# Iterated collapsing phenomenon on $G_{2}$-manifolds

Yang Li

###### Abstract

We propose a new collapsing mechanism for $G_{2}$-metrics, with the generic
region admitting a circle bundle structure over a K3 fibration over a Riemann
surface. The adiabatic description involves a weighted version of the maximal
submanifold equation. In a local smooth setting we prove the existence of
formal power series solutions, and the problem of compactification is
discussed at a heuristic level.

##  1 Introduction

The purpose of this paper is to introduce a formal differential geometric
mechanism for the degeneration of $G_{2}$-metrics, which we hope will
evantually lead to the construction of new $G_{2}$-metrics. In the generic
region the manifold is approximately a circle fibration over a 6-dimensional
manifold/orbifold which itself admits a K3 fibration over a Riemann surface
$S$ (with boundary), where the sizes of the circle and the K3 fibres are
shrinking at a fine tuned rate. This collapsing mechanism exhibits a plethora
of phenomena:

  * •

It can be viewed as a degenerate case of circle collapsing over a Calabi-Yau
3-fold studied by Foscolo-Haskins-Nordström [10]. However, unlike in [10], by
tuning the collapsing rate of the circle and K3 fibres, our setup maintains
some nonlinearity of the Apostolov-Salamon equation, namely the
$S^{1}$-reduction of the $G_{2}$-holonomy condition.

  * •

It has close analogy with Donaldson’s proposal of adiabatic Kovalev-Lefschetz
fibration over a 3-dimensional base [8]. Unlike in [8], the new feature is
that K3 collapsing can happen without being the fibres of a coassociative
fibration.

  * •

It generalizes in $G_{2}$ geometry some aspects of the ‘small complex
structure limit’ of Calabi-Yau metrics studied by Sun and Zhang [22]. In
particular, the Gibbons-Hawking ansatz and the Ooguri-Vafa type metrics in
[22] play a crucial role here in describing wall crossing behaviour.

  * •

It involves fibrations of small ALF gravitational instantons along some
3-dimensional locus which itself admits a fibration over $S^{1}$ with small
Riemann surface fibres. The picture has close analogy with ALE fibrations
discussed in the physics literature, and the gluing construction of Joyce and
Karigiannis [14].

Our central philosophy, much like in Donaldson’s proposal [8], is to encode
the degenerating $G_{2}$-structures into certain adiabatic data. Roughly
speaking, the topology is encoded into a local system over the Riemann surface
$S$ with fibres isomorphic to $H^{2}(K3)$, and the $G_{2}$-structure is
encoded by a section of this local system satisfying a number of linear
algebraic constraints coming from the topology of the $S^{1}$-fibration. The
$G_{2}$-holonomy condition then requires this section to be the critical point
of a weighted area functional, analogous to the maximal submanifold equation
proposed in [8]. This local picture is put on solid foundation, as we
demonstrate how to reconstruct a formal power series solution in the
collapsing parameter $\epsilon$, starting from the data of a weighted maximal
submanifold (_cf._ section 3). This is accomplished by a quite intricate
induction scheme, after taking into account several gauge fixing issues.

Global questions will be discussed at a heuristic level, such as Lefschetz
fibration, wall crossing phenomena, boundary behaviour of the Riemann surface,
ALF fibrations, etc. The walls indicate certain jumping discontinuities, and
the positions of the walls are not known a priori, but should instead be
solved along with the PDE, so the proposed global weighted maximal submanifold
equation will have the nature of a free boundary problem.

The following problems still need to be resolved to turn our proposal into
actual constructions of $G_{2}$-metrics on compact manifolds:

  * •

Find the topological data over the Riemann surface $S$, using the global
Torelli theorem for K3 surfaces and lattice theory. This problem is similar to
the ‘matching problem’ in the Kovalev twisted sum construction [15][7].

  * •

Solve the free boundary problem to obtain the adiabatic data, prove the
regularity of solutions, and check certain genericity assumptions.

  * •

Perform the gluing construction. This will require substantial work, and our
more limited goal here is to discuss some likely geometric ingredients.

The ansatz in the $G_{2}$-setting also has a natural $Spin(7)$ analogue: a
small circle bundle over a 7-manifold with a closed $G_{2}$-structure
admitting a collapsing K3 fibration. It turns out to be encoded into a
3-dimensional version of the weighted maximal submanifold equation. This
provides a unifying viewpoint for circle collapsing of $Spin(7)$ manifolds,
and the adiabatic coassociative K3 fibration proposed by Donaldson [8]. In
particular, the $Spin(7)$ version of our ansatz generalizes the Donaldson
ansatz.

######  Acknowledgement.

The author is a 2020 Clay Research Fellow, based at MIT. This work was
partially done at the IAS. The author thanks Simon Donaldson, Song Sun and
Mark Haskins for discussions.

##  2 Local differential geometry

###  2.1 Circle collapsing and Apostolov-Salamon equations

We briefly review the construction in [10]. A $G_{2}$-structure $(M,\phi)$
with a free $S^{1}$-symmetry action can be described in terms of an
$SU(3)$-structure $(\omega,\Omega,g_{M/S^{1}})$ on $M/S^{1}$, together with a
function $h:M/S^{1}\to\mathbb{R}_{+}$ encoding the length of the Killing
vector field, and an $S^{1}$-connection 1-form $\vartheta$ on the principal
circle bundle $M\to M/S^{1}$. Explicitly the $SU(3)$-structure is specified by
a non-degenerate real 2-form $\omega$ and a complex volume form $\Omega$
satisfying

$\omega^{3}=\frac{3}{2}\text{Re}\Omega\wedge\text{Im}\Omega,\quad\omega\wedge\Omega=0.$

and the $G_{2}$-structure is

$\phi=\vartheta\wedge\omega+h^{3/4}\text{Re}\Omega,\quad*_{\phi}\phi=-h^{1/4}\vartheta\wedge\text{Im}\Omega+\frac{1}{2}h\omega^{2},\quad
g_{\phi}=h^{1/2}g_{M/S^{1}}+h^{-1}\vartheta^{2}.$

The torsion free condition for the $G_{2}$-structure is equivalent to the
Apostolov-Salamon equation

$\begin{cases}d\omega=0,\quad
d(h^{3/4}\text{Re}\Omega)=-d\vartheta\wedge\omega,\\\
d(h^{1/4}\text{Im}\Omega)=0,\quad\frac{1}{2}dh\wedge\omega^{2}=h^{1/4}d\vartheta\wedge\text{Im}\Omega.\end{cases}$

In particular $[d\vartheta]\wedge[\omega]=0\in H^{4}(M)$, where
$\frac{1}{2\pi}[d\vartheta]$ represents the first Chern class of the principal
circle bundle.

While this is in general a highly nonlinear coupled system, the key
observation of Foscolo-Haskins-Nordström [10] is that it formally linearizes
in the adiabatic limit where the $S^{1}$-fibres are much smaller compared to
the size of $M/S^{1}$. They consider a family of $S^{1}$-invariant torsion
free $G_{2}$-structures, written in a rescaled convention

$\phi_{\epsilon}=\epsilon\vartheta\wedge\omega+h^{3/4}\text{Re}\Omega,\quad
g_{\phi_{\epsilon}}=h^{1/2}g_{M/S^{1}}+\epsilon^{2}h^{-1}\vartheta^{2},$ (1)

where the data $(\omega,\Omega,h,\vartheta)$ depend on $\epsilon$. The torsion
free condition then reads

$\begin{cases}d\omega=0,\quad\frac{1}{2}dh\wedge\omega^{2}=\epsilon
h^{1/4}d\vartheta\wedge\text{Im}\Omega,\\\
d\text{Re}\Omega=-\frac{3}{4}h^{-1}dh\wedge\text{Re}\Omega-\epsilon
h^{-3/4}d\vartheta\wedge\omega,\quad
d\text{Im}\Omega=-\frac{1}{4}h^{-1}dh\wedge\text{Im}\Omega.\end{cases}$ (2)

Combining with the $SU(3)$-structure condition, we see

$d\vartheta\wedge\omega^{2}=-\epsilon^{-1}d(h^{3/4}\text{Re}\Omega\wedge\omega)=0.$

Assuming $(\omega,\Omega,h,\vartheta)$ all have smooth limits as $\epsilon\to
0$, we find that $h$ is constant to leading order, which we can normalize to
be one. Then to leading order

$\omega=\omega_{CY}+O(\epsilon),\quad\Omega=\Omega_{CY}+O(\epsilon),\quad
d\omega_{CY}=0,\quad d\Omega_{CY}=0,$

namely the $SU(3)$-structure is approximately Calabi-Yau. Writing

$h=1+\epsilon\mathfrak{h}+O(\epsilon^{2}),\quad\vartheta=\vartheta_{0}+O(\epsilon),$

we obtain the linear limiting equations

$d\vartheta_{0}\wedge\omega_{CY}^{2}=0,\quad\frac{1}{2}d\mathfrak{h}\wedge\omega_{CY}^{2}=d\vartheta_{0}\wedge\text{Im}\Omega_{CY}.$
(3)

Equivalently $(\mathfrak{h},\vartheta_{0})$ satisfies the Calabi-Yau monopole
equation

$d\vartheta_{0}\wedge\omega_{CY}^{2}=0,\quad
d\mathfrak{h}=*(d\vartheta_{0}\wedge\text{Re}\Omega_{CY}).$

In particular $d^{*}d\mathfrak{h}=0$. In the special case $\mathfrak{h}=0$,
this reduces to the Hermitian Yang-Mills condition on the $U(1)$-connection
$\vartheta_{0}$. Pluging this back into (2) has the effect of deforming the
$SU(3)$-structure to being only approximately Calabi-Yau. Assuming there are
no singular fibres, and under the cohomological condition
$[d\vartheta]\wedge[\omega]=0$, then one can solve (2) iteratively to obtain a
convergent expansion in $\epsilon$, and thereby produce infinitely many
families of collapsing $G_{2}$-metrics on noncompact manifolds [10].

On compact manifolds there are no nontrivial examples of $G_{2}$-manifolds
with $S^{1}$-symmetry, because any Killing vector field on a compact Ricci
flat manifold is parallel. A folklore construction strategy, which is not
fully carried out in the literature, is to incorporate distributional effects
into the Calabi-Yau monopole equation:

$d(d\vartheta_{0})=2\pi\sum_{i}k_{i}L_{i}$

where $L_{i}$ represent the currents defined by disjoint 3-dimensional
submanifolds, and $k_{i}\in\mathbb{Z}$. Compatibility with the Calabi-Yau
monopole condition requires $L_{i}$ to be special Lagrangians, namely

$\omega_{CY}|_{L_{i}}=0,\quad\text{Im}\Omega_{CY}|_{L_{i}}=0.$

The cohomological condition $\sum k_{i}[L_{i}]=0$ is known as charge
conservation. Solving the Calabi-Yau monopole equation, $d*d\mathfrak{h}$ is
equal to the signed measure $v\mapsto-2\pi\sum
k_{i}\int_{L_{i}}v\text{Re}\Omega_{CY}$ supported on the 3-cycles, so
$\mathfrak{h}\sim\frac{k_{i}}{2\text{dist}(L_{i},\cdot)}$ near $L_{i}$. The
curvature form $d\vartheta$ integrates to $2\pi k_{i}$ on the suitably
oriented 2-spheres linking $L_{i}$.

As an important variant, we allow for Calabi-Yau orbifolds with
$\mathbb{Z}_{2}$ quotient singularities locally modelled on the fixed point
set of an antiholomorphic involution, so the orbifold singular sets $L_{i}$
are special Lagrangians. In such cases the modification is that
$\mathfrak{h}\sim\frac{2k_{i}}{2\text{dist}(L_{i},\cdot)}$.

Geometrically, the asymptotes of $(\mathfrak{h},\vartheta_{0})$ near $L_{i}$
means that the $G_{2}$-structure $\phi_{\epsilon}$ transverse to $L_{i}$
matches approximately with the asymptotes for ALF gravitational instantons, of
$A_{k_{i}-1}$-type for $k_{i}\geq 0$ in the manifold locus case, and
respectively of $D_{k_{i}+2}$-type for $k_{i}\geq-2$ in the orbifold locus
case. The strategy is then to desingularize the neighbourhood of $L_{i}$ by
gluing in a suitable fibration of ALF instantons over $L_{i}$. The meaning of
some special values of $k_{i}$ are as follows:

  * •

The $A_{-1}$-type ALF instanton is the flat product
$S^{1}\times\mathbb{R}^{3}$. In this case $k_{i}=0$, so the singularity does
not actualy exist.

  * •

The $A_{0}$-type ALF instanton is the Taub-NUT metric, which has no
deformation once we fix the asymptotic circle length. One expects the ALF
fibration to be essentially uniquely determined by the asymptotic matching
requirement. Similarly with the $D_{-2}$-type ALF instanton, also known as the
Atiyah-Hitchin metric.

  * •

In other cases, the ALF instantons have nontrivial moduli, so one expects
internal degrees of freedoms to arise in the ALF fibration, similar to the
harmonic one-forms appearing in the work of Joyce and Karigiannis [14].

Notice charge conservation necessitates the appearance of some $L_{i}$ with
$k_{i}<0$. The corresponding dihedral type ALF gravitational instantons then
break the $S^{1}$-symmetry of the $G_{2}$-structure, thus making the compact
examples possible.

###  2.2 Iterated Collapse I: fast circle collapsing

We consider now the degenerate situation where the Calabi-Yau 3-fold $M/S^{1}$
itself admits a holomorphic K3 fibration over a Riemann surface $S$, whose
fibres shrink down as we vary the Calabi-Yau structure. It is natural to
expect that if the circle collapsing happens at a much faster rate than the
shrinking of the K3 fibres, then the Foscolo-Haskins-Nordström picture of
$S^{1}$-invariant collapsing $G_{2}$-metrics should still be valid. We shall
make a formal analysis for the range of parameters

$\text{diam}(S)\sim 1,\quad\text{diam}(K3)\sim t\ll
1,\quad\text{diam}(S^{1})\sim\epsilon\ll t^{2},$

and over a local region of $S$ where all K3 fibres are smooth.

The leading order behaviour of the Calabi-Yau metric on $M/S^{1}$ is

$\omega\approx\omega_{S}+t^{2}\omega_{y},\quad\Omega\approx
t^{2}dy\wedge\Omega_{y},$

where $y$ is a local holomorphic coordinate of $S$, and
$\omega_{y},\Omega_{y}$ are a family of Kähler metrics and holomorphic 2-forms
on the K3 fibres parametrized by $y$. We are also implicitly using the
horizontal distribition defined by the orthogonal complements of the tangent
space of the K3 fibres, which induces a horizontal-vertical type decomposition
on differential forms over $M/S^{1}$. In the $t\to 0$ limit $\omega_{y}$ is
the Calabi-Yau metric on the K3 fibres in the class $t^{-2}[\omega]\in
H^{2}(K3)$, so $\omega_{y},\Omega_{y}$ define a hyperkähler structure. Without
loss of generality $t^{-4}\int_{K3}[\omega]^{2}=1$.

Assuming that $d\vartheta$ has a smooth limit $d\vartheta_{0}$ as $t\to 0$,
the condition $d\vartheta\wedge\omega^{2}=0$ becomes in the limit

$d\vartheta_{0}\wedge\omega_{y}=0$

on all K3 fibres, and in particular $[d\vartheta]\wedge[\omega]=0\in
H^{4}(K3)$. We then analyze the condition
$\frac{1}{2}dh\wedge\omega^{2}=\epsilon
h^{1/4}d\vartheta\wedge\text{Im}\Omega$ by decomposition into horizontal-
vertical types:

  * •

The vertical derivative of $h$ along the K3 surfaces is of order
$O(\epsilon)$. This allows us to treat $h$ as if it only depends on $y\in Y$.
We can write $y=y_{1}+\sqrt{-1}y_{2}$, and
$\Omega_{y}=\omega_{2}+\sqrt{-1}\omega_{1}$, so
$t^{-2}\text{Im}\Omega=\omega_{1}dy_{1}+\omega_{2}dy_{2}$.

  * •

On the K3 surfaces,

$\frac{1}{2}\frac{\partial h}{\partial y_{i}}\omega_{y}^{2}\approx\epsilon
t^{-2}hd\vartheta_{0}\wedge\omega_{i},\quad i=1,2.$

We see that the key assumption $h\approx 1$ in [10] is only valid if
$\epsilon\ll t^{2}$. When this holds, then

$dh\approx 2\epsilon t^{-2}\sum dy_{i}\int_{K3}[d\vartheta]\wedge\omega_{i},$

and the restriction of $d\vartheta_{0}$ to the K3 fibres are the unique
harmonic 2-form in the fixed class $[d\vartheta]$, because the self dual part
is prescribed above.

We now seek special Lagrangians in the adiabatic setting. For simplicity
consider a 3-manifold $L$ inside $M/S^{1}$ fibred over a curve $l$ in $S$,
whose fibres are 2-spheres $L_{y}$ representing a $(-2)$-class $\sigma\in
H^{2}(K3)$. We require

$\omega_{y}|_{L_{y}}=0,\quad\text{Im}\Omega|_{L}=0,$

so along $l$,

$[\omega]\cdot\sigma=0,\quad\sigma\cdot\sum_{i}[\omega_{i}]dy_{i}=0,$ (4)

meaning $\sigma$ is of type $(1,1)$ for a particular choice of complex
structure on the K3 surfaces. Setting $L_{y}$ as the $(-2)$-curve in the class
$\sigma$, we obtain $L=\cup_{y\in l}L_{y}$.

###  2.3 Iterated Collapse II: fine tuned circle collapsing

We now turned to the fine tuned scaling $\epsilon=t^{2}$, which is the main
setting of this paper (the case of $\epsilon/t^{2}=\text{const}$ can be
reduced to this by redefining $h$). The basic distance scales are

$\text{diam}(S)\sim 1,\quad\text{diam}(K3)\sim t\ll
1,\quad\text{diam}(S^{1})\sim\epsilon=t^{2}.$

Our goal is to find an ansatz solving the Apostolov-Salamon equation (2)
approximately, and encode it by adiabatic data. This procedure has strong
analogy with Donaldson’s proposal about coassociative K3 fibrations [8]. In
this section the base $S$ is local.

In analogy with the $\epsilon\ll t^{2}$ case, we wish to maintain the
following features in the adiabatic limit:

  * •

The 6-fold $M/S^{1}$ with $SU(3)$-structure $(\omega,\Omega)$ is fibred by K3
surfaces over a surface $S$, and the tangent spaces of the K3 fibres are
preserved by the almost complex structure. We write $y_{1},y_{2}$ as the local
real coordinates on $S$, and

$\begin{cases}t^{-2}\text{Im}\Omega=\omega_{1}dy_{1}+\omega_{2}dy_{2},\\\
\omega=t^{2}\omega_{y}+\omega_{S},\end{cases}$

using the horizontal distribution induced by the $SU(3)$-structure. We impose
the cohomological normalisation $\int_{K3}[\omega]^{2}=t^{4}$, namely
$[\omega_{y}]^{2}=1$.

  * •

The positive valued function $h$ depends only on $y_{1},y_{2}$ to leading
order, and the vertical derivative along K3 fibres are negligible.

We will derive the ansatz by formal calculations using (2):

  * •

By $d\text{Im}\Omega=-\frac{1}{4}h^{-1}dh\wedge\text{Im}\Omega$, the 2-forms
$\omega_{1},\omega_{2}$ restricted to the K3 fibres are closed, and

$\frac{\partial}{\partial y_{2}}(h^{1/4}[\omega_{1}])=\frac{\partial}{\partial
y_{1}}(h^{1/4}[\omega_{2}]).$

Over the local base, we find a function $H:S\to H^{2}(K3)$, with

$\frac{\partial H}{\partial y_{i}}=h^{1/4}[\omega_{i}],\quad i=1,2.$ (5)

The $SU(3)$-condition implies $\omega_{y}\wedge\omega_{i}=0$ on K3 surfaces,
so $\frac{\partial H}{\partial y_{i}}\wedge[\omega]=0\in H^{4}(K3)$, and up to
choosing an additive constant $H\wedge[\omega]=0$.

  * •

By $d(h^{3/4}\text{Re}\Omega)=-\epsilon d\vartheta\wedge\omega$, writing
$t^{-2}\text{Re}\Omega=\Theta_{1}dy_{1}+\Theta_{2}dy_{2}$, we deduce to
leading order on the K3 fibres,

$d\Theta_{1}=0,\quad d\Theta_{2}=0.$

From the SU(3)-structure $\omega_{y}\wedge\Theta_{i}=0$, and $\Theta_{i}$ are
closed self dual 2-forms on the K3 fibres. Since $\omega_{1},\omega_{2}$ are
also closed self dual 2-forms orthogonal to $\omega_{y}$, it follows that
$\Theta_{1},\Theta_{2}$ are $\mathbb{R}$-linear combinations of
$\omega_{1},\omega_{2}$. The linear coefficients can be determined by a
cohomological calculation. Denote

$g_{ij}=\int_{K3}\omega_{i}\wedge\omega_{j},\quad i,j=1,2,\quad\det
g=\det(g_{ij}),$

then the $SU(3)$-structure requirement implies

$t^{-2}\text{Re}\Omega=\sqrt{\det
g}\\{dy_{1}(g^{12}\omega_{1}+g^{22}\omega_{2})-dy_{2}(g^{11}\omega_{1}+g^{21}\omega_{2})\\}.$
(6)

As a simple check, if $y_{1},y_{2}$ are isothermal coordinates so that
$g_{ij}\propto\delta_{ij}$, then

$t^{-2}\text{Re}\Omega=dy_{1}\omega_{2}-dy_{2}\omega_{1},\quad
t^{-2}\Omega=(\omega_{2}+\sqrt{-1}\omega_{1})(dy_{1}+\sqrt{-1}dy_{2})$

as expected. In particular, the K3 fibres are endowed with a hyperkähler
structure via $(\omega_{y},\omega_{2}+\sqrt{-1}\omega_{1})$. Since
$\omega^{3}=\frac{3}{2}\text{Re}\Omega\wedge\text{Im}\Omega$, to leading order

$3t^{4}\omega_{S}\wedge\omega_{y}^{2}=\frac{3}{2}\text{Re}\Omega\wedge\text{Im}\Omega=3t^{4}\sqrt{\det
g}dy_{1}\wedge dy_{2}\wedge\omega_{y}^{2},$

hence $\omega_{S}\approx\sqrt{\det g}dy_{1}\wedge dy_{2}$.

  * •

By $\frac{1}{2}dh\wedge\omega^{2}=\epsilon
h^{1/4}d\vartheta\wedge\text{Im}\Omega$, we see on the K3 fibres

$\frac{\partial h}{\partial
y_{i}}\omega_{y}^{2}=2h^{1/4}d\vartheta\wedge\omega_{i},\quad i=1,2.$

Taking the cohomology classes,

$\frac{\partial h}{\partial
y_{i}}=2h^{1/4}\int_{K3}[d\vartheta]\wedge[\omega_{i}]=2\int_{K3}\frac{\partial
H}{\partial y_{i}}\wedge[d\vartheta],$

so by a choice of additive constant for $H$,

$h=\begin{cases}2[d\vartheta]\cdot H,\quad&[d\vartheta]\neq 0,\\\
\text{const}>0,\quad&[d\vartheta]=0.\end{cases}$ (7)

This additive choice is compatible with $H\cdot[\omega]=0$ because
$[d\vartheta]\cdot[\omega]=0$. We require $h>0$.

  * •

By $d\vartheta\wedge\omega^{2}=0$, to leading order
$d\vartheta\wedge\omega_{y}=0$ on the K3 fibres. We see that the self dual
part of $d\vartheta$ on the K3 fibres are completely prescribed, so
$d\vartheta$ is the unique harmonic 2-form in its class with respect to the
hyperkähler structure on the K3 fibre.

  * •

The function $H$ maps to the polarized subspace $[\omega]^{\perp}\subset
H^{2}(K3)$, which is of signature $(2,19)$. Now the metric induced on $S$ by
the immersion $H$ is

$\tilde{g}=\frac{\partial H}{\partial y_{i}}\cdot\frac{\partial H}{\partial
y_{j}}dy_{i}dy_{j}=h^{1/2}g_{ij}dy_{i}dy_{j},$ (8)

which is positive definite. Thus we call $H$ a polarized positive section, in
close analogy with Donaldson’s proposal [8].

  * •

Since in this local discussion we assume there is no singular fibre,
cohomologically we require there is no $(-2)$-class $\sigma$ with

$\frac{\partial H}{\partial y_{i}}\wedge\sigma=0,\quad
i=1,2,\quad[\omega_{y}]\wedge\sigma=0$

at any point on our local base. We say the polarized positive section $H$
avoids excess $(-2)$-classes.

  * •

We take a closer look at $d(h^{3/4}\text{Re}\Omega)=-\epsilon
d\vartheta\wedge\omega$. Our previous requirements imply
$\epsilon^{-1}d(h^{3/4}\text{Re}\Omega)$ has horizontal-vertical type $(2,2)$,
so it defines an $H^{2}(K3)$-valued 2-form on $S$,

$\\{\frac{\partial}{\partial
y_{2}}(h^{3/4}[\Theta_{1}])-\frac{\partial}{\partial
y_{1}}(h^{3/4}[\Theta_{2}])\\}dy_{2}\wedge dy_{1}=\frac{\partial}{\partial
y_{j}}(h^{1/2}g^{ij}\sqrt{\det g}\frac{\partial H}{\partial
y_{i}})dy_{2}\wedge dy_{1}.$

Now $d\vartheta\wedge\omega$ is to leading order $d\theta\wedge\omega_{S}$, so
defines the $H^{2}(K3)$-valued 2-form

$[d\vartheta]\omega_{S}=[d\vartheta]\sqrt{\det g}dy_{1}\wedge dy_{2}.$

Comparing the two expressions,

$\frac{\partial}{\partial y_{j}}(h^{1/2}g^{ij}\sqrt{\det g}\frac{\partial
H}{\partial y_{i}})=[d\vartheta]\sqrt{\det g}.$ (9)

This is a PDE on the polarized positive section $H$, and we call it the
weighted maximal submanifold equation in view of its close analogy with the
maximal submanifold equation in [8]. In fact, when $[d\vartheta]=0$, then $h$
is a positive constant, and this is the 2-dimensional version of the usual
maximal submanifold equation, namely the image of $H$ has mean curvature zero.

We now discuss the reconstruction problem: given a local polarized positive
section $H:S\to H^{2}(K3)$ avoiding excess $(-2)$-classes, so that

$H\cdot[\omega_{0}]=0,\quad
h=2H\cdot[d\vartheta]>0,\quad\tilde{g}=\frac{\partial H}{\partial
y_{i}}\cdot\frac{\partial H}{\partial y_{j}}dy_{i}dy_{j}\text{ Riemannian},$

for some $[\omega_{0}]\in H^{2}(K3)$ with $[\omega_{0}]^{2}>0$, and given an
integral class $\frac{1}{2\pi}[d\vartheta]\in H^{2}(K3)$ with
$[\omega_{0}]\cdot[d\vartheta]=0$, how can we reverse engineer the data
$(\omega,\Omega,h,\vartheta)$ in the adiabatic limit? (If $[d\vartheta]=0$,
then we should modify to $h=\text{const}>0$ instead, and $H$ is just a
positive section constrained to be orthogonal to $[\omega_{0}]$.) We will
describe such a procedure that is nearly canonical.

  * •

Since we are working over a local base $S$, topologically the K3 fibration is
just the trivial fibration. The $S^{1}$-bundle is topologically determined by
its Chern class $\frac{1}{2\pi}[d\vartheta]$.

  * •

Suppose we are given the polarized positive section $H:S\to H^{2}(K3)$. As a
first step, we use the global Torelli theorem on K3 surfaces to construct the
hyperkähler structure on the K3 surfaces, such that

$[\omega_{y}]=[\omega_{0}],\quad[\omega_{1}]=h^{-1/4}\frac{\partial
H}{\partial y_{1}},\quad[\omega_{2}]=h^{-1/4}\frac{\partial H}{\partial
y_{2}}.$

This makes sense because the 3-subspace spanned by
$[\omega_{y}],[\omega_{1}],[\omega_{2}]$ is a positive 3-subspace inside
$H^{2}(K3)$, by the polarized positive section condition. The resulting
hyperkähler structures vary smoothly with the K3 fibres. We have
$[\omega_{i}]\cdot[\omega_{j}]=g_{ij}=h^{-1/2}\tilde{g}_{ij}$. This allows us
to construct $\Theta_{1},\Theta_{2}$ as linear combinations of
$\omega_{1},\omega_{2}$ via (6).

  * •

Now to regard $\omega_{y},\omega_{1},\omega_{2}$ as forms on the 6-fold
$M/S^{1}$ requires us to specify a horizontal distribution. The difference
between two horizontal distributions is given by a 1-form on $S$ with values
in the space of vector fields on the fibre K3 surfaces, which modifies
$\omega_{y},\omega_{i}$ (modulo terms which involve $dy_{1}\wedge dy_{2}$) by

$\omega_{y}\to\omega_{y}+\iota_{v_{1}}\omega_{y}dy_{1}+\iota_{v_{2}}\omega_{y}dy_{2},\quad\omega_{i}\to\omega_{i}+\iota_{v_{1}}\omega_{i}dy_{1}+\iota_{v_{2}}\omega_{i}dy_{2},\quad
i=1,2.$

We start with an arbitrary horizontal distribution, to get a preliminary
choice of $\omega_{y},\text{Im}\Omega,\text{Re}\Omega$. Using the above
modification, we can make $d\omega_{y}=0$ modulo terms involving $dy_{1}\wedge
dy_{2}$. This reduces the ambiguity of the horizontal distribution to

$\iota_{v_{i}}\omega_{y}=da_{i},\quad i=1,2,$

for real valued functions $a_{1},a_{2}$ on the K3 surfaces. We claim we can
spend up this freedom to make $d(h^{1/4}\text{Im}\Omega)=0$.

Notice first that by construction $d(h^{1/4}\text{Im}\Omega)$ is a 2-form on
$S$ valued in the space of exact 2-forms on the K3 fibres; there is no
horizontal-vertical type $(1,3)$-piece because $\omega_{1},\omega_{2}$ are
closed on the K3, and the 2-form on the K3 fibres are exact instead of closed
because

$\frac{\partial}{\partial y_{2}}(h^{1/4}[\omega_{1}])=\frac{\partial}{\partial
y_{1}}(h^{1/4}[\omega_{2}]).$

Morever, since $\omega_{y}\wedge\text{Im}\Omega=0$, we get

$\omega_{y}\wedge d(h^{1/4}\text{Im}\Omega)=d(\omega_{y}\wedge
h^{1/4}\text{Im}\Omega)=0,$

so the exact 2-forms are fibrewise orthogonal to $\omega_{y}$. It then
suffices to show that by adjusting the horizontal distribution, we can impose
the extra conditions

$d(h^{1/4}\text{Im}\Omega)\wedge\omega_{i}=0,\quad i=1,2.$

This would force $d(h^{1/4}\text{Im}\Omega)=0$ because an exact anti-self-dual
2-form is zero.

To see this new claim, notice under the modification of the horizontal
distribution with $\iota_{v_{i}}\omega_{y}=da_{i}$,

$\text{Im}\Omega\to\text{Im}\Omega+(\iota_{v_{1}}\omega_{2}-\iota_{v_{2}}\omega_{1})dy_{1}\wedge
dy_{2}.$

We need to choose $a_{1},a_{2}$ so that the exact 4-forms on the K3 fibres
appearing in $d(h^{1/4}\text{Im}\Omega)\wedge\omega_{i}$ are cancelled by

$d(\iota_{v_{1}}\omega_{2}-\iota_{v_{2}}\omega_{1})\wedge\omega_{i},\quad
i=1,2.$

This problem is easiest to analyze in isothermal coordinates using the
hyperkähler structure on the K3 surfaces, where up to constants

$d(\iota_{v_{1}}\omega_{2}-\iota_{v_{2}}\omega_{1})\wedge\omega_{i}\propto
d*da_{i},$

which can indeed realize any pair of exact 4-forms by a suitable choice of
$a_{1},a_{2}$. We have precisely fixed all the ambiguity of the horizontal
distribution.

  * •

We can write $t^{-2}d(h^{3/4}\text{Re}\Omega)=dy_{2}\wedge
dy_{1}\wedge\Theta_{3}$ since the horizontal-vertical type (1,3) component
vanishes. The 2-form $\Theta_{3}$ is closed on each K3 fibre, and lies in the
cohomology class $\frac{\partial}{\partial y_{j}}(h^{1/2}g^{ij}\sqrt{\det
g}\frac{\partial H}{\partial y_{i}})$. We claim that on each fibre it is the
unique harmonic 2-form in this class.

We start with the fact that on a 6-manifold, if $\text{Re}\Omega$ and
$\text{Im}\Omega$ are related as the real and imaginary parts of a complex
3-form, then for any vector field $w$,

$\iota_{w}d\text{Im}\Omega\wedge\text{Re}\Omega=\iota_{w}\text{Im}\Omega\wedge
d\text{Re}\Omega.$

Applying this to $h^{1/4}\text{Re}\Omega$ and $h^{1/4}\text{Im}\Omega$, and
taking $w$ as the horizontal lifts of $\frac{\partial}{\partial y_{i}}$, we
see from $d(h^{1/4}\text{Im}\Omega)=0$ that
$d(h^{1/4}\text{Re}\Omega)\wedge\omega_{i}=0.$ Thus

$d(h^{3/4}\text{Re}\Omega)\wedge\omega_{i}=\frac{1}{2}h^{-1/4}dh\wedge(\text{Re}\Omega\wedge\omega_{i})=\frac{t^{2}}{2}h^{-1/4}\frac{\partial
h}{\partial y_{i}}\sqrt{\det g}dy_{2}\wedge dy_{1}\wedge\omega_{y}^{2},$

or equivalently $\Theta_{3}\wedge\omega_{i}=\frac{1}{2}h^{-1/4}\frac{\partial
h}{\partial y_{i}}\sqrt{\det g}\omega_{y}^{2}$ on each K3 fibre.

Another relation comes from $\text{Re}\Omega\wedge\omega_{y}=0$, and gives
upon differentiation that $\Theta_{3}\wedge\omega_{y}=0$ on each K3 fibre.
These relations completely prescribe the self dual part of $\Theta_{3}$,
implying it is the harmonic form.

  * •

Notice $d\omega_{y}$ is a 2-form on $S$ valued in the space of closed 1-forms
on the K3 fibres. Since $H^{1}(K3)=0$, the 1-forms are exact, so up to
adjusting $\omega_{y}$ by a function times $dy_{1}\wedge dy_{2}$, we can make
$\omega_{y}$ closed on $M/S^{1}$. Unlike all previous constructions, this step
is not fully canonically determined by the topological data and the polarized
positive section $H$. The ambiguity in $\omega_{y}$ is a 2-form pulled back
from $S$.

  * •

Let $\omega_{S}=\sqrt{\det g}dy_{1}\wedge dy_{2}$, and set
$\omega=t^{2}\omega_{y}+\omega_{S}$. Then $d\omega=0$ on $M/S^{1}$, and
$(\omega,\Omega)$ satisfy all other requirements of the $SU(3)$-structure
precisely, except for the normalisation condition which is only approximate:

$\omega^{3}=\frac{3}{2}(1+O(\epsilon))\text{Re}\Omega\wedge\text{Im}\Omega.$

Notice the construction of $\Omega$ is completely canonical, and $\omega$ is
canonical up to $\epsilon$ times a 2-form pulled back from $S$; we use this
freedom to make the $O(\epsilon)$ term have zero fibrewise integral.

  * •

The restrictions of $d\vartheta$ to the K3 fibres are prescribed to be the
harmonic 2-forms in the class $[d\vartheta]$. By construction, when restricted
to the K3 fibres

$d\vartheta\wedge\omega_{y}=0,\quad
2h^{1/4}d\vartheta\wedge\omega_{i}=\frac{\partial h}{\partial
y_{i}}\omega_{y}^{2}.$

This determines $\vartheta$ up to $f_{1}dy_{1}+f_{2}dy_{2}$ for some arbitrary
smooth functions $f_{1},f_{2}$, and gauge equivalence.

  * •

Notice the quantity

$\frac{1}{2}dh\wedge\omega^{2}-\epsilon
h^{1/4}d\vartheta\wedge\text{Im}\Omega=\frac{\epsilon^{2}}{2}dh\wedge\omega_{y}^{2}-\epsilon^{2}h^{1/4}d\vartheta\wedge(\omega_{1}dy_{1}+\omega_{2}dy_{2})$

can be written as $\epsilon^{2}dy_{1}\wedge dy_{2}\wedge\Theta_{4}$ where
$\Theta_{4}$ defines fibrewise 3-forms on K3 surfaces, because by construction
the horizontal-vertical (1,4)-type component vanishes. Since

$d\omega=0,\quad d(h^{1/4}\text{Im}\Omega)=0,$

we see $d\Theta_{4}=0$ on the K3 fibres, and by $H^{3}(K3)=0$ the $\Theta_{4}$
is exact. Any such $\Theta_{4}$ on a hyperkähler K3 can be written as

$\Theta_{4}=df_{1}\wedge\omega_{1}+df_{2}\wedge\omega_{2}+df_{3}\wedge\omega_{y}.$

Spending up the freedom in $d\vartheta$, we can cancel out the $f_{1},f_{2}$
terms, and demand $\Theta_{4}=df_{3}\wedge\omega_{y}$ for some function
$f_{3}$.

Thus

$\frac{1}{2}dh\wedge\omega^{2}-\epsilon
h^{1/4}d\vartheta\wedge\text{Im}\Omega=\epsilon^{2}dy_{1}\wedge
dy_{2}\wedge\omega_{y}\wedge df_{3},$

which is suppressed by the factor $\epsilon^{2}$. With respect to the metric
defined by the $G_{2}$-structure $\phi$, this quantity has small magnitude of
order $O(t)$. This measures the error of the coclosed condition of the
$G_{2}$-structure.

  * •

Now suppose the polarized positive section $H$ satisfies the weighted maximal
submanifold equation (9). By construction,

$d(h^{3/4}\text{Re}\Omega)=\epsilon\Theta_{3}\wedge dy_{2}\wedge
dy_{1}=\epsilon d\vartheta\wedge\sqrt{\det g}dy_{2}\wedge dy_{1}=-\epsilon
d\vartheta\wedge\omega_{S}.$

Then the quantity

$d(h^{3/4}\text{Re}\Omega)+\epsilon
d\vartheta\wedge\omega=\epsilon^{2}d\vartheta\wedge\omega_{y}$

is of type $(1,3)+(2,2)$, and its magnitude with respect to the metric defined
by the $G_{2}$-structure $\phi$ is small of order $O(t)$. This measures the
error of the closed condition of the $G_{2}$-structure. We emphasize that only
in this last step do we bring in the weighted maximal submanifold equation.

In conclusion, we produced an $S^{1}$-invariant ansatz $G_{2}$-structure
$\phi$ from the data of a polarized positive section satisfying the weighted
maximal submanifold equation, which is approximately closed and coclosed.

######  Remark 2.1.

The adiabatic special Lagrangian condition (4) reads in this $\epsilon=t^{2}$
context

$[\omega]\cdot\sigma=0,\quad\sigma\cdot H=\text{const}.$

It makes sense even though $M/S^{1}$ loses the interpretation as a Calabi-Yau
3-fold.

######  Remark 2.2.

The geometrical significance of our ansatz is that it is a plausible
approximation for collapsing $G_{2}$-manifolds in the generic region. When we
deform the cohomology class of the $G_{2}$-structure, so that the circle
collapsing happens at an even slower rate $\epsilon\gg t^{2}$ compared to the
K3 collapsing, then plausibly the $S^{1}$-symmetry reduction hypothesis will
be invalidated, and higher Fourier modes in the $S^{1}$-direction start to be
significant.

###  2.4 Variational formulation

We now explain how to see the weighted maximal submanifold equation as an
Euler Lagrange equation. Given a polarized positive section $H:S\to
H^{2}(K3)$, we define the weighted area functional as

$\mathcal{A}_{w}(H)=\int_{S}h^{1/2}\sqrt{\det\tilde{g}}dy_{1}\wedge
dy_{2},\quad h=2[d\vartheta]\cdot H,\quad\tilde{g}_{ij}=\frac{\partial
H}{\partial y_{i}}\cdot\frac{\partial H}{\partial y_{j}}.$

We calculate the first variation. Given a variation $\delta H=f$, then

$\delta h=2[d\vartheta]\cdot f,\quad\delta\tilde{g}_{ij}=\frac{\partial
H}{\partial y_{i}}\cdot\frac{\partial f}{\partial y_{j}}+\frac{\partial
H}{\partial y_{j}}\cdot\frac{\partial f}{\partial y_{i}},$
$\delta\sqrt{\det(\tilde{g})}=\frac{1}{2}\sqrt{\det(\tilde{g})}\tilde{g}^{ij}\delta\tilde{g}_{ij}=\sqrt{\det(\tilde{g})}\tilde{g}^{ij}\frac{\partial
H}{\partial y_{i}}\cdot\frac{\partial f}{\partial y_{j}},$

Hence the first variation formula

$\begin{split}\delta\mathcal{A}_{w}=&\int
h^{1/2}\delta\sqrt{\det(\tilde{g})}dy_{1}dy_{2}+\int\frac{1}{2}h^{-1/2}\delta
h\sqrt{\det(\tilde{g})}dy_{1}dy_{2}\\\ =&\int h^{1/2}\sqrt{\det
g}{g}^{ij}\frac{\partial H}{\partial y_{i}}\cdot\frac{\partial f}{\partial
y_{j}}dy_{1}dy_{2}+\int[d\vartheta]\cdot f\sqrt{\det g}dy_{1}dy_{2}\\\
=&\int\\{[d\vartheta]\sqrt{\det g}-\frac{\partial}{\partial
y_{j}}(h^{1/2}\sqrt{\det g}{g}^{ij}\frac{\partial H}{\partial y_{i}})\\}\cdot
fdy_{1}dy_{2}.\end{split}$ (10)

The critical points exactly reproduce the weighted maximal submanifold
equation.

######  Remark 2.3.

The terminology ‘weighted area’ is of course based on the area functional

$\mathcal{A}(H)=\int_{S}\sqrt{\det\tilde{g}}dy_{1}\wedge
dy_{2},\quad\tilde{g}_{ij}=\frac{\partial H}{\partial
y_{i}}\cdot\frac{\partial H}{\partial y_{j}},$

whose critical points are called maximal submanifolds, because these are local
maxima of the area functional due to the signature of the setup [8][17]. Such
objects (over a 3-manifold) are central to Donaldson’s proposal to adiabatic
coassociative K3 fibrations [8]. In fact, when $[d\vartheta]=0$, then in the
discussion above we should use $h=\text{const}>0$, and then $\mathcal{A}_{w}$
is just up to a multiplicative constant the area functional.

######  Remark 2.4.

In the ansatz of section 2.3, the volume form is approximately

$dvol_{\phi}\sim\frac{1}{2}\epsilon^{3}h^{1/2}\sqrt{\det\tilde{g}}\vartheta\wedge
dy_{1}\wedge dy_{2}\wedge\omega_{y}^{2},$

so the volume is approximately
$\text{Vol}\sim\pi\epsilon^{3}\mathcal{A}_{w}(H).$ The variational formulation
can be then expected from Hitchin’s characterisation of the torsion free
$G_{2}$-structures in terms of the critical points of the volume functional.

We now look at the second variation $\mathcal{L}$ of the weighted area
functional at some critical point $H$, namely

$\mathcal{A}_{w}(H+sf)=\mathcal{A}_{w}(H)+\frac{s^{2}}{2}\int_{S}f\mathcal{L}f+O(s^{3}),\quad
s\to 0.$

Here $\mathcal{L}f$ takes value in $H^{2}(K3)$-valued 2-forms. This is
naturally expected to be important in the deformation theory of our collapsing
$G_{2}$-manifolds. Since $\mathcal{A}_{w}$ is diffeomorphism invariant on $S$,
we know that only the normal component of $f$ is relevant for the second
variation, and $\mathcal{L}f$ is in particular normal.

######  Lemma 2.5.

Let $S$ be a 2-manifold with boundary. Then $\int_{S}f\mathcal{L}f$ is
negative definite as a symmetric form on normal vector fields $f$ vanishing on
the boundary.

###### Proof.

Consider the images of a family of maps $H+sN$ for $N$ some normal vector
field to the image of $H$, inducing the metrics $\tilde{g}(s)$ from the
immersion into $H^{2}(K3)$. Standard calculation from submanifold theory gives

$\frac{\partial}{\partial s}|_{s=0}\sqrt{\det\tilde{g}}dy_{1}\wedge
dy_{2}=-\langle\mathfrak{m},N\rangle\sqrt{\det\tilde{g}}dy_{1}\wedge dy_{2}$

where $\mathfrak{m}=(\sum_{i}\nabla_{e_{i}}e_{i})^{\perp}$ is the mean
curvature, and $e_{i}$ stands for an orthonormal frame, and $\perp$ means the
normal projection. The second derivative is

$\begin{split}&\frac{\partial^{2}}{\partial
s^{2}}|_{s=0}\sqrt{\det\tilde{g}(s)}dy_{1}\wedge dy_{2}\\\
&=(-\operatorname{Tr}(II_{N}^{2})+\sum_{i}\langle(\nabla_{e_{i}}N)^{\perp},(\nabla_{e_{i}}N)^{\perp}\rangle+\langle\mathfrak{m},N\rangle^{2})\sqrt{\det\tilde{g}}dy_{1}\wedge
dy_{2}\end{split}$

where $II_{T}(e_{i},e_{j})=\langle N,\nabla_{e_{i}}e_{j}\rangle$ is the
$N$-component of the second fundamental form. Morever,

$\frac{\partial}{\partial s}|_{s=0}h(s)^{1/2}=h^{-1/2}\langle
N,[d\vartheta]\rangle,$ $\frac{\partial^{2}}{\partial
s^{2}}|_{s=0}h(s)^{1/2}=-h^{-3/2}\langle N,[d\vartheta]\rangle^{2}.$

Since the first variation of $h^{1/2}\sqrt{\det\tilde{g}}$ is zero at $H$, we
have

$-h^{1/2}\langle\mathfrak{m},N\rangle+h^{-1/2}\langle
N,[d\vartheta]\rangle=0.$

Thus by the Leibniz rule

$\begin{split}&\frac{\partial^{2}}{\partial
s^{2}}|_{s=0}(h(s)^{1/2}\sqrt{\det\tilde{g}(s)}dy_{1}\wedge dy_{2})\\\
=&h^{1/2}(-\operatorname{Tr}(II_{N}^{2})+\sum_{i}\langle(\nabla_{e_{i}}N)^{\perp},(\nabla_{e_{i}}N)^{\perp}\rangle-2\langle\mathfrak{m},N\rangle^{2})\sqrt{\det\tilde{g}}dy_{1}\wedge
dy_{2}.\end{split}$

The second variation formula is then

$\int_{S}N\mathcal{L}N=\int_{S}(-\operatorname{Tr}(II_{N}^{2})+\sum_{i}\langle(\nabla_{e_{i}}N)^{\perp},(\nabla_{e_{i}}N)^{\perp}\rangle-2\langle\mathfrak{m},N\rangle^{2})h^{1/2}\sqrt{\det\tilde{g}}dy_{1}\wedge
dy_{2}.$ (11)

Crucially, in the normal direction, the ambient metric is negative definite,
so all the 3 terms are negative. The equality requires $II_{N}=0$ and
$(\nabla_{e_{i}}N)^{\perp}=0$, so $\nabla_{e_{i}}N=0$, and $N$ is parallel
along $S$. The zero boundary condition then forces $N$ to be zero. ∎

In particular, the boundary value problem of the weighted maximal submanifold
equation will have unobstructed deformation under the change of boundary data.

###  2.5 Instrinsic geometry of the weighted maximal submanifold

As $\epsilon\to 0$, the family of (approximate) $G_{2}$-metrics collapse down
to the base $S$ in the Gromov-Hausdorff sense, whose natural structure is a
metric measure space. Since $g_{\phi}\approx
h^{1/2}g_{M/S^{1}}+\epsilon^{2}h^{-1}\vartheta^{2}$, the limiting metric is

$\tilde{g}=h^{1/2}g=\frac{\partial H}{\partial y_{i}}\cdot\frac{\partial
H}{\partial y_{j}}dy_{i}\otimes dy_{j},$

namely the induced metric from the immersion $H:S\to H^{2}(K3)$. The natural
measure is the one corresponding to the weighted area functional, namely
$h^{1/2}\sqrt{\det\tilde{g}}dy_{1}\wedge dy_{2}$. Notice this is different
from the metric area form by a factor $h^{1/2}$. This is natural from the
perspective of low energy effective action: when $\epsilon\to 0$, the low
energy modes of

$\frac{1}{2}\int|du|_{g_{\phi}}^{2}dvol_{g_{\phi}}$

correspond to $u$ being the pullback of a function on $S$, in which case the
action is approximately

$\frac{\pi\epsilon^{3}}{2}\int_{S}|du|_{\tilde{g}}^{2}h^{1/2}\sqrt{\det\tilde{g}}dy_{1}\wedge
dy_{2},$

proportional to

$\frac{1}{2}\int_{S}|du|_{\tilde{g}}^{2}h^{1/2}\sqrt{\det\tilde{g}}dy_{1}\wedge
dy_{2}.$

For a general Riemannian metric $g$ with a measure form $e^{-f}dvol_{g}$,
there is a notion called the Bakry-Émery Ricci curvature [2]:

$Ric_{f}=Ric_{g}+Hess(f).$

We briefly explain the motivations. First, the $f$-Laplacian is defined as

$\Delta_{f}=\Delta-\nabla f\cdot\nabla,$

which satisfies the integration by part formula

$\int\langle\nabla u,\nabla v\rangle e^{-f}dvol_{g}=\int
u\Delta_{f}ve^{-f}dvol_{g}.$

The Bakry-Émery Ricci curvature naturally appears in the Bochner formula

$\frac{1}{2}\Delta_{f}|\nabla u|^{2}=|Hess(u)|^{2}+\langle\nabla
u,\nabla\Delta_{f}u\rangle+Ric_{f}(\nabla u,\nabla u).$

######  Lemma 2.6.

On the metric measure space $S$ with
$(\tilde{g},h^{1/2}\sqrt{\det\tilde{g}}dy_{1}dy_{2})$, the Bakry-Émery Ricci
curvature is positive semi-definite.

###### Proof.

The usual Ricci curvature of $\tilde{g}$ can be expressed by the second
fundamental form via the Gauss equation:

$Ric_{\tilde{g}}(e_{i},e_{j})=\langle
II(e_{i},e_{j}),\mathfrak{m}\rangle-\sum_{k}\langle
II(e_{k},e_{j}),II(e_{i},e_{k})\rangle,$

where $\mathfrak{m}=\sum(\nabla_{e_{k}}e_{k})^{\perp}$, and
$II(e_{i},e_{j})=(\nabla_{e_{i}}e_{j})^{\perp}$. From the weighted maximal
submanifold equation, by the proof of Lemma 2.5,

$\mathfrak{m}=h^{-1}[d\vartheta]^{\perp}.$

Next we compute the Hessian of $\log h$. We have

$\nabla_{e_{i}}h=2\langle[d\vartheta],e_{i}\rangle,$

so $\nabla h=2([d\vartheta]-[d\vartheta]^{\perp})$. The Hessian matrix is

$\langle\nabla_{e_{i}}\nabla h,e_{j}\rangle=2\langle
II(e_{i},e_{j}),[d\vartheta]^{\perp}\rangle.$

Then $\nabla\log h=h^{-1}\nabla h$, and

$\begin{split}&\langle\nabla_{e_{i}}\nabla\log
h,e_{j}\rangle=-h^{-2}\nabla_{e_{i}}h\cdot\nabla_{e_{j}}h+h^{-1}\langle\nabla_{e_{i}}\nabla
h,e_{j}\rangle\\\
=&-4h^{-2}\langle[d\vartheta],e_{i}\rangle\langle[d\vartheta],e_{j}\rangle+2h^{-1}\langle
II(e_{i},e_{j}),[d\vartheta]^{\perp}\rangle\\\
=&-4h^{-2}\langle[d\vartheta],e_{i}\rangle\langle[d\vartheta],e_{j}\rangle+2\langle
II(e_{i},e_{j}),\mathfrak{m}\rangle.\end{split}$

The Bakry-Émery Ricci curvature is
$Ric_{BE}=Ric_{\tilde{g}}-\frac{1}{2}\nabla^{2}\log h$. Combining the above,
the mean curvature term cancels, and

$Ric_{BE}(e_{i},e_{j})=-\sum_{k}\langle
II(e_{k},e_{j}),II(e_{i},e_{k})\rangle+2h^{-2}\langle[d\vartheta],e_{i}\rangle\langle[d\vartheta],e_{j}\rangle,$

or equivalently for any tangent vector $V$,

$Ric_{BE}(V,V)=-\sum_{k}\langle
II(e_{k},V),II(e_{k},V)\rangle+2h^{-2}\langle[d\vartheta],V\rangle^{2}.$ (12)

Crucially the ambient metric in the normal direction is negative definite, so
both terms are nonnegative, as required. ∎

######  Remark 2.7.

When $[d\vartheta]=0$, so $h=\text{const}>0$, the above reduces to the non-
negativity of the usual Ricci curvature on maximal submanifolds. This has been
observed in [8].

###  2.6 Dimensional reduction to Calabi-Yau case

There is a different way the Apostolov-Salamon framework relates to Calabi-Yau
3-folds: we can start with a (non-compact) Calabi-Yau 3-fold $X$ with
$S^{1}$-symmetry, and take the product with $\mathbb{R}$. Notice here the
Calabi-Yau does not appear as $M/S^{1}$, and such examples are nontrivial in
the sense that the $S^{1}$-bundle is not flat.

We first briefly review the basic setup of Calabi-Yau metrics with holomorphic
and Hamiltonian $S^{1}$-actions [22, chapter 2]. We have the
$S^{1}$-connection $\vartheta$ and the moment coordinate $\mu$, which
parametrizes a family of Kähler quotients $D=X//S^{1}$, carrying a family of
induced Kähler forms $\tilde{\omega}(\mu)$ and a fixed holomorphic volume form
$\Omega_{D}$ on $D$. In our later case of interest $X$ will be a Calabi-Yau
3-fold, and $D$ will be a K3 surface. We can write the Kähler structure on $X$
as

$\begin{cases}\omega^{\prime}=\vartheta\wedge d\mu+\tilde{\omega},\\\
\Omega^{\prime}=(hd\mu-\sqrt{-1}\vartheta)\wedge\Omega_{D}.\end{cases}$

Here $h^{-1}$ is the norm squared of the Killing vector field. (In our
convention $\vartheta$ corresponds to $-\Theta$ in [22].) The Kähler condition
in this formalism leads to

$d\vartheta=-\partial_{\mu}\tilde{\omega}-d_{D}^{c}h\wedge d\mu,$

where $d_{D}$ is the exterior derivative along $D$, and $d_{D}^{c}f=Jd_{D}f$
on $D$. This requires an integrability condition

$\partial^{2}_{\mu}\tilde{\omega}+d_{D}d_{D}^{c}h=0\quad\text{on $D$}.$

In fact, when the $S^{1}$-action is allowed to have fixed points, then one
should also incorporate distributional terms supported on the discriminant
locus, to be discussed later. Finally, the Calabi-Yau condition reads

$\frac{\tilde{\omega}^{n-1}}{(n-1)!}=\frac{\sqrt{-1}^{(n-1)^{2}}}{2^{n-1}}h\Omega_{D}\wedge\overline{\Omega}_{D}.$

In the 3-fold case this specializes to

$\tilde{\omega}^{2}=\frac{1}{2}h\Omega_{D}\wedge\overline{\Omega}_{D}.$

After eliminating variables, we arrive at a key equation in [22]

$\partial^{2}_{\mu}\tilde{\omega}+d_{D}d_{D}^{c}\frac{2\tilde{\omega}^{2}}{\Omega_{D}\wedge\overline{\Omega}_{D}}=0\quad\text{on
$D$}.$

######  Example 2.8.

The prototype example is the case of complex dimension two, where this
discussion specializes to the standard Gibbons-Hawking ansatz:

$\begin{cases}\omega^{\prime}=\vartheta\wedge
d\mu+h\frac{\sqrt{-1}}{2}d\zeta\wedge d\bar{\zeta},\\\
\Omega^{\prime}=(hd\mu-\sqrt{-1}\vartheta)\wedge d\zeta,\\\
d\vartheta=-\partial_{\mu}h\frac{\sqrt{-1}}{2}d\zeta\wedge
d\bar{\zeta}-d_{D}^{c}h\wedge d\mu,\end{cases}$

where $\zeta$ is an $S^{1}$-invariant holomorphic coordinate, and $h$
satisfies the Laplace equation except on some isolated points:

$(\partial_{\mu}^{2}+4\partial_{\zeta}\partial_{\bar{\zeta}})h=0.$

For instance,

$h=A+\frac{1}{2r},\quad r^{2}=|\mu|^{2}+|\zeta|^{2},\quad A=\text{const}>0$

corresponds to the Taub-NUT metric. The formula

$h=A+\frac{k+1}{2r},\quad r^{2}=|\mu|^{2}+|\zeta|^{2},\quad
A=\text{const}>0,\quad k=-1,0,1,\ldots$

describes the asymptote at infinity of $A_{k}$ type ALF gravitational
instantons.

A small variant of the construction is to take the $\mathbb{Z}_{2}$-quotient
under $(\mu,\zeta)\to(-\mu,-\zeta)$. The Gibbons-Hawking ansatz with

$h=A+\frac{2m-4}{2r},\quad r^{2}=|\mu|^{2}+|\zeta|^{2},\quad
A=\text{const}>0,\quad m=0,1,2,\ldots$

describes the asymptote at infinity of type $D_{m}$ ALF gravitational
instantons. The special case $m=0$ is known as the Atiyah-Hitchin metric, and
$m=1$ is known as the Dancer metrics. The crucial difference with the $A_{k}$
case is that the $S^{1}$-symmetry is not global but only asymptotic in the
$D_{m}$ case. A good survey for the Gibbons-Hawking ansatz and gravitational
instantons is [9, section 3].

######  Example 2.9.

Now move to complex dimension 3. Let $\omega_{D,CY}$ be a Calabi-Yau metric on
$D$ with $\omega_{D,CY}^{2}=\frac{1}{2}\Omega_{D}\wedge\overline{\Omega}_{D}$.
The special solution

$\begin{cases}\tilde{\omega}=c\mu\omega_{D,CY},\\\ h=c^{2}\mu^{2},\\\
d\vartheta=-c\omega_{D,CY}\end{cases}$

corresponds to the Calabi ansatz. See section 4.5 for another perspective.

The explicit conversion from the Calabi-Yau case to the Apostolov-Salamon
setting is as follows. The data of $h,\vartheta$ are of course the same in
both settings. We add the flat 7-th dimension parametrized by the coordinate
$\tau$. Let the $SU(3)$-structure be

$\begin{cases}\omega=d\mu\wedge d\tau+\text{Im}\Omega_{D},\\\
h^{1/4}\Omega=(h^{-1/2}d\tau-\sqrt{-1}d\mu)(\tilde{\omega}+\sqrt{-1}h^{1/2}\text{Re}\Omega_{D}),\end{cases}$

so the $G_{2}$-structure is

$\begin{cases}\phi=\omega^{\prime}d\tau+\text{Re}\Omega^{\prime},\\\
*_{\phi}\phi=-d\tau\wedge\text{Im}\Omega^{\prime}+\frac{1}{2}\omega^{\prime
2},\end{cases}$

satisfying the Apostolov-Salamon equation before the rescaling.

To make a more direct comparison with the collapsing setup we also record the
rescaled version: write the Calabi-Yau structure as

$\begin{cases}\omega^{\prime}=\epsilon\vartheta\wedge
d\mu+\epsilon\tilde{\omega},\\\
\Omega^{\prime}=\epsilon(hd\mu-\sqrt{-1}\epsilon\vartheta)\wedge\Omega_{D}.\end{cases}$

The family of Kähler metrics satisfy

$\begin{cases}\partial^{2}_{\mu}\tilde{\omega}+\epsilon^{-1}d_{D}d_{D}^{c}h=0\quad\text{on
$D$},\\\
d\vartheta=-\partial_{\mu}\tilde{\omega}-\epsilon^{-1}d_{D}^{c}h\wedge
d\mu,\\\
\tilde{\omega}^{2}=\frac{1}{2}h\Omega_{D}\wedge\overline{\Omega}_{D}.\end{cases}$

To convert this to the $SU(3)$-structure on $M/S^{1}$, write

$\begin{cases}\omega=d\mu\wedge d\tau+\epsilon\text{Im}\Omega_{D},\\\
h^{1/4}\Omega=\epsilon(h^{-1/2}d\tau-\sqrt{-1}d\mu)(\tilde{\omega}+\sqrt{-1}h^{1/2}\text{Re}\Omega_{D}),\end{cases}$
(13)

so upon substituting

$\begin{cases}\phi=\epsilon\vartheta\wedge\omega+h^{3/4}\text{Re}\Omega,\\\
*_{\phi}\phi=-\epsilon
h^{1/4}\vartheta\wedge\text{Im}\Omega+\frac{1}{2}h\omega^{2},\end{cases}$

compatibly with the rescaled convention (1), then

$\begin{cases}\phi=\omega^{\prime}d\tau+\text{Re}\Omega^{\prime},\\\
*_{\phi}\phi=-d\tau\wedge\text{Im}\Omega^{\prime}+\frac{1}{2}\omega^{\prime
2}.\end{cases}$

Thus the Calabi-Yau condition will lead to the rescaled Apostolov-Salamon
equation (2). In particular, the complex 3-dimensional case of the degenerate
Calabi-Yau metrics studied in [22] can be viewed as a dimensionally reduced
example of our proposal.

It is instructive to reexamine the polarized positive section $H$ in this
dimensionally reduced setting. We have

$\frac{\partial
H}{\partial\tau}=[\text{Re}\Omega_{D}]=const,\quad\frac{\partial
H}{\partial\mu}=-[\tilde{\omega}],\quad[\omega_{y}]=[\text{Im}\Omega_{D}].$

Combined with $[d\vartheta]=-\partial_{\mu}[\tilde{\omega}]$, we see $H$ can
only be affine linear in $\tau$ and quadratic in $\mu$. A special instance is
the Calabi ansatz in example 2.9, where

$H=[\text{Re}\Omega_{D}]\tau+\frac{1}{2}[d\vartheta]\mu^{2}.$

The fact that such examples are so restrictive suggests that the landscape of
7-dimensional examples is not well captured by the dimensional reduction.
Rather the merit of the Calabi-Yau case is to guide our speculations about how
to compactify the collapsing $G_{2}$-manifolds.

##  3 Formal power series solution

The goal of this section is to show that over a given local base $S$ without
singular fibres, the approximate solution introduced in section 2.3 can be
perturbed into a formal power series solution to the Apostolov-Salamon
equation (2):

$\begin{cases}&h_{\epsilon}=h+\epsilon h^{(1)}+\epsilon^{2}h^{(2)}+\ldots,\\\
&\vartheta_{\epsilon}=\vartheta+\epsilon\theta^{(2)}+\epsilon^{2}\theta^{(3)}+\ldots,\\\
&\omega_{\epsilon}=\omega_{S}+\epsilon\omega_{y}+\epsilon^{2}\omega^{(1)}+\epsilon^{3}\omega^{(2)}+\ldots,\\\
&h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon}=\epsilon
h^{1/4}(\omega_{1}dy_{1}+\omega_{2}dy_{2})+\epsilon^{2}\rho^{(1)}+\epsilon^{3}\rho^{(2)}+\ldots,\\\
&H_{\epsilon}=H+\epsilon H^{(1)}+\epsilon^{2}H^{(2)}+\ldots.\end{cases}$

######  Remark 3.1.

Here $H,h,\vartheta,\omega_{S},\omega_{y},\omega_{1},\omega_{2}$ together with
the horizontal distribution are constructed in section 2.3. The superscripts
suggest the number of iterations in an inductive construction. The
$\theta^{(k)}$ terms are 1-forms instead of connections. The term
$h^{1/4}\text{Im}\Omega$ is viewed as an independent variable, in favour of
$\text{Im}\Omega$, because we wish to keep this term $d$-closed. The power
series solution is only formal because the inductive steps involve higher
derivatives in the horizontal directions, coupled to higher powers of
$\epsilon$. However, this whole discussion presumes all K3 fibres are smooth,
so all functions involved are $C^{\infty}$.

The key feature of the Apostolov-Salamon system is that it contains the
Calabi-Yau monopole system which (almost) decouples from the rest of the
equations (_cf._ section 3.3). Then given the solution to the Calabi-Yau
monopole system, one tries to solve for
$\omega_{\epsilon},h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon}$ as an
overdetermined system, where the existence of solution crucially relies on the
Calabi-Yau monopole system as an integrability condition.

###  3.1 Gauge fixing issues

While in the formal limit the K3 surfaces are ‘complex submanifolds’ in the
sense that $\Omega|_{K3}=0$ (equivalently $\text{Im}\Omega|_{K3}=0$), this can
not be expected to hold in the finite $\epsilon$ setting. To explain this
heuristically, given that $d(h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon})=0$,
we wish to deform the K3 fibres such that
$h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon}|_{K3}=0$. The closedness
condition enables us to write fibrewise

$h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon}|_{K3}=df_{1}\wedge\omega_{1}+df_{2}\wedge\omega_{2}+df_{3}\wedge\omega_{y},$

and deformation of the fibre along a vector field $v$ gives a first order
correction

$\mathcal{L}_{v}(h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon})=d(\iota_{v}(h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon}))\approx
d(a\omega_{1}+b\omega_{2})$

for suitable functions $a,b$ corresponding to the components of $v$. This can
be used to cancel the $df_{1},df_{2}$ terms in
$h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon}|_{K3}$, but insufficient to
remove the $df_{3}$ term. From this discussion, we see it is reasonable to
impose

$h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon}|_{K3}=df\wedge\omega_{y}$ (14)

for some function $f$, as a compatibility condition between the K3 fibration
structure and the Apostolov-Salamon equation, which is a conceptual substitute
for the calibration condition. Its chief effect is to rigidify the fibration.

The $S^{1}$-bundle causes a gauge ambiguity for $\vartheta_{\epsilon}$, but
this can be largely ignored because only the curvature $d\vartheta_{\epsilon}$
enters into the Apostolov-Salamon equation, and gauge equivalent choices give
isomorphic constructions.

Next, we seek an optimal way to represent the Apostolov-Salamon solution with
respect to the diffeomorphism action preserving the fibration structure. Given
the K3 fibration structure, and the condition $d\omega_{\epsilon}=0$, with
fixed class $\epsilon^{-1}[\omega_{\epsilon}]$, the standard Moser’s trick
allows us to assume that $\epsilon^{-1}\omega_{\epsilon}|_{K3}$ is fixed to be
$\omega_{y}$. This reduces the fibrewise diffeomorphism group to the fibrewise
symplectic group; to remove this, we can impose further that fibrewise

$(\omega_{1}dy_{1}+\omega_{2}dy_{2})\wedge
h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon}=\text{const}(y,\epsilon)\omega_{y}^{2}\wedge\omega_{S}.$
(15)

We have thus removed the fibrewise diffeomorphism group. The remaining gauge
freedom is the fibration preserving diffeomorphisms moving the fibres around,
alternatively thought as the diffeomorphisms of $S$, which are symmetries of
the weighted maximal submanifold equation. We first associate a polarized
positive section $H_{\epsilon}$ to a solution of the Apostolov-Salamon system.
Notice that modulo $d(f\omega_{y})$, the closed form
$h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon}$ vanishes on K3 fibres, so
defines a closed $H^{2}(K3)$-valued 1-form on $S$. If we impose $H^{1}(S)=0$,
or if we demand all $\rho^{(k)}$ to be exact (as we will later do), then this
is exact, so can be written as $dH_{\epsilon}$ for some $H_{\epsilon}:S\to
H^{2}(K3)$. Now if $f$ is chosen appropriately, then $H_{\epsilon}$ can be
made orthogonal to $[\omega_{y}]$ pointwise on $S$. Morever, since
$H_{\epsilon}$ is well approximated by $H$, it is also a positive section. Our
gauge fixing condition is that $H_{\epsilon}$ differs from $H$ by a normal
vector field to the image of $H$. This makes $H_{\epsilon}$ canonically
defined.

######  Remark 3.2.

The gauge fixing is convenient for our induction scheme, but the condition
(14) seems less fundamental compared to the coassociative K3 fibration
analgoue. The iterated fibration picture relies on the $S^{1}$-symmetry
reduction assumption, which cannot persist on nontrivial compact examples.
Instead, it is expected to be a good effective description up to exponentially
suppressed error, in the generic region of compact examples near the adiabatic
limit.

###  3.2 Iterative scheme setup

We write

$\begin{cases}&h^{[k]}=h+\epsilon
h^{(1)}+\epsilon^{2}h^{(2)}+\ldots+\epsilon^{k}h^{(k)},\\\
&\vartheta^{[k]}=\vartheta+\epsilon\theta^{(2)}+\epsilon^{2}\theta^{(3)}+\ldots+\epsilon^{k-1}\theta^{(k)},\\\
&\omega^{[k]}=\omega_{S}+\epsilon\omega_{y}+\epsilon^{2}\omega^{(1)}+\epsilon^{3}\omega^{(2)}+\ldots+\epsilon^{(k+1)}\omega^{(k)},\\\
&(h^{1/4}\text{Im}\Omega)^{[k]}=\epsilon
h^{1/4}(\omega_{1}dy_{1}+\omega_{2}dy_{2})+\epsilon^{2}\rho^{(1)}+\epsilon^{3}\rho^{(2)}+\ldots+\epsilon^{(k+1)}\rho^{(k)},\\\
&H^{[k]}=H+\epsilon
H^{(1)}+\epsilon^{2}H^{(2)}+\ldots+\epsilon^{k}H^{(k)}.\end{cases}$

These are best thought as representatives of formal power series modulo higher
powers of $\epsilon$. They are required to satisfy the approximate Apostolov-
Salamon equation

$\begin{cases}&d\omega^{[k]}=0,\\\
&\frac{1}{2}dh^{[k]}\wedge(\omega^{[k]})^{2}=\epsilon
d\vartheta^{[k]}\wedge(h^{1/4}\text{Im}\Omega)^{[k]}+O(\epsilon^{k+2}),\\\
&d(h^{3/4}\text{Re}\Omega)^{[k]}=-\epsilon
d\vartheta^{[k]}\wedge\omega^{[k]}+O(\epsilon^{k+2}),\\\
&d(h^{1/4}\text{Im}\Omega)^{[k]}=0,\end{cases}$ (16)

and the approximate $SU(3)$-structure constraints

$\begin{cases}\omega^{[k]}\wedge\text{Im}\Omega^{[k]}=O(\epsilon^{k+3}),\\\
(\omega^{[k]})^{3}=\frac{3}{2}\text{Re}\Omega^{[k]}\wedge\text{Im}\Omega^{[k]}(1+O(\epsilon^{k+1}))\end{cases}$
(17)

Here $\text{Re}\Omega^{[k]}$ is canonically determined by
$\text{Im}\Omega^{[k]}$ so that $\Omega^{[k]}$ is a complex volume form.
Notice we require at the onset that $\omega^{[k]}$ and
$(h^{1/4}\text{Im}\Omega)^{[k]}$ remain closed.

The following version of the gauge fixing condition will be used:

$\begin{cases}\epsilon^{-1}\omega^{[k]}|_{K3}=\omega_{y},\\\
(h^{1/4}\text{Im}\Omega)^{[k]}|_{K3}=\epsilon^{2}d(\text{function})\wedge\omega_{y},\\\
(\omega_{1}dy_{1}+\omega_{2}dy_{2})\wedge(h^{1/4}\text{Im}\Omega)^{[k]}=\epsilon^{2}(\text{function
of $y,\epsilon$}+O(\epsilon^{k}))\omega_{y}^{2}\wedge\omega_{S},\\\
H^{[k]}-H\perp[\omega_{1}],[\omega_{2}],[\omega_{y}].\end{cases}$

We shall also make use of a number of normalisation conditions during the
induction below. For instance, we will require

  * •

The correction terms $\omega^{(k)}$ and $\rho^{(k)}$ are exact for all $k$.

  * •

The boundary value of $H^{[k]}$ on $\partial S$ is fixed to be $H$.

  * •

The total volume
$\int(h^{[k]})^{1/2}(\omega^{[k]})^{3}=3\epsilon^{2}\int_{S}h^{1/2}\omega_{S}(1+O(\epsilon^{k+1}))$.

Another more technical auxiliary condition will appear in the inductive
hypothesis in section 3.3.

######  Remark 3.3.

By the $O(\epsilon^{k})$ notation, we mean $\epsilon^{k}$ times a formal power
series in $\epsilon$ of smooth forms. In particular, if a quantity is
$O(\epsilon^{k})$, then so are its derivatives in this convention.

######  Theorem 3.4.

Given a solution of the weighted maximal submanifold equation $H$ on a surface
$S$ with boundary, avoiding excess $(-2)$-classes. Then there exists a unique
formal power series solution
$(h_{\epsilon},\vartheta_{\epsilon},\omega_{\epsilon},\Omega_{\epsilon})$
solving the Apostolov-Salamon equation, the $SU(3)$-structure condition, the
gauge fixing conditions, and the various normalisation conditions.

In the sequel we will concentrate on the existence part; the uniqueness is
more or less a byproduct. We shall assume the conditions are satisfied up to
$k-1$, and the induction step is roughly to solve the equations in step $k$.
The actual order in which the equations are solved is a little more intricate,
due to the fact that the hierarchy of equations is not only organized in order
of $\epsilon$ but also according to the filtration. That is, when we solve for
$h^{(k)},\vartheta^{(k)},\omega^{(k)},\rho^{(k)},H^{(k)}$, we assume
$h^{[k-1]}$ etc to be fixed by induction, except that $\rho^{(k-1)}$ can still
be modified by $\omega_{S}\wedge d(\text{function})$, and $\omega^{(k-1)}$ can
still be modified by the pullback of a 2-form from $S$. We shall describe how
to solve the system from an algorithmic viewpoint, namely we repeatedly update
the choices to satisfy more and more conditions, thus making the error
functions take increasingly special forms, often keeping the same notation.

###  3.3 Calabi-Yau monopole system

The Calabi-Yau monopole system refers to the integrability conditions

$\begin{cases}d\vartheta^{[k]}\wedge(\omega^{[k-1]})^{2}=O(\epsilon^{k+1}),\\\
\frac{1}{2}dh^{[k]}\wedge(\omega^{[k-1]})^{2}=\epsilon
d\vartheta^{[k]}\wedge(h^{1/4}\text{Im}\Omega)^{[k-1]}+O(\epsilon^{k+2}).\end{cases}$
(18)

The word ‘Calabi-Yau’ comes from the analogy with [10], even though in our
setting the 6-manifold needs not be Calabi-Yau. Rewriting the equations in
terms of the unknown functions $\theta^{(k)},h^{(k)}$,

$\begin{cases}&2d\theta^{(k)}\wedge\omega_{y}\wedge\omega_{S}=-\epsilon^{-k}d\vartheta^{[k-1]}\wedge(\omega^{[k-1]})^{2}+O(\epsilon)=\alpha^{(k)}+O(\epsilon)\\\
&dh^{(k)}\wedge\omega_{y}\wedge\omega_{S}-d\theta^{(k)}\wedge(\omega_{1}dy_{1}+\omega_{2}dy_{2})\\\
&=\epsilon^{-k-1}(\frac{1}{2}dh^{[k-1]}\wedge(\omega^{[k-1]})^{2}-\epsilon
d\vartheta^{[k-1]}\wedge(h^{1/4}\text{Im}\Omega)^{[k-1]})+O(\epsilon)=\beta^{(k)}+O(\epsilon).\end{cases}$

From the induction hypothesis, the $\alpha^{(k)},\beta^{(k)}$ are closed forms
of order $O(1)$, and furthermore we assume the following cohomological
requirement by induction

  * •

$\alpha^{(k)}=\omega_{S}\wedge$(exact 4-form on K3 fibres),

  * •

$\beta^{(k)}=dy_{1}\wedge$(exact 4-form on K3 fibres)$+dy_{2}$(exact 4-form on
K3 fibres) modulo $dy_{1}\wedge dy_{2}$.

We determine the restriction of $d\theta^{(k)}$ to K3 fibres by prescribing
$d\theta^{(k)}\wedge\omega_{1},d\theta^{(k)}\wedge\omega_{2}$, and
$d\theta^{(k)}\wedge\omega_{y}$, which amounts to prescribing the self dual
part of an exact 2-form on the hyperkähler K3, and can be done precisely by
the cohomological condition on $\alpha^{(k)},\beta^{(k)}$. Notice that the
ambiguity in $\rho^{(k-1)},\omega^{(k-1)}$ so far plays no role.

After this step we have reduced to $\alpha^{(k)}=0$ and
$\beta^{(k)}=dy_{1}dy_{2}\wedge$(closed 3-form on K3 fibres). The modification
of $\rho^{(k-1)}$ takes place immediately after this, using the rest of the
Apostolov-Salamon system, see section 3.4 below. Granted that $\rho^{(k-1)}$
is fixed, we proceed with the rest of the Calabi-Yau monopole system. Writing
the closed 3-forms as

$df_{1}\wedge\omega_{1}+df_{2}\wedge\omega_{2}+df_{3}\wedge\omega_{y},$

we see that the $df_{1},df_{2}$ terms can be cancelled by modifying
$\theta^{(k)}$ by $ady_{1}+bdy_{2}$ depending linearly on $f_{1},f_{2}$, and
$df_{3}$ can be cancelled by adjusting $h^{(k)}$. We have thus satisfied (18),
and the remaining ambiguity is

  * •

$h^{(k)}$ is determined up to a fibrewise constant,

  * •

$\theta^{(k)}$ is determined up to the pullback of a 2-form on $S$.

We shall see that these are fixed by cohomological requirements, up to a
global constant on $h^{(k)}$. Notice the ambiguity of $\omega^{(k-1)}$ has not
played any role.

Notice that because of the smallness of the correction induced by the yet
unknown $\omega^{(k)},\rho^{(k)}$, (18) is equivalent to

$\begin{cases}d\vartheta^{[k]}\wedge(\omega^{[k]})^{2}=O(\epsilon^{k+1}),\\\
\frac{1}{2}dh^{[k]}\wedge(\omega^{[k]})^{2}=\epsilon
d\vartheta^{[k]}\wedge(h^{1/4}\text{Im}\Omega)^{[k]}+O(\epsilon^{k+2}).\end{cases}$

This is of course the statement that $\alpha^{(k+1)},\beta^{(k+1)}$ are of
order $O(1)$. It is not sensitive to the ambiguity of $\omega^{(k-1)}$ caused
by the pullback of a 2-form on $S$. The inductive construction of
$\omega^{[k]},(h^{1/4}\text{Im}\Omega)^{[k]}$ guarantees this is exact. Our
induction relies on the cohomological requirement that

  * •

$\alpha^{(k+1)}=\omega_{S}\wedge$(exact 4-form on K3 fibres),

  * •

$\beta^{(k+1)}=dy_{1}\wedge$(exact 4-form on K3 fibres)$+dy_{2}$(exact 4-form
on K3 fibres) modulo $dy_{1}\wedge dy_{2}$.

These exactness conditions amount to conditions on fibrewise integrals. This
requirement on $\alpha^{(k+1)}$ can be achieved precisely by adjusting
$\theta^{(k)}$ by a 2-form pulled back from $S$. As for $\beta^{(k)}$, it
defines an exact $H^{4}(K3)$-valued 1-form on $S$. Now adjusting $h^{(k)}$ by
a function of $y$ we can cancel this exact 1-form, thus achieving the
cohomological requirement. Later in our construction, this cohomological
condition is not sensitive to corrections caused by $\omega^{(k)}$, but
$\rho^{(k)}$ will damage it in a very specific way. We shall later return to
this issue and correct $h^{(k)}$ by a further function depending only on $y\in
S$ (_cf._ section 3.7).

The above only concerns $dh^{(k)}$, rather than $h^{(k)}$ itself. The global
constant for $h^{(k)}$ is specified by the total volume normalisation

$\int
h^{[k]}(\omega^{[k-1]})^{3}=3\epsilon^{2}\int_{S}h\omega_{S}(1+O(\epsilon^{k+1})).$

This condition is not sensitive to $\omega^{(k)}$. The place to take care of
this global constant is in section 3.5.

######  Remark 3.5.

The above discussion is uniform for $k\geq 2$. For $k=1$ this story is
basically the same as the determination of $\vartheta$ we saw in section 2.3,
the small difference being that $\vartheta$ is a connection while the higher
corrections are 1-forms, and the role of the cohomological requirement for
$k=1$ is instead played by the prescription of the first Chern class
$\frac{1}{2\pi}[d\vartheta]$.

######  Remark 3.6.

Once the algorithm fixes $\omega^{(k-1)}$ evantually, we can keep $h^{[k]}$
fixed and make an adjustment to $\theta^{(k)}$ with a further $\epsilon$ power
in front. By one more iteration of some steps above, we can thus ensure a
better condition

$d\vartheta^{[k]}\wedge(\omega^{[k-1]})^{2}=O(\epsilon^{k+2}),$

while keeping all others. Notice that independent of the unknown
$\omega^{(k)}$, we will still have

$d\vartheta^{[k]}\wedge(\omega^{[k]})^{2}=O(\epsilon^{k+2}),$

using that $\omega^{(k)}$ is required to lie in the filtration with at least
one $dy_{i}$ term.

###  3.4 Deformation of the $SU(3)$-structure I: first filtration

The rest of the Apostolov-Salamon system that remains to be solved involves

$d(h^{3/4}\text{Re}\Omega)^{[k]}=-\epsilon
d\vartheta^{[k]}\wedge\omega^{[k]}+O(\epsilon^{k+2}),$

together with the $SU(3)$-structure requirements, and the gauge fixing
conditions. Essentially the degrees of freedom at our disposal is to vary
$\omega^{[k]}$ keeping the closedness condition and the prescribed restriction
to K3 fibres, and to vary $(h^{1/4}\text{Im}\Omega)^{[k]}$ keeping the
closedness condition. We shall see the problem is rather overdetermined. Our
strategy involves solving the equation according to the Leray filtration
determined by the K3 fibration structure. In this section we discuss the first
filtred piece of the equations, which is tied up with the issue of modifying
$\rho^{(k-1)}$. This step takes place after we have specified
$\theta^{(k)}|_{K3}$, before we solve the rest of the Calabi-Yau monopole
system.

######  Remark 3.7.

In the discussion below $\rho^{(k)}$ plays a much more major role than
$\omega^{(k)}$, which does not affect most equations. The reader can also keep
in mind that the increment $\epsilon^{k+1}\omega^{(k)}$ will not be added
until almost the end of the algorithm.

We write

$\epsilon^{-k-1}(d(h^{3/4}\text{Re}\Omega)^{[k]}+\epsilon
d\vartheta^{[k]}\wedge\omega^{[k]})=h^{1/2}d(h^{1/4}\text{Re}\Omega)^{(k)}+\gamma_{k}+O(\epsilon)$

where $\epsilon^{k+1}(h^{1/4}\text{Re}\Omega)^{(k)}$ denotes the small
increment to $(h^{1/4}\text{Re}\Omega)^{[k]}$ induced by the small increment
$\epsilon^{k+1}\rho^{(k)}$ for $(h^{1/4}\text{Im}\Omega)^{[k]}$, and

$\gamma_{k}=\epsilon^{-k-1}(d(h^{3/4}\text{Re}\Omega)^{[k-1]}+\epsilon
d\vartheta^{[k]}\wedge\omega^{[k-1]}).$

Clearly $\gamma_{k}$ is closed. Morever, restricted to the K3 fibres,

$[\gamma_{k}]=\epsilon^{-k}[d\vartheta]\wedge[\omega_{y}]=0\in H^{4}(K3).$

Notice the restriction of $\gamma_{k}$ to K3 fibres depends only on
$\theta^{(k)}$ restricted to the K3 fibres, not the full solution to the
Calabi-Yau monopole system. We are allowed to modify $\rho^{(k)}$ by a term
proportional to $d(f\omega_{y})$ compatible with the gauge condition (14) and
the closedness of $(h^{1/4}\text{Im}\Omega)^{[k]}$, where $f$ is a function to
be determined. Modulo higher order in $\epsilon$ and restricted to K3 fibres,
the corresponding change to $(h^{1/4}\text{Re}\Omega)^{(k)}$ is
$-Jdf\wedge\omega_{y}$, where $J$ is the complex structure on the K3 fibres.
We recognize the fibrewise equation

$d(Jdf\wedge\omega_{y})=\gamma_{k}|_{K3}$

is just the Laplace equation, so by the cohomological condition can be solved
fibrewise up to fibrewise constants. After this step, we can assume
$\gamma_{k}|_{K3}=0$. To fix $f$ as a function of $y$, we notice
$\omega^{[k-1]}\wedge(h^{1/4}\text{Im}\Omega)^{[k-1]}$ is an exact 5-form so
defines an exact $H^{4}(K3)$-valued 1-form, hence can be made to vanish
precisely by utilizing $f$ as a function of $y$. This determines $f$ up to a
global constant, which does not affect $\rho^{(k)}$.

The subtle issue is that while by induction

$\omega^{[k-1]}\wedge(h^{1/4}\text{Im}\Omega)^{[k-1]}=O(\epsilon^{k+2}),$

the term $df\wedge\omega_{y}$ in $\rho^{(k)}$ contributes to
$\omega^{[k]}\wedge(h^{1/4}\text{Im}\Omega)^{[k]}$ by
$\epsilon^{k+1}df\wedge\omega_{y}\wedge\omega_{S}$. This needs to be
compensated by modifying $\rho^{(k-1)}$ by $-d(f\omega_{S})$. This fixes the
little ambiguity of $\rho^{(k-1)}$. In the algorithm, we return to the Calabi-
Yau monopole system and solve for $h^{(k)},\theta^{(k)}$. In the remaining
discussions, we shall take $h^{[k]},\vartheta^{[k]}$ as fixed except when
announced otherwise, and we have

$\omega^{[k]}\wedge\text{Im}\Omega^{[k]}=O(\epsilon^{k+2}),$

which we seek to improve.

###  3.5 Deformation of the $SU(3)$-structure II: second filtration

Next we consider the filtration with $dy_{1}$ or $dy_{2}$ terms. We are
allowed to modify $\rho^{(k)}$ by a term proportional to

$dy_{1}\wedge d(\text{1-form on K3})+dy_{2}\wedge d(\text{1-form on K3}).$

An efficient way to prescribe an exact 2-form on K3 fibres is to prescribe its
self dual part:

$f_{11}\omega_{1}+f_{12}\omega_{2}+f_{13}\omega_{y},\quad
f_{21}\omega_{1}+f_{22}\omega_{2}+f_{23}\omega_{y}$

where $\int_{K3}f_{ij}\omega_{y}^{2}=0$. We write the increment of
$\rho^{(k)}$ modulo $dy_{1}\wedge dy_{2}$ term as

$dy_{1}\wedge(f_{11}\omega_{1}+f_{12}\omega_{2}+f_{13}\omega_{y}+ASD_{1})+dy_{2}\wedge(f_{21}\omega_{1}+f_{22}\omega_{2}+f_{23}\omega_{y}+ASD_{2})$

Working at a given fibre in the coodinates with $g_{ij}=\delta_{ij}$, the
corresponding change to $(h^{1/4}\text{Re}\Omega)^{(k)}$ modulo $dy_{1}dy_{2}$
term is

$dy_{1}(ASD_{2}+f_{23}\omega_{y}+f_{11}\omega_{2}-f_{12}\omega_{1})+dy_{2}(-ASD_{1}-f_{13}\omega_{y}+f_{21}\omega_{2}-f_{22}\omega_{1}).$

Using $d\rho^{(k)}=0$ to eliminate the ASD terms, the change to
$d(h^{1/4}\text{Re}\Omega)^{(k)}$ modulo $dy_{1}dy_{2}$ term is

$d(f_{11}-f_{22})\wedge(\omega_{2}dy_{1}+\omega_{1}dy_{2})+d(f_{12}+f_{21})\wedge(\omega_{2}dy_{2}-\omega_{1}dy_{1}).$

Notice the answer depends only on two combination of the 6 functions $f_{ij}$,
an indication that integrability is at play.

The roles of the other 4 functions are as follows:

  * •

The horizontal-vertical $(1,4)$ component of
$\omega^{[k]}\wedge\text{Im}\Omega^{[k]}$ is controlled precisely by
$f_{13},f_{23}$. By adjusting $f_{13},f_{23}$ we can make
$\omega^{[k]}\wedge(h^{1/4}\text{Im}\Omega)^{[k]}=O(\epsilon^{k+3})$ modulo
$dy_{1}\wedge dy_{2}$ term. Here notice that the yet unknown $\omega^{(k)}$
has no effect to $O(\epsilon^{k+2})$ order due to our choice to fix
$\omega_{y}$ as the restriction to K3 fibres. Notice also that the condition
$\int_{K3}f_{i3}\omega_{y}^{2}=0$ is precisely compatible with our previous
cohomological normalisation on
$\omega^{[k-1]}\wedge(h^{1/4}\text{Im}\Omega)^{[k-1]}$.

  * •

The volume normalisation on $\text{Re}\Omega^{[k]}\wedge\text{Im}\Omega^{[k]}$
is controlled by $f_{11}+f_{22}$. By possibly modifying $\omega^{(k-1)}$ by
the pullback of a 2-form on $S$, which does not affect the inductive
hypotheses, we can ensure that

$(h^{[k]})^{1/2}(\omega^{[k-1]})^{3}=\frac{3}{2}(1+O(\epsilon^{k}))(h^{1/4}\text{Re}\Omega)^{[k-1]}\wedge(h^{1/4}\text{Im}\Omega)^{[k-1]},$

where the $O(\epsilon^{k})$ function has _fibrewise integral zero_ with
respect to $\omega_{y}^{2}$. By allowing $h^{(k)}$ to drift by a global
constant, accompanied by a corresponding change in $\omega^{(k-1)}$ so as to
preserve the fibrewise integral zero condition, we can enforce the total
volume normalisation on

$\int h^{[k]}(\omega^{[k-1]})^{3}$

as promised in section 3.3.

Now the $f_{11}+f_{12}$ term gives rise to a term proportional to
$\text{Im}\Omega$ in $\rho^{(k)}$, whose effect is to modify
$(h^{1/4}\text{Re}\Omega)^{[k]}\wedge(h^{1/4}\text{Im}\Omega)^{[k]}$, and we
can use this to cancel the $O(\epsilon^{k})$ term. This ensures

$(\omega^{[k]})^{3}=\frac{3}{2}(1+O(\epsilon^{k+1}))\text{Re}\Omega^{[k]}\wedge\text{Im}\Omega^{[k]},$

since $\omega^{(k)}$ does not give rise to corrections to volume of relative
order $O(\epsilon^{k})$. Notice again that we need the cohomological
normalisation to be compatible with $\int_{K3}f_{ij}\omega_{y}^{2}=0$.

  * •

The combination $f_{12}-f_{21}$ controls the gauge fixing condition on
$(h^{1/4}\text{Im}\Omega)^{[k]}\wedge(\omega_{1}dy_{1}+\omega_{2}dy_{2})$. It
almost renders the expression zero modulo $O(\epsilon^{k+2})$, except that the
integral normalisation $\int_{K3}f_{ij}\omega_{y}^{2}=0$ means that we cannot
remove a fibrewise constant depending on $y\in S$.

To summarize, while keeping the closedness of
$(h^{1/4}\text{Im}\Omega)^{[k]}$, there are six free functions at our
disposal, which we can use to achieve three constraints in the
$SU(3)$-structure condition, one gauge fixing condition, and the two others
can be used to adjust $\gamma_{k}$ by

$da\wedge(\omega_{2}dy_{1}+\omega_{1}dy_{2})+db\wedge(\omega_{2}dy_{2}-\omega_{1}dy_{1})\quad\mod
dy_{1}\wedge dy_{2}$

for any two prescribed function $a,b$ on the fibres.

###  3.6 Deformation of the $SU(3)$-structure III: integrability

The crucial feature of the Apostolov-Salamon system is that while the
equations governing the deformation of $h^{1/4}\text{Im}\Omega$ are quite
overdetermined, there are additional integrability from the Calabi-Yau
monopole system that comes to the rescue. To explain the key ideas, we will
present the argument without keeping track of the power of $\epsilon$ error
everywhere.

We begin with some basic linear algebra of $SU(3)$-structures:

######  Lemma 3.8.

Let $(\omega,\Omega)$ be an $SU(3)$-structure, and $v$ be a vector field. Then

$\begin{cases}\omega^{2}\wedge\iota_{v}\omega=-\text{Re}\Omega\wedge\iota_{v}\text{Im}\Omega,\\\
\iota_{v}\text{Re}\Omega=\iota_{Jv}\text{Im}\Omega,\\\
\text{Im}\Omega\wedge\iota_{Jv}\omega=\omega\wedge\iota_{v}\text{Re}\Omega,\\\
\iota_{v}d\text{Im}\Omega\wedge\text{Re}\Omega=\iota_{v}\text{Im}\Omega\wedge
d\text{Re}\Omega\end{cases}$

Here the last identity comes from Hitchin’s stable 3-forms, and expresses the
relation between the 6-dimensional components of
$d\text{Re}\Omega,d\text{Im}\Omega$.

######  Proposition 3.9.

Assume $(\omega_{\epsilon},\Omega_{\epsilon})$ is an $SU(3)$-structure,
$h_{\epsilon}$ a positive valued function, and $d\vartheta_{\epsilon}$ a
closed 2-form, satisfying

$\begin{cases}\epsilon
h_{\epsilon}^{1/4}d\vartheta_{\epsilon}\wedge\text{Im}\Omega_{\epsilon}=\frac{1}{2}dh_{\epsilon}\wedge\omega_{\epsilon}^{2},\\\
d(h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon})=0,\end{cases}$

then for any vector field $v$,

$(\epsilon
d\vartheta_{\epsilon}\wedge\omega_{\epsilon}+d(\text{Re}\Omega_{\epsilon}h_{\epsilon}^{3/4}))\wedge\iota_{v}\text{Re}\Omega_{\epsilon}=0,$

namely the 6-dimensional component of $\epsilon
d\vartheta_{\epsilon}\wedge\omega_{\epsilon}+d(\text{Re}\Omega_{\epsilon}h_{\epsilon}^{3/4})$
vanishes automatically.

###### Proof.

We compute using the first equation

$\epsilon
d\vartheta_{\epsilon}\wedge\omega_{\epsilon}\wedge\iota_{v}\text{Re}\Omega_{\epsilon}=\epsilon
d\vartheta_{\epsilon}\wedge\text{Im}\Omega_{\epsilon}\wedge\iota_{Jv}\omega=\frac{1}{2}h_{\epsilon}^{-1/4}dh_{\epsilon}\wedge\omega_{\epsilon}^{2}\wedge\iota_{Jv}\omega_{\epsilon}.$

By the second equation and Hitchin’s stable form identity applied to
$h_{\epsilon}^{1/4}\text{Im}\Omega_{\epsilon}$,

$0=d(\text{Im}\Omega_{\epsilon}h_{\epsilon}^{1/4})\wedge\iota_{Jv}\text{Re}\Omega_{\epsilon}=-\iota_{Jv}\text{Im}\Omega_{\epsilon}\wedge
d(\text{Re}\Omega_{\epsilon}h_{\epsilon}^{1/4})=-\iota_{v}\text{Re}\Omega_{\epsilon}\wedge
d(\text{Re}\Omega_{\epsilon}h_{\epsilon}^{1/4}),$

whence

$d(\text{Re}\Omega_{\epsilon}h_{\epsilon}^{3/4})\wedge\iota_{v}\text{Re}\Omega_{\epsilon}=\frac{1}{2}h_{\epsilon}^{-1/4}dh_{\epsilon}\wedge\text{Re}\Omega_{\epsilon}\wedge\iota_{v}\text{Re}\Omega_{\epsilon}.$

The claim then follows from

$\omega_{\epsilon}^{2}\wedge\iota_{Jv}\omega_{\epsilon}=-\text{Re}\Omega_{\epsilon}\wedge\iota_{Jv}\text{Im}\Omega_{\epsilon}=-\text{Re}\Omega_{\epsilon}\wedge\iota_{v}\text{Re}\Omega_{\epsilon}.$

∎

In our application, we use $h^{[k]},\vartheta^{[k]}$, satisfying the Calabi-
Yau monopole system (18). The $(h^{1/4}\text{Im}\Omega)^{[k]}$ term is always
closed. The error to the $SU(3)$-structure, after our initial stages of
correction, is

$\begin{cases}(\omega^{[k]})^{3}=\frac{3}{2}(1+O(\epsilon^{k+1}))\text{Re}\Omega^{[k]}\wedge\text{Im}\Omega^{[k]},\\\
\omega^{[k]}\wedge\text{Im}\Omega^{[k]}=O(\epsilon^{k+3})\quad\mod
dy_{1}\wedge dy_{2},\\\
\omega^{[k]}\wedge\text{Im}\Omega^{[k]}=O(\epsilon^{k+2})\end{cases}$

Restoring the powers of $\epsilon$ into the above argument, we reach an
important conclusion: modulo $O(\epsilon)$, the error function $\gamma_{k}$
satisfies

$\begin{cases}\gamma_{k}\wedge\iota_{v}(\omega_{1}dy_{1}+\omega_{2}dy_{2})=0,\quad\text{for
any vertical vector $v$},\\\
\gamma_{k}\wedge\omega_{1}=0,\quad\gamma_{k}\wedge\omega_{2}=0.\end{cases}$
(19)

Now $d\gamma_{k}=0$ by construction, and at this stage $\gamma_{k}$ lies in
the filtration with at least one $dy_{i}$ term, so modulo $dy_{1}dy_{2}$ terms
we can write

$\gamma_{k}=(da_{11}\wedge\omega_{1}+da_{12}\wedge\omega_{2}+da_{13}\omega_{y})dy_{1}+(da_{21}\wedge\omega_{1}+da_{22}\wedge\omega_{2}+da_{23}\wedge\omega_{y})dy_{2}$

for some functions $a_{ij}$ with fibre average zero. At a given fibre we work
in the coordinate with $g_{ij}=\delta_{ij}$. From the first part of the
integrability condition (19),

$(da_{11}\wedge\omega_{1}+da_{12}\wedge\omega_{2}+da_{13}\wedge\omega_{y})=J(da_{21}\wedge\omega_{1}+da_{22}\wedge\omega_{2}+da_{23}\wedge\omega_{y})$

on the K3 fibre. Taking $d$ on both sides gives $a_{23}=0$, so by symmetry
$a_{13}=0$. Now

$da_{11}\wedge\omega_{1}+da_{12}\wedge\omega_{2}=J(da_{21}\wedge\omega_{1}+da_{22}\wedge\omega_{2})=da_{21}\wedge\omega_{2}-da_{22}\wedge\omega_{1},$

so by uniqueness of such a decomposition $a_{11}=-a_{22}$ and $a_{12}=a_{21}$.
Thus

$\gamma_{k}=da_{11}\wedge(\omega_{1}dy_{1}-\omega_{2}dy_{2})+da_{12}(\omega_{2}dy_{1}-\omega_{1}dy_{2})\quad\mod
dy_{1}dy_{2}.$

But this is precisely of the form we can cancel by adjusting $\rho^{(k)}$, as
discussed in section 3.5. The conclusion is that we can now assume
$\gamma_{k}$ is a multiple of $dy_{1}\wedge dy_{2}$.

###  3.7 Weighted maximal submanifold system

At this stage

$\gamma_{k}=dy_{1}\wedge dy_{2}\wedge(\text{closed 2-forms on K3})$

Thus $\gamma_{k}$ defines an $H^{2}(K3)$-valued 2-form on $S$, which we denote
as $[\gamma_{k}]$. There is a natural decomposition of $[\gamma_{k}]$ into
fibrewise self dual and ASD parts, or equivalently the tangential and normal
component to the image of $H:S\to H^{2}(K3)$. Our next goal is to remove the
ASD part of $[\gamma_{k}]$.

To correct this error, we reflect on the degrees of freedom not yet used. We
have already adjusted $\rho^{(k)}$ by corrections of the form

$dy_{1}(\text{exact 2-form on K3})+dy_{2}(\text{exact 2-form on K3})\quad\mod
dy_{1}\wedge dy_{2}.$

However, the actual constraint is that $\rho^{(k)}$ should be exact, which
leaves the possibility of adjusting by harmonic 2-forms in place of exact
2-forms, corresponding to the deformation of $H^{[k]}$ (_cf._ the last part of
section 3.1). Given a vector field $H^{(k)}:S\to H^{2}(K3)$ normal to the
weighted maximal submanifold defined by the image of $H$, we can view
$\frac{\partial H^{(k)}}{\partial y_{i}}$ for $i=1,2$ as specifying the data
of the families of harmonic 2-forms. The normality condition can be thought as
a gauge choice, related to the diffeomorphism of $S$. The requirement that the
two families of harmonic forms come from the gradient of a single $H^{(k)}$,
means that we can arrange $\rho^{(k)}$ to be exact, instead of just being
closed modulo $dy_{1}\wedge dy_{2}$.

One can then calculate the response of $(h^{1/4}\text{Re}\Omega)^{(k)}$. By
reexamining the calculations about the exact forms in section 3.5, one sees
that the change in $d(h^{1/4}\text{Re}\Omega)^{(k)}$ lies in the filtration
with $dy_{1}\wedge dy_{2}$ terms, so defines an $H^{2}(K3)$-valued 2-form on
$S$. However, we need to be careful not to ruin the various conditions we
already achieved. The fact that $H^{(k)}$ is orthogonal to the constant class
$[\omega_{y}]$ implies that for the change in $\rho^{(k)}$ induced by
$H^{(k)}$,

$\rho^{(k)}\wedge\omega_{y}=O(\epsilon),\quad\mod dy_{1}\wedge dy_{2},$

so we still have

$\omega^{[k]}\wedge\text{Im}\Omega^{[k]}=O(\epsilon^{k+3})\quad\mod
dy_{1}\wedge dy_{2}.$

A more subtle effect happens to the Calabi-Yau monopole system. In section 3.3
we required the cohomological condition that the $H^{4}(K3)$-valued 1-form on
$S$ defined by $\beta^{(k+1)}$ vanishes. This forces us to adjust $h^{(k)}$ by
a function depending only on $y$, satisfying an equation on $S$

$\frac{1}{2}dh^{(k)}[\omega_{y}]^{2}=[d\vartheta]\wedge dH^{(k)}.$

We take $h^{(k)}=2[d\vartheta]\cdot H^{(k)}$ compatible with (7); the ultimate
reason for this choice has to do with total volume normalisation, see below.
Thus the response of $d(h^{3/4}\text{Re}\Omega)^{(k)}$ is still in the
$dy_{1}\wedge dy_{2}$ filtration, and defines an $H^{2}(K3)$-valued 2-form on
$S$. This can be identified as follows: consider the map

$\tilde{H}\mapsto\frac{\partial}{\partial
y_{j}}(\tilde{h}^{1/2}\tilde{g}^{ij}\sqrt{\det\tilde{g}}\frac{\partial\tilde{H}}{\partial
y_{i}})dy_{2}\wedge
dy_{1},\quad\tilde{g}_{ij}=\frac{\partial\tilde{H}}{\partial
y_{i}}\cdot\frac{\partial\tilde{H}}{\partial
y_{j}},\quad\tilde{h}=2[d\vartheta]\cdot\tilde{H},$

and denotes its first variation at the map $H$ as $\mathcal{L}_{1}$. Then the
$H^{2}(K3)$-valued 2-form induced by the variation $H^{(k)}$ is
$\mathcal{L}_{1}(H^{(k)})$. This can be seen by reexamining the derivation of
the weighted maximal submanifold equation in section 2.3.

The next subtle effect happens to the volume normalisation condition in the
$SU(3)$-structure. To leading order, the increment of
$(h^{[k]})^{1/2}(\omega^{[k]})^{3}$ induced by adjusting $h^{(k)}$ is

$3\epsilon^{k+2}([d\vartheta]\cdot
H^{(k)})h^{-1/2}\omega_{y}^{2}\wedge\omega_{S},$

and the increment of
$(h^{1/4}\text{Re}\Omega)^{[k]}\wedge(h^{1/4}\text{Im}\Omega)^{[k]}$ induced
by $\rho^{(k)}$ is after an instructive exercise

$2\epsilon^{k+2}\omega_{S}\wedge\omega_{y}^{2}\left(g^{ij}\frac{\partial
H}{\partial y_{i}}\cdot\frac{\partial H^{(k)}}{\partial y_{j}}\right).$

To preserve the volume normalisation

$(h^{[k]})^{1/2}(\omega^{[k]})^{3}=\frac{3}{2}(h^{1/4}\text{Re}\Omega)^{[k]}\wedge(h^{1/4}\text{Im}\Omega)^{[k]}(1+O(\epsilon^{k+1})),$

we need to adjust $\omega^{(k-1)}$ by the pullback of a 2-form on $S$

$\mathcal{L}_{2}(H^{(k)})=\omega_{S}\left(\tilde{g}^{ij}\frac{\partial
H}{\partial y_{i}}\cdot\frac{\partial H^{(k)}}{\partial
y_{j}}-h^{-1}[d\vartheta]\cdot H^{(k)}\right).$

This provides the mechanism that will ultimately fix the ambiguity of
$\omega^{(k-1)}$. This operator $\mathcal{L}_{2}$ has the pleasant
interpretation of being the first variation at $H$ of the mapping

$\tilde{H}\mapsto\tilde{h}^{-1/2}\sqrt{\det(\tilde{g}_{ij})}dy_{1}\wedge
dy_{2},\quad\tilde{g}_{ij}=\frac{\partial\tilde{H}}{\partial
y_{i}}\cdot\frac{\partial\tilde{H}}{\partial
y_{j}},\quad\tilde{h}=2[d\vartheta]\cdot\tilde{H}.$

Combining the above, the response to $H^{(k)}$ of the $H^{2}(K3)$-valued
2-form $[\gamma_{k}]$ is

$\mathcal{L}_{1}(H^{(k)})+[d\vartheta]\wedge\mathcal{L}_{2}(H^{(k)}),$

which by comparison with section 2.4 is identified as $\mathcal{L}(H^{(k)})$,
where $\mathcal{L}$ is the variation of the operator $\delta\mathcal{A}_{w}$,
or equivalently the Hessian operator of the weighted area functional. Our
initial problem in this section is now rephrased as solving for $H^{(k)}$ so
that $\mathcal{L}(H^{(k)})$ cancels the ASD part of the initial error
$[\gamma_{k}]$, which defines a normal vector field to the image of $H:S\to
H^{2}(K3)$. But by Lemma 2.5 the Hessian operator $\mathcal{L}$ is negative
definite on normal vector fields with zero boundary data, so $H^{(k)}$ can
always be solved uniquely.

We now reexamine the total volume normalisation on $\int
h^{[k]}(\omega^{[k]})^{3}$. In the above procedure, the change in the total
volume corresponds to the change in the weighted area $\mathcal{A}_{w}$, which
is zero to leading order because $H$ is a critical point. This ensures the
total volume normalisation is preserved.

Now that we have solved for $H^{(k)}$, so as to be able to gain the condition
$[\gamma_{k}]=0$, we can tie up a few loose ends. Notice $\omega^{(k-1)}$ is
now fixed. Then we can follow Remark 3.6 to modify $\theta^{(k)}$ by a higher
order in $\epsilon$ term to achieve

$d\vartheta^{[k]}\wedge(\omega^{[k]})^{2}=O(\epsilon^{k+2}).$

Also, while the steps taken in this section damage the gauge fixing condition
on $(h^{1/4}\text{Im}\Omega)^{[k]}\wedge(\omega_{1}dy_{1}+\omega_{2}dy_{2})$,
we can repeat some parts in section 3.5 to restore the gauge condition. Since
this only involve exact 2-forms, it has no effect on the class $[\gamma_{k}]$.
Repeating some steps in section 3.4 for one order higher in $\epsilon$, we can
ensure $\gamma_{k}$ lies in the filtration with at least one $dy_{i}$ term
modulo $O(\epsilon^{2})$.

###  3.8 Deformation of the $SU(3)$-structure IV: linear algebraic constraints

We now start to modify $\omega^{[k]}$ to improve the $SU(3)$-structure
condition. Modulo $O(\epsilon^{k+3})$, at this stage

$\omega^{[k-1]}\wedge(h^{1/4}\text{Im}\Omega)^{[k]}=\epsilon^{k+2}dy_{1}\wedge
dy_{2}\wedge\text{closed 3-form on K3}$

Write the closed 3-form as
$df_{1}\wedge\omega_{1}+df_{2}\wedge\omega_{2}+df_{3}\wedge\omega_{y}$. The
$df_{3}$ term can be removed by adjusting $\rho^{(k)}$ by
$d(f_{3}\omega_{S})$. Now the freedom to adjust $\omega^{(k)}$ is
$d(ady_{1}+bdy_{2})$. Choosing the functions $a,b$ depending linearly on
$f_{1},f_{2}$, we can also remove the $df_{1},df_{2}$ terms. After this
adjustment,

$\omega^{[k]}\wedge\text{Im}\Omega^{[k]}=O(\epsilon^{k+3}).$

The $H^{2}(K3)$-valued 2-form $[\gamma_{k}]$ is not affected in this step.

Recall the integrability condition (19) gives
$\omega_{1}\wedge\gamma_{k}=\omega_{2}\wedge\gamma_{k}=0$. There is one extra
piece of integrability we have not used: recall (_cf._ Remark 3.6)

$d\vartheta^{[k]}\wedge(\omega^{[k]})^{2}=O(\epsilon^{k+2}),\quad
d\omega^{[k]}=0,$

hence

$\gamma_{k}\wedge\omega_{y}=\gamma_{k}\wedge\epsilon^{-1}\omega^{[k]}=\epsilon^{-k-2}\\{\epsilon
d\vartheta^{[k]}\wedge(\omega^{[k]})^{2}+d((h^{[k]})^{3/4}\text{Re}\Omega^{[k]}\wedge\omega^{[k]})\\}=O(\epsilon)$

Namely up to $O(\epsilon)$ error $\gamma_{k}$ defines a family of ASD harmonic
forms on the K3 fibres. But the ASD component of $[\gamma_{k}]$ has been
removed in section 3.7, so $\gamma_{k}=0$, and we have completed the
induction.

##  4 Global aspects

We now discuss the phenomena that arise when trying to compactify the iterated
fibration structure. These involve both topological and metric aspects. In
particular, since any Killing vector field on a compact Ricci flat manifold
must be parallel, we need to display mechanisms to break the circle symmetry
in order to make nontrivial compact examples possible. The emphasis here is on
the geometric picture, without the more serious analytical work.

###  4.1 Lefschetz fibration

Lefschetz fibration is a phenomenon which has relatively little to do with the
small circle in $M$, but instead happens to the 6-manifold $M/S^{1}$. The
motivation is that in a global setting, excess $(-2)$-classes are quite
inevitable.

Consider first the simpler problem: let $X$ be a Calabi-Yau 3-fold with a
holomophic K3 fibration $\pi:X\to\mathbb{P}^{1}$, where all critical points of
$\pi$ have the local complex analytic description as
$\pi=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}$ in local coordinates, known as Lefschetz
fibrations. Given a Kähler class $[\omega_{X}]$ on $X$, and a Kähler class
$[\omega_{\mathbb{P}^{1}}]$ on $\mathbb{P}^{1}$, the problem is to describe
the limiting behaviour of the Calabi-Yau metric $\omega_{CY,\epsilon}$ in the
class $[\epsilon\omega_{X}+\omega_{\mathbb{P}^{1}}]$ as $\epsilon\to 0$. This
can be viewed as a special case, when $M=X\times S^{1}$ decouples the small
circle factor.

This problem has been satisfactorily analyzed in [18][19][20]. There are two
general lessons:

  * •

(Multiscaledness) The metrics exhibit different characteristic behaviours at
different length scales.

  * •

(Universality) The microscopic behaviour of the metric depends essentially
only on the local geometry, not on the global information of the Calabi-Yau
3-fold or the Lefschetz fibration.

Globally, the Calabi-Yau metrics converge in Gromov-Hausdorff sense to a
singular metric on $\mathbb{P}^{1}$ which can be explicitly written down via
period integrals. The characteristic length scale of the base is $O(1)$.
Around any fixed smooth $K3$ fibre, the rescaled Calabi-Yau metrics
$\epsilon^{-1}\omega_{CY,\epsilon}$ converge smoothly to $K3\times\mathbb{C}$
with the product metric, where the K3 factor carries the unique Calabi-Yau
metric in class $[\omega_{X}]$. Around a singular K3 fibre (which has
$\mathbb{C}^{2}/\mathbb{Z}_{2}$ orbifold singularity by the Lefschetz
condition), the convergence to $K3\times\mathbb{C}$ happens only in a Gromov-
Hausdorff sense; here the K3 factor carries the orbifold Calabi-Yau metric.
The convergence cannot be smooth because $K3\times\mathbb{C}$ has
$\mathbb{C}^{2}/\mathbb{Z}_{2}\times\mathbb{C}$ local singularity. The
characteristic length scale of the K3 fibre is $O(\epsilon^{1/2})$.

For our current purpose the most interesting aspect is what happens around the
critical points of the fibration. The answer involves an exotic complete
Calabi-Yau metric $\omega_{\mathbb{C}^{3}}$ on $\mathbb{C}^{3}$, whose tangent
cone at infinity is $\mathbb{C}^{2}/\mathbb{Z}_{2}\times\mathbb{C}$ with the
flat orbifold metric. In particular the volume of geodesic balls in
$\omega_{\mathbb{C}^{3}}$ is of order $\sim r^{6}$, known as maximal volume
growth, but $\omega_{\mathbb{C}^{3}}$ is not the flat Euclidean metric. This
exotic metric is constructed by writing down an approximate metric ansatz near
the spatial infinity, and using a noncompact version of the Calabi conjecture
to perturb it into a Calabi-Yau metric. It is proved in [20] that after
suitable rescaling by a magnification factor $O(\epsilon^{-4/3})$, the
collapsing Calabi-Yau metrics $\epsilon^{-4/3}\omega_{CY,\epsilon}$ around the
critical point converge to $\omega_{\mathbb{C}^{3}}$ as $\epsilon\to 0$. In
particular, the characteristic length scale of the bubble around the critical
point is $O(\epsilon^{2/3})$, much smaller compared to the characteristic
length scale $O(\epsilon^{1/2})$ of the K3 fibres. The local metric behaviour
is captured by the canonical object $\omega_{\mathbb{C}^{3}}$, independent of
the source of the global Calabi-Yau 3-fold.

Back to the 7-dimensional problem where $M/S^{1}$ around a particular K3 fibre
looks approximately like a Lefschetz K3 fibration, we discuss the heuristic
reason why the presence of the extra small cicle does not significantly affect
the geometry near the critical point, as long as $[d\vartheta]$ is orthogonal
to the $(-2)$-class of the vanishing sphere. The effect of this extra $S^{1}$
comes from its curvature $d\vartheta$. Around the singular K3 fibre, from our
ansatz construction in section 2.3, the leading order behaviour of
$d\vartheta$ is the harmonic 2-form on the singular K3 in the appropriate
class. Since $[d\vartheta]$ is orthogonal to the $(-2)$-class, there is no
cohomological incentive for the curvature to concentrate significantly near
the critical point. The characteristic curvature scale is $O(1)$. Now when we
apply our magnifying glass to see the geometry of the exotic $\mathbb{C}^{3}$,
the curvature effect of $d\vartheta$ is washed away by the scaling factor,
_i.e._ microscopically the circle bundle behaves as if it is flat. A further
piece of evidence is that $h$ is to leading order constant on fibres. Thus we
expect the local geometry is approximately a product of the exotic metric on
$\mathbb{C}^{3}$ with the small $S^{1}$ factor. In particular, this does not
provide a mechanism to break the $S^{1}$-symmetry.

The monodromy of the Lefschetz fibration is controlled by the Picard-Lefschetz
formula. This will require a slight modification to our local discussion of
polarized positive sections; instead of taking value in $H^{2}(K3)$, the
section $H$ really takes value in an affine bundle with fibres isomorphic to
$H^{2}(K3)$ (_cf._ section 4.6 below).

###  4.2 Wall crossing

Wall crossing refers here to the following phenomenon. Let $\sigma\in
H^{2}(K3)$ be a fixed class orthogonal to $[\omega_{y}]$. We propose a
mechanism for $S$ to be locally divided into two components by a curve
$l\subset S$ satisfying $\sigma\cdot H=\text{const}$, such that the first
Chern class $\frac{1}{2\pi}[d\vartheta]$ jumps by $\sigma$ when we cross $l$.
The mechanism involves allowing the $S^{1}$-bundle to degenerate along a
3-dimensional discriminant locus fibred over $l$.

######  Remark 4.1.

Curiously, the condition $\sigma\cdot H=\text{const}$ encodes special
Lagrangians in the fast circle collapsing setting (_cf._ section 2.2) where
$M/S^{1}$ is approximately Calabi-Yau. In the fine tuned circle collapsing
setting we are studying $M/S^{1}$ is not close to being Calabi-Yau, so such an
interpretation is not available, but the intuition is still useful.

Let $\tau$ be a coordinate along $l$. Our interest is in the behaviour of the
$G_{2}$-structure transverse to $l$. Since the characteristic length scale of
the transition (which turns out to be $O(\epsilon)$) is much smaller than the
characteristic scale of $l$ which is $O(1)$, it is reasonable to suppose that
the geometry is locally approximately independent of $\tau$. A natural
suggestion is that the transverse behaviour is modelled on the dimensionally
reduced case of Calabi-Yau 3-folds with $S^{1}$-symmetry, sketched in section
2.6. What we will now describe is essentially the same as [22, chapter 4], in
a slightly different normalisation convention, using the rescaled notation in
section 2.6.

Since $\sigma\cdot dH=0$ along $l$ and $\sigma\cdot[\omega_{y}]=0$, and
noticing how the conversion formula (13) mixes up the hyperkähler structure on
K3 surface $D$, along $l$ we have

$[\text{Im}\Omega_{D}]\cdot\sigma=0,\quad[\text{Re}\Omega_{D}]\cdot\sigma=0.$

We assume we can consistently find smooth holomorphic curves $C$ in the class
$\sigma\in H^{2}(K3)$ with respect to the complex structure defined by
$\Omega_{D}$; this can be sometimes guaranteed by cohomological criterions,
for example in the case of $(-2)$-classes. From the 7-dimensional perspective,
the discriminant locus of the $S^{1}$-fibration will be the union of these
holomorphic curves along $l$. From the dimensional reduction viewpoint, the
$\tau$ variable will be dropped, and we think of a Calabi-Yau 3-fold with
$S^{1}$-action, whose fixed points lie over $C$ inside $D\times\\{0\\}\subset
D\times\mathbb{R}_{\mu}$.

The Calabi-Yau requirement leads to

$\begin{cases}d\vartheta=-\partial_{\mu}\tilde{\omega}-\epsilon^{-1}d_{D}^{c}h\wedge
d\mu,\\\ (\partial^{2}_{\mu}\tilde{\omega}+\epsilon^{-1}d_{D}d_{D}^{c}h)\wedge
d\mu=-d(d\vartheta)=-2\pi\delta_{C}\\\
\tilde{\omega}^{2}=\frac{1}{2}h\Omega_{D}\wedge\overline{\Omega}_{D}.\end{cases}$

This is the same as in section 2.6 except for the codimension 3 distributional
current $\delta_{C}$ along $C\subset D\times\mathbb{R}_{\mu}$, which arises
from the requirement that the integral of $d\vartheta$ on a 2-sphere linking
$C$ in $D\times\mathbb{R}_{\mu}$ should have total integral $2\pi$ to maintain
the smoothness of the Calabi-Yau 3-fold. As $\mu$ increases to cross
$0\in\mathbb{R}$, the first Chern class $\frac{1}{2\pi}[d\vartheta]$ increases
by $\sigma\in H^{2}(K3)$.

The system so far is highly nonlinear. Fortunately, since the $S^{1}$-fibres
are near the collapsing limit, perturbation techniques are available. We
impose the ansatz

$\begin{cases}h=h_{0}+\epsilon\mathfrak{h},\quad h_{0}=const,\\\
\tilde{\omega}=\tilde{\omega}_{D}+\epsilon\psi,\quad\tilde{\omega}_{D}^{2}=\frac{h_{0}}{2}\Omega_{D}\wedge\overline{\Omega}_{D}.\end{cases}$

This means unless we are very close to the discriminant locus, then $h$ should
be locally approximately constant, and $\tilde{\omega}$ should approximately
agree with the Calabi-Yau metric on $D$, in line with the basic assumptions in
section 2.3. This motivates the linearization of the above system:

$\begin{cases}\tilde{\omega}_{D}\wedge\psi=\frac{1}{4}\mathfrak{h}\Omega_{D}\wedge\overline{\Omega}_{D},\quad\emph{i.e.}\quad\mathfrak{h}=h_{0}\operatorname{Tr}_{\tilde{\omega}_{D}}\psi,\\\
\epsilon\partial_{\mu}^{2}\psi+d_{D}d_{D}^{c}\mathfrak{h}=-2\pi\delta_{C},\end{cases}$

hence the Laplace equation

$(\epsilon\partial_{\mu}^{2}-h_{0}d_{D}d_{D}^{*})\psi=-2\pi\delta_{C}.$

The leading singularity along $C$ is

$\begin{cases}\psi\sim\frac{\sqrt{-1}}{4r}\eta\wedge\bar{\eta},\\\
\mathfrak{h}\sim\frac{1}{2r},\\\ r\sim\sqrt{\epsilon
h_{0}^{-1}\text{dist}_{\tilde{\omega}_{D}}(\cdot,C)^{2}+|\mu|^{2}},\end{cases}$

where $\eta$ denotes a $(1,0)$-form on $D$ normal to $C$, with
$\operatorname{Tr}_{\tilde{\omega}_{D}}\frac{h_{0}\sqrt{-1}}{2}\eta\wedge\bar{\eta}=1$.

To give the geometric interpretation, first observe that the correction effect
is significant only when $r\lesssim\epsilon$, corresponding to length scale
$O(\epsilon)$, which is comparable to the average length of $S^{1}$-fibres and
much smaller compared to the length scale $O(\epsilon^{1/2})$ of the K3
surfaces.

The correction is dominant in the small neighbourhood around $C$, and its
effect is to make the geometry transverse to $C$ look approximately like Taub-
NUT metrics. In fact, we have a metric asymptote near $C$

$g^{\prime}\sim\epsilon\tilde{g}_{D}+\frac{\epsilon^{2}}{2r}\eta\otimes\eta+(h_{0}+\frac{\epsilon}{2r})(d\mu)^{2}+(h_{0}+\frac{\epsilon}{2r})\epsilon^{2}\vartheta^{2},$

and the component of the K3 metric $\tilde{g}_{D}$ normal to $C$ is
$h_{0}\eta\otimes\eta$ by construction. We recognize the metric component
transverse to $C$

$(h_{0}+\frac{\epsilon}{2r})\\{(d\mu)^{2}+\epsilon\eta\otimes\eta\\}+(h_{0}+\frac{\epsilon}{2r})\epsilon^{2}\vartheta^{2}$

is up to scaling just Taub-NUT metric in disguise. In particular, while one
may a priori worry about the validity of linear approximations when the
correction is large, it turns out that after adding the linear correction, the
structure we obtain is close to being Calabi-Yau. This ansatz is used in [22]
to describe the neck region for certain degenerations of Calabi-Yau metrics.
Our proposal here is that the same picture occurs for iterated collapsing of
$G_{2}$-manifolds.

In this mechanism, the $S^{1}$-bundle becomes degenerate over some
discriminant locus, but the global $S^{1}$-symmetry is still not broken; the
singularity just corresponds to the fixed points of the $S^{1}$-action.

######  Remark 4.2.

Under additional hypotheses, we can incorporate wall crossing phenomenon with
two transversely intersecting walls $l_{1},l_{2}$. The intersection $l_{1}\cap
l_{2}$ corresponds to a K3 fibre containing holomorphic curves $C_{1},C_{2}$
in the class $\sigma_{1}$, $\sigma_{2}$ respectively. One needs to be careful
that the two complex structures on the K3 surface corresponding to
$l_{1},l_{2}$ are different. We now assume that $C_{1}$ and $C_{2}$ do not
intersect; this is in general stronger than the homological condition
$[\sigma_{1}]\cdot[\sigma_{2}]=0$. We can then expect that the wall crossing
across $l_{1},l_{2}$ behave independently. The reason is that the
characteristic length scale of the wall crossing in $O(\epsilon)$, which is
much less than the characteristic scale of the K3 fibre $O(\epsilon^{1/2})$,
so the linear correction induced by the distributional terms supported on
$C_{1},C_{2}$ have neglegible influence on each other.

If $C_{1}$ and $C_{2}$ are intersecting, then the local situation is
supposedly modelled on a collapsing $G_{2}$-manifold with $S^{1}$-fibration
over Euclidean $\mathbb{C}^{3}$ whose discriminant locus is the union of two
intersecting special Lagrangian 3-planes. This conjectural metric model is a
folklore prediction of Atiyah and Witten [3] and its unsolved existence
problem is a major difficulty in the Foscolo-Haskins-Nordström picture [10].

###  4.3 ALF fibration

Instead of the Taub-NUT metric, we can modify the distributional equation to

$(\partial^{2}_{\mu}\tilde{\omega}+\epsilon^{-1}d_{D}d_{D}^{c}h)\wedge
d\mu=-d(d\vartheta)=-2\pi(k+1)\delta_{C},$

so the solution is modified into

$\begin{cases}\psi\sim\frac{\sqrt{-1}(k+1)}{4r}\eta\wedge\bar{\eta},\\\
\mathfrak{h}\sim\frac{k+1}{2r},\\\ r\sim\sqrt{\epsilon
h_{0}^{-1}\text{dist}_{\tilde{\omega}_{D}}(\cdot,C)^{2}+|\mu|^{2}},\end{cases}$

The transverse structure to $C$ matches with the asymptote of the $A_{k}$ type
ALF gravitational instantons. Since ALF gravitational instantons in general
have nontrivial moduli, one expects internal degrees of freedom supported on
the discriminant locus to arise when we attempt to glue in an ALF fibration,
similar to the harmonic 1-forms appearing in Joyce and Karigiannis [14]. We
will be brief on issues already discussed extensively for ALE fibrations in
the physics literature [21][4][5][13].

The basic situation is that we are given a 3-manifold with a given Riemannian
metric, and we wish to construct an approximate $G_{2}$-metric on a fibration
over the 3-manifold with fibres diffeomorphic to $A_{k}$-type ALF space, whose
asymptotic behaviour near the fibrewise infinity is prescribed, and the metric
restricted to each fibre is approximately isomorphic to some choice of
$A_{k}$-type ALF space with characteristic length scale $O(\epsilon)$ much
smaller compared to the length scale of the 3-manifold. Morever, for
simplicity we assume that the asymptotic circle length of the ALF fibres is
approximately constant over the region of the 3-manifold under consideration,
so that the variation of the ALF structures can be captured completely by
period integrals. In our case of interest the 3-manifold is the discriminant
locus fibred over $l$ with fibre $C$ of length scale $O(\epsilon^{1/2})$, and
the asymptotic circle length is indeed almost constant within length
$O(\epsilon^{1/2})$, although not necessarily so over length scale $O(1)$.

Taking an orthonormal frame $e_{1},e_{2},e_{3}$ on the 3-manifold, we can
write the ansatz for the $G_{2}$-structure as

$\phi\sim\epsilon^{2}(\omega_{1}^{ALF}\wedge e_{1}^{*}+\omega_{2}^{ALF}\wedge
e_{2}^{*}+\omega_{3}^{ALF}\wedge e_{3}^{*})-e_{1}^{*}\wedge e_{2}^{*}\wedge
e_{3}^{*},$

where $\omega_{i}^{ALF}$ are fibrewise hyperkähler triples of $A_{k}$-type ALF
metrics with prescribed asymptote at fibre infinity. (Strictly speaking, to
make sense of $\omega_{i}^{ALF}$ on the total space, we need the data of a
horizontal distribution; its determination is similar to Donaldson’s proposal
of adiabatic K3 fibration [8]).

Now working in geodesic coordinates around a given point, in the adiabatic
limit $\epsilon\to 0$, the requirement for $d\phi=0$ leads to that the
$H^{2}(ALF)$-valued 1-form $\lambda=\sum_{i}[\omega_{i}]e_{i}^{*}$ is closed,
and the requirement for $d*_{\phi}\phi=0$ leads to $d*\lambda=0$ on the
3-manifold. To summarize, as long as the adiabatic approximation holds and the
fibre asymptotic behaviour stays approximately constant, then the fibration is
governed by an $H^{2}(ALF)$-valued harmonic 1-form locally on the 3-manifold.

Thinking more globally on the 3-manifold, $H^{2}(ALF)$ becomes a local system,
alternatively viewed as a bundle with a flat connection, and $\lambda$ is a
harmonic 1-form valued in this bundle. Physicists view such data as an Abelian
solution to the BPS equation [13, section 2.3], also related to spectral
covers [21, section 2.4]. In our case of interest, the 3-manifold is itself
fibred over $l$ with very small fibres $C$, and to leading order we get bundle
valued harmonic 1-forms over the Riemann surfaces $C$.

Some sources of difficulty are:

  * •

In elementary examples of the 3-manifolds, typically there are no nontrivial
harmonic 1-forms. It has been suggested that we should relax the harmonic
1-form condition along some collection of knots inside the 3-manifold, and
build in nontrivial monodromy for the local system around these knots. The
geometric situation seems to correspond to the Kovalev-Lefschetz fibrations
discussed by Donaldson [8], adapted to the ALF setting. It is also suggested
that we may consider branched covers over the 3-manifold, so the harmonic
1-forms may be multivalued [4, section 9]. Geometrically, the small ALF
instantons are arranged in several sheets.

  * •

On a generic 3-manifold, it is commonly expected that when the
$H^{2}(ALF)$-valued 1-form evaluated on a (-2)-curve has a zero, the
$G_{2}$-manifold should develop an isolated singularity modelled on the
Bryant-Salamon cone over $\mathbb{CP}^{3}$, producing chiral matter in the
physics literature, but the metric local model seems difficult to construct
rigorously.

The fact that our 3-manifold is itself highly collapsed somewhat alters this
picture. Thinking in terms of the dimensional reduction to Calabi-Yau 3-fold,
we suggest the following picture may be relevant for the simplest case, and is
accessible within current technology. Complex geometrically, consider the
quadric cone

$\\{z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}=0\\}\subset\mathbb{C}^{4},$

admitting a fibration

$\\{z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}=0\\}\xrightarrow{z_{4}}\mathbb{C},$

with fibres being affine quadric surfaces. These agree with the underlying
space of $A_{1}$-type ALF gravitational instantons, so we can hope to
construct an ansatz Kähler metric asymptotic to an ALF fibration near
infinity. One expects that by a noncompact version of the Calabi conjecture,
the ansatz metric can be corrected into a Calabi-Yau metric. The result of
Hein and Sun [11] then suggests that the Calabi-Yau metric is asymptotic to
the Stenzel cone at the origin. Thus this Calabi-Yau metric will furnish the
transition between the asymptotic ALF fibration picture and the local Stenzel
cone.

  * •

We also wish to know the long range behaviour over $l$. One expects that the
variation of the asymptotic circle length, and the moduli of Abelian solutions
of the BPS equation to play a role. Morever, assuming the above Calabi-Yau
model metric can be constructed, then naïvely we have a codimension 6
singularity along $l$ transversely modelled on the Stenzel cone. Gao Chen [6]
has studied the deformation theory of such singularities, and found an
infinite dimensional obstruction. The expectation is that the codimension 6
singularity breaks up into isolated codimension 7 singularities modelled on
the cone over $S^{3}\times S^{3}$.

Of course, these rather formidable difficulties only arise if $k\geq 1$. The
$k=0$ case with the Taub-NUT metric does not have internal moduli, so does not
exhibit these difficult yet rich phenomena.

###  4.4 Orientifold and boundary behaviour

We now discuss an analogue of the wall crossing phenomenon in section 4.2
involving orbifold behaviour, which will give rise to boundaries in $S$.

Since the wall crossing phenomenon has characteristic length scale
$O(\epsilon)$, the variation of the hyperkähler structure on K3 fibres is
negligible. We will assume the existence of a $\mathbb{Z}_{2}$-symmetry acting
as a non-symplectic involution on the K3 surface

$\rho^{*}\tilde{\omega}_{D}=\tilde{\omega}_{D},\quad\rho^{*}\Omega_{D}=-\Omega_{D}.$

There is a classification for K3 surfaces with non-symplectic involutions in
terms of K3 lattice theory (_cf._ [16, section 3] for a short survey). In
particular, the fixed locus of $\rho$ is a disjoint union of smooth
holomorphic curves $C_{i}$ in $D$. This $\rho$ extends to an action

$\rho^{*}\tilde{\omega}_{D}=\tilde{\omega}_{D},\quad\rho^{*}h=h,\quad\rho^{*}\Omega_{D}=-\Omega_{D},\quad\rho^{*}\vartheta=-\vartheta,\quad\rho^{*}\tau=\tau,\quad\rho^{*}\mu=-\mu.$

In terms of the Apostolov-Salamon $SU(3)$-structure,

$\rho^{*}\omega=-\omega,\quad\rho^{*}\Omega=\overline{\Omega}.$

This behaves like an antiholomorphic involution. Similar to section 4.3, we
modify the distributional equation of section 4.2 to

$(\partial^{2}_{\mu}\tilde{\omega}+\epsilon^{-1}d_{D}d_{D}^{c}h)\wedge
d\mu=-d(d\vartheta)=-2\pi\sum_{i}(2m_{i}-4)\delta_{C_{i}},$

so the linear correction term is modified into

$\begin{cases}\psi\sim\frac{\sqrt{-1}(2m_{i}-4)}{4r}\eta\wedge\bar{\eta},\\\
\mathfrak{h}\sim\frac{2m_{i}-4}{2r},\\\ r\sim\sqrt{\epsilon
h_{0}^{-1}\text{dist}_{\tilde{\omega}_{D}}(\cdot,C_{i})^{2}+|\mu|^{2}}.\end{cases}$

The $\mathbb{Z}_{2}$ symmetry constrains $[d\vartheta]$. To see this, notice
wall crossing predicts that as $\mu$ increases across zero,
$\frac{1}{2\pi}[d\vartheta]$ increases by $\sum_{i}(2m_{i}-4)[C_{i}]\in
H^{2}(K3)$. Compatibility with $\rho^{*}\vartheta=-\vartheta$ then implies

$\frac{1}{2\pi}[d\vartheta]=\begin{cases}\sum(2-m_{i})[C_{i}],\quad\mu<0,\\\
\sum(m_{i}-2)[C_{i}],\quad\mu>0.\end{cases}$ (20)

Morever, we have the cohomological constraints

$\rho^{*}[\omega_{y}]=-[\omega_{y}],\quad\rho^{*}\frac{\partial
H}{\partial\tau}=-\frac{\partial H}{\partial\tau},\quad\rho^{*}\frac{\partial
H}{\partial\mu}=\frac{\partial H}{\partial\mu},$ (21)

where $\frac{\partial}{\partial\tau}$ is tangent to $l$ and
$\frac{\partial}{\partial\mu}$ is normal to $l$ with respect to the induced
metric $g$ on $S$.

We now take the $\mathbb{Z}_{2}$ quotient of the construction. The structure
transverse to $C_{i}$ now matches precisely with the asymptote at infinity of
$D_{m_{i}}$ type ALF gravitational instantons. This suggests that we should
glue in a family of $D_{m_{i}}$-type ALF gravitational instantons fibred over
$C_{i}$ in the Calabi-Yau case (or the 3-manifold swept out by $C_{i}$ over
$l$ in the $G_{2}$-case). For $m=0$, the $D_{0}$-type ALF gravitational
instanton is the Atiyah-Hitchin manifold, which has no deformation except for
scaling, so we expect that the matching condition essentially determines this
ALF fibration up to small error. For $m>0$, the $D_{m}$ instantons have
internal moduli, so we expect extra data to be necessary in specifying the ALF
fibration, in the same spirit as in section 4.3.

######  Remark 4.3.

We saw in section 4.3 that zeros of harmonic 1-forms typically lead to
difficult singularity problems. There is one case where we can hope the
$D$-type is better behaved: the $D_{1}$-type corresponds to deformations of
the double cover of the Atiyah-Hitchin manifold. In particular, the zero
harmonic 1-form corresponds to taking a fibration by the the double cover of
Atiyah-Hitchin manifold over the 3-dimensional discriminant locus, without
creating any singularity.

An important conceptual point is that unlike the $A_{k}$-type gravitational
instantons, the $D_{m}$-type can break the global $S^{1}$-symmetry, as must be
the case for nontrivial compact $G_{2}$-manifolds.

Another conceptual feature is the effect of $\mathbb{Z}_{2}$-quotient. While
the quotient results in codimension 4 orbifold singularity, this is resolved
when we glue in the ALF fibration, so $M$ is still smooth. (In some cases such
as $m_{i}=2$, we can also choose to not glue in the ALF fibration, which will
result in codimension 4 orbifold singularity on $M$). Nevertheless, the
$\mathbb{Z}_{2}$-quotient identifies $\mu$ with $-\mu$, so instead of two
components of $S$ separated by $l$, there is only one component with boundary
$l$. Even though we ultimately want to compactify $M$ into a closed manifold,
we should allow $S$ to be a manifold with boundary instead.

######  Remark 4.4.

It may be asked what happens when two boundary curves $l$ intersect. One
possible guess is to introduce corners to the Riemann surface with boundary,
but the $G_{2}$-geometry seems unclear in general.

###  4.5 Tian-Yau region

There is yet another mechanism to break the global $S^{1}$-symmetry, again
coming from the dimensional reduction to Calabi-Yau 3-folds. This plays a
prominent role in [22] to describe the end regions of certain degenerating
Calabi-Yau metrics. More backgrounds can be found in [12, section 3].

Let $Y$ be an $n$-dimensional Fano manifold, $D$ a smooth anticanonical
divisor in $Y$, and denote $Z=Y\setminus D$. By adjuntion $D$ is itself a
Calabi-Yau manifold. Fix a defining section $S$ of $D$, and view $S^{-1}$ as a
holomorphic $n$-form $\Omega_{Z}$ on $Z$ with a simple pole along $D$, whose
Poincaré residue defines a holomorphic volume form $\Omega_{D}$ on $D$. Under
suitable normalisation

$\frac{n!}{2^{n}}\int_{D}\sqrt{-1}^{(n-1)^{2}}\Omega_{D}\wedge\overline{\Omega}_{D}=(2\pi
c_{1}(K_{Y}^{-1}|_{D}))^{n-1}.$

By Yau’s theorem, there is a unique Calabi-Yau metric $\omega_{D}$ in the
class $2\pi c_{1}(K_{Y}^{-1}|_{D})$. The form $-\sqrt{-1}\omega_{D}$ is the
curvature form of a Hermitian metric $\left\lVert\cdot\right\rVert$ on
$K_{Y}^{-1}|_{D}$; we fix a smooth extension of the Hermitian metric to $Z$,
preserving the positivity of the curvature.

The form

$\omega_{Z}=\frac{n}{n+1}\sqrt{-1}\partial\bar{\partial}(-\log\left\lVert
S\right\rVert^{2})^{\frac{n+1}{n}}$

is Kähler on a neighbourhood of infinity of $Z$. By a noncompact version of
the Calabi conjecture, one finds the Tian-Yau metric
$\omega_{TY}=\omega_{Z}+\sqrt{-1}\partial\bar{\partial}\phi$ on $Z$ asymptotic
to $\omega_{Z}$ at infinity, satisfying the Calabi-Yau condition

$\omega_{TY}^{n}=\frac{n!}{2^{n}}\sqrt{-1}^{n^{2}}\Omega_{Z}\wedge\overline{\Omega}_{Z}.$

The asymptotic geometry is identified as the Calabi ansatz. Complex
geometrically, the neighbourhood of $D$ inside $Y$ is well approximated by the
normal bundle of $D$, namely the total space of $K_{Y}^{-1}|_{D}\to D$. The
holomorphic volume form $\Omega_{Z}$ is approximately
$d\log\xi\wedge\Omega_{D}$, where $\xi$ denotes a local holomorphic variable
on the fibres of $K_{Y}^{-1}|_{D}\to D$. The asymptotic metric $\omega_{Z}$ is
approximated by

$\omega_{Calabi}=\frac{n}{n+1}\sqrt{-1}\partial\bar{\partial}(-\log\left\lVert\xi\right\rVert^{2})^{\frac{n+1}{n}}$

using the Hermitian metric on $K_{Y}^{-1}|_{D}$. This setup is invariant under
the $S^{1}$-action on the fibres of the line bundle. The moment map is

$\mu=(-\log\left\lVert\xi\right\rVert^{2})^{1/n},$

and the Calabi ansatz can be written in the framework of section 2.6 as

$\begin{cases}\omega_{Calabi}=\vartheta\wedge d\mu+\mu\omega_{D},\\\
\Omega_{Calabi}=d\log\xi\wedge\Omega_{D}=-(\frac{n}{2}(-\log\left\lVert\xi\right\rVert)^{(n-1)/n}d\mu-\sqrt{-1}\vartheta)\wedge\Omega_{D},\\\
d\vartheta=-\omega_{D},\\\
h_{Calabi}=\frac{n}{2}(-\log\left\lVert\xi\right\rVert)^{(n-1)/n}=\frac{n}{2}\mu^{n-1}.\end{cases}$

For our purpose the case of interest is $n=3$ and $D=K3$. Then the above
description agrees with Example 2.9, up to suitable scaling factors. The
significance is that at least in the dimensionally reduced setting, up to
rescaling the asymptote of the Tian-Yau metric matches the adiabatic ansatz
with

$H=[\text{Re}\Omega_{D}]\tau+\frac{1}{2}[d\vartheta]\mu^{2}$

(compare the end of section 2.6), so that by gluing in the Tian-Yau metric we
can desingularize the $\mu=0$ boundary locus of the adiabatic ansatz where
$h=0$. The characteristic length scale of the Tian-Yau bubble region is
$O(\epsilon^{2/3})$. Thus we can at least hope that the Tian-Yau regions
contribute to another type of boundary for $S$. More evidence is needed to
test this gluing proposal, since the weighted maximal submanifold equation may
have poor regularity near the $h=0$ boundary, caused by the degeneracy of
ellipticity.

###  4.6 Global topology

We now discuss some topology in order to set up a global version of the
weighted maximal submanifold equation. The base $S$ will be a Riemann surface
with boundary, with isolated interior points $S_{sing}$ over which the K3
fibres are supposed to develop Lefschetz singularities. The complement of
these points in $S$ is denoted $S_{sm}$, where the K3 fibres will be smooth.
The interior of $S$ can also contain a number of walls, which are unions of
disjoint circles, not touching the boundary or $S_{sing}$. (These requirements
are meant to be tentative, presumably relaxable if we have better
understanding of local metric models.) The precise location of the walls
should not be fixed a priori, and behave instead as in a free boundary
problem. The topological setup will come directly from abstracting the
cohomological/lattice theoretic aspects of the Lefschetz fibration, wall
crossing, and the orientifold boundary behaviour. It is conceivable that other
local mechanisms could be incorporated into this formalism.

First we consider the smooth K3 fibration $(M/S^{1})_{sm}\to S_{sm}\subset S$,
and seek cohomological descriptions of the adiabatic solution to the
Apostolov-Salamon equation. The fibration induces an $H^{2}(K3,\mathbb{Z})$
local system $\Gamma$ over the smooth locus $S_{sm}$. The symplectic form
$\omega$ defines a constant section $[\omega_{y}]$ of
$\Gamma_{\mathbb{R}}=\Gamma\otimes_{\mathbb{Z}}\mathbb{R}$.

On a local chart $U_{\alpha}$, the information of $h^{1/4}\text{Im}\Omega$ is
encoded by a polarized positive section $H_{\alpha}:U_{\alpha}\to H^{2}(K3)$,
determined up to constant. On the overlaps of two charts $U_{\alpha}$ and
$U_{\beta}$, the difference $H_{\alpha}-H_{\beta}$ is locally constant. The
collection $\\{H_{\alpha}-H_{\beta}\\}$ defines a Cech cocycle in
$H^{1}(S_{sm},H^{2}(K3))$. More invariantly, the Leray spectral sequence gives

$H^{3}((M/S^{1})_{sm})\simeq H^{1}(S_{sm},\Gamma_{\mathbb{R}})$

and the class of $h^{1/4}\text{Im}\Omega$ is mapped to a class in
$H^{1}(S_{sm},\Gamma_{\mathbb{R}})$. Via the cocycle, we can regard the
collection $\\{H_{\alpha}\\}$ as defining a section $H$ of an affine bundle
$\tilde{\Gamma}_{\mathbb{R}}$, whose associated vector bundle is
$\Gamma_{\mathbb{R}}$.

The class $\frac{1}{2\pi}[d\vartheta]$ is viewed as a discontinuous section of
$\Gamma$ with precise jumping behaviours across the walls. The boundary value
of $[d\vartheta]$ is prescribed in (20). More precisely, we equip each
boundary circle of $S$ with an involution $\rho^{*}$ acting on the fibres of
$\Gamma|_{S^{1}}$, which is the cohomological version of non-symplectic
involutions. This $\rho^{*}$ determines the class of the components $C_{i}$ of
the fixed point locus of the non-symplectic involution. With additional choice
of integers $m_{i}$ we prescribe $[d\vartheta]$ according to the formula (20).
On the complement of the walls, the class $[d\vartheta]$ is locally constant.
Each (oriented) wall is associated with a class $\sigma\in\Gamma$ orthogonal
to $[\omega_{y}]$, and $\frac{1}{2\pi}[d\vartheta]$ jumps by $\sigma$ across
the wall.

In each local chart $H_{\alpha}$ is a polarized positive section, namely

$H_{\alpha}\cdot[\omega_{y}]=0,\quad
h=2H_{\alpha}\cdot[d\vartheta]+\text{const}(\alpha)>0,\quad\tilde{g}=\frac{\partial
H_{\alpha}}{\partial y_{i}}\cdot\frac{\partial H_{\alpha}}{\partial
y_{j}}dy_{i}dy_{j}\text{ Riemannian}.$

The constants here arise from the affine ambiguity of $H_{\alpha}$. The
gradient of $H_{\alpha}$ should be continuous across the wall, and $h$ should
be independent of the charts as it is related to the circle length. To be
compatible, we must require along the wall

$H_{\alpha}\cdot\sigma=\text{const}(\alpha).$

The precise position of the walls are thus tied up with the unknown $H$, and
must be solved alongside $H$ rather than prescribed a priori. Furthermore,
over $S_{sm}$ there are no singular K3 fibres by assumption, so
cohomologically we require there is no excess $(-2)$-class for $H$ over
$S_{sm}$.

We now turn to the boundary of $S$. For simplicity we only consider
orientifold type boundary described in section 4.4. The boundary condition is
specified by (21), namely we constrain the tangential derivative of $H$ to the
$-1$ eigenspace of $\rho^{*}$, and the normal derivative of $H$ to the $+1$
eigenspace of $\rho^{*}$, where the normal direction is with respect to the
metric induced from immersion in $H^{2}(K3)$. In particular, in some preferred
local chart $H$ restricted to the boundary lands in the $-1$ eigenspace. These
are a mixture of Dirichlet and Neumann boundary conditions of complementary
dimensions.

Next we deal with the singular K3 fibres. We shall assume that the K3
fibration is Lefschetz. To incorporate the Picard-Lefschetz monodromy, we
consider $S$ as an orbifold locally modelled on
$\mathbb{C}/\mathbb{Z}_{2}=\mathbb{C}/\iota$ near $p\in S_{sing}$, and require
that $\tilde{\Gamma}_{\mathbb{R}}$ is a flat orbifold affine bundle. The local
sections of $\tilde{\Gamma}_{\mathbb{R}}$ in an open ball
$U_{\alpha}=\tilde{U}_{\alpha}/\iota$ around $p$ are concretely written in
some chart as maps $\tilde{H}_{\alpha}:\tilde{U}_{\alpha}\to H^{2}(K3)$
satisfying an equivariance condition

$\tilde{H}_{\alpha}(\iota(z))=\tau_{\alpha}(\tilde{H}_{\alpha}(z)),\quad\tau_{\alpha}(v)=v+(\delta\cdot
v)\delta,\quad z\in\tilde{U}_{\alpha},$

where $\delta$ is the $(-2)$-class of the vanishing sphere, orthogonal to
$[\omega_{y}]$. The class $[d\vartheta]$ is required to be monodromy
invariant. To incorporate Lefschetz fibrations into the notion of polarized
positive sections, we require that around $p$, up to the ambiguity of an
affine constant,

$\tilde{H}_{\alpha}(z)=v_{1}\text{Re}(z^{2})+v_{2}\text{Im}(z^{2})+\text{Im}(bz^{3})\delta+O(z^{4}),$

where $v_{1},v_{2}\in H^{2}(K3)$ satisfy

$v_{i}\perp[\omega_{y}],\quad v_{i}\perp\delta,\quad(v_{i}\cdot
v_{j})=\lambda\delta_{ij},\quad\lambda>0,$

and $b\in\mathbb{C}$ is nonzero, corresponding to the nondegeneracy of the
Lefschetz singularity. The function $h$ is required to be positive and at
least $C^{1}$.

The problem to solve the global weighted maximal submanifold equation means to
find the configuration of the walls and boundaries, such that the PDE (9) is
satisfied on the complement of the walls and boundaries, the mixed boundary
condition is satisfied on $\partial S$, the jumping behaviour across the walls
and the local behaviour near the Lefschetz singularities are as prescribed,
and there are no excess $(-2)$-classes so that no other singular K3 fibres can
appear.

The topological consistency of our setup has an interesting consequence
closely related to charge conservation (_cf._ section 2.2). Recall that the
walls carry $\Gamma$-valued sections $\sigma$ prescribing the jumping of
$[d\vartheta]$, so define $\Gamma$-valued 1-clycles. Similarly with the
boundary components of $S$, carrying the $\Gamma$-valued 1-cycles coming from
(20). We claim the sum of all these $\Gamma$-valued 1-cycles equals zero in
homology. This is because $[d\vartheta]$ is locally constant in each domain
$S_{i}$ bounded by walls and boundaries, and when we take the boundary of
$\sum_{i}[d\vartheta]S_{i}$, the wall contributions cancel in pairs to yield
the desired linear relation.

###  4.7 Variational viewpoint

The local version of the weighted maximal submanifold equation has the
interpretation as the critical point condition of a weighted area functional.
We now fix all the global topological data, and compute formally the first
variation of the weighted area functional

$\mathcal{A}_{w}(H)=\int_{S}h^{1/2}\sqrt{\det\tilde{g}}dy_{1}\wedge dy_{2},$

allowing for the deformation of the walls and boundaries subject to the
various boundary conditions. The local discussion in section 2.4 already shows
that the weighted maximal submanifold (9) holds for critical points, so the
task here is to examine boundary terms.

In each chamber $S_{i}$ bounded by some walls and boundaries, given an
arbitrary variation $f=\delta H$ which is a local section of
$\Gamma_{\mathbb{R}}$, and assuming (9), the contribution to
$\delta\mathcal{A}_{w}$ is the boundary line integral

$\int_{\partial S_{i}}h^{1/2}\frac{\partial H}{\partial\nu}\cdot fdl,$

where $\frac{\partial H}{\partial\nu}$ denotes the normal derivative with
respect to the induced metric, and $dl$ denotes the induced line element on
$\partial S_{i}$. Notice the affine ambiguity of $H$ is eliminated in taking
the derivative. The Lefschetz singularities occur over isolated points, and
give no contribution to these integrals.

The boundaries of $S_{i}$ can be the walls or the boundaries of $S$. Since
$f,h$ and the gradient of $H$ are required to be continuous across the walls,
the wall contributions cancel in pairs. As for the boundary of $S$, our mixed
boundary condition requires the normal derivative of $H$ to be in the $+1$
eigenspace of the cohomological involution $\rho^{*}$, and in some preferred
chart the boundary value of $H$ lands in the $-1$ eigenspace. For the first
variation to be compatible with the Dirichlet part of the boundary conditions,
the boundary value of $f$ needs to land in the $-1$ eigenspace. But the two
eigenspaces of $\rho^{*}$ are orthogonal, so $f$ is orthogonal to
$\frac{\partial H}{\partial\nu}$ on the boundary of $S$, hence

$\sum\int_{\partial S_{i}}h^{1/2}\frac{\partial H}{\partial\nu}\cdot fdl=0.$

This shows the boundary conditions are compatible with the critical point
interpretation.

###  4.8 Open questions

In the program to ultimately construct new compact examples of
$G_{2}$-manifolds, there are 3 types of questions to be addressed:

  * •

Find consistent topological data, involving the Riemann surface, the
topological configuration of walls, boundaries and singularities, the affine
bundle $\tilde{\Gamma}_{\mathbb{R}}$, the lattice theoretic data on the
boundary and the walls, the polarization class $[\omega_{y}]$, and the first
Chern class $[d\vartheta]$ in each chamber. These data should allow for the
existence of a global polarized positive section $H$. This problem has a
flavour analogous to K3 matching problems involved in twisted connected sum
constructions.

  * •

Solve a free boundary type problem to find a solution $H$ to the global
weighted maximal submanifold equation, along with the position of the walls.
There should be no excess $(-2)$-classes, so the only singular K3 fibres are
those nodal fibres of the Lefschetz fibration. This step deals with the
nonlinearity of the Apostolov-Salamon equation, and the variational
interpretation is likely useful. One also needs to develop a good regularity
theory for the weighted maximal submanifold equation; most of this difficulty
comes from the walls and boundaries, where the jumping behaviour of
$[d\vartheta]$ indicates that $H$ cannot be smooth.

  * •

Perform a gluing construction to construct the $G_{2}$-metric, taking into
account the various geometric ingredients. As discussed, this combines
features of many known constructions, but there are still quite substantial
challenges. For instance, one needs to develop a good deformation theory for
the local $G_{2}$-metrics on the ALF fibrations.

Some other natural questions include:

  * •