathcal{O}(-\mathcal{X}_{0})).$

Now the global sections of $H^{0}(X_{\Delta},kL)$ are spanned by the monomial
sections, which correspond to the integral points on $k\Delta_{\mu}$, while
$H^{0}(X_{\Delta},kL\otimes\mathcal{O}(-\mathcal{X}_{0}))$ are spanned by
those monomials which vanish on the toric boundary of $X_{\Delta}$. Hence
their difference counts the integral points on $\partial(k\Delta_{\mu})$. Now
these correspond to the rational points in

$\frac{1}{k}\partial(k\Delta_{\mu})\cap\frac{1}{k}\mathbb{Z}^{d+1}=\partial\Delta_{\mu}\cap\frac{1}{k}\mathbb{Z}^{d+1}.$

and as $k\to+\infty$, they equidistribute with respect to the Lebesgue measure
on $\partial\Delta_{\mu}$. Hence

$h^{0}(X_{t},kL)=\\#(\partial\Delta_{\mu}\cap\frac{1}{k}\mathbb{Z}^{d+1})\sim
k^{d}\int_{\partial\Delta_{\mu}}dp.$

Comparing the asymptotes proves the claim.

Applying the claim to (11), we see the identification of the NA MA measure
with $(L^{d})\mu_{0}$, over all the open faces of
$Sk(X)\simeq\partial\Delta^{\vee}$. On the other hand, the total measure of
the semipositive metric $\Phi$ on $L\to X^{an}$ is $(L^{d})$. This forces the
open faces to contain the full measure. ∎

Now by the uniqueness of the NA MA equation, the semipositive metric $\Phi$
must agree with the Boucksom-Favre-Jonsson solution
$\left\lVert\cdot\right\rVert_{CY}$. Since $\Phi$ is known to satisfy the weak
comparison property, by applying Thm. 3.4 we deduce that the SYZ conjecture
holds for the family $X_{t}$, whence Theorem 1.2 is proved.

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