# Survey on the metric SYZ conjecture and non-archimedean geometry

Yang Li

###### Abstract

We survey the metric aspects of the Strominger-Yau-Zaslow conjecture on the
existence of special Lagrangian fibrations on Calabi-Yau manifolds near the
large complex structure limit. We will discuss the diverse motivations for the
conjectural picture, what the best hopes are, and a number of subtleties. The
bulk of the survey highlights the role of pluripotential theory, and non-
archimedean geometry in particular, with a list of open questions.

##  1 Overview

We survey the recent progress on the metric aspect of the Stominger-Yau-Zaslow
conjecture, which concerns the existence of special Lagrangian fibrations on
Calabi-Yau manifolds near the large complex structure limit. The paper is
based substantially on [52][53][51] and various subsequent talks. A rough
outline of the contents of chapter 2,3,4,5,6 is as follows:

  * •

We trace the diverse motivations of the SYZ conjecture from mirror symmetry,
minimal surfaces, Riemannian geometry, complex and non-archimedean geometry.
We discuss the most optimistic interpretation and its difficulties, before
moving on to a more cautious weak version.

  * •

We review the meaning of the large complex structure and essential skeleton
from a complex geometric perspective, before a brief survey on the Kontsevich-
Soibelman conjecture.

  * •

We explain the various analytic ingredients. A brief overview of Yau’s
solution of the Calabi conjecture is given, to explain why new ideas are
needed to understand the large complex structure limit. Our primary focus is
on complex pluripotential theory, which is the analytic core of [52][53],
alongside Savin’s small perturbation theorem in elliptic PDE, and the
regularity theory of real Monge-Ampère equation.

  * •

We include a minimalistic overview of non-archimedean pluripotential theory:
the notion of Berkovich spaces, semipositive metrices, and the non-archimedean
Calabi metric. We also explain the intuition of the conjectural ‘comparison
property’, which is needed to give a differential geometric interpretation of
the non-archimedean Calabi metric.

  * •

We outline the proof of the recent progress [52][53], emphasizing on the
intuition, the subtleties, and the open problems.

######  Acknowledgement.

The author is a current Clay Research Fellow and a MIT CLE Moore Instructor.
He thanks Léonard Pille-Schneider for comments.

##  2 Metric SYZ conjecture

The Strominger-Yau-Zaslow conjecture is originally motivated by a combination
of physical and differential geometric considerations, and stands at the
crossroad of mirror symmetry, minimal surface theory, Riemannian geometry,
complex Kähler geometry, and non-archimedean geometry. We will trace some
historically significant developments, to see how the gradual realization of
the analytic difficulties, has led to eclectic interpretations of the
conjecture.

###  2.1 The genesis of the SYZ conjecture

An $n$-dimensional Calabi-Yau (CY) manifold $(X,\omega,\Omega)$ is a Kähler
manifold with a nowhere vanishing holomorphic volume form $\Omega$, satisfying
the complex Monge-Ampère (MA) equation

$\omega^{n}=\text{const}\Omega\wedge\overline{\Omega},$ (1)

which implies the Ricci flatness of the metric. Furthermore, such manifolds
admit parallel spinors, hence are candidates for the target space metrics of
supersymmetric type II string theories. Special Lagrangians of phase $\theta$
are $n$-dimensional submanifolds $L$ satisfying

$\omega|_{L}=0,\quad\text{Im}(e^{-i\theta}\Omega)|_{L}=0.$ (2)

These are absolute minimizers within their homology classes, due to the
calibration inequality

$\int_{L}\text{Re}(e^{-i\theta}\Omega)\leq\int_{L}dvol_{L}=\text{Vol}(L)$ (3)

saturated precisely by the special Lagrangians. Physically, these correspond
to the support of BPS D-branes.

The Strominger-Yau-Zaslow conjecture [67] in its primitive form asks:

######  Conjecture 2.1.

Given a compact Calabi-Yau manifold $X$ near the large complex structure
limit, can we find a _special Lagrangian torus fibration_ on $X$?

The physical origin of the SYZ conjecture [67] comes largely from mirror
symmetry, and a very brief sketch is as follows. From homological mirror
symmetry, one expects a compact Calabi-Yau manifold $X$ admits a mirror
$X^{\vee}$, such that the category of D-branes on both sides are identified.
On the holomorphic side $X^{\vee}$ (‘B-side’), the points $x\in X^{\vee}$
support skyscrapper sheaves $\mathcal{O}_{x}$, which should correspond to
certain Lagrangian branes inside the symplectic side $X$ (‘A-side’). The
extension groups $\text{Ext}^{*}(\mathcal{O}_{x},\mathcal{O}_{x})\simeq
H^{*}(T^{n})$, which suggests the Lagrangian branes are torus objects. For
$x\neq y$, the Ext groups between $\mathcal{O}_{x},\mathcal{O}_{y}$ would
vanish, which suggests (inconclusively111The vanishing of Floer cohomology
does not imply the vanishing of Floer cochain spaces, and there seems to be no
strong argument to rule out intersecting special Lagrangians.) that the tori
are disjoint, leading to the speculation of the Lagrangian fibration
structure. The assertion about _special_ Lagrangians, is however beyond mere
homological mirror symmetry, and comes from the BPS condition on the D-branes.
The moduli space of all $\mathcal{O}_{x}$ is the mirror manifold $X^{\vee}$,
which should then be identified with the moduli space of BPS branes supported
on the special Lagrangian tori. This moduli interpretation gives rise to a
zeroth order approximation of Kähler structure on the mirror manifold
$X^{\vee}$, subject to the higher order corrections related to the holomorphic
discs (‘instanton corrections’), whose effect is supposedly exponentially
suppressed except near the singular fibres. Ignoring the subtleties of
singular fibres, then the SYZ picture offers a program to reconstruct the
mirror, and interpret homological mirror symmetry as a version of Fourier-
Mukai transform (‘Mirror symmetry is T-duality’).

######  Notation.

Our convention is $d=\partial+\bar{\partial}$,
$d^{c}=\frac{\sqrt{-1}}{2\pi}(-\partial+\bar{\partial})$, so
$dd^{c}=\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}$. The relation between
Kähler potentials and Kähler metrics is $\omega_{\phi}=\omega+dd^{c}\phi$.
Alternatively, we think of a Kähler metric in terms of local absolute
potentials, meaning $\omega=dd^{c}\varphi$ for locally defined psh functions
$\varphi$. Given a Hermitian metric $h$ on a line bundle $L$, its curvature
form is $-dd^{c}\log h^{1/2}$ in the class $c_{1}(L)$.

Differential geometrically, the main evidence presented in the SYZ paper is
the semiflat metrics. Consider the logarithm map
$\text{Log}_{t}:(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$ over some open convex
subset $U\subset\mathbb{R}^{n}$,

$\text{Log}_{t}(z_{1},\ldots
z_{n})=\frac{1}{\log|t|}(\log|z_{1}|,\ldots\log|z_{n}|).$

Imposing $T^{n}$ symmetry, then by an elementary Hessian computation, $\phi$
is a smooth strictly convex function downstairs on $U$ if and only if its
pullback to $\text{Log}_{t}^{-1}(U)$ is a smooth Kähler potential, and the
Calabi-Yau condition

$(dd^{c}\phi\circ\text{Log}_{t})^{n}=\frac{\text{const}}{|\log|t||^{2n}}\prod
d\log z_{i}\wedge d\overline{\log z_{i}}$

is equivalent to the real Monge-Ampère equation

$\det(D^{2}\phi)=\text{const}.$

In this setting, the metric $dd^{c}\phi\circ\text{Log}_{t}$ is called
semiflat, because its restriction to the $T^{n}$ fibres are Euclidean, due to
$T^{n}$-symmetry. With respect to the Calabi-Yau structure

$\omega=dd^{c}(\phi\circ\text{Log}_{t}),\quad\Omega=\sqrt{-1}^{n}\prod d\log
z_{i},$

the $T^{n}$ fibres are _special Lagrangians_ of phase zero. The purpose of
introducing the normalization parameter $t$, is that as $t\to 0$, the
$T^{n}$-fibres shrink down to zero size, and the Calabi-Yau metrics
$dd^{c}(\phi\circ\text{Log}_{t})$ converge to the real Monge-Ampère metric on
$U\subset\mathbb{R}^{n}$

$g_{0}=\frac{1}{2\pi}\sum_{i,j}\frac{\partial^{2}\phi}{\partial x_{i}\partial
x_{j}}dx_{i}dx_{j},\quad\det(D^{2}\phi)=\text{const}.$ (4)

Now near the large complex structure limit, which is a certain limiting
situation for a family of Calabi-Yau metrics, it is expected that the semiflat
metrics emerge as an asymptotic description of the degenerating Calabi-Yau
metrics, in the _generic region_ of the Calabi-Yau manifolds. In the SYZ
picture, the generic region heuristically means away from the singular special
Lagrangian fibres. The main point is that in the generic region, the special
Lagrangians are just small perturbations of the logarithm maps in local toric
charts. As we approach the large complex structure limit, the percentage of
the Calabi-Yau volume measure occupied by the generic region should tend to
$100\%$.

This recount of this SYZ heuristic reasoning underlines a few precautions:

  * •

The metric predictions are more compelling in the _generic region_. The non-
generic region is subject to instanton correction effects, whose metric
significance is much more difficult to analyze.

  * •

The SYZ fibration is likely an _emergent_ behaviour near the large complex
structure limit. In particular, the conjecture is concerned with a _family_ of
Calabi-Yau manifolds, and features such as semiflat metrics would only appear
sufficiently close to the limit.

###  2.2 Further motivations

Aside from its importance in mirror symmetry, the SYZ conjecture is
interesting for other diverse fields such as minimal surface theory,
Riemannian geometry, Kähler and algebraic geometry.

  * •

Special Lagrangians are minimal submanifolds, and in fact calibrated
submanifolds. Currently, there are few methods for producing special
Lagrangians in sufficiently large supply of Calabi-Yau manifolds, although
there is a series of conjectures initiated by Thomas-Yau [75][74] and further
developed in [43][57].

  * •

The behaviour of a family of Einstein metrics depends strongly on whether the
volume of geodesic balls satisfies the noncollapsing condition for some
uniform constant $\kappa>0$.

$\text{Vol}_{g}(B_{g}(r))\geq\kappa r^{\dim_{\mathbb{R}}X},\quad\forall
0<r\leq\text{diam}(X).$

A good convergence and regularity theory is available in the non-collapsing
case [13]. On the other hand, metric degeneration in the collapsing case is
largely terra incognita in Riemannian geometry, and the semiflat metric
asymptote is a highly nontrivial emergent feature for a collapsing family of
Calabi-Yau metrics.

  * •

A recurring theme of Kähler geometry is the interplay between metric and
complex geometry. Käher-Einstein metrics in the non-collapsing case is tied to
projective geometry [23]. The large complex structure limit is a very severe
kind of polarized degeneration, whose transcendental behaviour (related to
exponential and logarithms) is not adequately captured by traditional
projective geometry, and instead non-archimedean geometry stands out as a
natural framework. One can then ask about the relation between Calabi-Yau
metrics and non-archimedean geometry, a problem that turns out to be related
to the SYZ conjecture.

###  2.3 Collapsing K3 surfaces with elliptic surfaces

An influential early work related to the SYZ conjecture is the gluing
description of Gross-Wilson [32] concerning the hyperkähler metric on K3
surfaces $X$ with elliptic fibrations $\pi:X\to\mathbb{CP}^{1}$.

By hyperkähler rotation, the special Lagrangian torus fibres can be viewed as
the holomorphic elliptic curves in a different complex structure. In the
generic situation, the elliptic fibration has 24 $I_{1}$-type singular fibres,
namely the local singularity in the fibration is modelled on
$(z_{1},z_{2})\mapsto z_{1}z_{2}$ complex geometrically. They fix Kähler
classes $[\omega_{X}]$ on the K3 surface, and $[\omega_{\mathbb{P}^{1}}]$ on
$\mathbb{CP}^{1}$, and describe the Calabi-Yau metrics $\omega_{\tau}$ in the
Kähler class $\tau[\omega_{X}]+\pi^{*}[\omega_{\mathbb{P}^{1}}]$ for
$0<\tau\ll 1$. Some key conceptual features are:

  * •

In the generic region, the metrics $\omega_{\tau}$ are up to exponentially
small errors modelled on _semiflat metrics_ , which can be explicitly
described via the periods integrals of the elliptic curves. The subset of the
K3 surface on which the semiflat metric asymptote breaks down, has length
scale $O(\tau^{1/2}|\log\tau|^{1/2})$.

  * •

As $\tau\to 0$, the metrics $\omega_{\tau}$ converge in the Gromov-Hausdorff
sense to a singular metric on $\mathbb{CP}^{1}$. The diameter of the K3
surfaces is of constant order $O(1)$. The length scale of generic elliptic
curve fibres is $O(\tau^{1/2})$, which shrinks to zero size as $\tau\to 0$.

  * •

In the neighbourhood of the $I_{1}$ type singular fibres, the metrics are
modelled on the _Ooguri-Vafa metrics_ , which are explicit $S^{1}$-invariant
incomplete hyperkähler metrics constructed by means of the _Gibbons-Hawking
ansatz_.

  * •

The asymptotic geometry of the Ooguri-Vafa metric matches with the semiflat
metric. This is a basic requirement for the gluing construction.

  * •

Near the nodal singular point of the $I_{1}$-fibre, the Ooguri-Vafa metric
contains a small region approximated by the _Taub-NUT metric_ , whose length
scale is $O(\tau^{1/2}|\log\tau|^{-1/2})$. These regions concentrate almost
the entire $L^{2}$-Riemannian curvature of the K3 surface.

The Gross-Wilson picture represents the _best hope_ on the SYZ
conjecture,222As a caveat sometimes overlooked in the literature, the
hyperkähler rotation of the Gross-Wilson setting is not quite a polarized
degeneration family, hence does not quite fit into our notion of large complex
structure limit. In our perspective, Gross-Wilson is an inspiration, rather
than an example of the SYZ conjecture.  which includes an almost explicit
description of the metric, and a special Lagrangian torus fibration exists
globally on $X$. It is partially generalized to higher dimensional hyperkähler
manifolds with holomorphic Lagrangian abelian variety fibrations. After
hyperkähler rotation, these can be regarded as special Lagrangian fibrations.
Tosatti et al. [72] [31] established the semiflat metric asymptote in the
generic region, and Gromov-Hausdorff collapse to the base manifold.

###  2.4 Best hope on Calabi-Yau 3-folds

For general information on this section, see [42, Chapter 8,9], and the
introduction in [54]. In the initial years following the SYZ proposal, there
was an overly optimistic belief based on the analogy with the hyperkähler
case, and based on topological and complex geometric considerations:

  * •

The special Lagrangian fibration exists globally and is defined by a
$C^{\infty}$ map, even though some fibres may be singular.

  * •

The discriminant locus on the base is codimension two. For Calabi-Yau 3-folds,
under suitable genericity assumption, the discriminant locus is a trivalent
graph, with two types of vertices, known as positive and negative
vertices.333The names ‘positive/negative vertex’ come from some old fashioned
topological models of the torus fibration where the most singular fibres have
Euler characteristics $\pm 1$ respectively.

  * •

Along the edges of the trivalent graph, the singularity of the SYZ fibration
is transversely modelled on the $I_{1}$ singularity.

  * •

The local region near the _positive vertex_ is complex geometrically a large
open subset inside

$\\{z_{0}z_{1}z_{2}=1-z_{3}\\}\subset\mathbb{C}^{3}\times\mathbb{C}^{*}_{z_{3}},\quad\Omega\propto\frac{1}{z_{3}}dz_{0}\wedge
dz_{1}\wedge dz_{2},$

while the _negative vertex_ region is modelled on a large open subset inside

$\\{z_{3}z_{4}=1-z_{1}-z_{2}\\}\subset(\mathbb{C}^{*})^{2}_{z_{1},z_{2}}\times\mathbb{C}^{2}_{z_{3},z_{4}},\quad\Omega\propto\frac{1}{z_{1}z_{2}}dz_{2}\wedge
dz_{3}\wedge dz_{4}.$

The naïvete was challenged by Joyce [41], based on his observations concerning
_special Lagrangian singularities_ , which are markedly different from those
visible in holomorphic fibrations. A global special Lagrangian fibration on
the compact Calabi-Yau manifolds near the large complex structure, should it
exist at all, is expected to have a much more subtle structure:

  * •

The special Lagrangian fibration is typically not defined by $C^{\infty}$
maps, but are at best piecewise smooth.

  * •

The discriminant locus on the base is typically not codimension two, but the
trivalent graph is expected to be thickened to a codimension one ‘ribbon’. The
amount of thickening probably tends to zero in the large complex structure
limit.

  * •

The $I_{1}\times\mathbb{R}$ local singularity model does not lead to a
Fredholm deformation theory for the singular special Lagrangian fibres, so
should be replaced by some other singularity models.

Stepping aside from the substantial difficulties of the special Lagrangian
local singularities, another major difficulty is to understand the Calabi-Yau
metrics near the large complex structure limit, in complex dimension three.
The optimistic expectations are inspired by the Gross-Wilson picture in
complex dimension two. Hypothetically, the Calabi-Yau 3-fold is Gromov-
Hausdorff close to a real 3-dimensional manifold, whose topology is believed
to be the 3-sphere444This expectation comes from the topology of the essential
skeleton, see section 3.3 below., containing a trivialent graph, such that

  * •

Away from the trivalent graph, the Calabi-Yau metric is semiflat up to
exponentially small errors.

  * •

Transverse to the edges in the trivalent graph, the metric is modelled on the
Ooguri-Vafa metric appearing in Gross and Wilson’s picture.

  * •

Near the positive and negative vertices of the trivalent graph, the local
metric is modelled on some generalization of the Ooguri-Vafa metrics.

It was recently realized that there exist almost canonical constructions of
Ooguri-Vafa type metrics in complex dimension three, with the predicted
topology and complex structure of the positive and negative vertices,
constructed from a (nonlinear) generalized Gibbons-Hawking ansatz [54]. The
asymptotic geometry of these Ooguri-Vafa type metrics matches with semiflat
metrics in the generic region. The positive vertex metric contains a local
region modelled on a generalized Taub-NUT type metric on $\mathbb{C}^{3}$,
analogous to the way the Ooguri-Vafa metric contains a region modelled on the
Taub-NUT metric.

The following questions are widely open:

######  Question 1.

Do such Ooguri-Vafa type metrics arise as blow up limits on any _compact_
Calabi-Yau manifolds near the large complex structure limit?

######  Question 2.

Can one give a gluing description of the CY metrics for 3-folds near the large
complex structure, eg. in the case of quintic hypersurfaces?

######  Question 3.

What kind of special Lagrangians can arise on $C^{\infty}$-small perturbations
of these Ooguri-Vafa type metrics?

###  2.5 Strong vs. weak SYZ conjecture

Due to the analytic difficulty of the SYZ conjecture, especially the
nongeneric regions with large Riemannian curvature, the literature has
developed many interpretations of the conjectures, with somewhat diverging
goals.

  * •

(Soft versions) For applications to homological mirror symmetry, one is
primarily interested in constructing Lagrangian fibrations without requiring
$\text{Im}(\Omega)|_{L}=0$, and therefrom build a mirror manifold $X^{\vee}$
and prove the categorical predictions $D^{b}Coh(X^{\vee})\simeq
D^{\pi}Fuk(X,\omega)$. This viewpoint separates the symplectic and holomorphic
data of the Calabi-Yau manifold, and has a topological/algebraic flavour.

######  Remark 1.

The _special Lagrangian_ condition usually left out of homological mirror
symmetry discussions, is supposedly related to _Bridgeland stability
conditions_ , which is a popular categorical interpretation of the BPS
condition on D-branes.

  * •

(Strong metric version) On compact Calabi-Yau manifolds near the large complex
structure limit, find a global special Lagrangian torus fibration.

  * •

(Weak metric version) Prove for a suitable class of compact Calabi-Yau
manifolds near the large complex structure limit that a special Lagrangian
$T^{n}$-fibration exists on a large subset with at least $99\%$ of the Calabi-
Yau measure. More precisely, the percentage converges to $100\%$ in the limit.

The metric versions are highly sensitive to the Calabi-Yau metric, which is a
much more rigid structure compared to the soft versions. They conform to the
PDE spirit of the SYZ paper, while the soft versions are closer to the mirror
symmetry motivations of the SYZ conjecture.

The strong metric version is perhaps the most faithful to the original
intention of the SYZ paper. Its main evidence comes from the special case of
hyperkähler metrics, mentioned in section 2.3. Other peripheral evidence comes
from the construction of Lagrangian fibrations in many examples, ignoring the
$\text{Im}(\Omega)|_{L}=0$ condition [58]. There are however many subtleties
besetting this strong version, mentioned in section 2.4, making the conjecture
very formidable, and by comparison the supporting evidence seems inadequate.
Notably, the special Lagrangian singularities are not sufficiently understood,
and we are not aware of any argument that definitively rules out special
Lagrangians intersecting each other in the non-generic regions with large
Riemannian curvature. As food for future thought, a somewhat weakened version
bypassing these possible objections is

######  Question 4.

Given a compact Calabi-Yau manifold $X$ sufficiently near the large complex
structure limit, is there an $n$-parameter family of special Lagrangian
currents, whose supports sweep out all points on $X$?

The weak metric version, on the other hand, concerns only the generic region,
which is more accessible than the strong version. It conforms to the more
cautious expectation, that the special Lagrangian fibration is only a limiting
phenomenon. The bulk of the survey will focus on this weak version.

##  3 Large complex structure limit

The SYZ paper does not make precise the notion of large complex structure
limit, and several non-equivalent interpretations are available in the current
literature. We shall place the large complex structure limit in the framework
of polarized degenerations. Intuitively, a polarized degeneration is when we
fix the symplectic structure inside an integral class, and vary the complex
structure in an algebraic one-parameter family so that it becomes singular in
the limit. The large complex structure limit is the additional requirement
that the degeneration is ‘as severe as possible’.

We work over $\mathbb{C}$. To set the scene,

  * •

Let $S$ be a smooth affine algebraic curve, with a point $0\in S$. An
_algebraic degeneration family_ is given by a submersive projective morphism
$\pi:X\to S\setminus\\{0\\}$ with smooth connected $n$-dimensional fibres
$X_{t}$ for $t\in S\setminus\\{0\\}$. This is in contrast with the _formal_
setting over the punctured formal disc $\text{Spec}(K)$ with
$K=\mathbb{C}(\\!(t)\\!)$. An algebraic degeneration induces a formal
degeneration by base change.

  * •

A _polarisation_ is given by an ample line bundle $L$ over $X$. This specifies
the Kähler class, up to rescaling conventions.

  * •

We say $\pi$ is a degeneration family of _Calabi-Yau_ manifolds if there is a
trivialising section $\Omega$ of the canonical bundle $K_{X}$. Over a small
disc $\mathbb{D}_{t}$ around $0\in S$, this induces holomorphic volume forms
$\Omega_{t}$ on $X_{t}$ via $\Omega=dt\wedge\Omega_{t}$. The _normalised
Calabi-Yau measure_ on $X_{t}$ is the probability measure

$d\mu_{t}=\frac{\Omega_{t}\wedge\overline{\Omega}_{t}}{\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}}.$
(5)

The Calabi-Yau metrics $\omega_{CY,t}$ on $X_{t}$ are the unique Kähler
metrics in the class $\frac{1}{|\log|t||}c_{1}(L)$ such that

$\frac{\omega_{CY,t}^{n}}{\int_{X_{t}}\omega_{CY,t}^{n}}=d\mu_{t}.$ (6)

  * •

We say $\pi:X\to S\setminus\\{0\\}$ is a large complex structure limit of
Calabi-Yau manifolds if the essential skeleton has the maximal dimension $n$
(to be explained below), and the degeneration family admits a semistable snc
model over $S$.

######  Remark 2.

Filling in the central fibre at $0\in S$ would involve the choice of a _model_
of $\pi:X\to S$, namely a normal flat projective $S$-scheme $\mathcal{X}$
together with an isomorphism with $X$ over the punctured curve
$S\setminus\\{0\\}$. It is called an _snc model_ if $\mathcal{X}$ is smooth,
and the central fibre over $0\in S$ is a simple normal crossing divisor in
$\mathcal{X}$, such that the intersections of divisors are irreducible or
empty. If furthermore the central fibre is reduced, it is called a _semistable
snc model_. Models can be analogously defined over the formal disc. The
existence of snc models is a consequence of Hironaka’s resolution theorem.
They are highly nonunique. By the semistable reduction theorem [44, chapter
2], after finite base change to another smooth algebraic curve $S^{\prime}$,
we can always find some semistable snc model for the degeneration family
$X\times_{S}(S^{\prime}\setminus\\{0\\})$, so the existence of a semistable
snc model is not a substantial assumption. Everything here is quasi-
projective. The choice of a model is very useful, but not intrinsic to the
degenerating CY metrics.

######  Example 3.1.

A typical example of large complex structure limit is the Fermat family of
Calabi-Yau hypersurfaces

$X_{t}=\\{Z_{0}Z_{1}\ldots
Z_{n+1}+t\sum_{0}^{n+1}Z_{i}^{n+2}=0\\}\subset\mathbb{CP}^{n+1}.$

As $t\to 0$, the algebraic limit is the union of $n+2$ projective planes.
Intuitively, the central fibre is highly reducible, and the degeneration is
very severe.

######  Example 3.2.

Consider a family of quartic K3 surfaces degenerating to a nodal K3 surface.
This is a typical example of a polarized degeneration which is _not_ a large
complex structure limit. In fact, the central fibre is irreducible, and the
nodal singularity is mild (Kawamata log terminal in the birational geometry
terminology).

###  3.1 Volume asymptote and essential skeleton

Consider an algebraic Calabi-Yau degeneration family $X\to S\setminus\\{0\\}$
as above. We follow [3] to consider the asymptote of
$\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$ as $t\to 0$. Along the
way, we will introduce the concept of dual intersection complexes and
essential skeletons, which are simplicial complexes encoding the intersection
patterns of divisors on the central fibre. An important lesson is that the
measure theoretic limit of the Calabi-Yau manifolds is closer to simplicial
complexes than algebraic varieties, indicating that the metric limit must be
significantly different from Fubini-Study metrics associated to projective
embeddings of bounded degree.

A very useful tool is to fill in the central fibre by choosing an _snc model_
(_cf._ Remark 2) $\mathcal{X}$ over $S$. The central fibre $\mathcal{X}_{0}$
is an snc divisor with components $E_{i}$ for $i\in I$, and we write
$\mathcal{X}_{0}=\sum_{i\in I}b_{i}E_{i}$. In the special case of _semistable
snc models_ $b_{i}=1$ for $i\in I$; this can always be achieved after finite
base change. The canonical divisor $K_{\mathcal{X}}$ is supported on
$\mathcal{X}_{0}$ as $K_{X}$ has a trivialising section $\Omega$. We may write
$K_{\mathcal{X}}=\sum_{i}(a_{i}+b_{i}-1)E_{i}$, so that the _relative log
canonical divisor_

$K^{log}_{\mathcal{X}/S}:=K_{\mathcal{X}}-K_{S}+\mathcal{X}_{0,red}-\mathcal{X}_{0}=\sum
a_{i}E_{i}.$

Shifting all $a_{i}$ by a constant $\kappa$ is equivalent to multiplying
$\Omega$ by $t^{\kappa}$, which gives an elementary factor $|t|^{2\kappa}$ to
$\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$. Thus we shall always
assume $\min a_{i}=0$.

It is useful to introduce a _quantitative stratification_ on $X_{t}$ according
to the intersection pattern of $E_{i}$. Let $E_{J}=\cap_{i\in J}E_{i}$ for
$J\subset I$, which is irreducible if nonempty. Using the distance function of
a fixed smooth background Kähler metric on $\mathcal{X}$, we can write

$E_{J}^{0}=\\{q\in X_{t}|d(q,E_{J})\ll 1\\}\setminus\\{q\in
X_{t}|d(q,E_{J^{\prime}})\ll 1,\quad\text{some }J^{\prime}\supsetneq J\\}.$

Around $\emptyset\neq E_{J}\subset\mathcal{X}$, we denote $p=|J|-1$, and
introduce local coordinates $z_{0},\ldots z_{n}$ on $\mathcal{X}$, such that
$z_{0},z_{1},\ldots,z_{p}$ are the defining equations of $E_{i}$ for $i\in J$.
The conditions on the divisors mean that away from deeper strata we may
arrange $t=\prod_{0}^{p}z_{i}^{b_{i}}$, and

$\Omega=u_{J}\prod_{0}^{p}z_{i}^{a_{i}+b_{i}}d\log
z_{i}\wedge\prod_{p+1}^{n}dz_{j}$

for some local nowhere vanishing holomorphic function $u_{J}$. By definition
$\Omega=dt\wedge\Omega_{t}$ along $X_{t}$, so on $E_{J}^{0}$

$\Omega_{t}=b_{0}^{-1}u_{J}z_{0}^{a_{0}}\ldots z_{p}^{a_{p}}\prod_{1}^{p}d\log
z_{i}\wedge\prod_{p+1}^{n}dz_{j},$
$\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}=|b_{0}|^{-2}|u_{J}|^{2}|z_{0}|^{2a_{0}}\ldots|z_{p}|^{2a_{p}}\prod_{1}^{p}\sqrt{-1}d\log
z_{i}\wedge d\log\bar{z}_{i}\wedge\prod_{p+1}^{n}\sqrt{-1}dz_{j}\wedge
d\bar{z}_{j}.$

Notice also that the local equation $t=\prod_{0}^{p}z_{i}^{b_{i}}$ has
$b_{J}=\gcd_{i\in J}b_{i}$ sheets of solutions. Using the polar coordinates by
$z_{i}=e^{x_{i}\log|t|+\sqrt{-1}\theta_{i}}$ for $i\in J$, ones sees that the
magnitude of
$\int_{E_{J}^{0}}\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}$ is
$O(|\log|t||^{l})$ for $l=|\\{j\in J:a_{j}=0\\}|-1$.

The local logarithmic variables $x_{i}=\frac{\log|z_{i}|}{\log|t|}$ lie on the
simplex

$\Delta_{J}=\\{\sum_{0}^{p}b_{i}x_{i}=1,\quad 0\leq x_{i}\leq 1\\}.$

These depend on the choice of $z_{i}$, but since the local defining equation
of divisors differ by a nowhere vanishing holomorphic function, the ambiguity
of $x_{i}$ is only $O(\frac{1}{|\log|t||})$ for $0<|t|\ll 1$. Taking a more
global viewpoint, the combinatorial pattern of how these simplices fit
together exactly reflects the intersection pattern of the divisors $E_{i}$.
Formally, this information is encoded in the dual intersection complex
$\Delta_{\mathcal{X}}$ for the snc model $\mathcal{X}$: this is the polyhedral
complex whose vertices $v_{i}$ correspond to $E_{i}$, and we assign a simplex
$\Delta_{J}$ with vertices $v_{i}$ for $i\in J$ if and only if $E_{J}\neq 0$.
The coodinates $x_{j}$ then define a _piecewise integral affine structure_ on
$\Delta_{\mathcal{X}}$. Up to the above $O(\frac{1}{|\log|t||})$ ambiguity, we
now have a _logarithm map_
$\text{Log}_{\mathcal{X}}:X_{t}\to\Delta_{\mathcal{X}}$, locally described by
$x_{i}=\frac{\log|z_{i}|}{\log|t|}$. Consequently, the ‘hybrid’ space
$X\sqcup\Delta_{\mathcal{X}}$ is equipped with a natural topology, so that a
sequence of points $z_{k}\in X_{t}$ converges to $x\in\Delta_{\mathcal{X}}$
iff $t\to 0$ and $\text{Log}_{\mathcal{X}}(z_{k})\to x$. The name ‘hybrid’
refers to the mixture of algebraic varieties with simplicial objects, which is
better suited for measure theoretic limits, than the algebraic family
$\mathcal{X}$.

The measure also singles out a distinguished subcomplex $Sk(\mathcal{X})$,
called the essential skeleton, consisting of the simplices in
$\Delta_{\mathcal{X}}$ whose vertices correspond to $E_{i}$ with $a_{i}=0$.
This is where the limit of the normalised CY measure is supported. The
dimension of $Sk(\mathcal{X})$ is a measurement of how transcendental the
degeneration $X$ is; it is reflected by the growth order of
$\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$. The largest possible
value for the dimension is $n$.

In the case of a large complex structure limit,
$\dim_{\mathbb{R}}Sk(\mathcal{X})=n$. Let us analyze the CY measure more
explicitly, in a semistable snc model. For $E_{J}$ corresponding to an
$n$-dimensional simplex in $Sk(\mathcal{X})$, on $E_{J}^{0}$

$\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}=|u_{J}|^{2}\prod_{1}^{n}\sqrt{-1}d\log
z_{i}\wedge d\log\bar{z}_{i}.$ (7)

Here $u_{J}$ limits to its value $u_{J}(E_{J})$ at the point stratum $E_{J}$,
which is called the Poincaré residue of $\Omega$, and is easily seen to be
independent of the choice of coordinates $z_{i}$. It is a consequence of the
residue theorem on Riemann surfaces that $|u_{J}(E_{J})|^{2}$ is independent
of such $J$ [3, Thm. 7.1]. Thus the pushforward to $\Delta_{\mathcal{X}}$ of
the normalised CY measure (5) converges smoothly in the interior of
$\Delta_{J}$ to a constant multiple of the _Lebesgue measure_ :

$\text{Log}_{\mathcal{X}*}d\mu_{t}=\text{Log}_{\mathcal{X}*}\frac{\Omega_{t}\wedge\overline{\Omega}_{t}}{\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}}\xrightarrow{t\to
0}d\mu_{0}:=\text{Const}\cdot dx_{1}\ldots dx_{n}.$ (8)

Notice $dx_{1}\ldots dx_{n}$ is canonically defined due to the presence of an
integral affine structure on $\Delta_{J}$. Viewed as a measure on
$\Delta_{\mathcal{X}}$, the limit $d\mu_{0}$ has null measure on the
complement of the $n$-dimensional faces of $Sk(\mathcal{X})$, as the integral
of $d\mu_{t}$ in the corresponding region is $O(\frac{1}{|\log|t||})$. The
constant in (8) is independent of $J$ and its sole purpose is to make
$d\mu_{0}$ a probability measure.

###  3.2 The effect of blow up

A major caveat is that snc models are highly non-unique, because one can
always blow up a given snc model along some locus inside the central fibre.
The intrinsic information concerning polarized degenerations, such as the
limiting behaviour of metrics and volume measures, are independent of
particular snc models.

The effect of blow up on the dual intersection complex and essential skeleton
is well understood (_cf._ [49, A.4]). The intuitive picture is as follows:

  * •

If we blow up an snc model $\mathcal{X}$ along a smooth irreducible subvariety
properly contained inside $E_{J}=\cap_{j\in J}E_{j}$, but not contained in any
smaller intersection stratum, then the new dual intersection complex contains
$\Delta_{\mathcal{X}}$, while introducing a new vertex, and some new wings
over certain faces of $\Delta_{\mathcal{X}}$. The essential skeleton remains
intact.

  * •

If we blow up an snc model $\mathcal{X}$ along some $E_{J}=\cap_{j\in
J}E_{j}$, then the new dual intersection complex is a subdivision of
$\Delta_{\mathcal{X}}$. If the simplex $\Delta_{J}$ corresponding to $E_{J}$
is a face of the essential skeleton $Sk(\mathcal{X})$, then there is an
induced subdivision on $Sk(\mathcal{X})$, and otherwise the essential skeleton
remains intact.

While $\Delta_{\mathcal{X}}$ generally becomes larger under blow ups, the
essential skeleton $Sk(\mathcal{X})$ is only subdivided, and its piecewise
affine structure (in particular its homeomorphism type) is a birational
invariant, denoted as $Sk(X)$. The birational invariance is not surprising:
the essential skeleton is the measure theoretic limit of $X_{t}$, a property
independent of the choice of models.

###  3.3 Kontsevich-Soibelman conjecture

In an attempt to extract limiting information from the SYZ picture, Kontsevich
and Soibelman [48][49] proposed the following picture. Let $X_{t}$ be a
polarized degeneration of Calabi-Yau manifolds near the large complex
structure limit, then

  * •

(Kähler class normalization) The rescaled metrics
$\omega_{CY,t}\in\frac{1}{|\log|t||}c_{1}(\mathcal{O}(1))$ have nontrivial
_finite diameter_ Gromov Hausdorff subsequential limits.

  * •

There is an _affine structure_ on the essential skeleton $Sk(X)$ away from a
Hausdorff _codimension two_ singular subset. On the smooth locus, there is a
Riemannian metric obtained as the Hessian of local solutions to the _real
Monge-Ampère equation_. This metric agrees with the _Gromov-Hausdorff limit_ ,
which is conjecturally independent of the subsequence.

  * •

(Topology) Under the strict Calabi-Yau condition $h^{p,0}(X_{t})=0$ for $p<n$,
the essential skeleton is homeomorphic to $S^{n}$.

The heuristic idea is that the essential skeleton should be the base of the
hypothetical SYZ fibration, and the SYZ fibration is approximated by logarithm
maps, at least in the generic region. We briefly comment on the status of the
Kontsevich-Soibelman conjecture (_cf._ also [53, section 4.5]):

  * •

The uniform diameter estimate independent of small $t$

$C^{-1}\leq\text{diam}(X,\omega_{CY,t})\leq C,$

is recently established in joint work with Tosatti [56], using primarily
Riemannian geometric methods. This together with Gromov compactness proves the
Kähler class normalization prediction.

  * •

The prediction about the existence of a metrically preferred affine structure
away from _codimension two_ on the base, is part of the general lore of the
SYZ conjecture (_cf._ section 2.4), even though there is insufficient
evidence. Generally speaking, the simplicial complex structure on $Sk(X)$
induces a piecewise affine structure, and improving it to an affine structure
away from codimension two, would require highly nontrivial choices. The author
is not aware of a completely satisfactory answer to the following elementary
question:

######  Question 5.

Given a one-parameter family of quartic K3 surfaces near the large complex
structure limit, without special symmetry, how can we determine the location
of singular points on $Sk(X)$? How can we write down the affine structure on
the regular locus explicitly?

  * •

The topology of the essential skeleton is an active research topic in
birational geometry. The homeomorphism between $Sk(X)$ with $S^{n}$ is
verified for many examples. In general, it is known [62][63] that $Sk(X)$ is a
‘pseudomanifold’, its rational homology groups agree with $S^{n}$, and its
fundamental group has trivial profinite completion, but the actual
homeomorphism type is still elusive, and supposedly requires appealing to the
Poincaré conjecture.

######  Remark 3.

Gromov-Hausdorff convergence is a standard way to make sense of weak limits,
but alternative weak notions are possible. Kontsevich and Soibelman [49][48]
aim to establish non-archimedean geometry as a suitable framework for studying
limits of Calabi-Yau manifolds and the consequences for mirror symmetry.
Kontsevich and Tscinkel [50] initiated the attempt to build up non-archimedean
pluripotential theory by imitating Kähler geometry, a task taken much further
by Boucksom et al. [4][3][6][5] (_cf._ section 5). Kontsevich and Soibelman
may have anticipated long before any rigorous definitions, that the Calabi-Yau
metrics should converge in some _potential theoretic_ sense to a non-
archimedean object, and the limiting information should be read off purely in
terms of data on the essential skeleton.

##  4 Analytical foundations

###  4.1 Yau’s solution to the Calabi conjecture

A cornerstone of Kähler geometry is Yau’s celebrated proof of the Calabi
conjecture, which implies

######  Theorem 4.1.

[77] A compact Kähler manifold $X$ with a nowhere vanishing holomorphic volume
form $\Omega$ admits a unique Calabi-Yau metric $(g,\omega,J,\Omega)$ within
any given Kähler class.

We now give a very sketchy account of (the modern view on) Yau’s proof of
Theorem 4.1. Fix a background Kähler metric $\omega$ in the given class, and
the task is to find a Kähler potential $\phi$ solving the complex Monge-Ampère
equation for any given density function $e^{f}$ satisfying the cohomological
constraint $\int_{X}e^{f}\omega^{n}=\int_{X}\omega^{n}$,

$\omega_{\phi}^{n}=(\omega+dd^{c}\phi)^{n}=e^{f}\omega^{n},\quad\int_{X}\phi\omega^{n}=0,$
(9)

In particular for
$e^{f}\omega^{n}=\text{const}\cdot\Omega\wedge\overline{\Omega}$ one obtains
the Calabi-Yau metric. The uniqueness follows from a simple integration by
parts argument.

The existence proof follows the continuity method, namely to deform through
the space of Kähler metrics as $f$ deforms from zero to the desired function.
Viewing (9) as defining a map from the potential to the density function, we
need to show the map is submersive and proper. The submersion property is a
consequence of standard Hodge theory on the Laplacian. To show properness one
needs a priori estimates on $\phi$ depending only on $X$ and bounds on $f$,
and for that matter we are free to impose any order of regularity on $f$.

The first step is to bound $\left\lVert\phi\right\rVert_{C^{0}}$ (known as
‘$C^{0}$-estimate’), using a technique called Moser iteration. We rewrite the
equation as

$(1-e^{f})\omega^{n}=-dd^{c}\phi\wedge(\omega^{n-1}+\omega^{n-1}\wedge\omega_{\phi}+\ldots+\omega_{\phi}^{n-1}).$

For any $p>1$, multiply the equation by $\phi|\phi|^{p-2}$ and integrate by
parts,

$\int_{X}|\nabla|\phi|^{p/2}|_{\omega}^{2}\omega^{n}\leq\frac{C(n,\left\lVert
f\right\rVert_{L^{\infty}})p^{2}}{p-1}\int_{X}|\phi|^{p-1}\omega^{n}.$

The key is that a weaker norm on RHS controls a stronger norm on LHS. This is
reverse to the direction of the Poincaré inequality

$\int_{X}u^{2}\omega^{n}\leq
C\int_{X}|du|_{\omega}^{2}\omega^{n},\quad\int_{X}u=0,$

and the Sobolev inequality

$\int_{X}|u|^{\frac{2n}{n-1}}\omega^{n}\leq
C(\int_{X}|du|^{2}_{\omega}\omega^{n}+\int_{X}|u|^{2}\omega^{n}).$

The combined force is that $\left\lVert\phi\right\rVert_{L^{2}_{1}}\leq C$,
and one can inductively bound $\left\lVert\phi\right\rVert_{L^{p}}$ for large
$p$, which turns out to be uniform in $p$. Taking the limit $p\to\infty$, this
gives a bound $\left\lVert\phi\right\rVert_{L^{\infty}}\leq C(X,\left\lVert
f\right\rVert_{L^{\infty}})$.

The second step is to bound $\left\lVert dd^{c}\phi\right\rVert_{C^{0}}$
(known as $C^{1,\bar{1}}$-estimate), _i.e._ to prove a uniform equivalence
between $\omega$ and $\omega_{\phi}$. The starting point is a differential
inequality by local calculations (in the analyst’s convention of Laplacian)

$\Delta_{\omega_{\phi}}\log\operatorname{Tr}_{\omega}\omega_{\phi}\geq-C\operatorname{Tr}_{\omega_{\phi}}\omega-C.$

Subtracting a large enough multiply of the identity

$\Delta_{\omega_{\phi}}\phi=n-\operatorname{Tr}_{\omega_{\phi}}\omega,$

we get a differential inequality

$\Delta_{\omega_{\phi}}(\log\operatorname{Tr}_{\omega}\omega_{\phi}-C\phi)\geq\operatorname{Tr}_{\omega_{\phi}}\omega-C^{\prime}.$

Using the a priori bound on $\phi$, an application of maximum principle then
shows $\operatorname{Tr}_{\omega}\omega_{\phi}\leq C$.

######  Remark 4.

This $C^{1,\bar{1}}$-estimate argument has global nature: if we only know (9)
on a standard unit ball in $\mathbb{C}^{n}$ with a given $C^{0}$-bound on
$\phi$, there are counterexamples for the $C^{1,\bar{1}}$-bound. The solution
can develop singularities.

The third step is to get higher order estimates. Standard elliptic theory
implies it is sufficient to have a $C^{2,\alpha}$ bound on $\phi$. Evans-
Krylov theory bridged the gap between $C^{1,\bar{1}}$-bound and
$C^{2,\alpha}$-bound (_cf._ [66] Chapter 2, Section 4 for details). This
argument is of local nature, and crucially uses that
$\log\det(\partial_{i}\partial_{\bar{j}}\phi)$ is a _concave_ function of the
matrix $(\partial_{i}\partial_{\bar{j}}\phi)$, and the main tool is a Harnack
inequality. The geometric insight is that higher order regularity is a
manifestation of the local Euclidean nature of Kähler manifolds.

Yau’s proof strategy is highly influential and permeates the vast majority of
works on CY metrics. The brief sketch above, however, highlights two reasons
why it is difficult to adapt to the setting of SYZ conjecture:

  * •

The CY metrics undergoing large complex structure limit are highly degenerate,
to the extent that the Sobolev constant becomes too big, so that the Moser
iteration technique does not give useful $C^{0}$-estimate on the potential.

  * •

The $C^{1,\bar{1}}$ bound has global nature, so in order to obtain useful
estimates on the Calabi-Yau metric only in the generic region, the design of
the maximum principle must hold globally. This is hard on highly degenerate
manifolds, where Riemannian curvature cannot be globally uniformly bounded.

For these reasons, our approach to the SYZ conjecture has significant
departure from Yau’s proof. The potential estimates use complex pluripotential
theory instead, and the metric estimate in the generic region uses a deep
theorem of Savin in elliptic PDE theory, bypassing Yau’s estimates.

###  4.2 Complex pluripotential theory

Complex pluripotential theory concerns the study of weak notions of Kähler
potentials, and their wider implications on algebraic geometry and PDEs. They
can be viewed as the generalization of potential theory on Riemann surfaces to
several complex variables. Some common themes include:

  * •

The weak compactness theory for the space of potentials. This is suited for
variational methods in Kähler geometry.

  * •

The relations between potential theory and algebraic geometry, via Hodge
theory, $\bar{\partial}$-operator, intersection theory, etc. This provides
transcendental methods to birational geometry.

  * •

The weak formulation of the complex Monge-Ampère equation, and a priori
potential estimates under weak assumptions on the volume density, or in the
presence of singularities for the ambient complex variety. This is useful for
studying singularity formation of canonical metrics.

Unlike Yau’s proof of the Calabi conjecture, whose techniques are typical of
elliptic PDEs, complex pluripotential theory is more akin to complex analysis,
and frequently provides stronger results. To build up the intuition, we will
first review the standard versions in the literature, before stating our
uniform versions, which are foundational to our approach to the SYZ
conjecture.

####  4.2.1 Skoda inequality

An upper semicontinuous $L^{1}$-function $\phi$ on a coordinate ball in
$\mathbb{C}^{n}$ is called plurisubharmonic (psh) if it satisfies the sub mean
value inequality when restricted to complex lines; this implies
$dd^{c}\phi\geq 0$. The basic intuition is that regularity properties for psh
functions in general dimensions are analogous to subharmonic functions on
Riemann surfaces. This is captured by the local version of the _Skoda
inequality_ :

######  Theorem 4.2.

(_cf._ [79, Thm 3.1]) If $\phi$ is psh on $B_{2}\subset\mathbb{C}^{n}$, with
$\int_{B_{2}}|\phi|\omega_{E}^{n}\leq 1$ with respect to the standard
Euclidean metric $\omega_{E}$, then there are dimensional constants $\alpha$,
$C$, such that

$\log\int_{B_{1}}e^{-\alpha\phi}\omega_{E}^{n}\leq C.$

######  Remark 5.

The Skoda inequality might be contrasted with subharmonic functions on the
unit ball in $\mathbb{R}^{2n}$ for $n>1$, for which exponential integrability
is far too much to expect.

On a compact Kähler manifold $(Y,\omega)$, we say an upper semicontinuous
$L^{1}$-function $\phi\in PSH(Y,\omega)$ if its sum with the local potential
of $\omega$ is psh, so that $\omega_{\phi}=\omega+dd^{c}\phi\geq 0$. This is
the generalised notion of Kähler potentials. The standard global analogue of
the Skoda inequality is:

######  Theorem 4.3.

[73] On a fixed $(X,\omega)$, there are positive constants $\alpha$, $C$
depending only on $X,\omega$, such that

$\int_{X}e^{-\alpha\phi}\omega^{n}\leq C,\quad\forall\phi\in
PSH(X,\omega)\text{ with }\sup\phi=0.$

In our applications, we need to work with a polarized algebraic degeneration
of Calabi-Yau manifolds $\pi:X\to S\setminus\\{0\\}$ near the large complex
structure limit, as in section 3. Let $\omega_{FS}$ be a fixed Fubini-Study
metric on $(X,c_{1}(L))$ induced by a projective embedding via the sections of
a high power of $L$, and use
$\omega_{FS,t}=\frac{1}{|\log|t||}\omega_{FS}|_{X_{t}}$ to define a family of
background metrics on $X_{t}$ in the class $\frac{1}{|\log|t||}c_{1}(L)$. The
normalization factor $\frac{1}{|\log|t||}$ is to ensure two different choices
of Fubini-Study reference metrics would differ by a potential with $C^{0}$
norm of order $O(1)$ independent of small $t$. Recall $d\mu_{t}$ is the
normalized Calabi-Yau measure.

We adapted the Skoda inequality to a uniform version [51]:

######  Theorem 4.4.

(Uniform Skoda estimate) There are uniform positive constants $\alpha,A$
independent of $t$ for $0<|t|\ll 1$, such that for the normalised Calabi-Yau
measures $d\mu_{t}$,

$\int_{X_{t}}e^{-\alpha u}d\mu_{t}\leq A,\quad\forall u\in
PSH(X_{t},\omega_{FS,t})\text{ with }\sup_{X_{t}}u=0.$

The proof involves covering $X_{t}$ by plenty of small regions which look like
standard balls in $\mathbb{C}^{n}$. An elementary but somewhat tricky
construction of test function allows one to estimate $L^{1}$ norms of the
local potentials. One then applies the local version of Skoda inequality to
each small region, and sum over all regions.

###  4.3 Estimate on pluripotentials

A basic problem in pluripotential theory is to estimate Kähler potentials.
Kolodziej pioneered a method to achieve the following effects. The basic
versions of his theorems work on a fixed ambient Kähler manifold $(Y,\omega)$,
and deal with Kähler potentials $\phi\in PSH(Y,\omega)$ normalized to
$\sup_{Y}\phi=0$.

  * •

(‘Potential estimate’) If the volume density of $\omega_{\phi}^{n}$ has some
integrability control such as an $L^{p}$-bound

$\int_{Y}|\frac{\omega_{\phi}^{n}}{\omega^{n}}|^{p}\leq C,\quad p>1,$ (10)

then $\phi$ has an $L^{\infty}$ bound depending only on $(X,\omega),n,p,C$
[45][20][24]. In fact this can be improved to a $C^{\alpha}$ Hölder bound on
$\phi$ [47]. (Notice $p=1$ would not suffice, as the $L^{1}$-bound
$\int_{Y}\omega_{\phi}^{n}\leq\int_{Y}\omega^{n}$ is automatic).

  * •

(‘$L^{1}$-stability estimates’) Suppose $\phi,\psi\in PSH(Y,\omega)$ are both
subject to the $L^{p}$ volume density integrability control (10), and the
normalization $\sup_{Y}\phi=\sup_{Y}\psi=0$. If

    1. 1.

Either $\phi,\psi$ are close together in the $L^{1}$-sense
$\int_{Y}|\psi-\phi|\omega^{n}\ll 1$, (‘$L^{1}$-potential stability’)

    2. 2.

Or the volume densities of $\omega_{\phi}^{n}$ and $\omega_{\psi}^{n}$ are
close together in the $L^{1}$-sense, namely the total variation
$\int_{Y}|\omega_{\phi}^{n}-\omega_{\psi}^{n}|\ll 1$, (‘$L^{1}$-volume
stability’)

Then $|\phi-\psi|$ is small in the $L^{\infty}$ sense with quantitative
estimates [46].

######  Remark 6.

To appreciate the strength of Kolodziej’s results, these should be contrasted
with the Poisson equation $\Delta u=f$ on a compact Riemannian manifold in
dimensions at least three. If $f\in L^{p}$ for some $p>1$, then we can only
deduce $u\in W^{2,p}$, and $W^{2,p}$ fails to embed into $L^{\infty}$ for $p$
close to one, so we cannot expect any a priori $L^{\infty}$ bound on $u$.
Ultimately, the _global positivity condition_ $\phi\in PSH(X,\omega)$ makes
the difference.

######  Remark 7.

Kolodziej’s proofs depend on his pluripotential theoretic ‘capacity decay’
argument. The author was informed by Freid Tong that a good part of
Kolodziej’s results have found new proofs [69][70], inspired by the recent
breakthrough of Chen and Cheng [15] on the constant scalar curvature Kähler
equation.

In our applications, we need to work with a family of Kähler manifolds, and
the estimates need to be uniform under very severe complex structure
degenerations, and allowing the total volume to collapse to zero. Some
subtleties are:

  * •

The Calabi-Yau volume measure is very different from the volume form of a
Fubini-Study metric (_cf._ section 3.1), so the idea of volume density with
respect to a Fubini-Study style ambient metric is no longer appropriate. The
replacement of (10) turns out to be a Skoda type estimate (11).

  * •

Unlike $L^{\infty}$ bounds, the Hölder norm depends strongly on the choice of
the ambient metric, which specifies a choice of distance function. We think it
is highly non-obvious how to make a semi-explicit choice uniformly in the
family, and therefore we do not attempt to generalize Hölder estimates.

  * •

A technical problem in our applications involving the comparison of two
potentials, only one potential has volume density control. Thus unlike the
$L^{1}$-stability estimates above, our version treats the two potentials
asymmetrically.

Our analogue of the potential estimate is

######  Theorem 4.5.

(_cf._ [52, section 2.2]) Let $(Y,\omega)$ be a compact Kähler manifold, and
$\phi\in PSH(Y,\omega)\cap C^{0}$, such that $\omega_{\phi}^{n}$ is an
absolutely continuous measure. Assume there are positive constants $\alpha,A$,
such that the Skoda type estimate holds with respect to $\omega_{\phi}^{n}$:

$\int_{Y}e^{-\alpha u}\frac{\omega_{\phi}^{n}}{\text{Vol}(Y)}\leq
A,\quad\forall u\in PSH(Y,\omega)\text{ with }\sup_{Y}u=0.$ (11)

Then we have

  * •

If $\sup_{Y}\phi=0$, then $\left\lVert\phi\right\rVert_{C^{0}}\leq
C(n,\alpha,A)$.

  * •

For fixed $n,\alpha,A$, there is number $B(n,\alpha,A)$, such that if
$\frac{\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(Y)}<(2B)^{-2n}$ for
some $t_{0}$, then $\min\phi\geq-t_{0}-4B(\frac{\int_{\phi\leq-
t_{0}}\omega_{\phi}^{n}}{\text{Vol}(Y)})^{1/2n}$.

The strength of this result is that it still applies when the complex/Kähler
structures are highly degenerate, as it distills the dependence on
$(Y,\omega)$ to only 3 constants $n,\alpha,A$. The relations to the more
standard version above can be explained as follows:

  * •

For a fixed ambient Kähler metric, by the Hölder inequality and the standard
Skoda inequality Theorem 4.3, under the density integrability assumption (10),
then for $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, and some small enough
$\beta>0$,

$\int_{Y}e^{-\beta
u}\frac{\omega_{\phi}^{n}}{\text{Vol}(Y)}\leq\frac{1}{\text{Vol}(Y)}\left(\int_{Y}|\frac{\omega_{\phi}^{n}}{\omega^{n}}|^{p}\omega^{n}\right)^{1/p}\left(\int_{Y}e^{-p^{\prime}\beta
u}\omega^{n}\right)^{1/p^{\prime}}\leq\text{const}.$

This means the Skoda type estimate (11) is a weaker assumption than (10), even
in the standard setting.

  * •

The first part of the conclusion recovers Kolodziej’s potential estimate.

  * •

The second part of the conclusion is about comparing the two potentials $\phi$
versus $-t_{0}$. The condition $\frac{\int_{\phi\leq-
t_{0}}\omega_{\phi}^{n}}{\text{Vol}(Y)}<(2B)^{-2n}$ means $\int_{\phi\leq-
t_{0}}\omega_{\phi}^{n}$ is small, which by the Hölder inequality and the
density integrability (10) follows from the smallness of $\int_{\phi\leq-
t_{0}}\omega^{n}$. This should be viewed as one half of the $L^{1}$-potential
stability condition $\int_{Y}|\phi+t_{0}|\omega^{n}\ll 1.$ The conclusion for
the lower bound on $\min\phi$, should be viewed as one half of a smallness
bound on the $C^{0}$-norm of $\phi+t_{0}$, namely the two potentials $\phi$
and $-t_{0}$ are close together in $C^{0}$-norm.

  * •

To generalize to the cases with $\psi$ not necessarily constant, it suffices
to replace $\omega$ by $\omega_{\psi}$, and $\phi$ by $\phi-\psi$ in Theorem
4.5.

Combining Thm. 4.4 with Thm. 4.5, we immediately obtain

######  Theorem 4.6.

[51, Thm. 1.4] (Uniform $L^{\infty}$-estimate) Given a large complex structure
limit of Calabi-Yau manifolds, the potential of the Calabi-Yau metrics
$\omega_{CY,t}$ in the class $\frac{1}{|\log|t||}c_{1}(L)$ relative to a fixed
Fubini-Study reference metrics $\omega_{FS,t}$, have uniform
$L^{\infty}$-estimate independent of $0<|t|\ll 1$, under suitable additive
normalization.

Our adaption of the $L^{1}$-volume stability estimate is

######  Theorem 4.7.

(_cf._ [53, Theorem 2.6]) (Uniform $L^{1}$-stability) Let $(Y,\omega)$ be a
compact Kähler manifold, and $\phi\in PSH(Y,\omega)\cap C^{0}$, satisfying the
complex MA equations

$\frac{\omega^{n}}{\text{Vol}(Y)}=d\mu,\quad\frac{\omega_{\phi}^{n}}{\text{Vol}(Y)}=d\nu$

for probability measures $d\mu$ and $d\nu$. Assume

  * •

There is a Skoda type estimate

$\int_{Y}e^{-\alpha u}d\mu\leq A,\quad\forall u\in PSH(Y,\omega)\text{ with
}\sup_{Y}u=0.$

  * •

The complement of $E_{0}=\\{\phi>0\\}$ has a mass lower bound

$\int_{E_{0}^{c}}d\mu\geq\lambda>0.$

  * •

($L^{1}$-stability assumption) The total variation $\int_{Y}|d\mu-d\nu|\leq
s^{2n+3}<1$.

  * •

$\phi$ is smooth away from a (possibly empty) closed subset $S$ with
$d\mu$-measure zero. Globally $\left\lVert\phi\right\rVert_{C^{0}}\leq
A^{\prime}$.

Then for $0<s<s_{0}(\lambda,n,\alpha,A,A^{\prime})\ll 1$, there is a uniform
estimate

$\sup_{Y}\phi\leq C(\lambda,n,\alpha,A,A^{\prime})s.$

Theorem 4.7 should be viewed as a one-sided version of $L^{1}$-volume
stability estimate. The goal is to compare the two potentials $\phi$ and $0$,
without loss of generality. The Skoda type estimate is the weakened version of
the volume density integrability assumption (10) as before. Assume for the
moment that this holds for _both measures_ $d\mu$ and $d\nu$. After adjusting
$\phi$ by an additive constant, we might as well assume
$\mu(E_{0})\approx\nu(E_{0})\approx\frac{1}{2}$, noticing that the total
variation between the two measures is small by assumption. Then we can reverse
the role of $\omega_{\phi}$ and $\omega$, to deduce a _two-sided_ smallness
bound on $\phi$, which is the content of the $L^{1}$-volume stability
estimate.

###  4.4 Savin’s small perturbation theorem

Savin [65] proved that for a large class of second order elliptic equations
satisfying certain structural conditions, any viscosity solution $C^{0}$-close
to a given smooth solution has interior $C^{2,\gamma}$-bound. In particular
this applies to complex MA equation. Combined with the Schauder estimate,

######  Theorem 4.8.

Fix $k\geq 2$ and $0<\gamma<1$. On the unit ball, let $v$ be a given smooth
solution to the complex Monge-Ampère equation $(dd^{c}v)^{n}=1$. Then there
are constants $0<\epsilon\ll 1$ and $C$ depending on $n,k,\gamma,\left\lVert
v\right\rVert_{C^{k,\gamma}}$, such that if

$(dd^{c}(u+v))^{n}=1+f,\quad\left\lVert
f\right\rVert_{C^{k-2,\gamma}}<\epsilon,$

and $\left\lVert u\right\rVert_{C^{0}}<\epsilon$, then $\left\lVert
u\right\rVert_{C^{k,\gamma}(B_{1/2})}\leq C\epsilon$.

Savin’s theorem has fully nonlinear nature, because the perturbative machinery
only applies once the solution has _a priori_ $C^{2}$ bound. His proof has two
main parts: first he shows a Harnack inequality by a nontrivial application of
Aleksandrov-Bakelman-Pucci estimates, and then uses a compactness argument to
prove $C^{2,\gamma}$ estimate, similar to De Giorgi’s almost flatness theorem
for minimal surfaces [21].

###  4.5 Regularity theory for real Monge-Ampère

There is an extensive literature on the local regularity theory for the real
Monge-Ampère equation, largely due to the Caffarelli school. All results
surveyed here can be found in [61].

Any convex function on an open set
$v:\Omega\subset\mathbb{R}^{n}\to\mathbb{R}$ has an associated Borel measure
called the Monge-Ampère measure, defined by

$MA(v)(E)=|\partial v(E)|,$

where $|\partial v(E)|$ denotes the Lebesgue measure of the image of the
subgradient map on $E\subset\Omega$. Given a Borel measure $\mu$, a solution
to $MA(v)=\mu$ is called an _Aleksandrov solution_ to $\det(D^{2}v)=\mu;$ if
$v\in C^{2}$, this is the classical real Monge-Ampère equation. We shall
assume a two-sided density bound

$\det(D^{2}v)=f\text{ in }B_{1},\quad 0<\Lambda_{1}\leq f\leq\Lambda_{2}.$

Let $B_{1}\setminus\Sigma$ be the set of strictly convex points of $v$, namely
there is a supporting hyperplane touching the graph of $v$ only at one point.
Then Caffarelli [7][8][9] shows

  * •

If $f\in C^{\gamma}(B_{1})$, then $v\in
C^{2,\gamma}_{loc}(B_{1}\setminus\Sigma)$. Then by Schauder theory, if $f$ is
smooth, then $v$ is smooth in $B_{1}\setminus\Sigma$.

  * •

If $L$ is a supporting affine linear function to $v$, such that the convex set
$\\{v=L\\}$ is not a point. Then $\\{v=L\\}$ has no extremal point in the
interior of $B_{1}$.

  * •

The above affine linear set $\\{v=L\\}$ has dimension $k<n/2$.

Mooney [61] shows further that

  * •

The singular set $\Sigma$ has $(n-1)$-Hausdorff measure zero. Consequently
$B_{1}\setminus\Sigma$ is path connected (because a generic path joining two
given points does not intersect a subset of zero $(n-1)$-Hausdorff measure).

  * •

The solution $v\in W^{2,1}_{loc}(B_{1})$ even if $\Sigma$ is nonempty.

######  Remark 8.

A classical counterexample of Pogorelov shows that for $n=3$, the singular set
$\Sigma$ can contain a line segment. This is generalised by Caffarelli [9],
who for any $k<n/2$ constructs examples where $f$ is smooth but $\Sigma$
contains a $k$-plane. A surprising example of Mooney [61] shows that the
Hausdorff dimension of $\Sigma$ can be larger than $n-1-\epsilon$ for any
small $\epsilon$. This means the local regularity theory surveyed above is
essentially optimal.

###  4.6 Special Lagrangian fibration

The classical result of McLean says that the deformation theory of special
Lagrangians with phase $\theta$ is unobstructed, and the first order
deformation space is isomorphic to $H^{1}(L,\mathbb{R})$. Thus if $L$ is
diffeomorphic to $T^{n}$, then the deformation space is $n$-dimensional,
compatible with the SYZ conjecture that $X$ admits a special Lagrangian
$T^{n}$-fibration. A sufficient condition to construct Slag fibrations, under
the very strong hypothesis of collapsing metric with locally bounded sectional
curvature, is obtained by Zhang [80, Thm 1.1].

The essence of Zhang’s result is a standard application of the implicit
function theorem, and we shall summarize the key points (_cf._ [80, section 4]
for more details). Denote $Y_{r}=T^{n}\times B(0,r)\subset
T^{n}_{x_{i}}\times\mathbb{R}^{n}_{y_{i}}\simeq T^{*}T^{n}$, where $r\gg 1$ is
fixed. The trivial example of a special Lagrangian fibration is the following:
the CY structure is the flat model

$g=\sum(dx_{i}^{2}+dy_{i}^{2}),\quad\omega=\sum dx_{i}\wedge
dy_{i},\quad\Omega=\bigwedge(dx_{j}+\sqrt{-1}dy_{j}),$

and the Slag fibration is just the projection to the $\mathbb{R}^{n}_{y_{i}}$
factor, namely the tori $T^{n}\times\\{y\\}$ are special Lagrangians. Zhang
considers a family of CY structures $(g_{k},\omega_{k},\Omega_{k})$ converging
to $(g,\omega,\Omega)$ in the $C^{\infty}$-sense on $Y_{2r}$ (which follows
from his bounded sectional curvature assumptions by elliptic bootstrap), such
that $\omega_{k}\in[\omega]\in H^{2}(Y_{2r},\mathbb{R})$. Small deformations
of the standard $T^{n}$ fibres can be represented as graphs on $T^{n}$: for
$y\in\mathbb{R}^{n}$ and a 1-form $\sigma$ on $T^{n}$ orthogonal to the
harmonic 1-forms $dx_{1},\ldots dx_{n}$, write

$L(y,\sigma)=\text{Graph}(x\mapsto y+\sigma(x))\subset T^{*}T^{n}.$

The condition for $L(y,\sigma)$ to be a special Lagrangian with respect to
$(g_{k},\omega_{k},\Omega_{k})$ is

$\omega_{k}|_{L(y,\sigma)}=0,\quad\text{Im}(e^{\sqrt{-1}\theta_{k}}\Omega_{k})|_{L(y,\sigma)}=0,$
(12)

where $\theta_{k}$ are chosen so that
$\int_{T^{n}}e^{\sqrt{-1}\theta_{k}}\Omega_{k}>0$. Zhang shows by perturbation
arguments that for each $y\in B(0,\frac{3r}{2})$ and $k\geq k_{0}\gg 1$, there
is a unique $\sigma=\sigma_{k,y}$ such that $L(y,\sigma_{k,y})$ solves (12)
with small norm bound $\left\lVert\sigma_{k,y}\right\rVert<\delta\ll 1$. He
then uses another implicit function argument to show that these special
Lagrangians indeed define a local special Lagrangian $T^{n}$-fibration on some
open subset of $Y_{3r/2}$ containing $Y_{r}$.

##  5 Nonarchimedean geometry

Non-archimedean (NA) pluripotential theory is a close analogue of Kähler
geometry. Impressionistically,

  * •

NA geometry offers a natural language to describe the degeneration of complex
manifolds into real simplicial/tropical objects.

  * •

It systematically encodes the combinatorics of tropical geometry.

  * •

Usual notions in Kähler geometry such as functions, line bundles, Kähler
metrics, complex Monge-Ampère measures, have natural (albeit exotic looking)
analogues in NA geometry.

  * •

An analogue of the Calabi conjecture holds in the NA context: one can solve
the NA Monge-Ampère equation.

  * •

Under additional hypotheses, the NA Monge-Ampère measure agrees with the real
Monge-Ampère measure.

We shall explain below that the basic concepts of NA pluripotential theory can
be motivated from the heuristic principle that _NA geometry is the limit of
complex geometry in the hybrid topology_. For some imprecise intuition, one
may imagine non-archimedean geometry approximately as the SYZ base, and the
hybric topology convergence roughly amounts to the collapse of an SYZ
fibration to its base. Our persepctive is heavily influenced by Boucksom et
al. [4][3][6][5].

###  5.1 Berkovich space, hybrid topology

We mentioned in section 3.2 that for a given polarized algebraic degeneration,
the choice of snc models is highly non-unique. There are two viewpoints on
extracting invariant information:

  * •

In birational geometry, one aims to find optimal representatives via the
minimal model program. Typically, this will leave the snc world, and require
divisorial log terminal models [62][63], but the minimal models may still be
non-unique.

  * •

In non-archimedean geometry, one looks simultaneously at the tower of all snc
models, and take the formal limit of their dual complexes, known as the
Berkovich space.

Good references can be found in [49, A] [3, Appendix][6, chapter 2,3].

An insight of Berkovich is that by thinking of points as multiplicative
seminorms, one obtains a kind of geometry analogous to complex manifolds. Let
$K\simeq\mathbb{C}(\\!(t)\\!)$ be equipped with its standard absolute value
$|\cdot|_{0}=e^{-ord_{t}}$ where $ord_{t}$ is the valuation defined by the
vanishing order. Its ultrametric property

$|f+g|_{0}\leq\max\\{|f|_{0},|g|_{0}\\}$

gives the name ‘non-archimedean’ to the subject. Let $X_{K}$ be a smooth,
geometrically connected, projective scheme over $\text{Spec}(K)$; the main
examples come from base changing an algebraic degeneration family $X$ over a
punctured curve. Choose a finite cover of $X_{K}$ by affine open sets of the
form $U=\text{Spec}(A)$, where $A$ is a finitely generated $K$-algebra. The
space $U^{an}$ is defined as the set of all multiplicative seminorms
$|\cdot|_{x}:A\to\mathbb{R}_{\geq 0}$ extending the absolute value of $K$,
endowed with the weakest topology so that the function $x\mapsto|f|_{x}$ is
continuous for any $f\in A$. The _Berkovich space_ $X_{K}^{an}$ is then
obtained by gluing together $U^{an}$; the notation stands for
‘analytification’. As a topological space $X_{K}^{an}$ is compact and
Hausdorff. In the CY case, the point-set description of $X_{K}^{an}$ is meant
to encode information about the base of the SYZ fibration; there is also a
natural structure sheaf which encodes information about the complex structure
[49].

Let $R\simeq\mathbb{C}[\\![t]\\!]$. The concept of models over
$\text{Spec}(R)$ is entirely analogous to the case over algebraic curves. The
dual intersection complexes $\Delta_{\mathcal{X}}$ for snc models over
$\text{Spec}(R)$ can be compared with $X_{K}^{an}$ through two natural maps:

  * •

There is a continuous _embedding map_ $emb:\Delta_{\mathcal{X}}\to
X_{K}^{an}$. Writing $\mathcal{X}_{0}=\sum b_{i}E_{i}$, each divisor $E_{i}$
defines $val_{E_{i}}=\frac{ord_{E_{i}}}{b_{i}}$ through the vanishing order
$ord_{E_{i}}$, so that $e^{-val_{E_{i}}}$ is a point in $X_{K}^{an}$, called a
_divisorial point_. More generally, given a point $x=(x_{0},\ldots x_{p})$ in
the interior of a face $\Delta_{J}\subset\Delta_{\mathcal{X}}$ corresponding
to $E_{J}=\cap_{0}^{p}E_{i}$, we can associate a _quasi-monomial valuation_ :
expanding any local function $f$ around $E_{J}$ in Taylor series,

$f=\sum_{\alpha\in\mathbb{N}^{p+1}}f_{\alpha}z_{0}^{\alpha_{0}}\ldots
z_{p}^{\alpha_{p}},\quad f_{\alpha}\in K(E_{J})$

then the quasi-monomial valuation is

$val_{x}(f)=\min\\{\sum_{0}^{p}\alpha_{i}x_{i}|f_{\alpha}\neq 0\\}.$

Thus $x$ gives rise to a point $e^{-val_{x}}\in X_{K}^{an}$. We shall regard
$\Delta_{\mathcal{X}}$ as a subset of $X_{K}^{an}$. In particular, the
essential skeleton embeds into $X_{K}^{an}$.

  * •

There is a continuous _retraction map_
$r_{\mathcal{X}}:X_{K}^{an}\to\Delta_{\mathcal{X}}$, which restricts to the
identity on $\Delta_{\mathcal{X}}\subset X^{an}$. Any point $e^{-v}\in
X_{K}^{an}$ admits a center on $\mathcal{X}$. This is the unique scheme
theoretic point $\xi\in X_{0}$ such that $|f|_{x}\leq 1$ for
$f\in\mathcal{O}_{\mathcal{X},\xi}$ and $|f|_{x}<1$ for $f\in
m_{\mathcal{X},\xi}$. Let $J\subset I$ be the maximal subset such that $\xi\in
E_{J}$. Then $r_{\mathcal{X}}(x)\in\Delta_{\mathcal{X}}$ corresponds to the
quasi-monomial valuation with the same value for $-\log|z_{j}|_{x},j\in J$.
Concretely, one should think the retraction map is about reading off
logarithmic coordinates.

For another perspective, if $\mathcal{X}^{\prime}$ is a blow up of
$\mathcal{X}$, then there is a natural simplicial map
$\Delta_{\mathcal{X}^{\prime}}\to\Delta_{\mathcal{X}}$, which is identity on
$\Delta_{\mathcal{X}}\subset\Delta_{\mathcal{X}^{\prime}}$. The retraction map
$X_{K}^{an}\to\Delta_{\mathcal{X}}$ can be viewed as a formal limit for very
large $\mathcal{X}^{\prime}$.

######  Remark 9.

The retraction map depends on the choice of the _model_. There are examples
where two models $\mathcal{X}$ and $\mathcal{X}^{\prime}$ define the same
$\Delta_{\mathcal{X}}$ as a subset of $X_{K}^{an}$, but the retraction maps
are different [35, Appendix].

With these comparison maps, the Berkovich space $X_{K}^{an}$ is homeomorphic
to the inverse limit of the dual intersection complexes of the snc models:

$X_{K}^{an}\simeq\varprojlim_{\text{snc models}}\Delta_{\mathcal{X}}$

Conceptually, _an snc model gives a finite approximation of the Berkovich
space_.

We now indicate how NA geometry is unified with complex geometry. Consider an
algebraic degeneration $X$ over a punctured curve. Let $|\cdot|$ denote the
usual absolute value for complex numbers. Given a $\mathbb{C}$-point $z\in
X_{t}$ for $0<|t|\ll 1$, inside some affine chart $U=\text{Spec}(A)$ of $X$,
we can define a multiplicative seminorm $A\to\mathbb{R}_{\geq 0}$ (not non-
archimedean!)

$f\mapsto e^{-\log|f(z)|/\log|t|}=|f(z)|^{1/|\log|t||}.$ (13)

As a sequence of points $z$ move towards $t\to 0$, for any given meromorphic
function $f=\sum a_{k}t^{k}$ on the base, $\lim_{t\to
0}\log|f(z)|/\log|t|=ord_{0}(f)$ which is the standard NA valuation on $K$.
Thus the points on $X_{K}^{an}$ are natural limits of the multiplicative
seminorms defined by $\mathbb{C}$-points on $X_{t}$. One can formalize this
notion by introducing a _hybrid topology_ on $X\sqcup X_{K}^{an}$, so that
$X_{K}^{an}$ takes the place of the central fibre [3, Appendix]. The functions
$f\in A$ then induce local continuous functions on $X\sqcup X_{K}^{an}$.

The ‘hybrid’ space $X\sqcup\Delta_{\mathcal{X}}$ discussed in section 3.1 can
be understood as a finite approximation. Given an snc model $\mathcal{X}$, and
take a sequence of $\mathbb{C}$-points $q_{k}$ tending to $e^{-v}\in X^{an}$,
whose image under the retraction map $r_{\mathcal{X}}$ is $x=(x_{0},\ldots
x_{p})\in\Delta_{J}\subset\Delta_{\mathcal{X}}$. Tautologically $q_{k}$
concentrate near $E_{J}$, and in the local coordinates $z_{0},\ldots z_{p}$,
we have $\log|z_{i}(q_{k})|/\log|t|\to v(z_{i})=x_{i}$, which is equivalent to
$\text{Log}_{\mathcal{X}}(z_{k})\to x=(x_{0},\ldots
x_{p})\in\Delta_{J}\subset\Delta_{\mathcal{X}}$. Formally, the topology on
$X\sqcup X_{K}^{an}$ is the inverse limit of $X\sqcup\Delta_{\mathcal{X}}$ by
taking all snc models.

###  5.2 Model functions, metrics, positivity

We now discuss functions, line bundles, and metrics on $X_{K}^{an}$ [6]. Given
a model $\mathcal{X}$ over $\text{Spec}(R)$ and a Cartier divisor $D$
supported on the central fibre $\mathcal{X}_{0}$, we can associate a
continuous function on $X_{K}^{an}$ by setting

$\phi_{D}(x)=\max\\{\log|f|_{x}:f\in\mathcal{O}_{\mathcal{X}}(D)\\},$

The association $D\mapsto\phi_{D}$ extends by $\mathbb{Q}$-linearity.
Functions obtained in the $\mathbb{Q}$-span using all such choices of models
and divisors are called _model functions_ on $X_{K}^{an}$, which form a dense
subset of $C^{0}(X_{K}^{an})$. The restrictions of such functions to dual
intersection complexes are _piecewise affine_.

To understand the complex geometric meaning, we think of models base changed
from snc models over an algebraic curve $S$. The divisor $D$ prescribes a
class of functions $\phi$ on the total space of the snc model with _analytic
singularities_ :

$\phi=\log|f|+C^{\infty}\text{ function},$

where $f$ is a local defining function of $D$. When we consider the rescaling
of the restrictions to $X_{t}$

$\phi_{t}=\frac{1}{\log|t|}\phi|_{X_{t}},$

only the singular term is relevant in the limit $t\to 0$, and $\phi_{t}$
converge to $-\phi_{D}$ in the hybrid topology.

We think about line bundles on $X_{K}^{an}$ via the GAGA principle: the line
bundles on $X_{K}^{an}$ correspond to the line bundles $L$ on the scheme
$X_{K}$. A _continuous metric_ on $L$ assigns to each local section $s$ a
nonnegative continuous local function $\left\lVert s\right\rVert$ on open
subsets of $X_{K}^{an}$, compatible with the sheaf structure, such that
$\left\lVert fs\right\rVert(x)=|f|_{x}\left\lVert s\right\rVert(x)$, and
$\left\lVert s\right\rVert>0$ if $s$ is a local frame of $L$. Given a
continuous metric, any other continuous metric on $L$ is of the form
$\left\lVert\cdot\right\rVert e^{-\phi}$ for some $\phi\in C^{0}(X^{an})$,
analogous to the usual relation between Hermitian metrics and Kähler
potentials. As such $\phi$ is referred to as a _potential_ function.

Given a model $\mathcal{X}$ for $X_{K}$, a model $\mathcal{L}$ of $L$ is a
line bundle $\mathcal{L}\to\mathcal{X}$ with $\mathcal{L}|_{X}=L$. To this
data we can associate a unique metric
$\left\lVert\cdot\right\rVert_{\mathcal{L}}$ on $L$ with the following
property: if $s$ is a nowhere vanishing local section of $\mathcal{L}$ on an
open set $\mathcal{U}\subset\mathcal{X}$, then $\left\lVert
s\right\rVert_{\mathcal{L}}\equiv 1$ on $\mathcal{U}\cap X_{K}$. This is well
defined because any two such sections differ by the multiplication of an
invertible function, whose NA absolute value equals the constant one. One can
extend this construction to $\mathbb{Q}$-line bundles, and the metrics arising
this way are called _model metrics_. They are dense within the continuous
metrics.

To see the complex geometric interpretation, we imagine a line bundle
$\mathcal{L}$ on some snc models $\mathcal{X}$ over an algebraic curve. Equip
$\mathcal{L}$ with any smooth Hermitian metric $h$. Given a local section $s$
of $\mathcal{L}$, the prescription compatible with (13) is to consider the
local functions on $X_{t}$

$z\mapsto|s(z)|_{h}^{1/|\log|t||}.$

Taking the limit as $t\to 0$, we precisely get the model metric. Notice the
ambiguity in the choice of the Hermitian metric is obliterated in the limit.

A paramount notion in Kähler geometry is the _positivity_ of the metric,
usually phrased in terms of psh properties of the potential. The above
discussion suggests that in the NA setting, namely $t\to 0$, such a notion
should be expressible as a numerical property of the line bundle.

######  Definition 5.1.

[4, Thm. 2.17] (Semipositivity I) Let $\left\lVert\cdot\right\rVert$ be a
model metric on $L$, associated to a $\mathbb{Q}$-line bundle $\mathcal{L}$ on
a model $\mathcal{X}$ of $X_{K}$. Then

  * •

the metric $\left\lVert\right\rVert$ is a semipositive model metric iff
$\mathcal{L}$ is nef, namely $\mathcal{L}\cdot C\geq 0$ for any projective
curve $C$ contained in $\mathcal{X}_{0}$;

  * •

a continuous metric $\left\lVert\right\rVert e^{-\phi}$ is _semipositive_ iff
it is the uniform limit of some sequence of semipositive model metrics on
$X_{K}^{an}$.

######  Remark 10.

The advantage of ‘nef’ instead of ‘ample’ is that if we blow up the model
further, the pullback of the model line bundle will stay nef, but ampleness
will be lost.

In Kähler geometry the definition of psh function is local in the complex
charts. Since the dual intersection complexes are simplicial objects, one
would expect the NA analogous notion to be related to convex functions. This
intuition is partially valid:

######  Proposition 5.2.

[6, Prop 5.9] Let $\mathcal{X}$ be an snc model for $X_{K}$, and
$\mathcal{L}\to\mathcal{X}$ be a model line bundle for $L\to X$, with
associated closed (1,1)-form $\theta$. Then the restriction of any continuous
$\theta$-psh function to any face of $\Delta_{\mathcal{X}}\subset X_{K}^{an}$
is convex.

The picture is that general $\theta$-psh functions define convex functions on
the faces of $\Delta_{\mathcal{X}}$, and among them the $\theta$-psh model
functions give piecewise affine approximations with finer and finer grids.

###  5.3 Approximation by Fubini-Study metrics

A fundamental result in Kähler geometry is that any Kähler metric in an
integral class can be approximated by Fubini-Study metrics associated with
projective embeddings. While the usual Fubini-Study metric depends on a choice
of a Hermitian inner product on the $\mathbb{C}$-vector space of global
sections, the NA analgoue depends on a NA norm on the $K$-vector space
$V=H^{0}(X_{K},mL)$ for $m\gg 1$, with the ultrametric property $\left\lVert
x+y\right\rVert_{V}\leq\max\\{\left\lVert x\right\rVert_{V},\left\lVert
y\right\rVert_{V}\\}$. In our case $K=\mathbb{C}(\\!(t)\\!)$ one can select a
$K$-basis $s_{0},s_{1},\ldots s_{N}$ for $V$, such that

$\left\lVert
a_{0}s_{0}+\ldots+a_{N}s_{N}\right\rVert_{V}=\max\\{|a_{0}|\left\lVert
s_{0}\right\rVert_{V},\ldots,|a_{N}|\left\lVert
s_{N}\right\rVert_{V}\\},\quad\forall a_{i}\in K.$

The NA Fubini-Study metric on $L\to X_{K}$ can be defined as

$\left\lVert s\right\rVert_{FS}(x)=\inf_{\tilde{s}\in
V,\tilde{s}(x)=s^{\otimes
m}(x)}\left\lVert\tilde{s}\right\rVert_{V}^{1/m},\quad\forall x\in
X_{K}^{an}.$

Concretely in the orthogonal basis, written in a local trivialisation,

$\left\lVert
s\right\rVert_{FS}(x)=\frac{|s(x)|}{\max_{j}\\{|s_{j}(x)|/\left\lVert
s_{j}\right\rVert_{V}\\}^{1/m}},\quad\forall x\in X^{an}.$ (14)

A NA analogue of the _Fubini-Study approximation theorem_ gives an alternative
view on semipositive metrics:

######  Proposition 5.3.

(Semipositivity II) [18] Assume $L\to X_{K}$ is ample. Then a continuous
metric on $L$ is semipositive iff it can be written as a uniform limit of
Fubini-Study metrics.

######  Remark 11.

In the approximation theorem we may assume $s_{i}$ to be finite Laurent
polynomials.

For the complex geometric interpretation, we assume as usual $X_{K}$ is the
base change of an algebraic degeneration family $X$, with an ample
polarisation line bundle $L$. For any given NA Fubini-Study metric (14), we
can associate a family of Fubini-Study metrics on $(X_{t},L)$:

$\left\lVert
s\right\rVert_{FS,t}(z)=\frac{|s(z)|}{\\{\\{\sum_{j}|s_{j}(z,t)|^{2}|t|^{2\log\left\lVert
s_{j}\right\rVert_{V}}\\}^{1/2m}},\quad\forall z\in X_{t}.$ (15)

Here $s_{j}$ make sense for finite $t$ because they are selected as finite
Laurent polynomials in $t$. As $t\to 0$, the Fubini-Study metrics converge to
the NA analogue, or more precisely $\left\lVert
s\right\rVert_{FS,t}^{1/|\log|t||}$ converges to (14) in the hybrid topology
on $X\sqcup X_{K}^{an}$.

###  5.4 NA Monge-Ampère measure

The NA Monge-Ampère measure [11] is defined through _intersection theory_ in a
somewhat counterintuitive manner. As a motivation, we consider the complex
analytic setting of an snc model $\mathcal{X}$ over an algebraic curve,
equipped with a Hermitian line bundle $(\mathcal{L},h)$ with curvature form
$\theta$ in the class $c_{1}(\mathcal{L})$. Then $\theta^{n}$ defines a family
of $n$-forms on $X_{t}$, such that $\int_{X_{t}}\theta^{n}$ equals the
intersection number $(L^{n})$. The question is to describe the limit of these
$n$-forms, when we view $X_{t}$ as converging to the dual intersection complex
$\Delta_{\mathcal{X}}$ (_cf._ section 3.1).

We write $X_{0}=\sum_{i\in I}b_{i}E_{i}$. Recall that the regions on $X_{t}$
corresponding to the faces in the dual intersection complex are from the
algebraic perspective only small neighbourhoods of $E_{J}$. Thus the limit of
$\theta^{n}|_{X_{t}}$ can only be supported at the vertices of
$\Delta_{\mathcal{X}}$, which correspond to the components $E_{i}$. The amount
of delta masses concentrated at the vertices are

$b_{i}\int_{E_{i}}\theta^{n}=b_{i}\mathcal{L}^{n}\cdot E_{i},$

where $b_{i}$ appears due to the multiplicity of the sheets. Reassuringly,

$\sum_{i}b_{i}\mathcal{L}^{n}\cdot E_{i}=(L^{n})$

gives the correct total mass.

Back to the NA setting, given a model $\mathbb{Q}$-line bundle
$\mathcal{L}\to\mathcal{X}$ for $L\to X_{K}$, we write
$\mathcal{X}_{0}=\sum_{i}b_{i}E_{i}$, and denote the divisorial points
associated to $E_{i}$ as $q_{i}$. We can then _define_ the NA Monge-Ampère
measure for the model metric $\left\lVert\cdot\right\rVert_{\mathcal{L}}$ as
the following signed atomic measure supported at $q_{i}\in X_{K}^{an}$:

$MA(\left\lVert\cdot\right\rVert_{\mathcal{L}})=\sum_{E_{i}}b_{i}(\mathcal{L}^{n}\cdot
E_{i})\delta_{q_{i}}$

This definition is compatible with pullback of line bundles by the projection
formula, and ensures the total mass is the intersection number $(L^{n})$. If
$\left\lVert\cdot\right\rVert_{\mathcal{L}}$ is furthermore semipositive, then
the intersection numbers are non-negative, so
$MA(\left\lVert\cdot\right\rVert_{\mathcal{L}})$ is a measure.

The theory of NA MA measures bears strong resemblance to the complex MA
measures [4][5]:

  * •

In the complex analytic world, one first define the complex MA for smooth
potentials. A general continuous semipositive potential in a Kähler class is
the uniform limit of smooth potentials, and its complex MA measure is then
determined by the weak continuity under $C^{0}$-convergence.

  * •

In the NA world, one first define the NA MA measure for the model metrics. A
general continuous semipositive metric on $L$ is the uniform limit of a
sequence of continuous semipositive model metrics [6, Cor. 8.8], and its NA MA
measure can be defined as the unique limiting Radon measure of the NA MA
measures for the sequence [4, Cor. 3.5].

Their main difference lies in the highly nonlocal appearance of the NA MA
measure. The recent result of Vilsmeier [76] offers a more concrete
perspective:

######  Proposition 5.4.

(NA MA-real MA comparison) Let $(\mathcal{X},\mathcal{L})$ be a semistable snc
model of $(X_{K},L)$, and $\text{Int}(\Delta_{J})$ be an $n$-dimensional open
face of $\Delta_{\mathcal{X}}$. Recall the retraction map
$r_{\mathcal{X}}:X_{K}^{an}\to\Delta_{\mathcal{X}}$. Let $\phi\in
C^{0}(X_{K}^{an})$ be the potential of a semipositive metric
$\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi}$, and suppose
$\phi=\phi\circ r_{\mathcal{X}}$ on $r_{\mathcal{X}}^{-1}(\Delta_{J})$, then
on $\text{Int}(\Delta_{J})$ the pushforward of the NA MA measure

$r_{\mathcal{X}*}MA(\left\lVert\cdot\right\rVert
e^{-\phi})=n!MA_{\mathbb{R}}(\phi|_{\text{Int}(\Delta_{J})})$

equals the real MA measure of the convex function
$\phi|_{\text{Int}(\Delta_{J})}$ up to a factor $n!$.

The rigorous proof of this comparison uses intersection theory, and the
following is a heuristic explanation. Consider an snc model $\mathcal{X}$ over
an algbebraic curve as in the motivation, and assume furthermore that it is
semistable. Recall our heuristic dictionary that a metric
$\left\lVert\cdot\right\rVert$ on $L\to X_{K}$ should encode a family of
Hermitian metrics $h_{t}$ on $L\to X_{t}$, such that
$h_{t}^{1/|\log|t||}\to\left\lVert\cdot\right\rVert^{2}$ in the hybrid
topology, and the NA MA measure of $\left\lVert\cdot\right\rVert$ should be
the limit of the measures associated to the curvature forms of $h|_{X_{t}}$.
We now focus on the neighbourhood of an $n$-dimensional open face
$\text{Int}(\Delta_{J})\subset\Delta_{\mathcal{X}}\subset\Delta_{\mathcal{X}}\sqcup
X$, where we have local coordinates $z_{0},\ldots z_{n}$ with
$\prod_{0}^{n}z_{i}=t$, and $x_{i}=\frac{\log|z_{i}|}{\log|t|}$. In the local
picture we identify metrics with potentials, so
$\left\lVert\cdot\right\rVert\sim e^{-\phi}$, and after ignoring
$C^{0}$-fluctuation effects $h_{t}^{1/|\log|t||}\sim
e^{-2\phi\circ\text{Log}_{\mathcal{X}}}$. Imposing more smoothness
assumptions, the curvature form of $h_{t}$ is approximately

$|\log|t||dd^{c}\phi\circ\text{Log}_{\mathcal{X}}=\frac{-1}{2\pi}\sum_{1\leq
i,j\leq n}\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}dx_{i}\wedge
d\text{arg}(z_{j}).$

The NA MA measure should agree with the limiting pushforward measure

$\lim_{t\to
0}\text{Log}_{\mathcal{X}*}(|\log|t||dd^{c}\phi\circ\text{Log}_{\mathcal{X}})^{n}=n!\det(D^{2}\phi)|dx_{1}\ldots
dx_{n}|=n!\text{MA}_{\mathbb{R}}(\phi)$

which equals the real MA measure up to the factor $n!$.

######  Remark 12.

In this heuristic calculation, the assumption for $\phi$ to factor through the
retraction map allows us to replace the hybrid space $X\sqcup X_{K}^{an}$ by
its finite approximation $X\sqcup\Delta_{\mathcal{X}}$.

###  5.5 NA Calabi conjecture

The central result of NA pluripotential theory is the solution to the NA
analogue of the Calabi conjecture. A good survey is [5].

######  Theorem 5.5.

[4] Let $X_{K}$ be a smooth projective K-scheme arising from the base change
of an algebraic degeneration family. Let $L$ be an ample line bundle on
$X_{K}$, and $d\mu$ be a Radon probability measure supported on the dual
intersection complex of some snc model of $X_{K}$. Then there is a unique
continuous semipositive metric $\left\lVert\cdot\right\rVert$ on $L$, such
that

$MA(\left\lVert\cdot\right\rVert)=(L^{n})d\mu.$

Their strategy uses a _variational method_. There is a concave _energy
functional_ $\mathcal{E}$ on the space of continuous semipositive metrics on
$L$ (equivalently viewed as continuous $\theta$-psh potentials $\phi$), whose
first variation is given by the NA MA measure. One seeks a maximizer of the
functional

$F_{\mu}(\phi)=\mathcal{E}(\phi)-(L^{n})\int_{X_{K}^{an}}\phi d\mu,$

by first enlarging the space of $\phi$ to a function space
$PSH(X_{K}^{an},\theta)$ which is _compact_ modulo the addition of a real
constant; this is analogous to the $L^{1}$-compactness of
$PSH(X,\omega)/\mathbb{R}$ in the Kähler setting. The notions of the NA MA
measure and the energy functional extend naturally to the _energy class_
functions inside $PSH(X_{K}^{an},\theta)$, much like in the complex
pluripotential theory setting. One then shows the maximizer is in fact a
_critical point_ , namely a weak solution to the NA MA equation. This is
subtle since small perturbations of functions in $PSH(X_{K}^{an},\theta)$ may
fall outside of the class by losing positivity. One proves the continuity of
the weak solution using analogues of Kolodziej’s estimates. The uniqueness of
the solution again relies on the concavity of $\mathcal{E}$.

While this strategy shares a very similar logical structure with the complex
analytic setting, the technical foundations are built upon intersection theory
and vanishing theorems in birational geometry, instead of differential
operators.

The main case of interest to us is when $X_{K}$ arises from a large complex
structure limit. Then NA pluripotential theory provides a unique solution to

$MA(\left\lVert\cdot\right\rVert_{CY})=(L^{n})d\mu_{0},$ (16)

where $d\mu_{0}$ is the Lebesgue measure supported on the essential skeleton
$Sk(X)\subset X_{K}^{an}$ (_cf._ section 3.1). We call
$\left\lVert\cdot\right\rVert_{CY}$ the _non-archimedean Calabi-Yau metric_.

###  5.6 Comparison property

Very little is proven about the non-archimedean CY metric
$\left\lVert\cdot\right\rVert_{CY}$ beyond existence and continuity. We now
discuss the meaning of the following conjectural NA MA-real MA comparison
property.

######  Definition 5.6.

We say $\left\lVert\cdot\right\rVert_{CY}$ satisfies the _NA MA-real MA
comparison property_ , if there exists a semistable snc model
$(\mathcal{X},\mathcal{L})$ of $(X,L)$ with the property that, the potential
$\phi_{0}$ defined by
$\left\lVert\cdot\right\rVert_{CY}=\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi_{0}}$
satisfies $\phi_{0}=\phi_{0}\circ r_{\mathcal{X}}$ on the preimages of the
retraction map over all the $n$-dimensional open faces
$\text{Int}(\Delta_{J})\subset Sk(X)$.

Notice $\text{Int}(\Delta_{J})\subset\Delta_{\mathcal{X}}$ inherits a natural
integral affine structures. Since the restriction of $\phi_{0}$ is convex on
these faces by Prop. 5.2, its real MA measure makes sense, and by Prop. 5.4 it
satisfies the _real MA equation_ on $\text{Int}(\Delta_{J})$

$\text{MA}_{\mathbb{R}}(\phi_{0})=\frac{(L^{n})}{n!}d\mu_{0}.$ (17)

A few comments are in order:

  * •

The comparison property is a conjecture in algebraic/non-archimedean geometry,
and does not _a priori_ involve PDE concepts. Its PDE implications come _a
posteriori_.

  * •

The Kontsevich-Soibelman picture (_cf._ section 3.3) expects that there is a
solution of the real MA equation on the essential skeleton, away from some
singular locus. The NA-MA equation via the comparison property is the only
known systematic method to produce solutions.

  * •

In the context of _toric invariant metrics_ on toric varieties, there are
comparison results between NA MA measure and real MA equation, _cf._ [28,
Prop. 4.4.4].

  * •

The reader may feel that NA pluripotential theory is a very long-winded way to
solve the real MA equation. However, surprisingly enough, it is not even known
how to formulate the real MA equation globally on $Sk(X)$ in general, not just
on the $n$-dimensional faces but also on the lower dimensional faces.

One problem is that $Sk(X)$ does not come with an obvious preferred affine
structure, but only a piecewise affine structure, so there is no obvious
coordinate independent definition of the real MA measure. It seems that the
affine structure conjectured by Kontsevich and Soibelman would need to be
solved simultaneously with the real MA equation, rather like free boundary PDE
problems.

Another problem is that solving the real MA equation requires first specifying
the class of convex functions to be admitted as potentials, just like solving
the complex Monge-Ampère equation requires first specifying the meaning of
Kähler potentials. We do not currently know any direct way of defining the
class of convex potentials on piecewise affine manifolds such as $Sk(X)$. The
semipositive metrics on the Berkovich space $X_{K}^{an}$, abstract as it may
be, is our only available substitute.

  * •

If one believes the NA Calabi-Yau metric $\left\lVert\cdot\right\rVert_{CY}$
is the potential theoretic limit of the Calabi-Yau metrics on $X_{t}$ in the
hybrid topology, and that the information of
$\left\lVert\cdot\right\rVert_{CY}$ can be recovered from data on $Sk(X)$,
then one may be inclined to think that the potential of
$\left\lVert\cdot\right\rVert_{CY}$ factors _globally_ through some retraction
map $X_{K}^{an}\to Sk(X)$, defined perhaps through some divisorial log
terminal minimal model.

Such a statement would need to confront the difficulty that the divisorial log
terminal model is not necessarily unique, and in principle the retraction map
depends on the choice of the model. It seems highly nontrivial how the NA MA
equation would select a preferred retraction map.

The formulation of the comparison property is more cautious than this. We
allow the dual intersection complex $\Delta_{\mathcal{X}}$ to be strictly
bigger than $Sk(X)$, and there is no assumption on the complement of the
$n$-dimensional faces of $Sk(X)$. Regarding the problem above, if one is
undecided between a finite number of candidate retraction maps, then one can
pass to a common snc resolution (and perhaps pass to finite base change, to
find a semistable snc resolution). Of course, the more we blow up the model
$\mathcal{X}$, the weaker is the comparison property.

######  Remark 13.

The very recent work of Pille-Schneider and Mazzon [59] proposes gluing the
retraction maps associated to several divisorial log terminal models to obtain
a map $X_{K}^{an}\to Sk(X)$. Their map still factors through the dual
intersection complex of some larger snc model, hence is compatible with the
comparison property.

  * •

Without the comparison property, it seems hard to give any differential
geometric interpretation to $\left\lVert\cdot\right\rVert_{CY}$ at all, since
$X_{K}^{an}$ contains arbitrarily large dual intersection complexes, and thus
is highly complicated. The heuristic intuition of this hypothetical scenario,
is that the potential theoretic limit of the Calabi-Yau metrics would require
_infinitely many blow ups_ to describe. This is not yet ruled out by a
theorem; we leave the reader to judge its plausibility.

The open question for algebraic geometers is

######  Question 6.

Can the comparison property be proven for a sufficiently large class of
examples, such as those from the Gross-Siebert program [29]?

##  6 Glimpse of proof strategy

We discuss some recent progress on the weak metric version of the SYZ
conjecture.

######  Theorem 6.1.

[53] Let $X\to S\setminus\\{0\\}$ be a large complex structure limit of
Calabi-Yau manifolds, with the polarization ample line bundle $L\to X$. Assume
the NA MA-real MA comparison property holds for $X$ (_cf._ section 5.6). For
sufficiently small $t\in S\setminus\\{0\\}$, there exists a special Lagrangian
$T^{n}$-fibration with respect to the Calabi-Yau structure
$(\omega_{CY,t},\Omega_{t})$ on an open subset of $X_{t}$ whose normalized
Calabi-Yau measure tends to $100\%$ as $t\to 0$.

There is a very particular family of projective Calabi-Yau hypersurfaces, for
which the NA MA-real MA comparison property can be bypassed, at the cost of
passing to subsequences:

$X_{t}=\\{Z_{0}Z_{1}\ldots
Z_{n+1}+t\sum_{i=0}^{n+1}Z_{i}^{n+2}=0\\}\subset\mathbb{CP}^{n+1},\quad
t\in\mathbb{R},0<t\ll 1.$ (18)

We call this the _Fermat family_ , on account of the famous Fermat polynomial
$\sum_{i=0}^{n+1}Z_{i}^{n+2}$.

######  Theorem 6.2.

[52] For the Fermat family, consider the Calabi-Yau metrics $\omega_{CY,t}$ on
$X_{t}$ in the polarisation class $\frac{1}{|\log|t||}c_{1}(\mathcal{O}(1))$.
Then for a subsequence of $X_{t}$ as $t\to 0$, there exists a special
Lagrangian $T^{n}$-fibration on an open subset of $X_{t}$, whose normalized
Calabi-Yau measure tends to $100\%$.

Our exposition will focus on the main line of thought and its many subtleties,
but not the full details of the proofs.

###  6.1 Reduction to potential estimates

The common part of the strategy is to reduce the existence question of special
Lagrangians to $C^{0}$-estimate on the potential.

Given a fixed snc model $\mathcal{X}\to S$, there is a logarithm map
$\text{Log}_{\mathcal{X}}:X_{t}\to\Delta_{\mathcal{X}}$ defined up to
$O(\frac{1}{|\log|t||})$ coordinate ambiguity. Given an $n$-dimensional face
$\Delta_{J}$ of $Sk(X)\subset\Delta_{\mathcal{X}}$, we consider the preimage
$U_{J,t}\subset X_{t}$ under the logarithm map, of a slightly shrinked version
of the interior of $\Delta_{J}$. We will take the liberty of shrinking
$U_{J,t}$ several times, as long as the deleted sets have negligible Calabi-
Yau measure in the $t\to 0$ limit. Since $U_{J,t}$ can be regarded as a torus
invariant subset of $(\mathbb{C}^{*})^{n}$, we can make sense of $C^{k}_{loc}$
norms uniformly in $t$, by passing to the universal cover with the coordinates
$\frac{\log z_{i}}{\log|t|}$.

The first main step is to improve $C^{0}$-estimate to $C^{\infty}$-estimate.

######  Proposition 6.3.

(_cf._ [53, section 4.5]) Let $\phi_{0}$ be an Alexandrov solution of the real
MA equation (17) on the interior of $\Delta_{J}$. Suppose the Calabi-Yau
metrics on $U_{J,t}$ admit local potential functions $\phi_{CY,J,t}$ such that
$\omega_{CY,t}=dd^{c}\phi_{CY,J,t}$, and
$\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{0}}\to
0$ as $t\to 0$. Then after slightly shrinking $U_{J,t}$, we have the
$C^{\infty}$-asymptote
$\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{k}_{loc}}\to
0.$

###### Proof.

(Sketch)

  * •

The first ingredient is that by the regularity theory of real MA equation
(_cf._ section 4.5), after deleting a subset of $\text{Int}(\Delta_{J})$ of
Hausdorff $(n-1)$-measure zero, then $\phi_{0}$ is smooth. After a slight
shrinking of the remaining open set, then $\phi_{0}$ has $C^{k}$ bounds.

  * •

The second ingredient is Savin’s small perturbation theorem (_cf._ section
4.8). After passing to the local universal cover, both $\phi_{CY,J,t}$ and
$\phi_{0}\circ\text{Log}_{\mathcal{X}}$ solve a complex Monge-Ampère equation.
The difference in their RHS vanishes in the $t\to 0$ limit in arbitrarily high
$C^{k}$ norm, as a consequence of the volume form asymptote in section 3.1.
Savin’s result then improves the $C^{0}$ closeness of $\phi_{CY,J,t}$ and
$\phi_{0}\circ\text{Log}_{\mathcal{X}}$ to $C^{\infty}_{loc}$ closeness, after
small shrinking of $U_{J,t}$.

∎

Prop. 6.3 implies the semiflat metric asymptote on the slightly shrinked
$U_{J,t}$, with $C^{\infty}$ small error in the $t\to 0$ limit:

$\omega_{CY,t}\sim
dd^{c}(\phi_{0}\circ\text{Log}_{\mathcal{X}})=\frac{\sqrt{-1}}{4\pi|\log|t||^{2}}\sum_{i,j}\frac{\partial^{2}\phi_{0}}{\partial
x_{i}\partial x_{j}}d\log z_{i}\wedge d\overline{\log z_{j}}.$ (19)

In terms of the Riemannian metric tensors,

$g_{CY,t}\sim\frac{1}{2\pi|\log|t||^{2}}\text{Re}\\{\sum_{1\leq i,j\leq
n}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}d\log z_{i}\otimes
d\log\bar{z}_{j}\\}.$ (20)

In particular the Riemannian curvature stays uniformly bounded in $U_{J,t}$.
By the perturbation theory of special Lagrangians reviewed in section 4.6, the
$T^{n}$ fibres of the logarithm map can be made into a special Lagrangian
fibration by a $C^{\infty}$ small perturbation, on a slightly shrinked subset.

Suppose the assumption of Prop. 6.3 holds on all the $n$-dimensional faces of
$Sk(X)$, then the union of all $U_{J,t}$ cover almost all the CY measure on
$X_{t}$, and the measure lost in the domain shrinking process is negligible.
The weak metric version of the SYZ conjecture then follows.

######  Remark 14.

A subtlety is that the _local regularity_ theory of real Monge-Ampère equation
allows for Hausdorff codimension $1+\epsilon$ singularities. This means the
codimension two singularity prediction in the Kontsevich-Soibelman conjecture
cannot follow simply from the above argument. One must find a more _global_
argument on $Sk(X)$, not just on the interior of its $n$-dimensional faces.

###  6.2 Strategy I: non-archimedean geometry

The remaining task is to achieve the local $C^{0}$-convergence of local
potentials to a solution of the real MA equation on the open $n$-dimensional
faces of $Sk(X)$ (_cf._ Prop. 6.3). The first strategy [53] is:

  * •

Solve the real MA equation on $Sk(X)$, independent of the CY metrics on
$X_{t}$.

  * •

Then attempt to compare the solution with the potential of the CY metrics on
$X_{t}$. First, one needs to produce a Kähler metric on $X_{t}$ whose local
potential is $C^{0}$-close to the real MA solution on $Sk(X)$ in some
topology. Then one needs some version of the $L^{1}$-volume stability estimate
(_cf._ section 4.3) to show the $C^{0}$-smallness of the relative potential
between this Kähler metric and the CY metric, at least in the generic region.

####  6.2.1 Motivation for NA geometry

The above strategy contains many problems:

  * •

As discussed in section 5.6, it is unknown how to directly formulate the real
MA equation on $Sk(X)$, nor do we know the precise class of convex functions
needed for such formulations.

  * •

The essential skeleton is a simplicial complex, and $X_{t}$ is a complex
manifold. These are conceptually very different objects, and we need a
topology to unify both sides.

  * •

Pluripotential theoretic arguments require the _global positivity_ (_i.e._ psh
property) of Kähler potentials (_cf._ Remark 6). Thus when we graft the real
MA solution from $Sk(X)$ to $X_{t}$, we must guarantee the global positivity.
The difficulty lies in the non-generic regions where the complex structure on
$X_{t}$ is highly singular.

These problems point naturally towards NA geometry:

  * •

The NA MA-real MA comparison property is a natural way to produce solutions.

  * •

The hybrid topology is a natural topology to compare $X_{t}$ with
$X_{K}^{an}$, which contains the essential skeleton.

  * •

The notion of semipositive metric is built into NA geometry.

######  Remark 15.

A byproduct of the non-archimedean approach, is that the limit of Calabi-Yau
local potentials is in fact independent of subsequence, since the non-
archimedean analogue of the Calabi-Yau metric is known to be unique.

####  6.2.2 Grafting the real MA solution

Let $(\mathcal{X},\mathcal{L})$ be a semistable snc model with
$\mathcal{L}|_{X}=L$. The NA pluripotential theory provides a continuous
semipositive metric
$\left\lVert\cdot\right\rVert_{CY}=\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi_{0}}$
on $L$ over $X_{K}^{an}$ solving the NA MA equation (16), which we assume
henceforth satisfies the NA MA-real MA comparison property, so $\phi_{0}$
solves the real MA equation over the $n$-dimensional open faces
$\text{Int}(\Delta_{J})$ of the essential skeleton $Sk(\mathcal{X})$ (_cf._
section 5.5).

######  Proposition 6.4.

[53, Lemma 4.1, 4.2] Given any $\epsilon\ll 1$, and let $t$ be small enough
depending on $\epsilon$. There is a Kähler metric $\omega_{\psi,t}$, such that

  * •

On $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))$, the local Kähler
potentials $\phi_{J,t}$ of $\omega_{\psi,t}$ can be chosen to satisfy
$|\phi_{J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}|<\epsilon$.

  * •

The total variation
$\int_{X_{t}}|\frac{|\log|t||^{n}\omega_{\psi,t}^{n}}{(L^{n})}-d\mu_{t}|<\epsilon.$

  * •

The Kähler potential of $\omega_{\psi,t}$ relative to a fixed Fubini-Study
reference metric, is uniformly bounded independent of $t,\epsilon$.

###### Proof.

(Sketch)

  * •

We first $C^{0}$ approximate the NA metric $\left\lVert\cdot\right\rVert_{CY}$
by some NA Fubini-Study metric, which arises naturally as a hybrid topology
limit of usual Fubini-Study metrics on $X_{t}$ (_cf._ section 5.3). The
Fubini-Study metrics are positive, and by construction their local potentials
differ from $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ by an arbitrarily small
amount in the $C^{0}$ sense.

  * •

We do not have direct control on the volume form of the Fubini-Study metrics;
the degrees of the associated projective embeddings are gigantic. In contrast,
the volume form of the local potential $\phi_{0}\circ\text{Log}_{\mathcal{X}}$
has negligible difference from $\frac{(L^{n})}{|\log|t||^{n}}d\mu_{t}$, by the
volume asymptote in section 3.1 and the real MA equation (17).

  * •

The idea is to perform a further regularization. We modify the Fubini-Study
metric in the generic region of $X_{t}$, so that it essentially agrees with
$\phi_{0}\circ\text{Log}_{\mathcal{X}}$ in the generic region up to
$C^{2}$-small error. In this step we appealed also to the regularity theory of
real MA equation. The end result is $\omega_{\psi,t}$, which is Kähler by
construction.

  * •

In the non-generic region, we do not perform regularization. Since the generic
region already takes up $99.9\%$ of the $\omega_{\psi,t}^{n}$ measure for
$|t|\ll 1$, the non-generic region has negligible total measure. We use this
to argue for the total variation bound.

∎

####  6.2.3 $C^{0}$-convergence of the potential

It remains to show

######  Proposition 6.5.

Up to slightly shrinking the domains, the Calabi-Yau metrics on $U_{J,t}$
admit local potential functions $\phi_{CY,J,t}$ such that
$\omega_{CY,t}=dd^{c}\phi_{CY,J,t}$, and
$\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{0}}\to
0$ as $t\to 0$.

Prop. 6.4 says that the local potential of $\omega_{\psi,t}$ and
$\phi_{0}\circ\text{Log}_{\mathcal{X}}$ differ negligibly in the $t\to 0$
limit in the $C^{0}$-sense. Ideally, one would like to use some version of
$L^{1}$-volume stability to conclude the $C^{0}$-smallness of the relative
potential between $\omega_{\psi,t}$ and $\omega_{CY,t}$. Unfortunately, due to
the difficulty of regularization in the non-generic region, there is very
little control on the volume density of $\omega_{\psi,t}$ in the non-generic
region, and we cannot conclude a Skoda type estimate like (11) for
$\omega_{\psi,t}$. This technical problem causes an _asymmetry_ between
$\omega_{\psi,t}$ and $\omega_{CY,t}$, and only ‘one half’ of the
$L^{1}$-volume stability estimate (_cf._ section 4.3) applies, which is why we
designed Theorem 4.7.

After the dust settles, Theorem 4.7 implies that
$\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}$ _concentrates near its
minimum value (normalized to be zero) on a subset with almost $100\%$ of the
Calabi-Yau measure_ (_cf._ [53, Prop 4.4, Cor. 4.6]). More precisely, for any
given small number $\kappa,\lambda\ll 1$, then for sufficiently small $t$, the
measure

$d\mu_{t}(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\geq\kappa/4)<\lambda.$
(21)

On a slightly shrinked version of $U_{J,t}$, this can be improved to the
$C^{0}$-control

$0\leq\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}<\kappa,$

by a slightly tricky application of the mean value inequality (_cf._ [53, Thm.
4.7]). Since $\kappa$ is arbitrary, this achieves Prop. 6.5, which verifies
the hypothesis of Prop. 6.5, whence the weak metric version of the SYZ
conjecture.

######  Remark 16.

The $C^{0}$ convergence statement only applies to the generic region. We do
not know the answer to

######  Question 7.

Do the potentials of the CY metrics on $X_{t}$ converge to the NA CY metric
$\left\lVert\cdot\right\rVert_{CY}$ on $X_{K}^{an}$ globally in the hybrid
topology?

###  6.3 Strategy II: a priori limit

The second strategy does not appeal to NA geometry, and is independent of
section 6.2.

  * •

Argue a priori that the local potential functions of the Calabi-Yau metrics on
$U_{J,t}\subset X_{t}$ converge subsequentially to some convex function on the
open $n$-dimensional faces of $Sk(X)$, in the $C^{0}$-norm.

  * •

Argue that the convex function satisfies the real MA equation.

The strategy is general, except for a delicate problem which we only solved in
the very special case for the Fermat family (_cf._ Theorem 6.2 [52]).

####  6.3.1 Producing convex functions

Recall the logarithm map
$\text{Log}_{t}:(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$.

$\text{Log}_{t}(z_{1},\ldots
z_{n})=\frac{1}{\log|t|}(\log|z_{1}|,\ldots\log|z_{n}|).$

Consider an open convex subset $U\subset\mathbb{R}^{n}$, and let $\phi$ be a
psh function on $\text{Log}_{t}^{-1}(U)\subset(\mathbb{C}^{*})^{n}$.

######  Lemma 6.6.

[52, Lemma 4.3] The fibrewise $T^{n}$ average function

$\bar{\phi}(x_{1},\ldots
x_{n})=\frac{1}{(2\pi)^{n}}\int_{T^{n}}\phi(e^{x_{1}\log|t|+i\theta_{1}},\ldots
e^{x_{n}\log|t|+i\theta_{n}})d\theta_{1}\ldots d\theta_{n}$

is a convex function in the variables $x_{1},\ldots x_{n}$.

###### Proof.

Since the function $\bar{\phi}$ is an average of psh functions, it is psh as a
$T^{n}$-invariant function on $\text{Log}_{t}^{-1}(U)$. Such functions
correspond to convex functions downstairs. ∎

An important intuition is that _on sufficiently collapsed toric regions inside
$(\mathbb{C}^{*})^{n}$, bounded Kähler potentials have a strong tendency to be
approximated by convex functions_.

######  Proposition 6.7.

Assume $\left\lVert\phi\right\rVert_{C^{0}}$ has a uniform bound independent
of $t$. Then after shrinking $U$ by a small amount independent of $t$, we have

  * •

The convex function $\bar{\phi}$ has a Lipschitz bound
$|\bar{\phi}(x)-\bar{\phi}(x^{\prime})|\leq C|x-x^{\prime}|.$

  * •

There is an upper bound $\phi-\bar{\phi}\leq\frac{C}{|\log|t||^{1/2}}$.

  * •

On each logarithmic dyadic scale $U_{a}=\\{a_{i}\leq\log|z_{i}|\leq
2a_{i},\forall i\\}\subset U$, the $L^{1}$-integral

$\int_{U_{a}}|\phi-\bar{\phi}|\prod\sqrt{-1}d\log z_{i}\wedge d\overline{\log
z_{i}}\leq\frac{C}{|\log|t||^{1/2}}.$

  * •

There is an _improved Skoda inequality_ with uniform constants $\alpha,C$
independent of $t$:

$\int_{U}e^{-\alpha|\log|t||^{1/2}(\phi-\bar{\phi})}d\mu_{t}\leq C.$

###### Proof.

(Sketch)

  * •

Bounded convex functions automatically have Lipschitz bound on slightly
shrinked convex domains.

  * •

The second item follows from a slightly tricky application of mean value
inequality for subharmonic functions, _cf._ [52, section 4.3].

  * •

The third item is because the function $\phi-\bar{\phi}$ has mean value zero,
so an upper bound implies an $L^{1}$-bound, _cf._ [52, section 4.3].

  * •

One first apply the basic Skoda estimate Thm. 4.2 to the function $\phi$ on
each logarithmic dyadic scale, where $\bar{\phi}$ is almost constant by the
Lipschitz bound. Then we sum over all the logarithmic dyadic scales (_cf._
[52, section 4.6]).

∎

######  Remark 17.

The improved Skoda estimate is one of the main discoveries in [52].
Intuitively, this means $\phi-\bar{\phi}$ can only be significantly _below_
$-\frac{Const}{|\log|t||^{1/2}}$ on sets with exponentially small measure.
Compounded with the upper bound
$\phi-\bar{\phi}\leq\frac{C}{|\log|t||^{1/2}}$, this means for sufficiently
small $t$, an arbitrary bounded psh function $\phi$ is very close to the
convex function $\bar{\phi}$ except on exponentially small measure.

In our applications, the psh functions $\phi$ arise from the local potentials
$\phi_{CY,J,t}$ of the Calabi-Yau metrics $\omega_{CY,t}$ on toric charts
inside $X_{t}$. Since $\omega_{CY,t}$ has uniformly bounded potential with
respect to Fubini-Study reference metrics (_cf._ Thm. 4.6), it is easy to
arrange the local potentials $\phi_{CY,J,t}$ on toric charts to be uniformly
bounded, whence the convex functions $\bar{\phi}_{CY,J,t}$ are also uniformly
bounded. Using the Lipschitz bound, by Arzela-Ascoli, we can extract a
collection of subsequential limits as $t\to 0$. By construction
$\bar{\phi}_{CY,J,t}\to\bar{\phi}_{J,0}$ in the $C^{0}_{loc}$ sense on the
interior of the $n$-dimensional faces of $Sk(X)$.

####  6.3.2 $C^{0}$-convergence of the potential and extension problem

We aim to show
$\left\lVert\phi_{CY,J,t}-\bar{\phi}_{J,0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{0}}$
on the slightly shrinked $U_{J,t}$ converges to zero along the subsequence. We
know $\left\lVert\bar{\phi}_{CY,t}-\bar{\phi}_{J,0}\right\rVert_{C^{0}}\to 0$
along the subsequence, and from Remark 17, we know
$|\phi_{CY,J,t}-\bar{\phi}_{CY,J,t}\circ\text{Log}_{\mathcal{X}}|$ is small
_except on a subset with small measure_. Removing this small measure problem,
is however rather subtle, and requires a global argument.

The strategy is:

  * •

(‘Extension problem’) Find a _global Kähler metric_ $\omega_{\psi,t}$ on
$X_{t}$ whose local potentials on $U_{J,t}$ agree with
$\bar{\phi}_{J,0}\circ\text{Log}_{\mathcal{X}}$ up to $C^{0}$ small error, and
whose potential with respect to a fixed Fubini-Study metric is bounded
independent of $t$.

  * •

(Potential stability estimate) We can then consider the potential
$\phi_{CY,rel}$ of the Calabi-Yau metric $\omega_{CY,t}$ relative to
$\omega_{\psi,t}$. A small upper bound for $\phi_{CY,rel}$ on $U_{J,t}$
follows from
$\phi_{CY,J,t}-\bar{\phi}_{CY,J,t}\circ\text{Log}_{\mathcal{X}}\leq\frac{C}{|\log|t||^{1/2}}$.
We also know a small lower bound on the $\phi_{CY,rel}$ holds except on a set
with very small measure, and then an application of Theorem 4.7 concludes a
small lower bound on $\inf\phi_{CY,rel}$. We emphasize that the _global
positivity_ of Kähler metrics is essential for this argument.

The net conclusion is that $\phi_{CY,rel}$ is $C^{0}$-small on a slightly
shrinked version of $U_{J,t}$. This amounts to the smallness of
$\left\lVert\phi_{CY,J,t}-\bar{\phi}_{J,0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{0}}$,
which is our goal.

The extension problem is about patching local potentials to global Kähler
potentials, and the difficulty is to achieve psh property in the non-generic
region. The core problem, which is not satisfactorily solved in general, is

######  Question 8.

Can we sufficiently explicitly characterize the class of convex potentials on
$Sk(X)$ that can be regarded as limits of Kähler potentials on $X_{t}$?

The extension problem is solved in an ad hoc way for the Fermat family, and
constitutes the most technical part of [52].555Technically, the paper [52]
does not use the language of dual complexes and essential skeletons, but
proceed via explicit charts controlled by tropical geometry. Recall the Fermat
family embeds into an ambient projective space $\mathbb{CP}^{n+1}$. Our
strategy is to produce the extension $\omega_{\psi,t}$ as a _toric Kähler
metric_ on $\mathbb{CP}^{n+1}$, and then restrict to $X_{t}$, which guarantees
the global positivity. Ensuring that $\omega_{\psi,t}$ agrees with the local
convex functions up to $C^{0}$-small error is a delicate matter, that involves
the explicit tropical hypersurface combinatorics, exploits the large amount of
discrete symmetry of the Fermat family, and uses a double Legendre transform
construction [52].

######  Remark 18.

The motivation for toric Kähler metrics on $\mathbb{CP}^{n+1}$ is as follows.
The toric property is a natural way to reduce general Kähler potentials to
convex functions. The idea of extension to an ambient space, is based on

######  Proposition 6.8.

([19, Thm. B]) Let $(X,\omega)$ be a projective manifold with a Kähler form
representing an integral class, and $Y$ be a smooth subvariety of $X$. Then
any $\phi\in PSH(Y,\omega|_{Y})$ extends to $\phi\in PSH(X,\omega)$.

####  6.3.3 Real MA metric

To complete the circle, we need

######  Lemma 6.9.

The limiting convex potentials $\bar{\phi}_{J,0}$ solve the _real MA equation_
(17) on the interior of $\Delta_{J}$.

The strategy is to pass the complex MA equation on $U_{J,t}$ to the limit.
This is feasible, morally because the complex MA operator is weakly continuous
under the $C^{0}$-convergence of potentials. In our setting, an extra subtlety
is that the sequence of potentials are defined on different manifolds, and the
modification of the usual arguments are carried out in [52, section 5.1].

####  6.3.4 Relation to NA geometry

The a priori limit strategy does not explicitly appeal to NA geometry. Its
principal remaining difficulty is the extension problem. Based on the
experience with the Fermat example, we anticipate that extension to toric
metrics on ambient toric varieties is a useful technique, and the problem may
have a substantially combinatorial aspect. As we emphasized in section 5.6, an
explicit class of convex potentials would also be essential for a direct
formulation of the real MA equation, which is likely needed for more refined
questions such as the affine structure and the singular set of the real MA
metric on $Sk(X)$ (_cf._ the Kontsevich-Soibelman conjecture in section 3.3).

In contrast, the NA pluripotential theory is built around the central concept
of NA semipositive metrics on $X_{K}^{an}$, which extend up to $C^{0}$-small
error to Kähler potentials on $X_{t}$ via the Fubini-Study approximation. In
that respect, NA pluripotential theory may be viewed as a disguised solution
of the extension problem. To make contact with differential geometric
applications, however, requires some additional hypothesis such as the NA MA-
real MA comparison property. Comparing the difficulties in the two strategies,
we speculate that proving the NA MA-real MA comparison property requires a
more concrete characterization of NA semipositive metrics, perhaps of
explicitly combinatorial nature.

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