Chapter 4 The Negative Vertex

4.1. First order approximate metric

We plan to construct an approximate Calabi-Yau metric using the generalised Gibbons-Hawking ansatz, on a singular $S^{1}$ -bundle $M^{-}$ over an open neighbourhood of the origin inside the real 5-dimensional base $\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}\times\mathbb{R}_{\mu}$ , whose discriminant locus is

S=\{z_{1}+z_{2}=1\}\subset\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}\times\{0\}\subset\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}\times\mathbb{R}_{\mu}.

Let

\eta_{1}=\frac{1}{2\pi\sqrt{-1}}\log z_{1},\quad\eta_{2}=\frac{1}{2\pi\sqrt{-1}}\log z_{2}

be complex variables with period 1, and denote $\eta_{p}=x_{p}+\sqrt{-1}y_{p}$ for $p=1,2$ . The topological situation is described in Section 1.1.5 and the expected complex structure is discussed in Section 1.1.6. A more historical view can be found in Section 1.1.4.

The basic heuristic idea is again to perturb the constant solution (cf. Example 1.6) while incorporating the topology. The information of the constant solution is contained in the base metric

(4.1)

g_{a}=\text{Re}(a_{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q})+A|d\mu|^{2},

where $(a_{p\bar{q}})$ is a Hermitian $2\times 2$ matrix referred to as coupling constants, with determinant $A=\det a$ , and $(a^{p\bar{q}})$ is the transposed inverse matrix such that $a_{j\bar{q}}a^{p\bar{q}}=\delta_{j}^{p}$ . The associated volume measure is

d\text{Vol}_{a}=A^{3/2}dx_{1}\wedge dy_{1}\wedge dx_{2}\wedge dy_{2}\wedge d\mu.

In order for the perturbative way of thinking to be effective, we impose

(4.2)

C^{-1}A^{1/2}\delta_{p\bar{q}}\leq a_{p\bar{q}}\leq CA^{1/2}\delta_{p\bar{q}},\quad A\gg 1.

In this Chapter all constants in estimates depend on $a_{p\bar{q}}$ only through the above scale-invariant uniform ellipticity constant.

Notation.

The $g_{a}$ -distance to the origin is $|(\eta_{1},\eta_{2},\mu)|_{a}=\sqrt{a_{p\bar{q}}\eta_{p}\bar{\eta}_{q}+A\mu^{2}}$ . A variant

\varrho=|(y_{1},y_{2},\mu)|_{a}^{\prime}=(\frac{A}{\mathbb{A}}a_{p\bar{q}}y_{p}y_{q}+A\mu^{2})^{1/2},\quad\mathbb{A}=A+|\text{Im}(a_{1\bar{2}})|^{2}.

stands for the distance function for the Euclidean metric $g_{a}^{\prime}$ on $\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}_{y}$

(4.3)

\begin{split}g_{a}^{\prime}=\frac{A}{\mathbb{A}}(a_{1\bar{1}}dy_{1}^{2}+2\text{Re}(a_{1\bar{2}})dy_{1}dy_{2}+a_{2\bar{2}}dy_{2}^{2})+A|d\mu|^{2}.\end{split}

Let $S$ is $R=\text{dist}_{g_{a}}(\cdot,S)$ . The parameter $R+A^{-1/2}$ is relevant for regularity scales.

Now in terms of the local potential $\Phi$ the Calabi-Yau condition (1.9) reads

\det(-4\frac{\partial^{2}\Phi}{\partial\eta_{p}\partial\bar{\eta}_{q}})=\frac{\partial^{2}\Phi}{\partial\mu\partial\mu},

whose linearised equation at the constant solution is the Laplace equation

\Delta_{a}\phi=A^{-1}\frac{\partial^{2}\phi}{\partial\mu\partial\mu}+4a^{p\bar{q}}\frac{\partial^{2}\phi}{\partial\eta_{p}\partial\bar{\eta}_{q}}=0.

Here $\Delta_{a}$ is unsurprisingly the Laplacian of $g_{a}$ . This suggests that at least away from the discriminant locus, the first order correction to $V$ and $W^{p\bar{q}}$ from the constant solution

(4.4)

v=\frac{\partial^{2}\phi}{\partial\mu\partial\mu},\quad w^{p\bar{q}}=-4\frac{\partial^{2}\phi}{\partial\eta_{p}\partial\bar{\eta}_{q}}

ought to be given by $\Delta_{a}$ -harmonic functions,

(4.5)

\Delta_{a}w^{p\bar{q}}=0,\quad\Delta_{a}v=0,\quad v=Aa^{p\bar{q}}w^{p\bar{q}}.

To incorporate the topology we recall the distributional equation (1.18). Since $v$ and $w^{p\bar{q}}$ are linearisations, it makes sense to require the equation on currents

(4.6)

-\frac{\sqrt{-1}}{4\pi}\left(\frac{\partial^{2}w^{p\bar{q}}}{\partial\mu\partial\mu}+4\frac{\partial^{2}v}{\partial\eta_{p}\partial\bar{\eta}_{q}}\right)d\mu\wedge d\eta_{p}\wedge d\bar{\eta}_{q}=S.

The task is to find a compatible solution to (4.4)(4.5)(4.6). As in the last two Chapters, the functions $v$ and $w^{p\bar{q}}$ are global quantities while $\phi$ is only locally defined. The existence of the local potential $\phi$ in (4.4) should be read as imposing some integrability on $v$ and $w^{p\bar{q}}$ (cf. (1.10)(1.11)).

Remark 4.1.

(Motivational Discussion on singularities) We denote

f_{S}=1-z_{1}-z_{2}=1-e^{2\pi i\eta_{1}}-e^{2\pi i\eta_{2}}

and write the 3-current $S$ as

S=\delta(f_{S})\frac{\sqrt{-1}}{4\pi^{2}}df_{S}\wedge d\bar{f}_{S}\wedge d\mu,

which defines a generalised function $\delta(f_{S})$ satisfying the measures identities:

\int\frac{\sqrt{-1}}{4\pi^{2}}d\eta_{p}\wedge d\bar{\eta}_{q}\wedge\delta(f_{S})\wedge df_{S}\wedge d\bar{f}_{S}\wedge d\mu=\int_{S}d\eta_{p}\wedge d\bar{\eta}_{q},

where the notation $\int_{S}d\eta_{p}\wedge d\bar{\eta}_{q}$ is the shorthand for the complex measure $f\mapsto\int_{S}fd\eta_{p}\wedge d\bar{\eta}_{q}$ , and similarly for the LHS. Now $df_{S}=-2\pi\sqrt{-1}(z_{1}d\eta_{1}+z_{2}d\eta_{2})$ , so

\begin{cases}-\int_{S}|z_{2}|^{2}\delta(f_{S})d\eta_{1}\wedge d\bar{\eta}_{1}\wedge d\eta_{2}\wedge d\bar{\eta}_{2}\wedge d\mu=\int_{S}\sqrt{-1}d\eta_{1}\wedge d\bar{\eta}_{1},\\ -\int_{S}|z_{1}|^{2}\delta(f_{S})d\eta_{1}\wedge d\bar{\eta}_{1}\wedge d\eta_{2}\wedge d\bar{\eta}_{2}\wedge d\mu=\int_{S}\sqrt{-1}d\eta_{2}\wedge d\bar{\eta}_{2},\\ \int_{S}\bar{z}_{1}z_{2}\delta(f_{S})d\eta_{1}\wedge d\bar{\eta}_{1}\wedge d\eta_{2}\wedge d\bar{\eta}_{2}\wedge d\mu=\int_{S}\sqrt{-1}d\eta_{1}\wedge d\bar{\eta}_{2},\\ \int_{S}\bar{z}_{2}z_{1}\delta(f_{S})d\eta_{1}\wedge d\bar{\eta}_{1}\wedge d\eta_{2}\wedge d\bar{\eta}_{2}\wedge d\mu=\int_{S}\sqrt{-1}d\eta_{2}\wedge d\bar{\eta}_{1}.\\ \end{cases}

The distributional equation (4.6) is written in components as

-\frac{1}{4\pi}\left(\frac{\partial^{2}w^{p\bar{q}}}{\partial\mu\partial\mu}+4\frac{\partial^{2}v}{\partial\eta_{p}\partial\bar{\eta}_{q}}\right)=\delta(f_{S})z_{p}\bar{z}_{q}.

Multiplying these equations by $a^{p\bar{q}}$ and summing up, we obtain

(4.7)

-\frac{1}{4\pi}\Delta_{a}v=\delta(f_{S})a^{p\bar{q}}z_{p}\bar{z}_{q},

or equivalently the measure equality

(\Delta_{a}v)d\text{Vol}_{a}=-\int_{S}\pi\sqrt{-1}A^{1/2}a_{p\bar{q}}d\eta_{p}\wedge d\bar{\eta}_{q}=-\int_{S}2\pi A^{1/2}d\mathcal{A},

where $d\mathcal{A}=\frac{\sqrt{-1}}{2}a_{p\bar{q}}d\eta_{p}\wedge d\bar{\eta}_{q}$ is the natural area form on $S$ . A natural guess for $w^{p\bar{q}}$ is then

(4.8)

-\frac{1}{4\pi}\Delta_{a}w^{p\bar{q}}=A^{-1}\delta(f_{S})z_{p}\bar{z}_{q},

or equivalently

(\Delta_{a}w^{p\bar{q}})d\text{Vol}_{a}=-\int_{S}2\pi\frac{A^{-1/2}z_{p}\bar{z}_{q}}{a^{i\bar{j}}z_{i}\bar{z}_{j}}d\mathcal{A},

where summation convention is used. The singularity around $S$ to leading order looks like (cf. Section 4.4 below)

v\sim\frac{A^{1/2}}{2R},\quad w^{p\bar{q}}\sim\frac{A^{-1/2}z_{p}\bar{z}_{q}}{2Ra^{i\bar{j}}z_{i}\bar{z}_{j}},\quad R\sim(\frac{|f_{S}|^{2}}{4\pi^{2}a^{i\bar{j}}z_{i}\bar{z}_{j}}+A\mu^{2})^{1/2},

which is compatible with the singularity in the distributional equation (4.6).

Now we move on to a more formal construction. The main idea is to write down the solution via a periodic version of Green’s representation. The series

(4.9)

\gamma(\eta_{1},\eta_{2},\mu)=-\frac{1}{8\pi^{2}}\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\frac{1}{|(\eta_{1}+n_{1},\eta_{2}+n_{2},\mu)|_{a}^{3}},

converges absolutely away from $\mathbb{Z}^{2}\times\{0\}\subset\mathbb{C}_{\eta_{1}}\times\mathbb{C}_{\eta_{2}}\times\mathbb{R}_{\mu}$ and is $\mathbb{Z}^{2}$ -periodic, so descends to a function on $\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}\times\mathbb{R}_{\mu}$ which is the periodic Newtonian potential. We shall extract the asymptote for $\gamma(\eta_{1},\eta_{2},\mu)$ :

Lemma 4.1.

For $\varrho\gtrsim A^{1/4}$ , we have

|\gamma(\eta_{1},\eta_{2},\mu)+\frac{1}{4\pi\varrho\sqrt{\mathbb{A}}}|\leq C\varrho^{-3}.

Proof.

We consider the closely related integral

\bar{\gamma}(y_{1},y_{2},\mu)=-\frac{1}{8\pi^{2}}\int\frac{1}{(a_{p\bar{q}}\eta_{p}\bar{\eta}_{q}+A\mu^{2})^{3/2}}dx_{1}dx_{2}.

After substituting the variables

\begin{cases}x_{1}^{\prime}=x_{1}-\frac{\text{Im}(a_{2\bar{1}})}{a_{1\bar{1}}}y_{2}+\frac{\text{Re}(a_{2\bar{1}})}{a_{1\bar{1}}}x_{2},\\ x_{2}^{\prime}=x_{2}+\frac{\text{Im}(a_{2\bar{1}})\text{Re}(a_{2\bar{1}})}{\mathbb{A}}y_{2}+\frac{a_{1\bar{1}}\text{Im}(a_{2\bar{1}})}{\mathbb{A}}y_{1},\end{cases}

we complete the square

a_{p\bar{q}}\eta_{p}\bar{\eta}_{q}+A\mu^{2}=a_{1\bar{1}}x_{1}^{\prime 2}+\frac{\mathbb{A}}{a_{1\bar{1}}}x_{2}^{\prime 2}+|(y_{1},y_{2},\mu)|_{a}^{\prime 2}=a_{1\bar{1}}x_{1}^{\prime 2}+\frac{\mathbb{A}}{a_{1\bar{1}}}x_{2}^{\prime 2}+\varrho^{2}.

This allows us to evaluate using polar coordinates

\bar{\gamma}=-\frac{1}{4\pi\varrho\sqrt{\mathbb{A}}}.

For fixed $y_{1},y_{2},\mu$ , we can compare the integral $\bar{\gamma}$ with the series $\gamma$ , by estimating the difference using the mean value inequality

\begin{split}&\frac{1}{|(\eta_{1},\eta_{2},\mu)|_{a}^{3}}-\int_{[x_{1}-\frac{1}{2},x_{1}+\frac{1}{2}]\times[x_{2}-\frac{1}{2},x_{2}+\frac{1}{2}]}\frac{1}{|(s_{1}+\sqrt{-1}y_{1},s_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{3}}ds_{1}ds_{2}\\ \leq&\frac{CA^{1/2}}{|(\eta_{1},\eta_{2},\mu)|_{a}^{5}}.\end{split}

Summing over all square regions, and applying Cauchy integral test,

\begin{split}|\gamma-\bar{\gamma}|&\leq CA^{1/2}\sum_{n,m}\frac{1}{|(\eta_{1}+n,\eta_{2}+m,\mu)|_{a}^{5}}\\ &\leq CA^{1/2}\int\frac{1}{|(s_{1}+\sqrt{-1}y_{1},s_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{5}}ds_{1}ds_{2}\\ &\leq C\varrho^{-3},\end{split}

as required. ∎

Lemma 4.2.

For $\varrho\lesssim A^{1/4}$ and $|x_{1}|,|x_{2}|\leq\frac{1}{2}$ we have

|\gamma(\eta_{1},\eta_{2},\mu)+\frac{1}{8\pi^{2}|(\eta_{1},\eta_{2},\mu)|_{a}^{3}}|\leq CA^{-3/4}.

Proof.

Modify the above proof to control the series for $(n_{1},n_{2})\in\mathbb{Z}^{2}\setminus\{0\}$ . ∎

Before proceeding further we recall that topologically $S$ is a thrice punctured 2-sphere. The 3 punctures correspond to 3 ends of $S$ :

\begin{cases}y_{2}>1,\quad\eta_{1}=\frac{1}{2\pi i}\log(1-e^{2\pi i\eta_{2}}),\\ y_{1}>1,\quad\eta_{2}=\frac{1}{2\pi i}\log(1-e^{2\pi i\eta_{1}}),\\ y_{1}<-1,\quad y_{2}<-1,\quad\eta_{2}-\eta_{1}=\frac{1}{2\pi i}\log(-1+e^{-2\pi i\eta_{1}}).\end{cases}

At infinity these are respectively asymptotic to $\mathfrak{D}_{1}\times S^{1}$ , $\mathfrak{D}_{2}\times S^{1}$ , $\mathfrak{D}_{3}\times S^{1}$ where

\mathfrak{D}_{1}=\{y_{1}=0,y_{2}>0,\mu=0\},\quad\mathfrak{D}_{2}=\{y_{2}=0,y_{1}>0,\mu=0\},\quad\mathfrak{D}_{3}=\{y_{1}=y_{2}<0,\mu=0\}.

The image of $S$ under the log map $\mathbb{C}^{2}_{z_{1},z_{2}}\to\mathbb{R}^{2}_{y_{1},y_{2}}$ (called the ‘amoeba’) is

(4.10)

\text{Image}(S)=\{e^{-2\pi y_{1}}+e^{-2\pi y_{2}}\geq 1,e^{-2\pi y_{1}}+1\geq e^{-2\pi y_{2}},e^{-2\pi y_{2}}+1\geq e^{-2\pi y_{1}}\},

which is a thickening of the trivalent graph $\mathfrak{D}=\mathfrak{D}_{1}\cup\mathfrak{D}_{2}\cup\mathfrak{D}_{3}\cup\{0\}.$ This is the simplest case of a general picture for amoebas of algebraic varieties [31].

We can now make the following definitions, involving a cutoff and limiting procedure for logarithmically divergent integrals.

(4.11)

\begin{cases}\gamma_{1}(\eta_{1},\eta_{2},\mu)=&-\text{Re}\lim_{\Lambda\to\infty}\{\pi A^{1/2}\int_{S\cap\{y_{2}^{\prime}<\Lambda\}}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\\ &\sqrt{-1}d\eta_{2}^{\prime}\wedge(d\bar{\eta}_{2}^{\prime}-d\bar{\eta}_{1}^{\prime})+\frac{1}{2\sqrt{a_{2\bar{2}}}}\log 2\Lambda\}\\ \gamma_{2}(\eta_{1},\eta_{2},\mu)=&-\text{Re}\lim_{\Lambda\to\infty}\{\pi A^{1/2}\int_{S\cap\{y_{1}^{\prime}<\Lambda\}}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\\ &\sqrt{-1}d\eta_{1}^{\prime}\wedge(d\bar{\eta}_{1}^{\prime}-d\bar{\eta}_{2}^{\prime})+\frac{1}{2\sqrt{a_{1\bar{1}}}}\log 2\Lambda\}\\ \gamma_{3}(\eta_{1},\eta_{2},\mu)=&-\text{Re}\lim_{\Lambda\to\infty}\{\pi A^{1/2}\int_{S\cap\{y_{1}^{\prime}>-\Lambda\}}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\\ &\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}+\frac{1}{2\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}}}\log 2\Lambda\}\\ \gamma_{4}(\eta_{1},\eta_{2},\mu)=&-\text{Im}\{\pi A^{1/2}\int_{S}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}\}.\end{cases}

The desired first order corrections $v$ and $w^{p\bar{q}}$ are constructed as linear combinations:

(4.12)

\begin{cases}v=Aa^{p\bar{q}}w^{p\bar{q}},\\ w^{1\bar{1}}=\gamma_{1}+\gamma_{3},\\ w^{1\bar{2}}=-(\gamma_{3}+\sqrt{-1}\gamma_{4}),\\ w^{2\bar{1}}=-(\gamma_{3}-\sqrt{-1}\gamma_{4}),\\ w^{2\bar{2}}=\gamma_{2}+\gamma_{3}.\end{cases}

The advantage of $\gamma_{i}$ is that they only involve divergence issues at one end. This is because the measures $\text{Re}(\sqrt{-1}d\eta_{2}^{\prime}\wedge(d\bar{\eta}_{2}^{\prime}-d\bar{\eta}_{1}^{\prime}))$ etc decay exponentially along all but one end, with respect to the Lebesgue measure on the three asymptotic cylinders.

Lemma 4.3.

The limits defining $\gamma_{i}$ converge as $\Lambda\to+\infty$ .

Proof.

We focus on $\gamma_{1}$ . The 2-form $\sqrt{-1}d\eta_{2}\wedge(d\bar{\eta}_{2}-d\bar{\eta}_{1})$ on $S$ is exponentially small along the $\mathfrak{D}_{2},\mathfrak{D}_{3}$ ends, so the only divergence problem happens at infinity along the $\mathfrak{D}_{1}$ end.

Applying Lemma 4.1 allows us to replace $\gamma$ by the much simpler function

-\frac{1}{4\pi\sqrt{\mathbb{A}}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}.

The integral

\int_{S\cap\{y_{2}^{\prime}<\Lambda\}}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\sqrt{-1}d\eta_{2}^{\prime}\wedge(d\bar{\eta}_{2}^{\prime}-d\bar{\eta}_{1}^{\prime})

has the same divergence behaviour as

\int^{\Lambda}-\frac{1}{2\pi\sqrt{\mathbb{A}}}|(y_{1},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}dy_{2}^{\prime}\sim-\frac{1}{2\pi\sqrt{Aa_{2\bar{2}}}}\log\Lambda,

which is cancelled by the log term we put in the limit. ∎

By the construction of the Green representations,

Proposition 4.4.

The functions $v$ and $w^{p\bar{q}}$ satisfy the decoupled Laplace equations with distributional terms (4.7) and (4.8).

However the original linearised equations we set off to solve is an overdetermined coupled system, not just the decoupled Laplace equations. We still need to check the integrability equation (4.4) and the distributional equation (4.6).

Lemma 4.5.

The following integrability condition is satisfied globally

\frac{\partial w^{p\bar{q}}}{\partial\eta_{r}}=\frac{\partial w^{rq}}{\partial\eta_{p}},\quad\frac{\partial w^{p\bar{q}}}{\partial\bar{\eta}_{r}}=\frac{\partial w^{pr}}{\partial\bar{\eta}_{q}},\quad p,q,r=1,2.

Proof.

We consider the Laplacian

\begin{split}\Delta_{a}(\frac{\partial w^{p\bar{q}}}{\partial\eta_{r}}-\frac{\partial w^{rq}}{\partial\eta_{p}})=&\frac{\partial}{\partial\eta_{r}}\Delta_{a}w^{p\bar{q}}-\frac{\partial}{\partial\eta_{p}}\Delta_{a}w^{rq}\\ =&-4\pi A^{-1}\{\frac{\partial}{\partial\eta_{r}}(\delta(f_{S})z_{p}\bar{z}_{q})-\frac{\partial}{\partial\eta_{p}}(\delta(f_{S})z_{r}\bar{z}_{q})\}\\ =&-4\pi A^{-1}\delta(f_{S})\{\frac{\partial}{\partial\eta_{r}}(z_{p}\bar{z}_{q})-\frac{\partial}{\partial\eta_{p}}(z_{r}\bar{z}_{q})\}=0,\end{split}

where we have crucially used that $S$ is an algebraic cycle to deduce $\partial\delta(f_{S})=0$ . Thus $\frac{\partial w^{p\bar{q}}}{\partial\eta_{r}}-\frac{\partial w^{rq}}{\partial\eta_{p}}=0$ would follow from a Liouville theorem argument, by checking some a priori growth condition

|\frac{\partial w^{p\bar{q}}}{\partial\eta_{r}}|\lesssim\begin{cases}R^{-1},\quad&\varrho\gtrsim A^{1/4},\\ A^{1/4}R^{-2},\quad&\varrho\lesssim A^{1/4},\end{cases}

which is easy to derive using the techniques in the previous lemmas in this Section. The $\bar{\eta}$ derivatives can be treated similarly. ∎

Corollary 4.6.

The distributional equation (4.6) is satisfied. In component form,

-\frac{1}{4\pi}(\frac{\partial^{2}w^{p\bar{q}}}{\partial\mu\partial\mu}+4\frac{\partial^{2}v}{\partial\eta_{p}\partial\bar{\eta}_{q}})=\delta(f_{S})z_{p}\bar{z}_{q}.

Proof.

Let’s focus on $p=q=1$ . By Proposition 4.4,

\frac{1}{4\pi}\frac{\partial^{2}w^{1\bar{1}}}{\partial\mu\partial\mu}+\frac{1}{\pi}Aa^{i\bar{j}}\frac{\partial^{2}w^{1\bar{1}}}{\partial\eta_{i}\partial\bar{\eta}_{j}}=-\delta(f_{S})|z_{1}|^{2}.

But by Lemma 4.6 we have $\frac{\partial^{2}w^{i\bar{j}}}{\partial\eta_{1}\partial\bar{\eta}_{1}}=\frac{\partial^{2}w^{1\bar{1}}}{\partial\eta_{i}\partial\bar{\eta}_{j}},$ so $Aa^{i\bar{j}}\frac{\partial^{2}w^{1\bar{1}}}{\partial\eta_{i}\partial\bar{\eta}_{j}}=\frac{\partial^{2}v}{\partial\eta_{1}\partial\bar{\eta}_{1}},$ hence the claim. ∎

Lemma 4.5 and Corollary 4.6 combine to imply the local existence of the potential away from $S$ as is required in (4.4). Taking stock of our progress,

Proposition 4.7.

(First order linearised solution) The functions $v$ and $w^{p\bar{q}}$ solve the integrability condition (4.4) and the harmonicity condition (4.5) away from $S$ , and the distributional equation (4.6) globally.

Remark 4.2.

It will turn out in the next few Sections that $v$ and $w^{p\bar{q}}$ have logarithmic growth at infinity bounded away from $S$ . If we restrict to solutions to (4.4)(4.5)(4.6) with the same growth properties, then $v$ and $w^{p\bar{q}}$ are unique up to additive constants. The choices of these constants are not completely canonical, related to the philosophy that the Ooguri-Vafa type metrics are only effective descriptions admitting a certain amount of small fluctuation.

We obtain by the generalised Gibbons-Hawking construction a Kähler ansatz $(g^{(1)},\omega^{(1)},J^{(1)},\Omega^{(1)})$ associated to

(4.13)

V_{(1)}=A+v,\quad W^{p\bar{q}}_{(1)}=a_{p\bar{q}}+w^{p\bar{q}}.

A subtlety here is that the $S^{1}$ -connection $\vartheta$ can be twisted by a flat connection. This choice is parametrised by $H^{1}((\mathbb{C}^{*})^{2}\times\mathbb{R}\setminus S,\mathbb{R}/\mathbb{Z})=H^{1}((\mathbb{C}^{*})^{2}\times\mathbb{R},\mathbb{R}/\mathbb{Z})=T^{2}$ , since the codimension 3 subset $S$ inside the base does not affect the fundamental group. We sometimes suppress mentioning this choice since it does not have a strong impact on the geometry, especially because we will exclusively work with $S^{1}$ -invariant tensors, which are rarely sensitive to the flat connection. The Kähler structure is well defined away from $S$ , over a bounded region where $V_{(1)}>0$ and $W^{p\bar{q}}_{(1)}$ is positive definite; the metric is incomplete. We will specify more precisely the ambient space $M^{-}$ of the Kähler ansatz once we obtain sufficiently accurate asymptotes on $V_{(1)}$ and $W^{p\bar{q}}_{(1)}$ to check positive definiteness (cf. Corollary 4.17).

The family of ansatzs admit $S_{3}$ -discrete symmetries, generated by

\eta_{1}\leftrightarrow\eta_{2},\quad\eta_{2}\leftrightarrow\eta_{2}-\eta_{1}+\frac{1}{2},\quad\eta_{1}\leftrightarrow\eta_{1}-\eta_{2}+\frac{1}{2},\quad\mu\to\mu.

These actions on $(\mathbb{C}^{*})^{2}\times\mathbb{R}_{\mu}$ preserve $S$ , and respectively interchange $\mathfrak{D}_{1}$ with $\mathfrak{D}_{2}$ , $\mathfrak{D}_{1}$ with $\mathfrak{D}_{3}$ , and $\mathfrak{D}_{2}$ with $\mathfrak{D}_{3}$ . The induced action on coupling constants permute $a_{1\bar{1}},a_{2\bar{2}},a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}$ , and act on $\text{Im}(a_{1\bar{2}})$ by $\pm 1$ depending on the sign of the permutation.

4.2. Asymptotic for the first order ansatz I

The following two Sections study the leading order behaviour of the Kähler ansatz away from $S$ at large distance. The region under consideration lies over

(4.14)

\begin{split}\{y_{2}>1,A^{1/4}|\mu|+|y_{1}|>e^{-2\pi y_{2}}\}\cup\{y_{1}>1,A^{1/4}|\mu|+|y_{2}|>e^{-2\pi y_{1}}\}\\ \cup\{y_{1}<-1,A^{1/4}|\mu|+|y_{2}-y_{1}|>e^{2\pi y_{1}}\}\cup\{|\mu|>A^{-1/4}\}.\end{split}

This is a quantitative way of asserting boundedness away from $S$ .

We define the average functions of $\gamma_{i}$ (cf. (4.11)) on $\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}_{\mu}$ by

(4.15)

\bar{\gamma}_{i}(y_{1},y_{2},\mu)=\int_{0}^{1}\int_{0}^{1}\gamma_{i}(x_{1}+\sqrt{-1}y_{1},x_{2}+\sqrt{-1}y_{2},\mu)dx_{1}dx_{2}.

This Section is concerned with describing the behaviour of $\bar{\gamma}_{i}$ , and next Section proves exponential decay estimate for $|\gamma_{i}-\bar{\gamma}_{i}|$ .

Recall from (4.3) the Euclidean metric $g_{a}^{\prime}$ on $\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}_{\mu}$ . Its volume measure is $d\text{Vol}_{a}^{\prime}=\frac{A^{3/2}}{\sqrt{\mathbb{A}}}dy_{1}dy_{2}d\mu,$ and the associated Laplacian is $\Delta_{a}^{\prime}=a^{p\bar{q}}\frac{\partial^{2}}{\partial y_{p}\partial y_{q}}+A^{-1}\frac{\partial^{2}}{\partial\mu\partial\mu}$ . Here some care is needed in the calculations regarding the difference between Hermitian and symmetric matrices.

Lemma 4.8.

(Harmonicity) In the region (4.14) inside $\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}_{\mu}$ the functions $\bar{\gamma}_{i}$ satisfy $\Delta_{a}^{\prime}\bar{\gamma}_{i}=0$ , or equivalently their pullbacks to $(\mathbb{C}^{*})^{2}_{\eta_{1},\eta_{2}}\times\mathbb{R}_{\mu}$ satisfy $\Delta_{a}\bar{\gamma}_{i}=0$ .

Proof.

The amoeba $\text{Im}(S)$ is disjoint from the region (4.14), so Proposition 4.4 asserts the $\Delta_{a}$ -harmonicity of $\gamma_{i}$ , whence the harmonicty of $\bar{\gamma}_{i}$ . ∎

Next we wish to write $\bar{\gamma}_{i}$ also in terms of a Green’s representation. From the calculation in Lemma 4.1,

(4.16)

\bar{\gamma}=\int_{0}^{1}\int_{0}^{1}\gamma(x_{1}+\sqrt{-1}y_{1},x_{2}+\sqrt{-1}y_{2},\mu)dx_{1}dx_{2}=-\frac{1}{4\pi\varrho\sqrt{\mathbb{A}}}.

Thus by integrating (4.11) in the $x_{1},x_{2}$ variables,

Corollary 4.9.

(Green’s representation formula for $\bar{\gamma}_{i}$ )

\begin{cases}\bar{\gamma}_{1}(y_{1},y_{2},\mu)=&\text{Re}\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{4\sqrt{\mathbb{A}}}\int_{S\cap\{y_{2}^{\prime}<\Lambda\}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}\\ &\sqrt{-1}d\eta_{2}^{\prime}\wedge(d\bar{\eta}_{2}^{\prime}-d\bar{\eta}_{1}^{\prime})-\frac{1}{2\sqrt{a_{2\bar{2}}}}\log 2\Lambda\}\\ \bar{\gamma}_{2}(y_{1},y_{2},\mu)=&\text{Re}\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{4\sqrt{\mathbb{A}}}\int_{S\cap\{y_{1}^{\prime}<\Lambda\}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}\\ &\sqrt{-1}d\eta_{1}^{\prime}\wedge(d\bar{\eta}_{1}^{\prime}-d\bar{\eta}_{2}^{\prime})-\frac{1}{2\sqrt{a_{1\bar{1}}}}\log 2\Lambda\}\\ \bar{\gamma}_{3}(y_{1},y_{2},\mu)=&\text{Re}\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{4\sqrt{\mathbb{A}}}\int_{S\cap\{y_{1}^{\prime}>-\Lambda\}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}\\ &\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}-\frac{1}{2\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}}}\log 2\Lambda\}\\ \bar{\gamma}_{4}(y_{1},y_{2},\mu)=&\text{Im}\{\frac{A^{1/2}}{4\sqrt{\mathbb{A}}}\int_{S}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}\}.\end{cases}

Our goal is to extract the leading order behvaiour in terms of an explicit elementary formula. For this purpose we essentially replace $S$ by its asymptotic cylinders $\mathfrak{D}_{i}\times S^{1}$ . Define

\begin{cases}\bar{\bar{\gamma}}_{1}(y_{1},y_{2},\mu)=&\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{2\sqrt{\mathbb{A}}}\int_{0}^{\Lambda}|(y_{1},y_{2}-s,\mu)|_{a}^{\prime-1}ds-\frac{1}{2\sqrt{a_{2\bar{2}}}}\log 2\Lambda\}\\ \bar{\bar{\gamma}}_{2}(y_{1},y_{2},\mu)=&\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{2\sqrt{\mathbb{A}}}\int_{0}^{\Lambda}|(y_{1}-s,y_{2},\mu)|_{a}^{\prime-1}ds-\frac{1}{2\sqrt{a_{1\bar{1}}}}\log 2\Lambda\}\\ \bar{\bar{\gamma}}_{3}(y_{1},y_{2},\mu)=&\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{2\sqrt{\mathbb{A}}}\int_{-\Lambda}^{0}|(y_{1}-s,y_{2}-s,\mu)|_{a}^{\prime-1}ds\\ &-\frac{1}{2\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}}}\log 2\Lambda\}.\end{cases}

By construction $\bar{\bar{\gamma}}_{i}$ is $\Delta_{a}^{\prime}$ -harmonic in the region (4.14). Morever,

Lemma 4.10.

(Estimate of remainder terms) In the region (4.14) we have $|\bar{\gamma}_{4}|\leq C\varrho^{-1}$ . For $i=1,2,3$ , in the subset of (4.14) where $\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{i})\gtrsim A^{1/4}$ we have $|\bar{\gamma}_{i}-\bar{\bar{\gamma}}_{i}|\leq C\varrho^{-1}$ .

Proof.

We use the Green representation of $\bar{\gamma}_{4}$ . The total measure

\int_{S}\text{Im}(\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime})\leq C,

so the contribution to $\bar{\gamma}_{4}$ from the ball $\{|(\eta_{1}^{\prime},\eta_{2}^{\prime},0)|_{a}\lesssim A^{1/4}\}\subset S$ is bounded by $C\varrho^{-1}$ . The contributions from the 3 ends are neglegible unless the point $(y_{1},y_{2},\mu)$ inside the region (4.14) is close to $S$ along some $\mathfrak{D}_{i}$ ; we focus on the case of $\mathfrak{D}_{1}$ . The key fact is the exponential decay of the measure: along $\mathfrak{D}_{1}$ we have

\text{Im}(\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime})\leq Ce^{-2\pi y_{2}^{\prime}}\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{2}^{\prime}.

Thus the contribution from the end $\{y_{2}^{\prime}>1\}\cap S$ is controlled by

\begin{split}C\int_{0}^{\infty}e^{-2\pi y_{2}^{\prime}}|(y_{1},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}dy_{2}^{\prime}\leq C\varrho^{-1}.\end{split}

which implies the estimates on $\bar{\gamma}_{4}$ .

For $\bar{\gamma}_{i}-\bar{\bar{\gamma}}_{i}$ , the main point is that $S$ approaches its asymptotic cylinder at an exponentially fast rate. The rest of the arguments are similar. ∎

Elementary integration gives

Lemma 4.11.

(Leading order asymptote) The formulae for $\bar{\bar{\gamma}}_{i}$ are given explicitly as

(4.17)

\begin{cases}\bar{\bar{\gamma}}_{1}=&-\frac{1}{2\sqrt{a_{2\bar{2}}}}\log(\frac{\mathbb{A}^{1/2}}{\sqrt{Aa_{2\bar{2}}}}|(y_{1},y_{2},\mu)|_{a}^{\prime}-y_{2}-\frac{\text{Re}(a_{1\bar{2}})}{a_{2\bar{2}}}y_{1})\\ \bar{\bar{\gamma}}_{2}=&-\frac{1}{2\sqrt{a_{1\bar{1}}}}\log(\frac{\mathbb{A}^{1/2}}{\sqrt{Aa_{1\bar{1}}}}|(y_{1},y_{2},\mu)|_{a}^{\prime}-y_{1}-\frac{\text{Re}(a_{1\bar{2}})}{a_{1\bar{1}}}y_{2})\\ \bar{\bar{\gamma}}_{3}=&-\frac{1}{2\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}}}\\ &\log(\frac{\mathbb{A}^{1/2}}{\sqrt{A(a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}})}}|(y_{1},y_{2},\mu)|_{a}^{\prime}+\frac{a_{1\bar{1}}y_{1}+\text{Re}(a_{1\bar{2}})y_{2}+\text{Re}(a_{2\bar{1}})y_{1}+a_{2\bar{2}}y_{2}}{a_{1\bar{1}}+2\text{Re}(a_{1\bar{2}})+a_{2\bar{2}}}).\end{cases}

Remark 4.3.

These formulae have strong similarity with $\bar{\alpha}_{i}$ in (3.6) except for the absence of an additive constant as in (3.6), which is an artefact of a non-canonical choice of constant in our definition of $\gamma_{i}$ (cf. Remark 4.2).

4.3. Asymptotic for the first order ansatz II

This Section proves exponential decay estimate for higher Fourier modes $\gamma_{i}-\bar{\gamma}_{i}$ in the region bounded away from $S$ . The main idea is that the Laplace equation together with the vanishing of the zeroth Fourier modes imply exponential decay through Fourier analysis; this discussion is parallel to Section 3.2.

Lemma 4.12.

In the region defined by (4.14) we have $|\gamma_{4}-\bar{\gamma}_{4}|\leq CA^{-1/4}$ . For $i=1,2,3$ , in the subset of the region (4.14) where $\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{i})\gtrsim A^{1/4}$ we have $|\gamma_{i}-\bar{\gamma}_{i}|\leq CA^{-1/4}$ .

Proof.

Consider $\gamma_{4}$ at a given point in the region (4.14). Its integral formula (4.11) can be split into two parts, corresponding to far away sources $|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime}\gtrsim A^{1/4}$ and nearby sources $|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime}\lesssim A^{1/4}$ .

For far away sources, we use Lemma 4.1 to write the integrand $\gamma$ as a dominant term $-\frac{1}{4\pi\sqrt{\mathbb{A}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime}}$ plus a remainder term estimated by $\frac{C}{|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime 3}}$ . The dominant term does not contribute to $\gamma_{4}-\bar{\gamma}_{4}$ because it is constant in the $x_{1},x_{2}$ direction. The remainder term contribution to $\gamma_{4}$ is bounded by

CA^{1/2}\text{Im}\int_{S\cap\{\text{dist}\gtrsim A^{1/4}\}}\frac{1}{|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime 3}}\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}\leq CA^{-1/4}.

The contribution from nearby sources only arises if our given point of interest is too close to $S$ along one of $\mathfrak{D}_{1}$ , $\mathfrak{D}_{2}$ or $\mathfrak{D}_{3}$ directions; we focus on $\mathfrak{D}_{1}$ . Lemma 4.2 allows us to write $\gamma$ as $-\frac{1}{8\pi^{2}|(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)|_{a}^{3}}$ plus a well controlled remainder term. By the exponential decay property of the measure $\text{Im}\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}$ ,

\begin{split}&CA^{1/2}e^{-2\pi y_{2}}\int_{S\cap\{\text{dist}\lesssim A^{1/4}\}}\frac{\sqrt{-1}d\eta_{1}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}}{|(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)|_{a}^{3}}\leq Ce^{-2\pi y_{2}}R^{-1}\leq CA^{-1/4}.\end{split}

Combining the above shows $|\gamma_{4}-\bar{\gamma}_{4}|\leq CA^{-1/4}$ .

All these arguments carry through to $\gamma_{1},\gamma_{2},\gamma_{3}$ except the exponential decay of the measure. This is compensated by staying sufficiently far from $\mathfrak{D}_{i}$ . ∎

Proposition 4.13.

(Exponential decay for higher Fourier modes in the first order ansatz) In the region where $\text{dist}_{g_{a}^{\prime}}(\cdot,\text{Im}(S))\gtrsim A^{1/4}$ ,

(4.18)

|\gamma_{i}-\bar{\gamma}_{i}|\leq CA^{-1/4}\tilde{\ell}e^{-\tilde{\ell}},\quad\tilde{\ell}=\kappa_{a}\text{dist}_{g_{a}^{\prime}}(\cdot,\text{Im}(S)),\quad i=1,2,3,4,

where $\kappa_{a}$ is the minimum of $k_{n_{1},n_{2}}=2\pi(\frac{Aa^{p\bar{q}}n_{p}n_{q}}{\mathbb{A}})^{1/2}$ for all $(n_{1},n_{2})\in\mathbb{Z}^{2}\setminus\{0\}$ .

Proof.

This proof is parallel to Proposition 3.5, so will be sketchy. We focus on $\gamma_{4}-\bar{\gamma}_{4}$ as the same arguments work for $\gamma_{i}-\bar{\gamma}_{i}$ .

We perform Fourier decomposition in the periodic variables $x_{1},x_{2}$ ,

\gamma_{4}-\bar{\gamma}_{4}=\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}\setminus\{0\}}h_{n_{1},n_{2}}(y_{1},y_{2},\mu)e^{2\pi in_{1}x_{1}+2\pi in_{2}x_{2}}.

The zeroth Fourier mode vanishes by construction. Parseval identity combined with Lemma 4.12 shows

\sum_{n_{1},n_{2}}|h_{n_{1},n_{2}}|^{2}=\int_{0}^{1}\int_{0}^{1}|\gamma_{4}-\bar{\gamma}_{4}|^{2}dx_{1}dx_{2}\leq CA^{-1/2}.

Over the region (4.14), according to Proposition 4.4 and (4.16)

\Delta_{a}(\gamma_{4}-\bar{\gamma}_{4})=0,

which translates into the Helmholtz type equations

\Delta_{a}^{\prime}h_{n_{1},n_{2}}-4\pi\sqrt{-1}\text{Im}(a^{1\bar{2}})(n_{1}\frac{\partial h_{n_{1},n_{2}}}{\partial y_{2}}-n_{2}\frac{\partial h_{n_{1},n_{2}}}{\partial y_{1}})-4\pi^{2}(a^{p\bar{q}}n_{p}n_{q})h_{n_{1},n_{2}}=0.

After the variable substitution

\tilde{h}_{n_{1},n_{2}}=h_{n_{1},n_{2}}\exp\left(\frac{2\pi i\text{Im}(a_{1\bar{2}})}{\mathbb{A}}(a_{1\bar{1}}n_{2}y_{1}-\text{Re}(a_{1\bar{2}})n_{1}y_{1}+\text{Re}(a_{1\bar{2}})n_{2}y_{2}-a_{2\bar{2}}n_{1}y_{2})\right),

these equations become the Helmholtz equations on $\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}_{\mu}$ ,

\Delta_{a}^{\prime}\tilde{h}_{n_{1},n_{2}}=k_{n_{1},n_{2}}^{2}\tilde{h}_{n_{1},n_{2}},

where we denote $k_{n_{1},n_{2}}=2\pi(a^{p\bar{q}}n_{p}n_{q})^{1/2}(\frac{A}{\mathbb{A}})^{1/2}$ .

The rest of the argument is substantially similar to Proposition 3.5. Observe that $|\text{dist}_{g_{a}^{\prime}}(\cdot,\text{Im}(S))-R|\leq CA^{1/4}$ . An upper barrier supersolution to the Helmholtz equation is directly constructed as

\begin{split}h_{n_{1},n_{2}}^{\prime}=A^{-1/4}&\{\int_{\mathfrak{D}_{1}}e^{-k_{n_{1},n_{2}}|(y_{1},y_{2}-s,\mu)|_{a}^{\prime}}ds+\int_{\mathfrak{D}_{2}}e^{-k_{n_{1},n_{2}}|(y_{1}-s,y_{2},\mu)|_{a}^{\prime}}ds\\ &+\int_{\mathfrak{D}_{3}}e^{-k_{n_{1},n_{2}}|(y_{1}-s,y_{2}-s,\mu)|_{a}^{\prime}}ds\},\end{split}

where $ds=dy_{2},dy_{1},-dy_{1}$ on $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ respectively. Comparing the real and imaginary parts of $\tilde{h}_{n_{1},n_{2}}$ with $C(|n_{1}|+|n_{2}|)h_{n_{1},n_{2}}^{\prime}$ yields the result. ∎

By the topological description (cf. Section 1.1.4 and 1.1.5), we can lift the $T^{2}\subset(\mathbb{C}^{*})^{2}\times\mathbb{R}_{\mu}$ to a $T^{2}$ -cycle on the total space of the singular $S^{1}$ -bundle, namely the monodromy invariant $T^{2}$ -cycle for the topological $T^{3}$ -fibration. As an application of the asymptotes above, we shall evaluate (up to sign) the integral of the closed 2-form $\omega^{(1)}$ on this $T^{2}$ -cycle.

Lemma 4.14.

The integral $\int_{T^{2}}\omega^{(1)}=\text{Im}(a_{2\bar{1}})$ .

Proof.

As a preliminary remark, although $\omega^{(1)}$ is a Kähler form only in a bounded region, it makes sense as a closed 2-form over the entire $(\mathbb{C}^{*})^{2}\times\mathbb{R}_{\mu}$ . The integral $\int_{T^{2}}\omega^{(1)}$ is a cohomological invariant, which can be evaluated asymptotically on a $T^{2}$ -cycle as $\eta_{1},\eta_{2}$ stay bounded and $\mu\to+\infty$ . By choosing the $T^{2}$ -cycle on which $\mu,y_{1},y_{2}$ are constants,

\begin{split}\int_{T^{2}}\omega^{(1)}=&\frac{\sqrt{-1}}{2}\int_{T^{2}}W^{p\bar{q}}_{(1)}d\eta_{p}\wedge d\bar{\eta}_{q}=\frac{\sqrt{-1}}{2}\int_{T^{2}}(W^{1\bar{2}}_{(1)}-W^{2\bar{1}}_{(1)})dx_{1}\wedge dx_{2}\\ =&\int_{T^{2}}\text{Im}(W^{2\bar{1}}_{(1)})dx_{1}\wedge dx_{2}=\text{Im}(a_{2\bar{1}})+\int_{T^{2}}\gamma_{4}dx_{1}\wedge dx_{2}.\end{split}

But by Lemma 4.10 and Proposition 4.13, the quantity $\gamma_{4}\to 0$ as $\mu\to+\infty$ , so the only contribution is $\int_{T^{2}}\omega^{(1)}=\text{Im}(a_{2\bar{1}}).$ ∎

4.4. Structure near the singular locus I

There exists a constant $0<C\ll 1$ such that discs of $g_{a}$ -radius $CA^{1/4}$ centred at points in $S$ do not intersect each other; their union defines a disc bundle over $S$ : $\{R\lesssim A^{1/4}\}\subset(\mathbb{C}^{*})^{2}\times\mathbb{R}_{\mu}$ . We need to understand the local singularity structure of $v$ and $w^{p\bar{q}}$ in this disc bundle.

The following general setting is a variant of the classical Green’s function asymptote for submanifolds. Take the Euclidean space $(\mathbb{R}^{5},\sum_{i=1}^{5}ds_{i}^{2})$ , containing the codimension 3 graphical submanifold

\Gamma=\{(s_{1},s_{2},s_{3},s_{4},s_{5})|s_{i}=f_{i}(s_{1},s_{2}),i=3,4,5,|s_{1}|^{2}+|s_{2}|^{2}<1\}

such that $|f_{i}|_{C^{2}}\leq C$ for $i=3,4,5$ . Let $\vec{n}_{1},\vec{n}_{2},\vec{n}_{3}$ be an orthonormal frame on $\Gamma$ and define a local parametrisation of a tubular neighbourhood of $\Gamma$ :

\Psi(s_{1},s_{2},t_{1},t_{2},t_{3})=\vec{h}(s_{1},s_{2})+\sum_{1}^{3}t_{i}\vec{n}_{i},\quad\vec{h}(s_{1},s_{2})=(s_{1},s_{2},f_{i}(s_{1},s_{2}))\in\Gamma.

such that $\Gamma$ is $\{t_{1}=t_{2}=t_{3}=0\}$ in the coordinates $s_{1},s_{2},t_{1},t_{2},t_{3}$ . Denote $R=\sqrt{\sum t_{i}^{2}}$ as the Euclidean distance to $\Gamma$ . Let $f=f(\vec{h}(s_{1},s_{2}))$ be a function on $\Gamma$ with bound $\left\lVert f\right\rVert_{C^{2}}\leq C$ . We need asymptotes for the Green integral

I_{\Gamma,f}(s)=\int_{\Gamma}-\frac{1}{8\pi^{2}|s-s^{\prime}|^{3}}f(s^{\prime})d\text{Area}_{\Gamma}(s^{\prime}),\quad s\in\mathbb{R}^{5}.

We view $I_{\Gamma,f}$ as a function of $s_{1},s_{2},t_{1},t_{2},t_{3}$ .

Lemma 4.15.

For $\sum s_{i}^{2}\lesssim 1$ ,

\begin{cases}|I_{\Gamma,f}(s_{1},s_{2},t_{1},t_{2},t_{3})+\frac{f(s_{1},s_{2})}{4\pi R}|\leq C,\\ |\frac{\partial I_{\Gamma,f}}{\partial t_{i}}-\frac{f(s_{1},s_{2})t_{i}}{4\pi R^{3}}(1+\frac{1}{5}(\sum_{j}t_{j}\vec{n}_{j})\cdot\vec{H})|\leq C,\quad&i=3,4,5,\\ |\frac{\partial I_{\Gamma,f}}{\partial s_{i}}+\frac{1}{4\pi R}\frac{\partial f}{\partial s_{i}}(s_{1},s_{2})|\leq C,\quad&i=1,2.\end{cases}

where $\vec{H}$ is the mean curvature vector of $\Gamma$ at $\vec{h}(s_{1},s_{2})$ . If morever $\left\lVert f\right\rVert_{C^{3}}\leq C$ , $\left\lVert f_{i}\right\rVert_{C^{3}}\leq C$ , then

|\frac{\partial^{2}I_{\Gamma,f}}{\partial t_{i}\partial t_{j}}-\frac{(\delta_{ij}R^{2}-3t_{i}t_{j})f}{4\pi R^{5}}|\leq\frac{C}{R^{2}},\quad|\frac{\partial^{2}I_{\Gamma,f}}{\partial t_{i}\partial s_{j}}-\frac{t_{i}}{4\pi R^{3}}\frac{\partial f}{\partial s_{j}}|\leq\frac{C}{R},\quad|\frac{\partial^{2}I_{\Gamma,f}}{\partial s_{i}\partial s_{j}}|\leq\frac{C}{R}.

If morever $f(0)=0$ , then at $s_{1}=s_{2}=0$ ,

|\frac{\partial^{2}I_{\Gamma,f}}{\partial t_{i}\partial t_{j}}|\leq\frac{C}{R},\quad|\frac{\partial^{2}I_{\Gamma,f}}{\partial s_{i}\partial s_{j}}+\frac{1}{4\pi R}\frac{\partial^{2}f}{\partial s_{i}\partial s_{j}}|\leq C.

If morever $df(0)=0$ , then $|\frac{\partial^{2}I_{\Gamma,f}}{\partial t_{i}\partial s_{j}}|\leq C$ for $s_{1}=s_{2}=0$ .

Proof.

(Sketch) Consider $s_{1}=s_{2}=0$ . The leading order asymptote of $I_{\Gamma,f}$ is obtained by replacing $f$ with the constant $f(0)$ and replacing $\Gamma$ with $\mathbb{R}^{2}_{s_{1},s_{2}}$ . At $s=\sum t_{i}\vec{n}_{i}\in\mathbb{R}^{5}$ ,

\int_{\mathbb{R}^{2}}-\frac{1}{8\pi^{2}|s-s^{\prime}|^{3}}f(0)ds_{1}^{\prime}ds_{2}^{\prime}=-\frac{f(0)}{4\pi R}.

We then need to estimate the deviation of $I_{\Gamma,f}$ from this leading asymptote. After writing the surface integral as an integral over $s_{1},s_{2}$ plane, we reduce to the flat graph case $f_{3}=f_{4}=f_{5}=0$ . Writing

f(s^{\prime})=f(0)+\sum_{i=1}^{2}\frac{\partial f}{\partial s_{i}^{\prime}}(0)s_{i}^{\prime}+O(|s^{\prime}|^{2}),

we observe that the linear term does not contribute to $I_{\Gamma,f}(s)$ by parity, and the $O(|s^{\prime}|^{2})$ contribution is bounded by

C\int\frac{|s^{\prime}|^{2}}{|s-s^{\prime}|^{3}}ds_{1}^{\prime}ds_{2}^{\prime}\leq C\int\frac{r^{2}}{(r^{2}+R^{2})^{3/2}}rdr\leq C.

We now consider the normal first derivative $\frac{\partial I_{\Gamma,f}}{\partial t_{i}}$ for $s_{1}=s_{2}=0$ assuming without loss of generality that $df_{i}(0)=0$ . After using the Taylor expansion and parity trick above, modulo bounded terms

\begin{split}\frac{\partial I_{\Gamma,f}}{\partial t_{i}}\sim&f(0)\frac{3t_{i}}{8\pi^{2}}\int\frac{1}{(s_{1}^{\prime 2}+s_{2}^{\prime 2}+\sum_{j}(t_{j}-f_{j})^{2})^{5/2}}ds_{1}^{\prime}ds_{2}^{\prime}\\ \sim&f(0)\frac{3t_{i}}{8\pi^{2}}\int\frac{1}{(s_{1}^{\prime 2}+s_{2}^{\prime 2}+R^{2})^{5/2}}(1+\frac{2\sum t_{j}f_{j}}{s_{1}^{\prime 2}+s_{2}^{\prime 2}+R^{2}})ds_{1}^{\prime}ds_{2}^{\prime}\\ \sim&f(0)\frac{t_{i}}{4\pi R^{3}}(1+\frac{1}{5}\sum_{j}t_{j}(\frac{\partial^{2}f_{i}}{\partial s_{1}^{\prime 2}}+\frac{\partial^{2}f_{i}}{\partial s_{2}^{\prime 2}}))\\ =&f(0)\frac{t_{i}}{4\pi R^{3}}(1+\frac{1}{5}(\sum_{j}t_{j}\vec{n}_{j})\cdot\vec{H}),\end{split}

where $\vec{H}$ is the mean curvature of $\Gamma$ at the origin.

In the same setup, the tangential first derivative $\frac{\partial I_{\Gamma,f}}{\partial s_{i}}$ is modulo bounded terms

\begin{split}\frac{\partial I_{\Gamma,f}}{\partial s_{i}}\sim&\int f\frac{\partial}{\partial s_{i}^{\prime}}\frac{1}{8\pi^{2}|s-s^{\prime}|^{3}}ds_{1}^{\prime}ds_{2}^{\prime}\\ \sim&\int\frac{-1}{8\pi^{2}|s-s^{\prime}|^{3}}\frac{\partial f}{\partial s_{i}^{\prime}}ds_{1}^{\prime}ds_{2}^{\prime}\sim-\frac{1}{4\pi R}\frac{\partial f}{\partial s_{i}^{\prime}}(0).\end{split}

The argument for second derivatives are similar. ∎

Around a point $P\in S$ of interest, we introduce linear change of coordinates,

\begin{cases}\xi^{\prime}_{1}=\frac{z_{1}(P)(\eta_{1}-\eta_{1}(P))+z_{2}(P)(\eta_{2}-\eta_{2}(P))}{A^{1/2}|a^{i\bar{j}}z_{i}(P)\bar{z}_{j}(P)|^{1/2}},\\ \xi_{2}=\frac{(a^{1\bar{1}}\bar{z}_{1}(P)+a^{1\bar{2}}\bar{z}_{2}(P))(\eta_{2}-\eta_{2}(P))-(a^{2\bar{1}}\bar{z}_{1}(P)+a^{2\bar{2}}\bar{z}_{2}(P))(\eta_{1}-\eta_{1}(P)))}{|a^{i\bar{j}}z_{i}(P)\bar{z}_{j}(P)|^{1/2}},\end{cases}

such that $T_{P}S=\{\xi^{\prime}_{1}=0,\mu=0\}$ , and $\xi_{2}=0$ on the normal 3-plane to $T_{P}S$ . In these new linear coordinates,

g_{a}=A(d\mu^{2}+|d\xi^{\prime}_{1}|^{2}+|d\xi_{2}|^{2}),\quad d\eta_{1}\wedge d\eta_{2}=A^{1/2}d\xi_{1}^{\prime}\wedge d\xi_{2}.

A problem is that the tangent planes tilts as $P$ moves along $S$ . We find a local smooth vector valued function $\vec{h}$ to represent $S$ locally as a graph

S\cap\{|\xi_{2}|\lesssim A^{-1/4}\}=\{(\xi_{1},\xi_{2},\mu)=\vec{h}(\xi_{2}),|\xi_{2}|\lesssim A^{-1/4}\}\subset(\mathbb{C}^{*})^{2}\times\mathbb{R}_{\mu}.

The $g_{a}$ -normal (1,0)-type vector to $S\subset(\mathbb{C}^{*})_{\eta_{1},\eta_{2}}^{2}$ at the point $\vec{h}(\xi_{2})\in S$ is

\vec{n}(\vec{h}(\xi_{2}))=\frac{a^{p\bar{q}}\bar{z}_{q}}{|a^{i\bar{j}}z_{i}\bar{z}_{j}|^{1/2}}(\vec{h}(\xi_{2}))\frac{\partial}{\partial\eta_{p}}.

Denote $r=\sqrt{A(|\xi_{1}|^{2}+|\xi_{2}|^{2}+\mu^{2})}$ . Define a local diffeomorphism $\Psi$ on the local chart $\{r\lesssim A^{1/4}\}$ ,

\Psi(\xi_{1},\xi_{2},\mu)=\vec{h}(\xi_{2})+A^{1/2}\xi_{1}\vec{n}(\vec{h}(\xi_{2}))+\mu\vec{e}_{\mu}\in(S^{1}\times\mathbb{R})_{\eta_{1},\eta_{2}}^{2}\times\mathbb{R}_{\mu},

where $\vec{e}_{\mu}$ denotes the basis vector with $g_{a}$ -magnitude $A^{1/2}$ corresponding to the $\mu$ -variable. The image of $\Psi$ is a tubular neighbourhood of a graphical subset of $S$ , and the straight degeneracy locus $\{\mu=\xi_{1}=0\}$ is identified with the curved degeneracy locus $S$ . The pullback function $\Psi^{*}R=A^{1/2}\sqrt{\mu^{2}+|\xi_{1}|^{2}}$ , and $\Psi^{*}g_{a}$ satisfies

(4.19)

\begin{cases}|\Psi^{*}g_{a}-A(|d\xi_{1}|^{2}+|d\xi_{2}|^{2}+d\mu^{2})|_{g_{a}}\leq CA^{1/4}|\xi_{2}|,\\ |\nabla^{k}_{g_{a}}\{\Psi^{*}g_{a}-A(|d\xi_{1}|^{2}+|d\xi_{2}|^{2}+d\mu^{2})\}|_{g_{a}}\leq CA^{-k/4},\quad k\geq 1.\end{cases}

Proposition 4.16.

(leading order asymptote near $S$ ) Via the local diffeomorphism $\Psi$ , on the chart $\{r\lesssim A^{1/4}\}$ ,

\begin{cases}|\Psi^{*}(w^{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q})-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}d\xi_{1}\otimes d\bar{\xi}_{1}|_{g_{a}}\leq CA^{-3/4}\max(1,\log(A^{-1/4}\varrho),\\ |\Psi^{*}v-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}|\leq CA^{1/4}\max(1,\log(A^{-1/4}\varrho),\end{cases}

Remark 4.4.

In the original coordinates, for $R\lesssim A^{1/4}$ ,

\begin{cases}|w^{p\bar{q}}-\frac{z_{p}\bar{z}_{q}}{2RA^{1/2}a^{i\bar{j}}z_{i}\bar{z}_{j}}|\leq CA^{-1/4}\max(1,\log(A^{-1/4}\varrho)),\\ |v-\frac{A^{1/2}}{2R}|\leq CA^{1/4}\max(1,\log(A^{-1/4}\varrho)).\end{cases}

Proof.

We focus on $w^{1\bar{1}}=\gamma_{1}+\gamma_{3}$ , where $\gamma_{1},\gamma_{3}$ admit the Green’s representation (4.11). We split the integral on $S$ into the short distance contribution from $S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}$ and the long distance contribution from $S\cap\{|\eta_{1}-\eta^{\prime}_{1}|\gtrsim\frac{1}{2}\text{ or }|\eta_{2}-\eta^{\prime}_{2}|\gtrsim\frac{1}{2}\}$ .

The short distance contribution to the integral $w^{1\bar{1}}$ is

-\pi A^{1/2}\int_{S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{2}^{\prime}.

Applying Lemma 4.2, we can replace the periodic Newtontian potential $\gamma$ by the ordinary Newtonian potential, so the short distance contribution is replaced by

-\pi A^{1/2}\int_{S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}}-\frac{1}{8\pi^{2}|(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)|_{a}^{3}}\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{2}^{\prime},

at a cost of a smooth error of order $O(A^{-1/4})$ . The measure $\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{2}^{\prime}$ is equal to $2\frac{|z_{1}^{\prime}|^{2}}{Aa^{i\bar{j}}z_{i}^{\prime}\bar{z}_{j}^{\prime}}d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime})$ , so the above expression is

\int_{S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}}-\frac{1}{8\pi^{2}|(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)|_{a}^{3}}\frac{-2\pi|z_{1}^{\prime}|^{2}}{A^{1/2}a^{i\bar{j}}z_{i}^{\prime}\bar{z}_{j}^{\prime}}d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime}).

We may assume the submanifold $S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}$ is graphical, so Lemma 4.15 applies after scaling. Thus the short distance contribution to $w^{1\bar{1}}$ is

-\frac{1}{4\pi R}\frac{-2\pi|z_{1}^{\prime}|^{2}}{A^{1/2}a^{i\bar{j}}z_{i}^{\prime}\bar{z}_{j}^{\prime}}+O(A^{-1/4})=\frac{|z_{1}^{\prime}|^{2}}{2RA^{1/2}a^{i\bar{j}}z_{i}^{\prime}\bar{z}_{j}^{\prime}}+O(A^{-1/4}),

where the complex coordinates $z_{i}^{\prime}$ are computed at $\vec{h}(\xi_{2})\in S$ . But the factor $\frac{|z_{1}|^{2}}{A^{1/2}a^{i\bar{j}}z_{i}\bar{z}_{j}}$ varies slowly, so we may as well compute it at $(\eta_{1},\eta_{2},\mu)$ .

The long distance contribution to $w^{1\bar{1}}$ is $O(A^{-1/4}\max(1,\log(A^{-1/4}\varrho)))$ by following the same steps as in Section 4.2, 4.3, using Lemma 4.1. Combining the two contributions,

|w^{1\bar{1}}-\frac{|z_{1}|^{2}}{2RA^{1/2}a^{i\bar{j}}z_{i}\bar{z}_{j}}|\leq CA^{-1/4}\max(1,\log(A^{-1/4}\varrho))).

The cases of $w^{p\bar{q}}$ and $v=Aa^{p\bar{q}}w^{p\bar{q}}$ are similar. ∎

Corollary 4.17.

Fix $0<\epsilon_{0}\ll 1$ , then on the total space

M^{-}=\{A^{-1/4}\varrho<\exp(\epsilon_{0}A^{3/4})\},

the function $V_{(1)}$ is positive and the matrix $W^{p\bar{q}}_{(1)}$ is positive definite.

4.5. Structure near the singular locus II

This Section interprets the metric structure transverse to the singular locus $S$ in terms of Taub-NUT metrics. First we emphasize that smooth topology of the $S^{1}$ -fibration is subtle.

•

The Kähler ansatz is only defined a priori on the complement of $S$ , and there are no transparent choices of smooth local coordinates near $S$ to exhibit the smooth extension of both the complex structure and the metric.
•

By the reasons stated in Remark 2.4, smooth topology of the singular $S^{1}$ -bundle is expected to be unstable under deformation.

Because of these difficulties, in the region $\{R\lesssim A^{-1/2}\}$ near the singular locus $S\cap M^{-}$ , we will be forced to work with tensor fields of low regularity.

We now define a model metric

(4.20)

\begin{cases}g_{\text{NUT}}=(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})(d\mu^{2}+|d\xi_{1}|^{2})+(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})^{-1}\vartheta_{\text{NUT}}^{2}+A|d\xi_{2}|^{2},\\ \omega_{\text{NUT}}=(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})(d\mu\wedge\vartheta_{\text{NUT}}+\frac{\sqrt{-1}}{2}d\xi_{1}\wedge d\bar{\xi}_{1})+\frac{A\sqrt{-1}}{2}d\xi_{2}\wedge d\bar{\xi}_{2},\\ \Omega_{\text{NUT}}=A^{1/2}(\vartheta_{\text{NUT}}-\sqrt{-1}(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})d\mu)\wedge d\xi_{1}\wedge d\xi_{2},\end{cases}

namely the product of the Taub-NUT metric with flat $\mathbb{R}^{2}$ . The weighted Hölder norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta}(g_{\text{NUT}})}$ for $S^{1}$ -invariant tensors $T$ on this model space is defined by

\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta}(g_{\text{NUT}})}=\sup_{p}(A^{1/2}\ell)^{-\delta}\{\sum_{j=0}^{k}|\ell^{j}\nabla^{j}T(p)|+\sup_{|p-p^{\prime}|_{a}\leq\ell}\frac{|\nabla^{k}T(p)-\nabla^{k}T(p^{\prime})|}{d_{g_{\text{NUT}}}(p,p^{\prime})^{\alpha}\ell^{-\alpha-k}}\}.

where $\ell\sim A^{1/2}\sqrt{\mu_{1}^{2}+|\xi_{1}|^{2}}+A^{-1/2}$ reflects the regularity scale of the model metric, and we compare $T(p)$ and $T(p^{\prime})$ using parallel transport along minimal geodesics. An estimate in this norm is thought as the Hölder version of $|T|=O((A^{1/2}\ell)^{\delta}).$

Proposition 4.18.

Via the local diffeomorphism $\Psi$ , on the chart $\{r\lesssim A^{1/4}\}$ ,

\begin{cases}|\nabla_{g_{a}}\{\Psi^{*}(w^{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q})-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}d\xi_{1}\otimes d\bar{\xi}_{1}\}|_{g_{a}}\leq CA^{-1},\\ |\nabla_{g_{a}}\{\Psi^{*}v-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}\}|_{g_{a}}\leq C.\end{cases}

Furthermore

\begin{cases}|\nabla^{2}_{g_{a}}\{\Psi^{*}(w^{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q})-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}d\xi_{1}\otimes d\bar{\xi}_{1}\}|_{g_{\text{NUT}}}\leq CA^{-3/4}\ell^{-2},\\ |\nabla^{2}_{g_{a}}\{\Psi^{*}v-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}\}|_{g_{\text{NUT}}}\leq CA^{1/4}\ell^{-2}.\end{cases}

Proof.

(Sketch) The method is the same as in Proposition 4.16, so we only mention the key points. The long distance contribution to the integral has improved decay, so there is no need for the log factor. The short distance contribution appeals to Lemma 4.15. In the first derivative estimates, notice the mean curvature vector vanishes because $S$ an algebraic curve.

For second derivative estimates, notice for $R\lesssim A^{-1/2}$ , the magnitudes of tensors are inhomogeneous:

|d\mu|_{g_{\text{NUT}}}\leq A^{-1/4}R^{1/2},\quad|d\xi_{1}|_{g_{\text{NUT}}}\leq A^{-1/4}R^{1/2},\quad|d\xi_{2}|_{g_{\text{NUT}}}\leq CA^{-1/2}.

These $R$ factors make the $g_{\text{NUT}}$ -magnitudes bounded even though the $g_{a}$ -magnitudes can be unbounded. Inside the tensor $\Psi^{*}(w^{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q})$ , the coeffient of $d\xi_{1}\otimes d\bar{\xi}_{2}$ and $d\xi_{2}\otimes d\bar{\xi}_{1}$ correspond to imposing $f(0)=0$ , and the coeffient of $d\xi_{2}\otimes d\bar{\xi}_{2}$ correspond to imposing $f(0)=0$ and $df(0)=0$ . ∎

Proposition 4.19.

(Transverse Taub-NUT metric) Fix $0<\alpha<1$ . Under suitable gauge choices for the $S^{1}$ -connection, the ansatz metric $g^{(1)}$ over the local chart $\{r\lesssim A^{1/4}\}$ is approximated by the model metric:

\begin{cases}|\Psi^{*}g^{(1)}-g_{\text{NUT}}|_{g_{\text{NUT}}}\leq CA^{-3/4}\max(\log(A^{-1/4}\varrho),1)+CA^{1/4}|\xi_{2}|,\\ |\nabla_{g_{\text{NUT}}}(\Psi^{*}g^{(1)}-g_{\text{NUT}})|_{C^{\alpha}_{-1}(r\lesssim A^{1/4})}\leq CA^{-1/4}\max(\log(A^{-1/4}\varrho),1).\end{cases}

and

\begin{cases}|\Psi^{*}\Omega^{(1)}-\Omega_{\text{NUT}}|_{g_{\text{NUT}}}\leq CA^{-3/4}\max(\log(A^{-1/4}\varrho),1)+CA^{1/4}|\xi_{2}|,\\ |\nabla_{\Omega_{\text{NUT}}}(\Psi^{*}g^{(1)}-\Omega_{\text{NUT}})|_{C^{\alpha}_{-1}(r\lesssim A^{1/4})}\leq CA^{-1/4}\max(\log(A^{-1/4}\varrho),1).\end{cases}

Proof.

Applying the asymptotes in Proposition 4.16 and 4.18, on the local chart $\{r\lesssim A^{1/4}\}$ , up to an error of order $O(A^{-3/4}\max(1,\log(A^{-1/4}\varrho)))$ , the ansatz metric admits asymptote

(4.21)

\begin{cases}\begin{split}g^{(1)}=&V_{(1)}d\mu^{2}+V_{(1)}^{-1}\vartheta^{2}+\text{Re}(W^{p\bar{q}}_{(1)}d\eta_{p}\otimes d\bar{\eta}_{q})\\ \sim&\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}(d\mu^{2}+|d\xi_{1}|^{2})+(A+\frac{1}{2\sqrt{\mu^{2}+|\xi|^{2}}})^{-1}\vartheta^{2}+g_{a},\end{split}\\ \omega^{(1)}\sim d\mu\wedge\vartheta+\frac{\sqrt{-1}}{2}(a_{p\bar{q}}d\eta_{p}\wedge d\bar{\eta}_{q}+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}d\xi_{1}\wedge d\bar{\xi}_{1}),\\ \Omega^{(1)}\sim A^{1/2}\{\vartheta-\sqrt{-1}(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})d\mu)\}\wedge d\eta_{1}\wedge d\eta_{2}.\end{cases}

The main issue then is to compare the connection $\vartheta$ with $\vartheta_{\text{NUT}}$ . The curvature of the Taub-NUT metric $d\vartheta_{\text{NUT}}$ has the explicit formula

\sqrt{-1}\{\frac{-\mu}{4(\mu^{2}+|\xi_{1}|^{2})^{3/2}}d\xi_{1}\wedge d\bar{\xi}_{1}+\frac{-\bar{\xi}_{1}}{4(\mu^{2}+|\xi_{1}|^{2})^{3/2}}d\mu\wedge d\xi_{1}-\frac{-\xi_{1}}{4(\mu^{2}+|\xi_{1}|^{2})^{3/2}}d\mu\wedge d\bar{\xi}_{1}\}.

The curvature $d\vartheta$ is prescribed by formula (1.6), involving the first derivatives of $v$ and $w^{p\bar{q}}$ . Applying the asymptotic from Proposition 4.18,

|(d\vartheta-d\vartheta_{\text{NUT}})|_{g_{a}}\leq CA^{-1/2},\quad|\nabla_{g_{a}}(d\vartheta-d\vartheta_{\text{NUT}})|_{g_{\text{NUT}}}\leq CA^{-1/4}\ell^{-2}.

In particular $\left\lVert(d\vartheta-d\vartheta_{\text{NUT}})\right\rVert_{C^{1}_{-1}(g_{\text{NUT}})}\leq CA^{1/4}$ . After suitable gauge fixing, we can find a 1-form $\vartheta-\vartheta_{\text{NUT}}$ on the base $\{r\lesssim A^{1/4}\}$ , with norm estimate upstairs $\left\lVert\vartheta-\vartheta_{\text{NUT}}\right\rVert_{C^{1,\alpha}_{0}(g_{\text{NUT}})}\leq CA^{-1/4}$ for any fixed $0<\alpha<1$ . This specifies a gauge choice of $\vartheta$ .

Thus up to an admissible amount of error we can replace $\vartheta$ by $\vartheta_{\text{NUT}}$ in the asymptote (4.21). The deviation between RHS of (4.21) and $g_{\text{NUT}}$ is an elementary term $\Psi^{*}g_{a}-A(d\mu^{2}+|d\xi_{1}|^{2}+|d\xi_{2}|^{2})$ controlled by (4.19). ∎

Remark 4.5.

For $R\gtrsim A^{-1/2}$ , one can use $\Delta_{a}$ -harmonicity to obtain weighted $C^{k,\alpha}$ -estimates for any large $k$ . For $R\lesssim A^{-1/2}$ , we exhibited a collection of local charts corresponding to a choice of $P\in S$ , such that the Kähler ansatz has $C^{1,\alpha}$ -regularity. Thus if $u$ is a $C^{2,\alpha}$ function on a chart, then its $g^{(1)}$ -Laplacian is $C^{\alpha}$ . This regularity is sufficient for setting up weighted Hölder analysis.

4.6. Complex geometric perspective

The goal of this Section is to identify the holomorphic structure of the Kähler ansatz $(J^{(1)},\Omega^{(1)})$ . Recall the (1,0)-form $\zeta=V_{(1)}d\mu+\sqrt{-1}\vartheta$ and formula (1.14) for its differential. The main idea is to produce holomorphic differentials

(4.22)

\begin{cases}\zeta_{3}=\zeta+\beta_{13}d\eta_{1}+\beta_{23}d\eta_{2},\\ \zeta_{4}=-\zeta+\beta_{14}d\eta_{1}+\beta_{24}d\eta_{2},\end{cases}

by solving for the unknown functions $\beta_{13},\beta_{23},\beta_{14},\beta_{24}$ . The requirement for $d\zeta_{3}=d\zeta_{4}=0$ translates into an overdetermined and underdetermined system of equations

(4.23)

\begin{cases}\frac{\partial\beta_{p3}}{\partial\bar{\eta}_{q}}=-\frac{1}{2}\frac{\partial w^{p\bar{q}}}{\partial\mu},\quad&\frac{\partial\beta_{p3}}{\partial\mu}=2\frac{\partial v}{\partial\eta_{p}},\\ \frac{\partial\beta_{p4}}{\partial\bar{\eta}_{q}}=\frac{1}{2}\frac{\partial w^{p\bar{q}}}{\partial\mu},\quad&\frac{\partial\beta_{p4}}{\partial\mu}=-2\frac{\partial v}{\partial\eta_{p}},\\ \frac{\partial\beta_{p3}}{\partial\eta_{q}}=\frac{\partial\beta_{q3}}{\partial\eta_{p}},\quad&\frac{\partial\beta_{p4}}{\partial\eta_{q}}=\frac{\partial\beta_{q4}}{\partial\eta_{p}},\quad p,q=1,2.\end{cases}

The overdetermined nature is closely related to the integrability of the complex structure. The underdetermined nature is related to the fact that we can add certain holomorphic functions of $\eta_{1},\eta_{2}$ to $\beta_{p3},\beta_{p4}$ and solve the same equations; to eliminate this ambiguity one has to impose more growth conditions. Our strategy for solving this system is a direct construction using integral representations, and the main technical difficulty is to extract finite expressions out of divergent integrals.

We use the shorthand notation $|\eta|_{a}=(a_{p\bar{q}}\eta_{p}\bar{\eta}_{q})^{1/2}$ and $|y|_{a}=(a_{p\bar{q}}y_{p}y_{q})^{1/2}$ . We introduce two auxiliary functions

\gamma_{\pm}(\eta_{1},\eta_{2},\mu)=\frac{1}{3}\frac{A^{1/2}\mu}{|(\eta_{1},\eta_{2},\mu)|_{a}^{3}|\eta|_{a}^{2}}+\frac{2}{3}\frac{1}{|\eta|_{a}^{4}}(\frac{A^{1/2}\mu}{|(\eta_{1},\eta_{2},\mu)|_{a}}\mp 1),

and define for $p=1,2$ the series

(4.24)

\begin{cases}\gamma_{p3}(\eta_{1},\eta_{2},\mu)=\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\frac{3a_{p\bar{q}}(\bar{\eta}_{q}+n_{q})}{8\pi^{2}A^{1/2}}\gamma_{+}(\eta_{1}+n_{1},\eta_{2}+n_{2},\mu),\\ \gamma_{p4}(\eta_{1},\eta_{2},\mu)=-\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\frac{3a_{p\bar{q}}(\bar{\eta}_{q}+n_{q})}{8\pi^{2}A^{1/2}}\gamma_{-}(\eta_{1}+n_{1},\eta_{2}+n_{2},\mu).\end{cases}

These series converge absolutely for $(\eta_{1},\eta_{2})\notin\mathbb{Z}^{2}$ , and are 1-periodic in $\eta_{1},\eta_{2}$ variables. When $\eta_{1}=\eta_{2}=0$ , the series are designed so that $\gamma_{+}$ and $\gamma_{p3}$ extend smoothly over $\{\mu>0\}$ , while $\gamma_{-}$ and $\gamma_{p4}$ extend smoothly over $\{\mu<0\}$ . We will later use $\gamma_{p3}$ and $\gamma_{p4}$ as integrands to construct $\beta_{p3}$ and $\beta_{p4}$ .

Lemma 4.20.

(Differential identities)

\frac{\partial\gamma_{p3}}{\partial\mu}=-\frac{\partial\gamma_{p4}}{\partial\mu}=2\frac{\partial\gamma}{\partial\eta_{p}},\quad p=1,2,

and morever

\frac{\partial\gamma_{p3}}{\partial\eta_{q}}=\frac{\partial\gamma_{q3}}{\partial\eta_{p}},\quad\frac{\partial\gamma_{p4}}{\partial\eta_{q}}=\frac{\partial\gamma_{q4}}{\partial\eta_{p}},\quad p,q=1,2.

Proof.

We differentiate the series definition (4.9) of $\gamma$ to get

\frac{\partial\gamma}{\partial\eta_{p}}=\frac{3}{16\pi^{2}}\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\frac{a_{p\bar{q}}(\bar{\eta}_{q}+n_{q})}{|(\eta_{1}+n_{1},\eta_{2}+n_{2},\mu)|_{a}^{5}}.

Using the elementary formula for indefinite integrals

\int\frac{1}{(s^{2}+t^{2})^{5/2}}ds=\frac{1}{3}\frac{s}{(s^{2}+t^{2})^{3/2}t^{2}}+\frac{2}{3}\frac{1}{t^{4}}(\frac{s}{\sqrt{s^{2}+t^{2}}}\mp 1),

we see

\int\frac{1}{|(\eta_{1},\eta_{2},\mu)|_{a}^{5}}d\mu=A^{-1/2}\gamma_{\pm},

or equivalently

\frac{\partial}{\partial\eta_{p}}\left(-\frac{1}{8\pi^{2}}\frac{1}{|(\eta_{1},\eta_{2},\mu)|_{a}^{3}}\right)=\frac{\partial}{\partial\mu}\left(\frac{3a_{p\bar{q}}\bar{\eta}_{q}}{16\pi^{2}A^{1/2}}\gamma_{\pm}\right).

Thus after summation

\frac{\partial\gamma}{\partial\eta_{p}}=\frac{\partial}{\partial\eta_{p}}\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\left(-\frac{1}{8\pi^{2}}\frac{1}{|(\eta_{1}+n_{1},\eta_{2}+n_{2},\mu)|_{a}^{3}}\right)=\frac{1}{2}\frac{\partial\gamma_{p3}}{\partial\mu}=-\frac{1}{2}\frac{\partial\gamma_{p4}}{\partial\mu}.

The ‘morever’ statement follows from summing over the elementary differential relations

(a_{p\bar{r}}\bar{\eta}_{r})\frac{\partial\gamma_{\pm}}{\partial\eta_{q}}(\eta_{1},\eta_{2},\mu)=(a_{q\bar{r}}\bar{\eta}_{r})\frac{\partial\gamma_{\pm}}{\partial\eta_{p}}(\eta_{1},\eta_{2},\mu),\quad p,q=1,2.

∎

By the periodicity of $\gamma_{p3},\gamma_{p4}$ , we may assume $|x_{1}|,|x_{2}|\leq\frac{1}{2}$ . In order to integrate $\gamma_{p3}$ and $\gamma_{p4}$ we need to bound these functions. It is convenient to introduce some closely related integrals:

(4.25)

\begin{cases}I_{01}(y_{1},y_{2},\mu)=\int_{\mathbb{R}^{2}}\frac{1}{|(s_{1}+\sqrt{-1}y_{1},s_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{3}a_{p\bar{q}}(s_{p}+\sqrt{-1}y_{p})(s_{q}-\sqrt{-1}y_{q})}ds_{1}ds_{2},\\ I_{02}(y_{1},y_{2},\mu)=\int_{\mathbb{R}^{2}}\frac{1}{|a_{p\bar{q}}(s_{p}+\sqrt{-1}y_{p})(s_{q}-\sqrt{-1}y_{q})|^{2}|(s_{1}+\sqrt{-1}y_{1},s_{2}+\sqrt{-1}y_{2},\mu)|_{a}}ds_{1}ds_{2},\\ I_{03}(y_{1},y_{2},\mu)=\int_{\mathbb{R}^{2}}\frac{1}{|a_{p\bar{q}}(s_{p}+\sqrt{-1}y_{p})(s_{q}-\sqrt{-1}y_{q})|^{2}}ds_{1}ds_{2},\end{cases}

and we can express

\begin{split}I_{p3}&=\frac{-3a_{p\bar{q}}\sqrt{-1}y_{q}}{8\pi^{2}A^{1/2}}\int_{\mathbb{R}^{2}}\gamma_{+}(s_{1}+\sqrt{-1}y_{1},s_{2}+\sqrt{-1}y_{2},\mu)ds_{1}ds_{2}\\ &=\frac{-a_{p\bar{q}}\sqrt{-1}y_{q}}{8\pi^{2}}(\mu I_{01}+2\mu I_{02}-2A^{-1/2}I_{03}),\end{split}

and

\begin{split}I_{p4}&=\frac{3a_{p\bar{q}}\sqrt{-1}y_{q}}{8\pi^{2}A^{1/2}}\int_{\mathbb{R}^{2}}\gamma_{-}(s_{1}+\sqrt{-1}y_{1},s_{2}+\sqrt{-1}y_{2},\mu)ds_{1}ds_{2}\\ &=\frac{a_{p\bar{q}}\sqrt{-1}y_{q}}{8\pi^{2}}(\mu I_{01}+2\mu I_{02}+2A^{-1/2}I_{03}).\end{split}

Lemma 4.21.

These integrals admit the simplified formulae:

\begin{cases}I_{01}=\frac{\pi}{\sqrt{\mathbb{A}}}\int_{\frac{A}{\mathbb{A}}|y|_{a}^{2}}^{\infty}\frac{1}{s(s+A\mu^{2})^{3/2}}ds,\\ I_{02}=-\frac{1}{2}I_{01}+\frac{\pi\sqrt{\mathbb{A}}}{A\varrho|y|_{a}^{2}}\\ I_{03}=\frac{\pi\sqrt{\mathbb{A}}}{A|y|_{a}^{2}}.\end{cases}

Consequently

\begin{cases}I_{p3}=\frac{a_{p\bar{q}}\sqrt{-1}y_{q}}{4\pi A^{1/2}\sqrt{\mathbb{A}}}\frac{1}{\varrho(\varrho+A^{1/2}\mu)},\\ I_{p4}=\frac{a_{p\bar{q}}\sqrt{-1}y_{q}}{4\pi A^{1/2}\sqrt{\mathbb{A}}}\frac{1}{\varrho(\varrho-A^{1/2}\mu)}.\end{cases}

Proof.

To evaluate these integrals, we introduce a radial variable

s=a_{p\bar{q}}(s_{p}+\sqrt{-1}y_{p})(s-\sqrt{-1}y_{q}),

and then elementary calculations in polar coordinates give

\begin{split}I_{01}=\frac{\pi}{\sqrt{\mathbb{A}}}\int_{\frac{A}{\mathbb{A}}|y|_{a}^{2}}^{\infty}\frac{1}{s(s+A\mu^{2})^{3/2}}ds,\end{split}

and similarly

\begin{split}I_{02}=\frac{\pi}{\sqrt{\mathbb{A}}}\int_{\frac{A}{\mathbb{A}}|y|_{a}^{2}}^{\infty}\frac{1}{s^{2}(s+A\mu^{2})^{1/2}}ds=-\frac{1}{2}I_{01}+\frac{\pi\sqrt{\mathbb{A}}}{A\varrho|y|_{a}^{2}},\end{split}

together with the formula for $I_{03}$ . The formulae for $I_{p3}$ and $I_{p4}$ follow from taking linear combinations. ∎

Lemma 4.22.

(Estimating integrands I) For $|y_{1}|+|y_{2}|\gtrsim 1$ and $|x_{1}|,|x_{2}|\leq\frac{1}{2}$ , we have the estimate

|\gamma_{p3}-\gamma_{p4}|\leq\frac{CA^{-1/4}|\mu|}{|y|_{a}\varrho}.

Morever there are improved estimates for $\gamma_{p3},\gamma_{p4}$ depending on the sign of $\mu$ :

\begin{cases}|\gamma_{p3}|\leq\frac{CA^{-3/4}|y|_{a}}{\varrho^{2}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})),\quad&\mu\geq 0,\\ |\gamma_{p4}|\leq\frac{CA^{-3/4}|y|_{a}}{\varrho^{2}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})),\quad&\mu\leq 0.\end{cases}

Proof.

Consider first the special case where $x_{1}=x_{2}=0,\eta_{p}=\sqrt{-1}y_{p}$ . By pairing $(n_{1},n_{2})$ with $(-n_{1},-n_{2})$ in the summation, we obtain

\begin{cases}\gamma_{p3}(\eta_{1},\eta_{2},\mu)=\frac{-3a_{p\bar{q}}\sqrt{-1}y_{q}}{8\pi^{2}A^{1/2}}\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\gamma_{+}(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu),\\ \gamma_{p4}(\eta_{1},\eta_{2},\mu)=\frac{3a_{p\bar{q}}\sqrt{-1}y_{q}}{8\pi^{2}A^{1/2}}\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\gamma_{-}(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu).\end{cases}

By the Cauchy integral test,

\begin{split}&\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\frac{1}{|(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{3}a_{p\bar{q}}(n_{p}+\sqrt{-1}y_{p})(n_{q}-\sqrt{-1}y_{q})}\\ \leq&CI_{01}\leq CA^{-1/2}\frac{1}{\varrho^{3}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})).\end{split}

Similarly,

\begin{split}&|\sum_{n_{1},n_{2}}\frac{1}{(a_{p\bar{q}}(n_{p}+\sqrt{-1}y_{p})(n_{q}-\sqrt{-1}y_{q}))^{2}|(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu)|_{a}}|\\ \leq&CI_{02}\leq CA^{-1/2}\frac{1}{|y|_{a}^{2}\varrho}.\end{split}

Combining these two estimates,

|\gamma_{p3}-\gamma_{p4}|(\eta_{1},\eta_{2},\mu)\leq CA^{-1/2}|a_{pq}y_{q}||\mu|\frac{1}{|y|_{a}^{2}\varrho}\leq\frac{CA^{-1/4}|\mu|}{|y|_{a}\varrho}.

Morever, when $\mu\geq 0$ ,

\begin{split}&\frac{1}{a_{p\bar{q}}(n_{p}+\sqrt{-1}y_{p})(n_{q}-\sqrt{-1}y_{q})}|\frac{A^{1/2}\mu}{|(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu)|_{a}}-1|\\ \leq&C\frac{1}{|(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{2}},\end{split}

whence

\begin{split}&|\sum_{n_{1},n_{2}}\frac{1}{|a_{p\bar{q}}(n_{p}+\sqrt{-1}y_{p})(n_{q}-\sqrt{-1}y_{q})|^{2}}\left(\frac{A^{1/2}\mu}{|(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu)|_{a}}-1\right)|\\ \leq&C\sum_{n_{1},n_{2}}\frac{1}{a_{p\bar{q}}(n_{p}+\sqrt{-1}y_{p})(n_{q}-\sqrt{-1}y_{q})|(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{2}}\\ \leq&C\int_{\mathbb{R}^{2}}\frac{1}{a_{p\bar{q}}(s_{p}+\sqrt{-1}y_{p})(s_{q}-\sqrt{-1}y_{q})|(s_{1}+\sqrt{-1}y_{1},s_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{2}}ds_{1}ds_{2}\\ \leq&CA^{-1/2}\frac{1}{\varrho^{2}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})).\end{split}

This leads to

|\gamma_{p3}(\eta_{1},\eta_{2},\mu)|\leq\frac{CA^{-1}|a_{p\bar{q}}y_{q}|}{\varrho^{2}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})),\quad\mu\geq 0.

Similarly

|\gamma_{p4}(\eta_{1},\eta_{2},\mu)|\leq\frac{CA^{-1}|a_{p\bar{q}}y_{q}|}{\varrho^{2}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})),\quad\mu\leq 0.

For general $|x_{1}|,|x_{2}|\leq\frac{1}{2}$ , the difference $\gamma_{p3}(\eta_{1},\eta_{2},\mu)-\gamma_{p3}(\sqrt{-1}y_{1},\sqrt{-1}y_{2},\mu)$ , respectively $\gamma_{p4}(\eta_{1},\eta_{2},\mu)-\gamma_{p4}(\sqrt{-1}y_{1},\sqrt{-1}y_{2},\mu)$ , can be estimated by termwise comparing the two series using the methods above. The result is

\begin{cases}|\gamma_{p3}(\eta_{1},\eta_{2},\mu)-\gamma_{p3}(\sqrt{-1}y_{1},\sqrt{-1}y_{2},\mu)|\leq\frac{C(|x_{1}|+|x_{2}|)}{A^{1/2}\varrho^{2}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})),\mu\geq 0,\\ |\gamma_{p4}(\eta_{1},\eta_{2},\mu)-\gamma_{p4}(\sqrt{-1}y_{1},\sqrt{-1}y_{2},\mu)|\leq\frac{C(|x_{1}|+|x_{2}|)}{A^{1/2}\varrho^{2}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})),\mu\leq 0,\end{cases}

and

\begin{split}&|(\gamma_{p3}-\gamma_{p4})(\eta_{1},\eta_{2},\mu)-(\gamma_{p3}-\gamma_{p4})(\sqrt{-1}y_{1},\sqrt{-1}y_{2},\mu)|\leq\frac{C|\mu|(|x_{1}|+|x_{2}|)}{|y|_{a}^{2}\varrho},\end{split}

so the claims in the Lemma reduces to the special case $x_{1}=x_{2}=0$ above. ∎

Next we examine

(4.26)

\begin{split}\gamma_{p3}+\gamma_{p4}=&-\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\frac{a_{p\bar{q}}(\bar{\eta}_{q}+n_{q})}{2\pi^{2}A^{1/2}|a_{i\bar{j}}(\eta_{i}+n_{i})(\bar{\eta}_{j}+n_{j})|^{2}}\\ =&\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\frac{\partial}{\partial\eta_{p}}\frac{1}{2\pi^{2}A^{1/2}a_{i\bar{j}}(\eta_{i}+n_{i})(\bar{\eta}_{j}+n_{j})}.\end{split}

Lemma 4.23.

(Estimating integrands II) For $|y_{1}|+|y_{2}|\gtrsim 1$ and $|x_{1}|,|x_{2}|\leq\frac{1}{2}$ , we have the estimate

|\gamma_{p3}+\gamma_{p4}-\frac{\sqrt{-1}a_{p\bar{q}}y_{q}\sqrt{\mathbb{A}}}{2\pi A^{3/2}|y|_{a}^{2}}|\leq\frac{C}{A^{1/2}|y|_{a}^{2}}.

Morever,

|\gamma_{p3}-I_{p3}|+|\gamma_{p4}-I_{p4}|\leq\frac{C|\mu|}{|y|_{a}^{2}\varrho}.

Proof.

Using the same strategy as in Lemma 4.22, we reduce to the special case $x_{1}=x_{2}=0,\eta_{p}=\sqrt{-1}y_{p}$ . Pairing $(n_{1},n_{2})$ with $(-n_{1},-n_{2})$ in the series (4.26),

(\gamma_{p3}+\gamma_{p4})(\eta_{1},\eta_{2},\mu)=\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\frac{\sqrt{-1}a_{p\bar{q}}y_{q}}{2\pi^{2}A^{1/2}|a_{i\bar{j}}(n_{i}+\sqrt{-1}y_{i})(n_{j}-\sqrt{-1}y_{j})|^{2}}.

We compare this series expression of $\gamma_{p3}+\gamma_{p4}$ to the closely related integral (cf. Lemma 4.21)

\begin{split}\frac{\sqrt{-1}a_{p\bar{q}}y_{q}}{2\pi^{2}A^{1/2}}I_{03}=\frac{\sqrt{-1}a_{p\bar{q}}y_{q}\sqrt{\mathbb{A}}}{2\pi A^{3/2}|y|_{a}^{2}}.\end{split}

The deviation between the series and the integral is bounded by

C|a_{p\bar{q}}y_{q}|A^{-1/4}\int_{\mathbb{R}^{2}}\frac{1}{|a_{i\bar{j}}(s_{i}+\sqrt{-1}y_{i})(s_{j}-\sqrt{-1}y_{j})|^{5/2}}ds_{1}ds_{2}\leq\frac{C}{A^{1/2}|y|_{a}^{2}},

using the same type of Cauchy integral test argument as Lemma 4.1.

The ‘morever’ statement is a minor variant of the proof of Lemma 4.22, where in the application of the Cauchy integral test we use the mean value inequality to estimate the difference between the series and the integral, similar to the argument in Lemma 4.1. ∎

We would like to use Lemma 4.22, 4.23 to construct functions $\beta_{p3},\beta_{p4}$ as integrals:

(4.27)

\begin{cases}\beta_{p3}(\eta_{1},\eta_{2},\mu)=-2\pi A^{1/2}\int_{S}\gamma_{p3}(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime}),\\ \beta_{p4}(\eta_{1},\eta_{2},\mu)=-2\pi A^{1/2}\int_{S}\gamma_{p4}(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime}).\end{cases}

where we recall $d\mathcal{A}$ is the area form on $S$ . The problem is that these integrals diverge at the three ends of $S$ , and we need to extract some convergent limit to make sense of $\beta_{p3},\beta_{p4}$ , in a fashion rather similar to (4.11).

The ends of $S$ are up to exponentially small errors approximately $\mathfrak{D}_{i}\times S^{1}$ for $i=1,2,3$ . By Lemma 4.22, the expression $\beta_{p3}-\beta_{p4}$ makes sense as an ordinary integral with integrand $\gamma_{p3}-\gamma_{p4}$ thanks to the convergence of $\int^{\infty}\frac{1}{y^{2}}dy$ . It suffices to makes sense of $\beta_{p3}+\beta_{p4}$ . We consider the integral over large bounded regions with a cutoff scale $\Lambda$ ,

\int_{S\cap\{|y^{\prime}|_{a}<\Lambda\}}(\gamma_{p3}+\gamma_{p4})(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime}).

Lemma 4.23 tells us the exact nature of divergence. At the end $\mathfrak{D}_{1}\times S^{1}$ ,

\gamma_{p3}+\gamma_{p4}\sim-\frac{\sqrt{-1}a_{p\bar{q}}y_{q}^{\prime}\sqrt{\mathbb{A}}}{2\pi A^{3/2}|y^{\prime}|_{a}^{2}}\sim-\frac{\sqrt{-1}a_{p\bar{2}}\sqrt{\mathbb{A}}}{2\pi A^{3/2}a_{2\bar{2}}y_{2}^{\prime}},\quad d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime})\sim a_{2\bar{2}}dx_{2}^{\prime}\wedge dy_{2}^{\prime},

so the divergence behaviour of the integral is $\sim-\frac{\sqrt{-1}a_{p\bar{2}}\sqrt{\mathbb{A}}}{2\pi A^{3/2}}\log\frac{\Lambda}{\sqrt{a_{2\bar{2}}}}$ at $\mathfrak{D}_{1}\times S^{1}$ . Similarly, the divergence behaviour is $\sim-\frac{\sqrt{-1}a_{p\bar{1}}\sqrt{\mathbb{A}}}{2\pi A^{3/2}}\log\frac{\Lambda}{\sqrt{a_{1\bar{1}}}}$ at $\mathfrak{D}_{2}\times S^{1}$ , and is $\sim\frac{\sqrt{-1}(a_{p\bar{1}}+a_{p\bar{2}})\sqrt{\mathbb{A}}}{2\pi A^{3/2}}\log\frac{\Lambda}{\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}}}$ at $\mathfrak{D}_{3}\times S^{1}$ . The remarkable fact is that the divergent parts cancel out so that

\lim_{\Lambda\to\infty}\int_{S\cap\{|y^{\prime}|_{a}<\Lambda\}}(\gamma_{p3}+\gamma_{p4})(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime})

converges; geometrically this cancellation comes from some balancing condition on the 3 directional vectors along $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ . The upshot is that $\beta_{p3}$ and $\beta_{p4}$ make sense as improper integrals. The domain of definition for $\beta_{p3}$ is $(\mathbb{C}^{*})^{2}\times\mathbb{R}_{\mu}\setminus\{f_{S}=0,\mu\leq 0\}$ , and for $\beta_{p4}$ it is $(\mathbb{C}^{*})^{2}\times\mathbb{R}_{\mu}\setminus\{f_{S}=0,\mu\geq 0\}$ .

Lemma 4.24.

(Asymptotes as $\mu\to\pm\infty$ ) For any fixed $\eta_{1},\eta_{2}$ ,

\begin{cases}\lim_{\mu\to+\infty}\beta_{p3}(\eta_{1},\eta_{2},\mu)=0,\\ \lim_{\mu\to-\infty}\beta_{p4}(\eta_{1},\eta_{2},\mu)=0.\end{cases}

Morever

\begin{cases}\lim_{\mu\to+\infty}\frac{\partial\beta_{p3}}{\partial\bar{\eta}_{p}}(\eta_{1},\eta_{2},\mu)=0,\\ \lim_{\mu\to-\infty}\frac{\partial\beta_{p4}}{\partial\bar{\eta}_{p}}(\eta_{1},\eta_{2},\mu)=0.\end{cases}

Proof.

We focus on the $\beta_{p3}$ case, and consider $A^{1/4}\mu\gg|\eta_{1}|+|\eta_{2}|+1$ . Using Lemma 4.22, the contribution to $\int\gamma_{p3}d\mathcal{A}$ from the region $\{|\eta_{1}-\eta_{1}^{\prime}|+|\eta_{2}-\eta_{2}^{\prime}|\leq(A^{1/4}\mu)^{1-\epsilon}\}\cap S$ is negligible, where $0<\epsilon\ll 1$ is any small given number. Outside this region $S$ is asymptotic to $\mathfrak{D}_{i}\times S^{1}$ along the three ends up to exponentially small error, and furthermore Lemma 4.23 allows us to replace $\gamma_{p3}$ by $I_{p3}$ without affecting the $\mu\to+\infty$ limit.

We are now left to consider the improper integral

\int_{\cup(\mathfrak{D}_{i}\times S^{1})}I_{p3}(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime}).

Using the formula of $I_{p3}$ in Lemma 4.21, we can simplify further by setting $y_{1}=y_{2}=0$ without affecting the $\mu\to+\infty$ limit. Along the $\mathfrak{D}_{1}\times S^{1}$ end,

\int_{\mathfrak{D}_{1}\times S^{1}\cap\{y_{2}^{\prime}<\frac{\Lambda}{\sqrt{a_{2\bar{2}}}}\}}I_{p3}(-y_{1}^{\prime},-y_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime})=\int_{0}^{\frac{\Lambda}{\sqrt{a_{2\bar{2}}}}}I_{p3}(0,-y_{2}^{\prime},\mu)a_{2\bar{2}}dy_{2}^{\prime},

which we compute as

\begin{split}&\frac{-a_{p\bar{2}}\sqrt{-1}}{4\pi A^{1/2}\sqrt{\mathbb{A}}}\int_{0}^{\frac{\Lambda}{\sqrt{a_{2\bar{2}}}}}a_{2\bar{2}}y_{2}^{\prime}dy_{2}^{\prime}\frac{1}{|(0,-y_{2}^{\prime},\mu)|_{a}^{\prime}(|(0,-y_{2}^{\prime},\mu)|_{a}^{\prime}+A^{1/2}\mu)}\\ =&\frac{-a_{p\bar{2}}\sqrt{-1}\sqrt{\mathbb{A}}}{8\pi A^{3/2}}\int_{A\mu^{2}}^{\frac{A}{\mathbb{A}}\Lambda^{2}+A\mu^{2}}\frac{ds}{s^{1/2}(s^{1/2}+A^{1/2}\mu)}\\ =&\frac{-a_{p\bar{2}}\sqrt{-1}\sqrt{\mathbb{A}}}{4\pi A^{3/2}}\log\left(\frac{(\mathbb{A}^{-1}\Lambda^{2}+\mu^{2})^{1/2}+\mu}{2\mu}\right).\end{split}

Similarly, the integrals from $\mathfrak{D}_{2}\times S^{1}$ and $\mathfrak{D}_{3}\times S^{1}$ are respectively

\frac{-a_{p\bar{1}}\sqrt{-1}\sqrt{\mathbb{A}}}{4\pi A^{3/2}}\log\left(\frac{(\mathbb{A}^{-1}\Lambda^{2}+\mu^{2})^{1/2}+\mu}{2\mu}\right)

and

\frac{(a_{p\bar{1}}+a_{p\bar{2}})\sqrt{-1}\sqrt{\mathbb{A}}}{4\pi A^{3/2}}\log\left(\frac{(\mathbb{A}^{-1}\Lambda^{2}+\mu^{2})^{1/2}+\mu}{2\mu}\right).

Summing over the three contributions and take the limit $\Lambda\to+\infty$ ,

\int_{\cup(\mathfrak{D}_{i}\times S^{1})}I_{p3}(-y_{1}^{\prime},-y_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime})=0.

This proves $\lim_{\mu\to+\infty}\beta_{p3}(\eta_{1},\eta_{2},\mu)=0$ . Likewise with the $\beta_{p4}$ case.

The ‘morever’ statement follows from a simpler argument. The key is that higher derivatives of the integrand have faster decay at large distance, so that the divergence issues do not arise. ∎

Lemma 4.25.

The explicit formula for $\beta_{p3}+\beta_{p4}$ is

\beta_{p3}+\beta_{p4}=\frac{-2\pi ie^{2\pi i\eta_{p}}}{1-e^{2\pi i\eta_{1}}-e^{2\pi i\eta_{2}}}+K_{p}(a)=\frac{-2\pi ie^{2\pi i\eta_{p}}}{f_{S}}+K_{p}(a),\quad p=1,2.

where the constant $K_{p}(a)$ is

\begin{split}K_{p}(a)=&\frac{\sqrt{-1}(a_{p\bar{2}}\text{Re}(a_{1\bar{2}})-a_{p\bar{1}}a_{2\bar{2}})}{A}(\frac{\pi}{2}+\arctan(\frac{a_{2\bar{2}}+\text{Re}(a_{1\bar{2}})}{\sqrt{\mathbb{A}}}))\\ +&\frac{\sqrt{-1}(a_{p\bar{1}}\text{Re}(a_{1\bar{2}})-a_{p\bar{2}}a_{1\bar{1}})}{A}(\frac{\pi}{2}+\arctan(\frac{a_{1\bar{1}}+\text{Re}(a_{1\bar{2}})}{\sqrt{\mathbb{A}}})).\end{split}

Proof.

The basic strategy is a Liouville theorem argument: we will construct a function with the same distributional $\Delta_{a}$ -Laplacian as $\beta_{p3}+\beta_{p4}$ , and then argue they must be equal.

We start with the Poincaré-Lelong formula

S=\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log|1-e^{2\pi i\eta_{1}}-e^{2\pi i\eta_{1}}|^{2}=\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log|f_{S}|^{2},

from which we obtain the equality of measures

\begin{split}\int_{S}d\mathcal{A}=&\int_{S}\frac{\sqrt{-1}}{2}a_{p\bar{q}}d\eta_{p}\wedge d\bar{\eta}_{q}\\ =&\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log|f_{S}|^{2}\wedge\frac{\sqrt{-1}}{2}a_{p\bar{q}}d\eta_{p}\wedge d\bar{\eta}_{q}\\ =&\frac{1}{4\pi}(\Delta_{a}\log|f_{S}|^{2})d\text{Vol}_{a}.\end{split}

The periodic Newtonian potential on $(\mathbb{C}^{*})^{2}$ with the $g_{a}$ -metric is

\gamma_{5}(\eta_{1},\eta_{2},\mu)=-\frac{1}{4\pi^{2}}\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\{\frac{1}{a_{p\bar{q}}(\eta_{p}+n_{p})(\bar{\eta}_{q}+n_{q})}-\frac{1}{a_{p\bar{q}}n_{p}n_{q}}\}.

Thus for any large cutoff scale $\Lambda$ , the Green’s representation

\int_{S\cap\{|y^{\prime}|_{a}<\Lambda\}}\gamma_{5}(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime})

has the same distributional $\Delta_{a}$ -Laplacian as that of $\frac{1}{4\pi}\log|f_{S}|^{2}$ in the large compact region. Taking the $\eta_{p}$ derivative and taking the $\Lambda\to\infty$ limit shows that the $\Delta_{a}$ -Laplacian of $\frac{-ie^{2\pi i\eta_{p}}}{2f_{S}}$ agrees with that of the improper integral

\int_{S}\frac{\partial}{\partial\eta_{p}}\gamma_{5}(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime}),

which by formula (4.26) is the same as the improper integral

-\frac{1}{2}A^{1/2}\int_{S}(\gamma_{p3}+\gamma_{p4})(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime})=\frac{1}{4\pi}(\beta_{p3}+\beta_{p4}).

The upshot is that $\beta_{p3}+\beta_{p4}$ differs from $\frac{-2\pi ie^{2\pi i\eta_{p}}}{f_{S}}$ by a globally smooth $\Delta_{a}$ -harmonic function on $(\mathbb{C}^{*})^{2}$ . It is also easy to show using techniques in this Section that this difference can have at most log growth in $y_{1},y_{2}$ variables. Thus it has to be a constant.

The rest of this proof is to pin down precisely this constant, by considering the limit $(\eta_{1},\eta_{2})=(\sqrt{-1}b,\sqrt{-1}b)$ for $b\to+\infty$ . This uses techniques similar to the proof of Lemma 4.24. Without affecting the limit, we can replace $S$ with $\cup\mathfrak{D}_{i}\times S^{1}$ and replace $(\gamma_{p3}+\gamma_{p4})$ with

\frac{\sqrt{-1}a_{p\bar{q}}(b-y_{q}^{\prime})\sqrt{\mathbb{A}}}{2\pi A^{3/2}|y-y^{\prime}|_{a}^{2}}.

This leads to an asymptotic expression for $b\gg 1$ ,

(\beta_{p3}+\beta_{p4})(\sqrt{-1}b,\sqrt{-1}b)\sim\int_{\cup\mathfrak{D}_{i}\times S^{1}}\frac{\sqrt{-1}a_{p\bar{q}}(-b+y_{q}^{\prime})\sqrt{\mathbb{A}}}{A|y-y^{\prime}|_{a}^{2}}d\mathcal{A},

where the RHS is understood as an improper integral. To evaluate this integral we fix $b$ and calculate the $\Lambda\to+\infty$ asymptotic expression of the integral over the large bounded domain $(\cup\mathfrak{D}_{i}\times S^{1})\cap\{|y^{\prime}|_{a}<\Lambda\}.$ The contribution from the end $\mathfrak{D}_{1}\times S^{1}$ is

\begin{split}&\frac{\sqrt{-1}a_{p\bar{2}}\sqrt{\mathbb{A}}}{2A}\log(\frac{\Lambda^{2}}{(a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}})b^{2}})\\ +&\frac{\sqrt{-1}(a_{p\bar{2}}\text{Re}a_{1\bar{2}}-a_{p\bar{1}}a_{2\bar{2}})}{A}(\frac{\pi}{2}+\arctan(\frac{a_{2\bar{2}}+\text{Re}(a_{1\bar{2}})}{\sqrt{\mathbb{A}}}))+o(1).\end{split}

The contribution from $\mathfrak{D}_{2}\times S^{1}$ is

\begin{split}&\frac{\sqrt{-1}a_{p\bar{1}}\sqrt{\mathbb{A}}}{2A}\log(\frac{\Lambda^{2}}{(a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}})b^{2}})\\ +&\frac{\sqrt{-1}(a_{p\bar{1}}\text{Re}a_{1\bar{2}}-a_{p\bar{2}}a_{1\bar{1}})}{A}(\frac{\pi}{2}+\arctan(\frac{a_{1\bar{1}}+\text{Re}(a_{1\bar{2}})}{\sqrt{\mathbb{A}}}))+o(1).\end{split}

The contribution from $\mathfrak{D}_{3}\times S^{1}$ is

-\frac{\sqrt{-1}\sqrt{\mathbb{A}}(a_{p\bar{1}}+a_{p\bar{2}})}{2A}\log(\frac{\Lambda^{2}}{(a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}})b^{2}})+o(1).

Summing up, the log terms cancel out, so the improper integral

\int_{\cup\mathfrak{D}_{i}\times S^{1}}\frac{\sqrt{-1}a_{p\bar{q}}(-b+y_{q}^{\prime})\sqrt{\mathbb{A}}}{A|y-y^{\prime}|_{a}^{2}}d\mathcal{A},

is equal to the constant $K_{p}(a)$ defined in the statement of the Lemma. This shows limiting value

\lim_{b\to+\infty}(\beta_{p3}+\beta_{p4})(\sqrt{-1}b,\sqrt{-1}b)=K_{p}(a).

Comparing this with

\lim_{b\to+\infty}\frac{-2\pi ie^{2\pi i\eta_{p}}}{f_{S}}(\sqrt{-1}b,\sqrt{-1}b)=0

determines the constant. ∎

Remark 4.6.

The trigonometric factors in $K_{p}(a)$ have elementary geometric interpretations. The Euclidean metric $g_{a}^{\prime}$ induces an inner product on $\mathbb{R}^{2}_{y_{1},y_{2}}$ . Then the angles between the asymptotic directions of $S$ are

\begin{cases}\angle(\mathfrak{D}_{1},\mathfrak{D}_{3})=\frac{\pi}{2}+\arctan(\frac{a_{2\bar{2}}+\text{Re}(a_{1\bar{2}})}{\sqrt{\mathbb{A}}}),\\ \angle(\mathfrak{D}_{2},\mathfrak{D}_{3})=\frac{\pi}{2}+\arctan(\frac{a_{1\bar{1}}+\text{Re}(a_{1\bar{2}})}{\sqrt{\mathbb{A}}}).\end{cases}

Remark 4.7.

We have chosen a special ray $(\eta_{1},\eta_{2})=(\sqrt{-1}b,\sqrt{-1}b)$ to calculate the asymptotic value of $\beta_{p3}+\beta_{p4}$ . More generally $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ divide the plane $\mathbb{R}^{2}_{y_{1},y_{2}}$ into three sectors, and the asymptotic value of function

\beta_{p3}+\beta_{p4}=\frac{-2\pi ie^{2\pi i\eta_{p}}}{1-e^{2\pi i\eta_{1}}-e^{2\pi i\eta_{2}}}+K_{p}(a)

along the ray $\{(y_{1},y_{2})=s\vec{e}\text{ for }s>0\}$ specified by a directional vector $\vec{e}$ depends on which sector $\vec{e}$ belongs to, and can have a jumping discontinuity as we cross $\mathfrak{D}_{i}$ . This is known as Stokes phenomenon in complex analysis.

Proposition 4.26.

The functions $\beta_{p3}$ and $\beta_{p4}$ solve the overdetermined system (4.23). Equivalently, the $(1,0)$ -forms $\zeta_{3},\zeta_{4}$ defined by (4.22) are holomorphic differentials.

Proof.

Starting from the definition of the function $v$ in terms of $\gamma_{i}$ (cf. (4.11)), we can differentiate with respect to $\eta_{p}$ to get

\frac{\partial v}{\partial\eta_{p}}=-2\pi A^{1/2}\int_{S}\frac{\partial\gamma}{\partial\eta_{p}}(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime}).

Using the differential relations in Lemma 4.20,

\begin{split}2\frac{\partial v}{\partial\eta_{p}}&=-2\pi A^{1/2}\int_{S}\frac{\partial\gamma_{p3}}{\partial\mu}(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime})\\ &=-2\pi A^{1/2}\frac{\partial}{\partial\mu}\int_{S}\gamma_{p3}(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime})\\ &=\frac{\partial\beta_{p3}}{\partial\mu},\end{split}

and similarly $2\frac{\partial v}{\partial\eta_{p}}=-\frac{\partial\beta_{p4}}{\partial\mu}.$

Next we study $\frac{\partial\beta_{p3}}{\partial\bar{\eta}_{q}}$ in the complement of $\{f_{S}=0,\mu\leq 0\}$ . We have

\frac{\partial^{2}\beta_{p3}}{\partial\bar{\eta}_{q}\partial\mu}=2\frac{\partial^{2}v}{\partial\eta_{p}\partial\bar{\eta}_{q}}=-\frac{1}{2}\frac{\partial^{2}w^{p\bar{q}}}{\partial\mu\partial\mu},

where the second equality uses the distributional equation (4.6). But by Lemma 4.24, for fixed $\eta_{1},\eta_{2}$ ,

\lim_{\mu\to+\infty}\frac{\partial\beta_{p3}}{\partial\bar{\eta}_{q}}(\eta_{1},\eta_{2},\mu)=0,

and the asymptotes we obtained in Section 4.2, 4.3 easily imply

\lim_{\mu\to+\infty}\frac{\partial w^{p\bar{q}}}{\partial\mu}(\eta_{1},\eta_{2},\mu)=0.

Thus we can integrate from $\mu=+\infty$ to obtain

\frac{\partial\beta_{p3}}{\partial\bar{\eta}_{q}}=-\frac{1}{2}\frac{\partial w^{p\bar{q}}}{\partial\mu}.

A completely parallel argument shows

\frac{\partial\beta_{p4}}{\partial\bar{\eta}_{q}}=\frac{1}{2}\frac{\partial w^{p\bar{q}}}{\partial\mu}.

Finally by integrating the second part of Lemma 4.20 we see

\frac{\partial\beta_{p3}}{\partial\eta_{q}}=\frac{\partial\beta_{q3}}{\partial\eta_{p}},\quad\frac{\partial\beta_{p4}}{\partial\eta_{q}}=\frac{\partial\beta_{q4}}{\partial\eta_{p}},\quad p,q=1,2.

∎

To compute the periods of the integrals $\int\zeta_{3}$ and $\int\zeta_{4}$ , we recall from the topological description (cf. review Section 1.1.4, 1.1.5) that there are 3 generating $S^{1}$ -cycles in $H_{1}(T^{3})$ , one of which is the $S^{1}$ -fibre, and the other two come from lifting $T^{2}\subset(\mathbb{C}^{*})^{2}\times\mathbb{R}_{\mu}$ to the total space, which involve monodromy issues.

Lemma 4.27.

For appropriate choices of constants $0\leq\theta_{31}^{\infty},\theta_{32}^{\infty}\leq 2\pi$ , the $T^{3}$ -periods of the holomorphic differentials

(4.28)

\begin{cases}d\log z_{3}=\zeta_{3}-\sqrt{-1}(\theta_{13}^{\infty}d\eta_{1}+\theta_{23}^{\infty}d\eta_{2})\\ d\log z_{4}=\zeta_{4}+\sqrt{-1}(\theta_{13}^{\infty}d\eta_{1}+\theta_{23}^{\infty}d\eta_{2})-(K_{1}(a)d\eta_{1}-K_{2}(a)d\eta_{2})\end{cases}

take values in $2\pi\sqrt{-1}\mathbb{Z}$ ; here $K_{p}(a)$ are the constants defined in Lemma 4.25. In particular, the holomorphic functions $z_{3}$ and $z_{4}$ are defined without multivalue issues. For a suitable choice of multiplicative normalisation on $z_{3},z_{4}$ we have the functional equation

(4.29)

z_{3}z_{4}=f_{S}=1-z_{1}-z_{2}=1-e^{2\pi i\eta_{1}}-e^{2\pi i\eta_{2}}.

Proof.

This Lemma is parallel to Lemma 3.10, so we will only highlight the key issues. The constants $\theta_{13}^{\infty}$ and $\theta_{23}^{\infty}$ are the asymptotic holonomy as $\mu\to+\infty$ of the $S^{1}$ -connection $\vartheta$ , along the $S^{1}$ -cycles in the base corresponding to the $x_{1}$ and $x_{2}$ variables respectively. These are introduced in order to cancel the twist of $\vartheta$ by a flat connection.

The functional equation follows from

\begin{split}\log z_{3}+\log z_{4}=&(\beta_{13}+\beta_{14}-K_{1}(a))d\eta_{1}+(\beta_{23}+\beta_{24}-K_{2}(a))d\eta_{2}\\ =&-2\pi i\frac{e^{2\pi i\eta_{1}}d\eta_{1}+e^{2\pi i\eta_{2}}d\eta_{2}}{f_{S}}\\ =&d\log f_{S}.\end{split}

which crucially uses Lemma 4.25. ∎

We have thus defined a holomorphic map away from the singular locus of the $S^{1}$ -fibration on the negative vertex $M^{-}$ :

M^{-}\setminus S\to\{z_{3}z_{4}=1-z_{1}-z_{2}\}\subset\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}\times\mathbb{C}^{2}_{z_{3},z_{4}}.

Here the functional equation allows us to extend the map holomorphically across $\{\mu\neq 0,f_{S}=0\}$ . However the complex structure on $M^{-}$ is not a priori defined along $S\cap M^{-}$ .

Lemma 4.28.

The holomorphic functions $z_{3},z_{4}$ on $M^{-}$ extend continuously over the singular locus $S$ where they attain the value zero. Morever $z_{3},z_{4}$ are $C^{2,\alpha}$ -regular with respect to $g^{(1)}$ -metric.

Proof.

By construction $\log|z_{3}|$ is a function of $\eta_{1},\eta_{2},\mu$ with differential

d\log|z_{3}|=V_{(1)}d\mu+\text{Re}(\beta_{13}d\eta_{1}+\beta_{23}d\eta_{2})+\text{Im}(\theta^{\infty}_{13}d\eta_{1}+\theta_{23}^{\infty}d\eta_{2}).

In particular the positivity of $V_{(1)}$ in $M^{-}$ means $\log|z_{3}|$ is increasing in $\mu$ . Around a given point $P\in S\cap M^{-}$ , we first show continuity of $z_{3}$ at $P$ . Observe

\log|z_{3}|(\eta_{1},\eta_{2},\mu)=\log|z_{3}|(\eta_{1},\eta_{2},A^{-1/4})+\int_{A^{-1/4}}^{\mu}V_{(1)}d\mu.

Here $\log|z_{3}|(\eta_{1},\eta_{2},\mu=A^{-1/4})$ is locally $L^{\infty}$ by smoothness of $\log z_{3}$ in $\{\mu>0\}$ . Applying Proposition 4.16 and neglecting all locally bounded terms, as $(\eta_{1},\eta_{2},\mu)\to(\eta_{1}(P),\eta_{2}(P),0)$ ,

\log|z_{3}|\sim\int_{A^{-1/4}}^{\mu}\frac{A^{1/2}}{2R}d\mu\sim\frac{1}{2}\log(\frac{A^{1/2}\mu+R}{A^{1/4}})\to-\infty,

or equivalently $|z_{3}|\to 0$ as required. The case of $z_{4}$ is completely analogous.

Since $(g^{(1)},\Omega^{(1)})$ is $C^{1,\alpha}$ -regular by Proposition 4.19, holomorphicity implies that $z_{3},z_{4}$ are $C^{2,\alpha}$ -regular in the local chart of Section 4.4. ∎

Proposition 4.29.

The map $M^{-}\to\{z_{3}z_{4}=1-z_{1}-z_{2}\}$ is a holomorphic open embedding. The $S^{1}$ -action is identified as

e^{i\theta}\cdot(z_{1},z_{2},z_{3},z_{4})=(z_{1},z_{2},e^{i\theta}z_{3},e^{-i\theta_{1}}z_{4}),

and the holomorphic volume form is $\Omega^{(1)}=-\frac{\sqrt{-1}}{4\pi^{2}z_{1}z_{2}}dz_{2}\wedge dz_{3}\wedge dz_{4}$ .

Proof.

The $S^{1}$ -action can be identified as in Proposition 2.11. The holomorphic volume form is characterised by $\iota_{\frac{\partial}{\partial\theta}}\Omega^{(1)}=d\eta_{1}\wedge d\eta_{2}$ , which is compared to

\begin{split}\iota_{\frac{\partial}{\partial\theta}}(\sqrt{-1}dz_{2}\wedge dz_{3}\wedge dz_{4})=&-d(z_{3}z_{4})\wedge dz_{2}=-d(1-z_{1}-z_{2})\wedge dz_{2}=dz_{1}\wedge dz_{2}\\ =&(2\pi\sqrt{-1}z_{1}d\eta_{1})\wedge(2\pi\sqrt{-1}z_{2}d\eta_{2})=-4\pi^{2}z_{1}z_{2}d\eta_{1}\wedge d\eta_{2},\end{split}

to yield $\Omega^{(1)}=-\frac{\sqrt{-1}}{4\pi^{2}z_{1}z_{2}}dz_{2}\wedge dz_{3}\wedge dz_{4}$ .

This holomorphic volume form formula in particular shows the map $M^{-}\to\{z_{3}z_{4}=1-z_{1}-z_{2}\}$ is a local biholomorphism wherever the complex structure is defined. We finally need to show this map is injective. Since both $M^{-}$ and $\{z_{3}z_{4}=1-z_{1}-z_{2}\}$ fibre over $\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}$ in a compatible way, it suffices to compare the $\mathbb{C}^{*}$ -fibres. The map between the fibres is equivariant with respect to the $S^{1}$ -action, so to conclude injectivity we only need to recall from the proof of Lemma 4.28 that $\log|z_{3}|$ is a monotone function of $\mu$ . ∎

4.7. Weighted Hölder norms and initial error estimate

The following few Sections are aimed at perturbing the Kähler ansatz into a Calabi-Yau metric. This Section sets up the weighted Hölder norms and measure the volume form error

(4.30)

E^{(1)}=\frac{\det(W^{p\bar{q}}_{(1)})}{V_{(1)}}-1=\frac{A+Aa^{p\bar{q}}w^{p\bar{q}}+\det(w^{p\bar{q}})}{A+v}-1=\frac{\det(w^{p\bar{q}})}{A+v}.

There are three weight parameters:

\varrho=|(y_{1},y_{2},\mu)|_{a}^{\prime},\quad R=\text{dist}_{g_{a}}(\cdot,S),\quad\tilde{\ell}=\kappa_{a}\text{dist}_{g_{a}^{\prime}}(\cdot,\text{Im}(S)).

The parameter $\tilde{\ell}$ is useful for measuring exponential decay rates (cf. Proposition 4.13). The following definitions are parallel to Section 3.4.

Let $\delta\leq 0$ . We shall define the weighted Hölder norms $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ for $S^{1}$ -invariant tensor fields $T$ on $M^{-}$ , by prescribing the norm on a number of overlapping regions up to uniform equivalence.

•

The region $\{R\lesssim A^{1/4}\}$ is covered by local charts $\{r\lesssim A^{1/4}\}$ introduced in Section 4.4 and 4.5, where the ansatz metric is approximated by $g_{\text{NUT}}$ . Let $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ be uniformly equivalent to the norm $\left\lVert\Psi^{*}T\right\rVert_{C^{k,\alpha}_{\delta}(g_{\text{NUT}})}$ in Section 4.5. Inside $\{R\lesssim A^{-1/2}\}$ the metric ansatz is $C^{1,\alpha}$ -regular, so correspondingly we should work with functions of at most $C^{2,\alpha}$ -regularity and tensors of at most $C^{1,\alpha}$ -regularity. Inside $\{R\gtrsim A^{-1/2}\}$ there is no restriction on regularity.
•

The region $\{\tilde{\ell}\gtrsim 1\}$ can be covered by subregions of diameter $\sim R$ , where the $S^{1}$ -bundle is topologically trivial. Over each subregion the metric is approximated by the periodic version of the constant solution $g_{\text{flat}}$ (cf. Section 2.2). The $x_{1},x_{2}$ variables define two periodic direction. We decompose $T$ into the part $\bar{T}$ independent of $x_{1},x_{2}$ (the ‘zeroth Fourier mode’) and the oscillatory part $T-\bar{T}$ (the ‘higher Fourier mode’), and define the weighted Hölder norm separately on the two parts:
•

On the zeroth Fourier mode, the norm $\left\lVert\bar{T}\right\rVert_{C^{k,\alpha}_{\delta,0}}$ is equivalent to

$A^{-3\delta/4}(\sum_{j=0}^{k}\left\lVert R^{j}\nabla^{j}\bar{T}\right\rVert_{L^{\infty}}+[R^{k}\nabla^{k}\bar{T}]_{\alpha}),$

where $[]_{\alpha}$ denotes the appropriately normalised Hölder seminorm.

•

On the higher Fourier modes we build in the exponential decay. Fix a parameter $0<\kappa<1$ . The norm $\left\lVert T-\bar{T}\right\rVert_{C^{k,\alpha}_{\delta,0}}$ in this region is equivalent to

A^{-3\delta/4}\sup_{\tilde{\ell}\gtrsim 1}e^{\kappa\tilde{\ell}}(\sum_{j=0}^{k}\left\lVert A^{j/4}\nabla^{j}(T-\bar{T})\right\rVert_{L^{\infty}}+A^{k/4}[\nabla^{k}(T-\bar{T})]_{\alpha}).

An estimate in this norm is the higher order version of $|T-\bar{T}|\leq CA^{3\delta/4}e^{-\kappa\tilde{\ell}}.$

Notation.

The norm $\left\lVert\cdot\right\rVert_{C^{k,\alpha}_{\delta,0}}$ can refer to any type of tensors depending on the context, such as functions, 1-forms, symmetric 2-tensors, and in some cases can refer to the norm computed in a subregion. Strictly speaking this norm depends on $\kappa$ , but we suppress this to avoid cluttering the notation.

We will also need a variant weighted Hölder norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta}}$ . The only difference from $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ is that in the region $\{\tilde{\ell}\gtrsim 1\}$ on the zeroth Fourier mode, $\left\lVert\bar{T}\right\rVert_{C^{k,\alpha}_{\delta}}$ is equivalent to

A^{-\delta/2}(\sum_{j=0}^{k}\left\lVert R^{j-\delta}\nabla^{j}\bar{T}\right\rVert_{L^{\infty}}+[R^{k-\delta}\nabla^{k}\bar{T}]_{\alpha}),

so an estimate in this norm is the higher order version of $|\bar{T}|=O(A^{3\delta/4}(A^{-1/4}R)^{\delta})$ . We have inserted an extra decay factor $(A^{-1/4}R)^{\delta}$ .

Notation.

For a parameter $\nu$ with $1\ll\nu<\epsilon_{0}A^{3/4}$ , define the subregion of $M^{-}$

M^{-}_{\nu}=\{A^{-1/4}\varrho<e^{\nu}\}\subset M^{-}.

Its base is $\mathcal{B}^{-}_{\nu}=\{A^{-1/4}\varrho<e^{\nu}\}\subset(\mathbb{C}^{*})^{2}_{\eta_{1},\eta_{2}}\times\mathbb{R}_{\mu}$ .

Lemma 4.30.

The volume form error $E^{(1)}$ satisfies the estimate on $M^{-}_{\nu}$ :

\left\lVert E^{(1)}\right\rVert_{C^{1,\alpha}_{-1,0}}\leq CA^{-3/4}\nu^{2}.

In the subregion $\{R\gtrsim A^{1/4}\}\subset M^{-}_{\nu}$ , and any fixed large $k$ ,

\left\lVert E^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq CA^{-3/4}\nu^{2}.

Proof.

(Sketch) In the region $\{R\lesssim A^{1/4}\}$ the volume form estimate follows from Proposition 4.16 and 4.18. In the region $\{R\gtrsim A^{1/4}\}$ the absolute estimate follows by combining Section 4.2, 4.3, notably the exponential decay estimate in Proposition 4.13, and the higher order estimate uses the $\Delta_{a}$ -harmonicity on $w^{p\bar{q}}$ . ∎

4.8. Harmonic analysis I: periodic Euclidean region

The refined mapping properties of the Euclidean Green operator $\Delta_{a}^{-1}$ on $(\mathbb{C}^{*})^{2}_{\eta_{1},\eta_{2}}\times\mathbb{R}_{\mu}$ follow Section 3.5 almost verbatim:

Proposition 4.31.

(Periodic Euclidean region) Let $-3<\delta<0$ . Let $f$ be a function compactly supported in $\mathcal{B}^{-}_{\nu}\cap\{R>A^{-1/2}\}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq 1$ (respectively $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta}}\leq 1$ ). Then $\Delta_{a}^{-1}f$ satisfies the $g_{a}$ -Hessian bound on $\mathcal{B}^{-}_{\nu}\cap\{R\gtrsim A^{-1/2}\}$ ,

\left\lVert\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq C\nu,\quad\text{resp. }\left\lVert\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta}}\leq C.

Remark 4.8.

The regularity of $\Delta_{a}^{-1}f$ in $\{R\lesssim A^{-1/2}\}$ is well controlled by $\Delta_{a}$ -harmonicity.

This allows us to correct the volume form error sufficiently away from $S$ as in proposition 3.23. From now on $1\ll\nu\ll A^{3/8}$ .

Proposition 4.32.

Let $1\ll\nu\ll A^{3/8}$ . Then there is a real valued function $\varphi_{1}$ on $\mathcal{B}^{-}_{\nu}$ , solving the generalised Gibbons-Hawking equation on $\mathcal{B}^{-}_{\nu}\cap\{\tilde{\ell}>2\}$

V_{(2)}=V_{(1)}+\frac{\partial^{2}\varphi_{1}}{\partial\mu\partial\mu},\quad W^{p\bar{q}}_{(2)}=W^{p\bar{q}}_{(1)}-4\frac{\partial^{2}\varphi_{1}}{\partial\eta_{p}\partial\bar{\eta}_{q}},\quad\det(W^{p\bar{q}}_{(2)})=V_{(2)}.

Morever $\varphi_{1}$ is $\Delta_{a}$ -harmonic on $\mathcal{B}^{-}_{\nu}\cap\{\tilde{\ell}<1\}$ , and

\left\lVert\nabla^{2}_{g_{a}}\varphi_{1}\right\rVert_{C^{k,\alpha}_{-1,0}(\mathcal{B}^{-}_{\nu}\cap\{\tilde{\ell}\gtrsim 1\})}\leq C\nu^{3}A^{-3/4},

and $|\nabla^{2}_{g_{a}}\varphi_{1}|_{g_{a}}\leq C\nu^{3}A^{-3/2}$ on $\mathcal{B}^{-}_{\nu}$ . In particular the matrix $(W^{p\bar{q}}_{(2)})$ is positive definite and $V_{(2)}$ is positive on $\mathcal{B}^{-}_{\nu}$ .

We obtain by the generalised Gibbons-Hawking construction $(g^{(2)},\omega^{(2)},J,\Omega)$ associated to the data $V_{(2)}$ and $W^{p\bar{q}}_{(2)}$ , and identify its ambient space as $M^{-}_{\nu}$ . The new $S^{1}$ -connection $\vartheta^{(2)}$ is related to $\vartheta$ by

\vartheta^{(2)}-\vartheta=\sqrt{-1}(\frac{\partial^{2}\varphi^{-}}{\partial\eta_{p}\partial\mu}d\eta_{p}-\frac{\partial^{2}\varphi^{-}}{\partial\bar{\eta}_{p}\partial\mu}d\bar{\eta}_{p}).

The new volume form error $E^{(2)}$ is supported in $\{\ell\lesssim 1\}$ with bound

(4.31)

E^{(2)}=\frac{\det(W^{p\bar{q}}_{(2)})}{V_{(2)}}-1,\quad\left\lVert E^{(2)}\right\rVert_{C^{1,\alpha}_{-1}(M^{-}_{\nu})}\leq CA^{-3/4}\nu^{2},

and in particular $\left\lVert E^{(2)}\right\rVert_{C^{\alpha}_{-1-\epsilon}(M^{-}_{\nu})}\leq CA^{-3/4}\nu^{2}$ .

Henceforth the holomorphic structures will be fixed, and can be identified building on results in Section 4.6. The new holomorphic differentials $d\log Z_{3},d\log Z_{4}$ are related to $d\log z_{3},d\log z_{4}$ by

\begin{cases}d\log Z_{3}=d\log z_{3}+d(\frac{\partial\varphi_{1}}{\partial\mu}),\\ d\log Z_{4}=d\log z_{4}-d(\frac{\partial\varphi_{1}}{\partial\mu}),\end{cases}

whence we find holomorphic coordinates by integration

(4.32)

Z_{3}=z_{3}\exp(\frac{\partial\varphi_{1}}{\partial\mu}),\quad Z_{4}=z_{4}\exp(-\frac{\partial\varphi_{1}}{\partial\mu}),

which satisfy the functional equation

Z_{3}Z_{4}=z_{3}z_{4}=1-z_{1}-z_{2}=1-e^{2\pi i\eta_{1}}-e^{2\pi i\eta_{2}}.

Following Lemma 4.28 and Proposition 4.29,

Proposition 4.33.

(Holomorphic structure) The map

M^{-}_{\nu}\to\{Z_{3}Z_{4}=1-z_{1}-z_{2}\}\subset\mathbb{C}^{2}_{Z_{3},Z_{4}}\times(\mathbb{C}^{*})^{2}_{z_{1},z_{2}}

extends continuously over the singular locus $S\cap M^{-}_{\nu}$ and defines a holomorphic open embedding under the complex structure $J$ . The $S^{1}$ -action is identified as

e^{i\theta}\cdot(z_{1},z_{2},Z_{3},Z_{4})=(z_{1},z_{2},e^{i\theta}Z_{3},e^{-i\theta}Z_{4}),

and the holomorphic volume form is $\Omega=-\frac{\sqrt{-1}}{4\pi^{2}z_{1}z_{2}}dz_{2}\wedge dZ_{3}\wedge dZ_{4}.$ The Kähler structure is $C^{1,\alpha}$ -regular near $S$ .

Proposition 4.34.

(Symplectic structure) The integral $\int_{T^{2}}\omega^{(2)}=\text{Im}(a_{2\bar{1}}).$

Proof.

The new Kähler form $\omega^{(2)}$ is cohomologous to $\omega^{(1)}$ by the formula

\omega_{-}=\omega^{(1)}+d(\sqrt{-1}\frac{\partial\varphi^{-}}{\partial\eta_{p}}d\eta_{p}-\sqrt{-1}\frac{\partial\varphi^{-}}{\partial\bar{\eta}_{p}}d\bar{\eta}_{p}).

The claim then follows from Lemma 4.14. ∎

4.9. Harmonic analysis II: Neighbourhood of $S$

This Section approximately inverts $\mathcal{L}$ for source functions supported in the vicinity of $S$ .

Recall the model metric $g_{\text{NUT}}$ defined in (4.20) as the product of the Taub-NUT metric with flat $\mathbb{C}$ . Denote $r=\sqrt{A(\mu_{1}^{2}+|\xi_{1}|^{2}+|\xi_{2}|^{2})}$ in the local chart around $P\in S$ , as in Section 4.4.

Lemma 4.35.

(Model Laplacian) Given $0<\epsilon\ll 1$ , let $f$ be an $S^{1}$ -invariant function on the model space supported in $\{r\lesssim A^{-1/2}\}$ , with bound $\left\lVert f\right\rVert_{C^{k,\alpha}_{0}(g_{\text{NUT}})}\leq 1$ . Then there is a function $u$ compactly supported in $\{r\lesssim A^{1/4}\}$ , such that on an annulus region at any given dyadic scale $r\sim r_{0}$ (or the ball region $r\lesssim A^{-1/2}$ ) we have a decay estimate

\left\lVert u\right\rVert_{C^{k+2,\alpha}_{0}(r\sim r_{0},g_{\text{NUT}})}\leq CA^{-1}(A^{1/2}r_{0}+1)^{-3+\epsilon},

and $\Delta_{g_{\text{NUT}}}u-f$ is only supported on one dyadic scale $\{r\sim A^{1/4}\}$ with the bound

\left\lVert\Delta_{g_{\text{NUT}}}u-f\right\rVert_{C^{k,\alpha}_{-2}(g_{\text{NUT}})}\leq CA^{3(-3+\epsilon)/4}.

Proof.

Let ${G}_{\text{NUT}}$ be the Green operator on the model space, so $u^{\prime}={G}_{\text{NUT}}f$ is an $S^{1}$ -invariant function. Applying Hein’s package on Poisson equations as in Corollary 2.17 with a simple scaling argument, the function $u^{\prime}$ decays like

|u^{\prime}|\leq CA^{-1}(A^{1/2}r+1)^{-3+\epsilon}.

The decay exponent $-3+\epsilon$ here comes from the quintic volume growth rate of $g_{\text{NUT}}$ . Since the model space is smooth, we can bootstrap this to a weighted $C^{2,\alpha}$ estimate on $u^{\prime}$ . The function $u$ is obtained by cutting off $u^{\prime}$ at a dyadic scale $r\sim A^{1/4}$ . The cutoff error is controlled by the Hessian estimate. ∎

The next Lemma patches together a large number of local parametrices to produce an approximate right inverse of $\Delta_{g^{(2)}}$ in a neighbourhood of $S$ , with sufficiently fast decay estimates. Its proof is similar to Lemma 2.20.

Lemma 4.36.

(Neighbourhood of $S$ ) Given $0<\epsilon\ll 1$ and $-3+\epsilon<\delta<0$ , let $f$ be an $S^{1}$ -invariant function compactly supported in $M^{-}_{\nu}\cap\{R\lesssim A^{-1/2}\}$ with bound $\left\lVert f\right\rVert_{C^{\alpha}_{\delta}}\leq 1$ . Then there is a function $u$ on $\mathcal{B}$ supported in $\{R\lesssim A^{1/4}\}$ , with decay estimates $\left\lVert u\right\rVert_{C^{2,\alpha}_{-1+\epsilon}}\leq CA^{-1},$ such that $\left\lVert\Delta_{g^{(2)}}u-f\right\rVert_{C^{\alpha}_{\delta}}\ll 1.$

Proof.

Take a large collection of points $\{P_{i}\}_{i=1}^{N}$ on $S\cap\mathcal{B}^{-}_{\nu}$ , such that for any point $P$ on $S\cap\mathcal{B}^{-}_{\nu}$ , the number of points $P_{i}$ in the collection within $g_{a}$ -distance $O(A^{-1/2})$ to $P$ is at least one but no more than $C$ . Then take cutoff functions $\chi_{i}$ on $\mathcal{B}^{-}_{\nu}$ supported in $\{\text{dist}_{g_{a}}(\cdot,P_{i})\lesssim A^{-1/2}\}$ such that $\sum\chi_{i}=1$ on $\mathcal{B}^{-}_{\nu}\cap\{R\lesssim A^{-1/2}\}$ . These allow us to decompose $f$ into a large number of localised contributions:

f=\sum_{i=1}^{N}\chi_{i}f,\quad\left\lVert\chi_{i}f\right\rVert_{C^{\alpha}_{0}}\leq C,

using the fact that the $C^{\alpha}_{\delta}$ -norm is not sensitive to $\delta$ inside $\{R\lesssim A^{-1/2}\}$ .

For each term $\chi_{i}f$ we apply Lemma 4.35 to produce an approximate local solution $u_{i}$ on $\{r_{P_{i}}\lesssim A^{1/4}\}$ , with bounds prescribed in Lemma 4.35. Here $r_{P_{i}}(Q)$ is uniformly equivalent to $|P_{i}-Q|_{g_{a}}$ . The candidate solution is

u=u_{1}+u_{2}+\ldots u_{N}.

By construction $u$ is supported in $\{R\lesssim A^{1/4}\}$ .

We now bound $u$ , focusing on the absolute estimate. Summing up the contributions

|u_{i}|\leq CA^{-1}(A^{1/2}r_{P_{i}}+1)^{-3+\epsilon}\left\lVert\chi_{i}f\right\rVert_{C^{\alpha}_{0}}\leq CA^{-1}(A^{1/2}r_{P_{i}}+1)^{-3+\epsilon},

we estimate at a point $Q$ :

\begin{split}|u(Q)|\leq&\sum_{|P_{i}-Q|_{g_{a}}\lesssim A^{1/4}}|u_{i}|(Q)\\ \leq&CA^{-1}\sum_{|P_{i}-Q|_{g_{a}}\lesssim A^{1/4}}(A^{1/2}|P_{i}-Q|_{g_{a}}+1)^{-3+\epsilon}\\ \leq&C\int_{S\cap\{|P-Q|_{g_{a}}\lesssim A^{1/4}\}}(A^{1/2}|P-Q|_{g_{a}}+1)^{-3+\epsilon}d\mathcal{A}(P)\\ \leq&CA^{-1}(A^{1/2}R(Q)+1)^{-1+\epsilon}.\end{split}

The higher order version is $\left\lVert u\right\rVert_{C^{2,\alpha}_{-1-\epsilon}}\leq CA^{-1},$ so $\left\lVert u\right\rVert_{C^{2,\alpha}_{\delta}}\leq CA^{-1}$ for $\delta>-3+\epsilon$ .

Next we estimate the error $\Delta_{g^{(2)}}u-f$ . The error $\Delta_{g^{(2)}}u_{i}-\chi_{i}f$ has two sources: the cutoff error supported on $\{r_{P_{i}}\sim A^{1/4}\}$ from Lemma 4.35

\left\lVert\Delta_{g_{\text{NUT}}}u_{i}-\chi_{i}f\right\rVert_{C^{\alpha}_{-2}(g_{\text{NUT}})}\leq CA^{3(-3+\epsilon)/4},

and the metric deviation error $\Delta_{g_{\text{NUT}}}u_{i}-\Delta_{g^{(2)}}f$ , which is controlled because by Proposition 4.19 the local diffeomorphism $\Psi$ is a $C^{1,\alpha}$ -approximate isometry between $g_{\text{NUT}}$ and $g^{(2)}$ , and $f$ has weighted $C^{2,\alpha}$ control. We focus on the absolute estimate:

|\Delta_{g^{(2)}}u_{i}-\Delta_{g_{\text{NUT}}}u_{i}|\leq CA^{-3/4}(A^{1/2}r_{P_{i}}+1)^{-3+\epsilon}(A^{1/2}R+1)^{-2}(A|\xi_{2}|+\nu),

so that

|\Delta_{g^{(2)}}u_{i}-\chi_{i}f|\leq CA^{-3/4}(A^{1/2}r_{P_{i}}+1)^{-3+\epsilon}(A^{1/2}R+1)^{-2}(A^{1/2}r_{P_{i}}+\nu).

Using

\begin{cases}\sum_{|P_{i}-Q|_{g_{a}}\lesssim A^{1/4}}(A^{1/2}r_{P_{i}}(Q)+1)^{-3+\epsilon}\leq(A^{1/2}R(Q)+1)^{-1+\epsilon},\\ \sum_{|P_{i}-Q|_{g_{a}}\lesssim A^{1/4}}(Ar_{P_{i}}(Q)+1)^{-2+\epsilon}\leq CA^{3\epsilon/4},\end{cases}

we sum up all contributions to deduce for $\delta>-3+\epsilon$ ,

\begin{split}&|\Delta_{g^{(2)}}u-f|\leq\sum_{|P_{i}-Q|_{g_{a}}\lesssim A^{1/4}}|\Delta_{g^{(2)}}u_{i}-\chi_{i}f|\\ \leq&CA^{-3/4}\{A^{3\epsilon/4}(A^{1/2}R+1)^{-2}+\nu(A^{1/2}R+1)^{-3+\epsilon}\}\ll(A^{1/2}R+1)^{\delta}.\end{split}

The Hölder version is $\left\lVert\Delta_{g^{(2)}}u-f\right\rVert_{C^{\alpha}_{\delta}}\ll 1$ as required. ∎

4.10. Harmonic analysis III: perturbation to Calabi-Yau metric

We now shift to the complex geometric perspective and solve the complex Monge-Ampère equation by perturbative methods. Below is main result of the linear theory, which is parallel to Proposition 2.23 and 3.32. The idea is to patch together as in Proposition 3.32 the local parametrices provided by Proposition 4.31 and 4.36.

Proposition 4.37.

Given $-3<\delta<-1$ and $1\ll\nu\ll A^{3/8}$ , let $f$ be an $S^{1}$ -invariant function compactly supported in $M^{-}_{\nu}$ with norm $\left\lVert f\right\rVert_{C^{\alpha}_{\delta}}=1$ . Then there is an $S^{1}$ -invariant function $u$ approximately solving the Poisson equation:

\left\lVert\Delta_{g^{(2)}}u-f\right\rVert_{C^{\alpha}_{\delta}(M^{-}_{\nu})}\ll 1,

with the Hessian bound

\left\lVert\nabla^{2}_{g^{(2)}}u\right\rVert_{C^{\alpha}_{\delta}(M^{-}_{\nu})}\leq C,\quad\left\lVert du\right\rVert_{C^{1,\alpha}_{\delta+1}(M^{-}_{\nu})}\leq CA^{-1/2}.

The constants depend only on $\delta,\alpha,\kappa$ and the scale invariant ellipticity bound on $a_{p\bar{q}}$ .

Combined with the initial error estimate (4.31) this allows us to set up a Banach iteration scheme to perturb $\omega^{(2)}$ to a Calabi-Yau metric, parallel to Theorem 3.33. This involves shrinking domain from $M^{-}_{\nu}$ to $M^{-}_{\nu-1}$ and changing $\nu$ to $\nu+1$ .

Theorem 4.38.

(Ooguri-Vafa type metric on the negative vertex) Fix $k,\alpha,\kappa$ and $0<\epsilon\ll 1$ , and let $1\ll\nu\ll A^{3/8}$ . Then there is an $S^{1}$ -invariant Calabi-Yau metric on $M^{-}_{\nu}$ with $S^{1}$ -invariant Kähler potential $\phi^{-}$ ,

\omega_{-}=\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}\phi^{-},\quad\omega_{-}^{3}=\frac{3}{4}\sqrt{-1}\Omega\wedge\overline{\Omega},

with metric deviation estimate $\left\lVert\nabla^{2}_{g^{(2)}}\phi^{-}\right\rVert_{C^{\alpha}_{-1-\epsilon}(\mathcal{B}^{-}_{\nu})}\leq C\nu^{2}A^{-3/4}.$ The constants depend only on $\alpha,\epsilon,\kappa$ and the scale invariant ellipticity bound on $a_{p\bar{q}}$ .

4.11. Ooguri-Vafa type metrics on the negative vertex

We discuss the geometric aspects of the Ooguri-Vafa type metric $(g_{-},\omega_{-},\Omega)$ .

Combining deviation estimates in Proposition 4.13, 4.32 and Theorem 4.38,

Corollary 4.39.

(Exponential decay to semiflat metric away from $S$ ) In the situation of Theorem 4.38, for $\tilde{\ell}\gtrsim 1$ the deviation of $\omega_{-}$ from its zeroth Fourier mode $\overline{\omega}_{-}$ decays exponentially:

|\omega_{-}-\overline{\omega}_{-}|\leq CA^{-3/4}\nu e^{-\kappa\tilde{\ell}}.

The transverse structure along $S$ follows immediately from Proposition 4.19 and the metric deviation estimates.

Corollary 4.40.

(Transverse Taub-NUT metrics) Under suitable gauge choices for the $S^{1}$ -connection, the metric $g_{-}$ restricted over the disc $\{\xi_{2}=0,R\lesssim A^{1/4}\}$ is approximated by the Taub-NUT metric:

|(g_{-}-g_{\text{NUT}})|_{\xi_{2}=0}|_{g_{\text{NUT}}}\leq CA^{-3/4}\nu.

The smooth topology emerges a posteriori after solving the Monge-Ampère equation as a consequence of regularity theory.

Proposition 4.41.

The metric structure $(g_{-},\omega_{-},J,\Omega)$ extends smoothly to a Calabi-Yau metric on $M^{-}_{\nu}$ .

Proof.

The holomorphic coordinates extend across $S$ , so induces a smooth structure on $M^{-}_{\nu}$ . The metric $\omega_{-}$ has $C^{\alpha}$ -regularity and is Calabi-Yau, so standard regularity theory of complex Monge-Ampère equation implies that it is smooth. ∎

4.12. Special Lagrangian geometry

This Section is an informal discussion concerning special Lagrangian 3-tori $L\simeq T^{3}$ on the negative vertex $M^{-}_{\nu}$ .

The generic special Lagrangian 3-tori $L$ are expected to be isotopic to the $T^{3}$ lying over the 2-tori in the 5-dimensional base defined by

(4.33)

(y_{1},y_{2},\mu)=\text{const}.

The Lagrangian requirement then imposes a homological constraint in the light of Proposition 4.34:

(4.34)

\int_{T^{2}}\omega_{-}=\text{Im}(a_{2\bar{1}})=0,

or equivalently $(a_{p\bar{q}})$ is real symmetric. The interpretation is that the Ooguri-Vafa type metrics we constructed on the negative vertex can be the metric model for the SYZ fibration only if the homological constraint is satisfied; when this fails, they may still be the local model for other types of degenerating 3-fold Calabi-Yau metrics which do not admit a global special Lagrangian 3-torus fibration.

From now on in this Section we assume the homological constraint, and proceed to speculate on the geometric features of the special Lagrangian 3-tori $L$ , without attempting to prove existence results.

First, notice that outside a tubular neighbourhood of the singular locus $S$ , the metric $g_{-}$ is a perturbation of the flat model $g_{\text{flat}}$ (cf. Example 1.6), namely the generalised Gibbons-Hawking construction applied to the constant solution

V=A,\quad W^{p\bar{q}}=a_{p\bar{q}}.

On the flat model it is elementary to check that the map to $\mathbb{R}^{3}$ defined by $(y_{1},y_{2},\mu)$ have special Lagrangian fibres, which are flat 3-tori invariant under the $S^{1}$ -action. In other words, to crudest approximation the map

(4.35)

M^{-}_{\nu}\xrightarrow{(y_{1},y_{2},\mu)}\mathbb{R}^{3}

is an approximate special Lagrangian fibration. Most of these $T^{3}$ -fibres stay far away from the curvature radius along $S$ , so it is likely that in the generic region these $T^{3}$ can be perturbed into a genuine special Lagrangian fibration with respect to the Calabi-Yau structure $(g_{-},\omega_{-},J,\Omega)$ , while maintaining the $S^{1}$ -invariance.

Near $S$ the features of the special Lagrangians $L$ have strong resonance with Joyce’s work [14] (cf. Section 1.1.5). It is natural to expect $L$ to be $S^{1}$ -invariant. Around $P\in S$ , the Calabi-Yau structure is transversely modelled on $g_{\text{NUT}}$ (cf. Section 4.5), so the $S^{1}$ -reduction of the special Lagrangian condition

\omega_{-}|_{L}=0,\quad\text{Im}(\Omega_{-})|_{L}=0

approximately reads:

(4.36)

\begin{cases}\mu=\text{constant},\\ \{(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})d\xi_{1}\wedge d\bar{\xi}_{1}+Ad\xi_{2}\wedge d\bar{\xi}_{2}\}|_{L/S^{1}}=0,\\ \text{Im}(d\xi_{1}\wedge d\xi_{2})|_{L/S^{1}}=0.\end{cases}

To render the analogy with Joyce [14] more transparent, we introduce real variables $\tilde{x}_{1},\tilde{x}_{2},\tilde{u}_{1},\tilde{u}_{2}$ such that $\xi_{1}=\tilde{x}_{1}-\sqrt{-1}\tilde{u}_{1},\xi_{2}=\tilde{x}_{2}+\sqrt{-1}\tilde{u}_{2}.$ Representing $L$ locally by

\tilde{u}_{1}=\tilde{u}_{1}(\tilde{x}_{1},\tilde{x}_{2}),\quad\tilde{u}_{2}=\tilde{u}_{2}(\tilde{x}_{1},\tilde{x}_{2}),\quad\mu=\text{constant},

then (4.36) takes the form of the nonlinear Cauchy-Riemann equation

(4.37)

\frac{\partial\tilde{u}_{1}}{\partial\tilde{x}_{1}}=\frac{\partial\tilde{u}_{2}}{\partial\tilde{x}_{2}},\quad\frac{\partial\tilde{u}_{1}}{\partial\tilde{x}_{2}}=-(1+\frac{1}{2A\sqrt{\mu^{2}+\tilde{x}_{1}^{2}+\tilde{u}_{1}^{2}}})\frac{\partial\tilde{u}_{1}}{\partial\tilde{x}_{2}},

which is very similar to the key equations in [14]. Morever $L$ should asymptotically match up with fibres of (4.35) at far distance from $S$ , described by the affine condition (4.33).

We can use this information to speculate on the nature of singularities in line with Joyce [14]. For $\mu\neq 0$ the equations are nonsingular, so the special Lagrangians $L$ will be smooth. When $\mu=0$ the $T^{3}$ -fibres of (4.35) intersect $S$ if and only if $(y_{1},y_{2})$ lies in the amoeba $\text{Im}(S)$ , and we expect a perturbation of such fibres to produce special Lagrangians $L$ with singularities. In the subcase where $(y_{1},y_{2})$ lies in the interior of the amoeba, there are two transverse intersection points $L\cap S$ , at which we expect to create a pair of special Lagrangian $T^{2}$ -cone singularities. At the boundary of the amoeba these two intersection points merge together, and the singularities disappear outside the amoeba.

The fine details of a special Lagrangian $L$ near a transverse intersection point with $S$ is conjecturally modelled by an entire solution to the nonlinear Cauchy-Riemann equation (4.37) over $\mathbb{R}^{2}_{\tilde{x}_{1},\tilde{x}_{2}}$ , which has a local special Lagrangian $T^{2}$ -cone singularity at the origin and is asymptotic to (4.33) at infinity. $L$ is then obtained by gluing this local picture to the corresponding fibres of (4.35) away from $S$ .

A salient feature of Joyce [14] is that the special Lagrangian fibration can fail to be defined by smooth maps (cf. Section 1.1.5). This is compatible with this Chapter. The key point is that the absence of an a priori smooth topology forces us to work with tensors of low regularity (cf. Section 4.5), and the smooth structure along $S$ only emerges a posteriori after solving the Monge-Ampère equation. Thus one neither expects to produce a model special Lagrangian fibration defined by smooth maps, nor expects smoothness properties to persist after perturbation inside function spaces of low regularity.

4.13. Incompleteness and running coupling

The incompleteness of the Ooguri-Vafa type metric on the negative vertex has a strong analogy with the positive vertex as discussed in detail in Section 3.10. The key point is that asymptotes of the first order corrections $v$ and $w^{p\bar{q}}$ lead naturally to a renormalisation flow equation, which in turn predicts the drifting of coupling constants $a_{p\bar{q}}$ over many log scales.

If we follow the discussion of Section 3.10, but replace the asymptote (3.6) by (4.17), then we find that in the following variables

p_{1}=\sqrt{a_{2\bar{2}}},\quad p_{2}=\sqrt{a_{1\bar{1}}},\quad p_{3}=\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}},

the renormalisation flow equation for the negative vertex is given as

(4.38)

\begin{cases}\frac{d}{d\lambda}p_{1}^{2}=-\frac{1}{2p_{2}}-\frac{1}{2p_{3}},\\ \frac{d}{d\lambda}p_{2}^{2}=-\frac{1}{2p_{1}}-\frac{1}{2p_{3}},\\ \frac{d}{d\lambda}p_{3}^{2}=-\frac{1}{2p_{1}}-\frac{1}{2p_{2}},\\ \frac{d}{d\lambda}\text{Im}(a_{2\bar{1}})=0,\end{cases}

where $\lambda$ is the log scale parameter. The rest of this Section is concerned with geometric interpretations.

The renormalisation flow equation implies that

\text{Im}(a_{2\bar{1}})=\text{constant}.

This is compatible with the fact that $\text{Im}(a_{2\bar{1}})$ is the cohomological invariant determined by integrating the Kähler form on the $T^{2}$ -cycle.

More interestingly, the evolution of $p_{1},p_{2},p_{3}$ is formally identical to the renormalisation flow equation (3.16) for the positive vertex. This can be explained in terms of semiflat mirror symmetry (cf. Section 1.1.1) as follows, assuming the homological constraint $\text{Im}(a_{2\bar{1}})=0$ , namely $a_{p\bar{q}}$ is a real symmetric matrix.

In general, given a semiflat SYZ fibration, the mirror SYZ fibration is obtained by replacing the torus fibres by their dual tori, interchanging the symplectic moment coordinates on the SYZ base with the complex affine coordinates on the SYZ base, and keeping the same Riemannian metric on the base. We apply this to the constant solution relevant to the positive vertex case (cf. Example 1.6)

V^{ij}=a_{ij},\quad W=A=\det(a_{ij}),

whose SYZ base is equipped with the Euclidean metric

a_{ij}d\mu_{i}\otimes d\mu_{j}+Ady^{2}

written in the two symplectic moment coordinates $\mu_{1},\mu_{2}$ and a complex affine coordinate $y=\text{Im}(\eta)$ . The SYZ mirror is the constant solution relevant to the negative vertex

W^{i\bar{j}}=a_{ij},\quad V=A,

whose SYZ base is equipped with the Euclidean metric

a_{ij}dy_{i}\otimes dy_{j}+Ad\mu^{2}

written in the two complex affine coordinate $y_{1}=\text{Im}(\eta_{1}),y_{2}=\text{Im}(\eta_{2})$ and a symplectic moment coordinate $\mu$ . The crucial point is that mirror symmetry means the matrices $a_{ij}$ appearing in both cases are the same.

Now the Ooguri-Vafa type metrics are perturbations of some constant solution at any given log scale, and the coupling constants $a_{ij}$ drift slowly according to the renormalisation flow as the log scale changes. The formal coincidence of the renormalisation flow equations for both the positive vertex and the negative vertex agrees with semiflat mirror symmetry.

Remark 4.9.

The exlusion of the natural possibility that $\text{Im}(a_{2\bar{1}})\neq 0$ suggests that there may be generalisations of semiflat mirror symmetry to situations where special Lagrangian fibrations cannot exist (cf. Section 4.12).

Remark 4.10.

The positive and the negative vertices have drastically different features at refined scales: for example the positive vertex contains a fully nonlinear region modelled on the Taub-NUT type metric on $\mathbb{C}^{3}$ , while the negative vertex metric is obtained by a perturbative analysis. Nonetheless they share the same renormalisation flow equation, which controls large scale behaviours. The insight is that mirror symmetry should govern metric behaviours at large scales, but not necessarily at refined scales. In this perspective mirror symmetry owes its predicative power to the fact that questions in algebraic or symplectic geometry are mostly insensitive to small scale metric fluctuations.

Acknowledgement. The author thanks his PhD supervisor Simon Donaldson and co-supervisor Mark Haskins for their inspirations, and Song Sun for discussions.

Chapter 4 The Negative Vertex [044F]