4 Potential estimates and SYZ fibration [00BB]

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4 Potential estimates and SYZ fibration

We consider a polarized algebraic maximal degeneration of Calabi-Yau manifolds $X\to S\setminus\{0\}$ over a smooth algebraic curve. Let $(\mathcal{X},\mathcal{L})$ be a semistable snc model with $\mathcal{L}|_{X}=L$ . The NA pluripotential theory provides a continuous semipositive metric $\left\lVert\cdot\right\rVert_{CY}=\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi_{0}}$ on $L$ over $X_{K}^{an}$ solving the NA MA equation (7), which we assume henceforth satisfies the NA MA-real MA comparison property, so $\phi_{0}$ solves the real MA equation over the $n$ -dimensional open faces $\text{Int}(\Delta_{J})$ of the essential skeleton $Sk(\mathcal{X})$ (cf. section 3.5).

4.1 Comparison Kähler metric I

We apply the Fubini-Study approximation to transfer the NA metric $\left\lVert\cdot\right\rVert_{CY}$ to the complex manifolds $X_{t}$ up to $C^{0}$ -small errors in the potential, while preserving the positivity of the metric.

Recall there is a logarithm map $\text{Log}_{\mathcal{X}}:X_{t}\to\Delta_{\mathcal{X}}$ , defined up to $O(\frac{1}{|\log|t||})$ coordinate ambiguity, so when we refer to $\text{Log}_{\mathcal{X}}$ over $\text{Int}(\Delta_{J})$ we will implicitly shrink $\text{Int}(\Delta_{J})$ by $O(\frac{1}{|\log|t||})$ to ensure the coordinate expression $x_{i}=\frac{\log|z_{i}|}{\log|t|}$ makes sense.

Lemma 4.1.

Given any $0<\epsilon\ll 1$ , then for sufficiently small $t$ depending on $\epsilon$ , there is a smooth Kähler metric $\omega_{FS,t}$ on $(X_{t},\frac{1}{|\log|t||}c_{1}(L))$ , such that

•

The relative Kähler potential for any two choices of $\epsilon$ is bounded uniformly independent of small $t$ .
•

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

Proof.

Let $\left\lVert\cdot\right\rVert_{FS}$ be a NA Fubini-Study approximation of $\left\lVert\cdot\right\rVert_{CY}$ , with potential difference less than $\epsilon/2$ . We construct the Fubini-Study metrics $\left\lVert\cdot\right\rVert_{FS,t}$ on $(X_{t},L)$ by the formula (10), so the curvature forms of $\left\lVert\cdot\right\rVert_{FS,t}^{1/|\log|t||}$ define the Kähler metrics $\omega_{FS,t}$ on $(X_{t},\frac{1}{|\log|t||}c_{1}(L))$ . By construction, for sufficiently small $t$ the local potentials are $C^{0}$ -close to that of $\left\lVert\cdot\right\rVert_{FS}$ , which is $\epsilon$ -close to the continuous metric $\left\lVert\cdot\right\rVert_{CY}$ , so the uniform boundedness of the potentials can be guaranteed.

By our NA MA-real MA comparison assumption, over $\text{Int}(\Delta_{J})$ the potential $\phi_{0}$ of $\left\lVert\cdot\right\rVert_{CY}$ equals the pullback $\phi_{0}\circ r_{\mathcal{X}}$ via the retraction map $r_{\mathcal{X}}$ . From our discussions on the hybrid topology in section 3.2, over $\text{Int}(\Delta_{J})$ , for $\phi_{J,t}$ to be $C^{0}$ -close to $\phi_{0}\circ r_{\mathcal{X}}$ means the same as saying $\phi_{J,t}$ is $C^{0}$ -close to $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ .

∎

4.2 Comparison Kähler metric II: Regularisation

We now improve the metric $\omega_{FS,t}$ on $(X_{t},\frac{1}{|\log|t||}c_{1}(L))$ to make the volume form approximately CY except on a set with a small percentage of the CY measure.

Let $0<\delta\ll 1$ . Since $\phi_{0}$ solves the real MA equation (8) on the $n$ -dimensional open faces $\text{Int}(\Delta_{J})$ , the regularity theory of real MA surveyed in section 2.5 applies. In particular, we can find finitely many small open balls $B_{l}=B(p_{l},2r(\delta))$ properly contained in $\cup_{J}\text{Int}(\Delta_{J})$ , such that $\phi_{0}$ has $C^{k}$ -norm uniformly bounded by $C(\delta)$ on any $B_{l}$ , and the complement of $W_{\delta}=\cup_{l}B(p_{l},r(\delta))$ has small $\mu_{0}$ -measure less than $\delta$ . The open sets $W_{\delta}$ form an exhaustion of $W_{0}=\cup_{\delta>0}W_{\delta}$ , which is the $C^{\infty}$ -locus of the real MA solution $\phi_{0}$ . Recall the normalised CY measure $d\mu_{t}$ from section 3.1.

Lemma 4.2.

(Regularisation) Let $\epsilon$ be suffiently small dependent on $\delta$ , and construct $\omega_{FS,t}$ as in Lemma 4.1, for $t$ sufficiently small dependent on $\epsilon$ and $\delta$ . There is a Lipschitz continuous function $\psi_{t}$ on $X_{t}$ with $\left\lVert\psi_{t}\right\rVert_{L^{\infty}}\leq 3\epsilon$ , such that

•

The function $\psi_{t}$ is smooth away from a closed subset with $d\mu_{t}$ -measure zero.
•

The (1,1)-current $\omega_{\psi,t}=\omega_{FS,t}+dd^{c}\psi_{t}\geq 0$ is positive on $X_{t}$ .
•

The metric estimate $(1-C\epsilon)dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}}\leq\omega_{\psi,t}\leq(1+C\epsilon)dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}}$ holds on the smooth locus of $\psi$ inside $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$ .
•

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

Proof.

Choose a smooth nonnegative bump function $\eta$ on $\mathbb{R}^{n}$ supported in $B(0,2)$ , and equals one on $B(0,1)$ , and construct $\eta_{i}=\eta(\frac{|x-p_{i}|}{r(\delta)})$ supported on $B_{i}$ . We calculate $\text{Log}_{\mathcal{X}}^{-1}(B_{l})$

\begin{split}dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}}=\frac{1}{4\pi|\log|t||^{2}}\sum_{1\leq i,j\leq n}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{j}\\ \gtrsim_{\delta}\frac{1}{|\log|t||^{2}}\sum\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{j},\end{split}

dd^{c}\eta_{l}\circ\text{Log}_{\mathcal{X}}\geq-\frac{C}{|\log|t||^{2}}\sum\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{j},

so if $0<\epsilon\ll 1$ is sufficiently small dependent on $\delta$ , we can ensure $(\phi_{0}+2\epsilon\eta_{l})\circ\text{Log}_{\mathcal{X}}$ is psh on $\text{Log}_{\mathcal{X}}^{-1}(B_{l})$ .

Consider the potential function on $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))$

\psi_{t}(z)=\max(0,\max_{\text{Log}_{\mathcal{X}}(z)\in B_{l}}\{(\phi_{0}+2\epsilon\eta_{l})\circ\text{Log}_{\mathcal{X}}-\epsilon-\phi_{J,t}\}),

with $\phi_{J,t}$ from Lemma 4.1.

Since $|\phi_{J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}|<\epsilon$ , we see that on $\partial B_{l}$ the maximum is never achieved by $(\phi_{0}+2\epsilon\eta_{l})\circ\text{Log}_{\mathcal{X}}-\epsilon-\phi_{J,t}$ , so the fact that this term is only locally defined causes no problem. By construction every term is $dd^{c}\phi_{J,t}$ -psh, so the maximum $\psi_{t}$ is also $dd^{c}\phi_{J,t}$ -psh. Near the boundary of $\text{Int}(\Delta_{J})$ the maximum is achieved by $\psi_{t}=0$ , so $\psi_{t}$ globalizes to define a Lipschitz continuous $\omega_{FS,t}$ -psh function on $X_{t}$ , with $\left\lVert\psi_{t}\right\rVert_{L^{\infty}}\leq 3\epsilon$ .

Morever, on $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$ the maximum of $\psi_{t}$ is strictly greater than zero. Perturbing the bump function in the construction if necessary, we may assume the locus on $X_{t}$ where the maximum is achieved by at least two terms is a subset of codimension one, and it is automatically closed. So a.e on $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$ , the metric $\omega_{\psi,t}=\omega_{FS,t}+dd^{c}\psi_{t}$ is smooth and equals $dd^{c}(\phi_{0}+2\epsilon\eta_{l})\circ\text{Log}_{\mathcal{X}}$ for some $l$ . We calculate using the regularity estimates on $dd^{c}\phi_{0}$ that

\omega_{\psi,t}^{n}=(dd^{c}(\phi_{0}+2\epsilon\eta_{l})\circ\text{Log}_{\mathcal{X}})^{n}=(1+O(\epsilon))(dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}})^{n}.

Using the real MA equation (8),

(dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}})^{n}=n!\det(D^{2}\phi_{0})\prod_{i}\frac{1}{4\pi|\log|t||^{2}}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{i},

where $\frac{(L^{n})}{n!}d\mu_{0}=\det(D^{2}\phi_{0})dx_{1}\ldots dx_{n}$ determines the constant $\det(D^{2}\phi_{0})$ . Comparing with section 3.1, the normalized CY measure $d\mu_{t}$ satisfies

\frac{(L^{n})}{n!}d\mu_{t}=(1+O(\frac{1}{|\log|t||}))\det(D^{2}\phi_{0})\prod_{i}\frac{1}{4\pi|\log|t||}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{i}.

For sufficiently small $t$ depending on $\epsilon$ and $\delta$ , we combine the above to deduce

\omega_{\psi,t}^{n}=(1+O(\epsilon))\frac{(L^{n})}{|\log|t||^{n}}d\mu_{t}.

The total complex MA measure is $\int_{X_{t}}\omega_{\psi,t}^{n}=\frac{(L^{n})}{|\log|t||^{n}}$ , and the contribution from the smooth region in $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$ is greater than $(1-\delta)\frac{(L^{n})}{|\log|t||^{n}}$ for very small $\epsilon$ dependent on $\delta$ . Thus the measure contribution from the complement must be less than $\delta\frac{(L^{n})}{|\log|t||^{n}}$ , namely $\delta$ -percent of the total measure. Thus the total variation of the signed measure $d\mu_{t}-\frac{|\log|t||^{n}}{(L^{n})}\omega_{\psi,t}^{n}$ is smaller than $\delta$ for small enough $\epsilon$ . ∎

4.3 Potential estimate I

We denote the CY metrics on $(X_{t},\frac{1}{|\log|t||}c_{1}(L))$ as

\omega_{CY,t}=\omega_{FS,t}+dd^{c}\phi_{CY,t}.

Using the local potentials $\phi_{J,t}$ of $\omega_{FS,t}$ , we can write $\omega_{CY,t}$ in terms of local absolute potentials on $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))$ :

\omega_{CY,t}=dd^{c}\phi_{CY,J,t},\quad\phi_{CY,J,t}=\phi_{J,t}+\phi_{CY,t}.

This depends on an implicit choice of $\omega_{FS,t}$ and $\phi_{J,t}$ in Lemma 4.1. Our goal is to find a suitable choice and show the smallness of $|\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}|$ in the generic region.

Proposition 4.3.

For $0<|t|\ll 1$ , the CY potentials have a uniform bound $\left\lVert\phi_{CY,t}\right\rVert_{L^{\infty}}\leq C$ under the normalisation $\sup_{X_{t}}\phi_{CY,t}=0$ .

Proof.

Combine the uniform $C^{0}$ -estimate Theorem 2.2 and the uniform Skoda estimate Theorem 2.1, we know the CY potential with respect to any fixed choice of Fubini-Study background metric is uniformly bounded for small $t$ . Here the implicit choice of $\epsilon$ in $\omega_{FS,t}$ does not matter because of Lemma 4.1. ∎

Proposition 4.4.

Given small numbers $0<\lambda,\kappa\ll 1$ , then for $\delta,\epsilon,t$ sufficiently small depending on $\lambda$ and $\kappa$ , the function $\phi_{CY,t}$ is near its minimum with large probability:

d\mu_{t}(\{\phi_{CY,t}-\min_{X_{t}}\phi_{CY,t}\geq\kappa/5\})<\lambda.

Proof.

We wish to compare $\omega_{CY,t}$ with $\omega_{\psi,t}$ from Lemma 4.2 by an $L^{1}$ -stability estimate. Pick a parameter $c$ such that

d\mu_{t}(\{\psi_{t}-\phi_{CY,t}-c\leq 0\})\geq\lambda.

Since the potential $\phi_{CY,t}$ has a uniform bound, Theorem 2.1 implies another uniform Skoda estimate with modified constants

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

We also have the $L^{1}$ -stability property for $\omega_{\psi,t}$ in Lemma 4.2:

\int_{X_{t}}|d\mu_{t}-\frac{\omega_{\psi,t}^{n}}{\text{Vol}(X_{t},\omega_{\psi,t})}|<\delta.

We then apply the uniform $L^{1}$ -stability estimate Theorem 2.6, with $Y=X_{t}$ , $\omega=\omega_{CY,t}$ and $\phi=\psi_{t}-\phi_{CY,t}-c$ . In this construction $\delta,\epsilon,t$ are sufficiently small, chosen successively depending on $\lambda$ and $\kappa$ . We conclude

\sup_{X_{t}}(\psi_{t}-\phi_{CY,t}-c)\leq C(\lambda)\delta^{1/(2n+3)}\ll\kappa.

Now $|\psi_{t}|\leq 3\epsilon\ll\kappa$ , so $\inf_{X_{t}}\phi_{CY,t}>-c-\kappa/10.$

Taking the contrapositive, if we choose $c=-\inf_{X_{t}}\phi_{CY,t}-\kappa/10$ , then

d\mu_{t}(\{\psi_{t}-\phi_{CY,t}-c\leq 0\})<\lambda,

whence for $\epsilon\ll\kappa$ , using again $|\psi_{t}|\leq 3\epsilon$ ,

d\mu_{t}(\{\phi_{CY,t}-\inf\phi_{CY,t}\geq\kappa/5\})<\lambda.

∎

Remark 4.5.

The reason we use an asymmetric version of the $L^{1}$ -stability estimate, is that we have no control on the density of the comparison metric $\omega_{\psi,t}$ away from the the generic region except for a small bound on the measure contribution there.

We can reformulate this in terms of the local potentials of the CY metrics, and thereby eliminate auxiliary choices of Fubini-Study metric and regularisation.

Corollary 4.6.

Given small numbers $0<\lambda,\kappa\ll 1$ , then for $t$ small enough depending on $\lambda,\kappa$ , there exist appropriately chosen local potentials $\phi_{CY,J,t}$ on $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))$ , normalized to $\inf(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})=0$ , satisfying

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

and $\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{L^{\infty}}\leq C$ independent of $\lambda,\kappa$ and small $t$ .

Proof.

By construction in Lemma 4.1, the local potential $\phi_{J,t}$ of $\omega_{FS,t}$ is $\epsilon$ -close to $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ , and since $\epsilon\ll\kappa$ these two are practically the same. Up to an overall normalisation constant, which is fixed by $\inf=0$ , we have $\phi_{CY,J,t}=\phi_{CY,t}+\phi_{J,t},$ so the measure bound follows from the previous result.

The uniform $L^{\infty}$ bound follows from Prop. 4.3 and Lemma 4.1 without any reference to $\lambda,\kappa$ . ∎

Given $0<\tau\ll 1$ , we consider the region obtained from shrinking the $n$ -dimensional faces near the boundary:

U_{J,t,\tau}=\text{Log}_{\mathcal{X}}^{-1}(\Delta_{J}\setminus\{|x_{i}|<\tau,|1-\sum_{1}^{n}x_{i}|<\tau\}).

The following theorem is a precise formulation for $C^{0}_{loc}$ -convergence of the local CY potentials to $\phi_{0}$ over the $n$ -dimensional open faces of $Sk(X)$ as $t\to 0$ .

Theorem 4.7.

( $C^{0}_{loc}$ -convergence estimate on the potential) Given $0<\tau,\kappa\ll 1$ , then for sufficiently small $t$ , on each $U_{J,t,\tau}$ there is a $C^{0}$ -bound

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

Proof.

We need to obtain upper bound on $\phi_{CY,J,t}$ . Consider $z\in U_{J,t,\tau}$ and $x=\text{Log}_{\mathcal{X}}(z)$ . Let $r\ll\tau$ be a parameter to be fixed, so $B(x,3r)\subset\text{Int}(\Delta_{J})$ . The function $\phi_{0}$ has an a priori Lipschitz estimate on $\Delta_{\mathcal{X}}$ , so the oscillation of $\phi_{0}$ on $B(x,3r)$ is less than $Cr\ll\kappa$ by choosing $r$ small enough.

Now we apply the mean value inequality to the psh function $\phi_{CY,J,t}$ on a ball in the local covering space of $U_{J,t,\tau}\subset(\mathbb{C}^{*})^{n}$ , which projects to $B(x,r)$ via $\text{Log}_{\mathcal{X}}$ . We have

\phi_{CY,J,t}(z)\leq\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{ball}\phi_{CY,J,t},

hence

\begin{split}(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})(z)&\leq\text{osc}_{B(r)}\phi_{0}+\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{ball}(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})\\ &\leq\frac{\kappa}{10}+\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{ball}(\phi_{CY,J,t}-\text{Log}_{\mathcal{X}}).\end{split}

But on the ball $\phi_{CY,J,t}-\text{Log}_{\mathcal{X}}<\kappa/4$ , except on a subset of the ball with $d\mu_{t}$ -percentage $\leq C\lambda r^{-n}$ , on which we use the coarser bound $\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\leq C$ . Combining these,

(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})(z)<\kappa/2+C\lambda r^{-n}<\kappa,

by choosing $\lambda$ sufficiently small depending on $\kappa,\tau$ . ∎

Remark 4.8.

The above estimates do not use the full strength of the regularisation lemma 4.2. We only use the $L^{1}$ -stability of the volume density, not the metric information.

4.4 Potential estimate II

Here we present a second strategy for the potential estimate, which aims to circumvent the uniform $L^{1}$ -stability estimate Theorem 2.6, and we explain why there is a difficulty with this second approach. Readers who wish to follow the main line of the proof may skip this section.

Lemma 4.9.

Given $0<\delta\ll 1$ , then for $0<\epsilon\ll 1$ depending on $\delta$ , and $t$ small enough depending on $\epsilon,\delta$ ,

\int_{\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})}d(\psi_{t}-\phi_{CY,t})\wedge d^{c}(\psi_{t}-\phi_{CY,t})\wedge(dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}})^{n-1}\leq\frac{C\delta}{|\log|t||^{n}},

where $C$ is independent of $\delta,\epsilon,t$ .

Proof.

Pretending everything is smooth, a standard integration by part gives

\begin{split}&\int_{X_{t}}(\psi_{t}-\phi_{CY,t})(\omega_{CY,t}^{n}-\omega_{\psi,t}^{n})\\ =&\int_{X_{t}}(\psi_{t}-\phi_{CY,t})dd^{c}(-\psi_{t}+\phi_{CY,t})\wedge(\omega_{CY,t}^{n-1}+\ldots+\omega_{\psi,t}^{n-1})\\ =&\int_{X_{t}}d(\psi_{t}-\phi_{CY,t})\wedge d^{c}(\psi_{t}-\phi_{CY,t})\wedge(\omega_{CY,t}^{n-1}+\ldots+\omega_{\psi,t}^{n-1})\\ \geq&\int_{X_{t}}d(\psi_{t}-\phi_{CY,t})\wedge d^{c}(\psi_{t}-\phi_{CY,t})\wedge\omega_{\psi,t}^{n-1}.\end{split}

The same calculations work for continuous $\omega_{FS,t}$ -psh functions by standard pluripotential theory.

Combine $\left\lVert\phi_{CY,t}\right\rVert_{L^{\infty}}\leq C$ with the total variation bound in Lemma 4.2,

\int_{X_{t}}|\frac{|\log|t||^{n}\omega_{\psi,t}^{n}}{(L^{n})}-d\mu_{t}|<\delta,

we get

\int_{X_{t}}d(\psi_{t}-\phi_{CY,t})\wedge d^{c}(\psi_{t}-\phi_{CY,t})\wedge\omega_{\psi,t}^{n-1}\leq C\int_{X_{t}}|\omega_{CY,t}^{n}-\omega_{\psi,t}^{n}|\leq\frac{C\delta}{|\log|t||^{n}}.

Again by Lemma 4.2, the metric $\omega_{\psi,t}$ is uniformly controlled a.e. on $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$ , so

\int_{\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})}d(\psi_{t}-\phi_{CY,t})\wedge d^{c}(\psi_{t}-\phi_{CY,t})\wedge(dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}})^{n-1}\leq\frac{C\delta}{|\log|t||^{n}}.

∎

An outline of this strategy is

•

Choose some suitable integral normalisation on $\phi_{CY,t}$ . Apply Poincaré inequality to prove the average $L^{2}$ -integral of $\psi_{t}-\phi_{CY,t}$ is small in the generic region $W_{\delta}$ , which occupies most of the $d\mu_{t}$ -measure.
•

Deduce the measure is small on the set where $\phi_{CY,t}-\psi_{t}$ is perceptibly negative.
•

Apply the stability estimate Cor. 2.3 to show $\phi_{CY,t}-\psi_{t}$ cannot be perceptibly negative. Since $\left\lVert\psi_{t}\right\rVert_{C^{0}}$ is negligible, it shows the minimum of $\phi_{CY,t}$ is almost zero.
•

Using the small $L^{2}$ -average bound on $\phi_{CY,t}$ , one applies the mean value inequality to derive a small upper bound on $\phi_{CY,t}$ in the generic region $W_{\delta}$ .

The problem lies in the fact that the $n$ -dimensional open faces $\text{Int}(\Delta_{J})$ of $Sk(X)$ are disconnected, so the average values on each face are a priori unrelated. Thus the Poincaré inequality can only imply a small bound on $\phi_{CY,t}-\psi_{t}$ with a priori different normalisations associated to each open face, which is not good enough to get a small bound on average $L^{2}$ -integral of $\psi_{t}-\phi_{CY,t}$ .

4.5 Metric convergence and SYZ fibration

Given the $C^{0}_{loc}$ -convergence estimate Theorem 4.7, then the metric SYZ conjecture would follow as explained in [32]. The most important step is the following $C^{\infty}_{loc}$ -convergence result in the generic region. Recall the exhaustion $W_{\delta}$ for the regular locus of the real MA solution $\phi_{0}$ . The open subsets $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$ occupy almost the full percentage of the $d\mu_{t}$ -measure on $X_{t}$ for small $\delta,t$ , and as such deserve the name ‘generic region’.

Theorem 4.10.

(Metric $C^{\infty}_{loc}$ -convergence in the generic region) For any given $0<\delta\ll 1$ , then as $t\to 0$ ,

\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{k}(\text{Log}_{\mathcal{X}}^{-1}(W_{\delta}))}\to 0,

where the $C^{k}$ -norm is defined by passing to the local universal cover of $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})\subset(\mathbb{C}^{*})^{n}$ with preferred coordinates $\zeta_{i}=\frac{1}{\log|t|}\log z_{i}$ for $i=1,2,\ldots,n$ .

Proof.

By the calculations in the proof of Lemma 4.2,

(dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}})^{n}=n!\det(D^{2}\phi_{0})\prod_{i}\frac{1}{4\pi}\sqrt{-1}d\zeta_{i}\wedge d\bar{\zeta}_{i},

while the CY condition gives (cf. section 3.1)

\begin{split}&(dd^{c}\phi_{CY,J,t})^{n}=\omega_{CY,t}^{n}=\frac{(L^{n})}{|\log|t||^{n}}d\mu_{t}=\frac{(L^{n})}{|\log|t||^{n}\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}\\ =&\frac{(L^{n})|\log|t||^{n}}{\int_{X_{t}}\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}}|u_{J}|^{2}\prod_{i}\sqrt{-1}d\zeta_{i}\wedge d\bar{\zeta}_{i},\end{split}

where $u_{J}$ is a holomorphic function of the defining functions $z_{0},\ldots z_{n}$ of the divisors $E_{i}$ , with limiting value $u_{J}(E_{J})\neq 0$ . The two expressions are matched by the condition that $\text{Log}_{\mathcal{X}*}d\mu_{t}$ converge to $d\mu_{0}$ as $t\to 0$ , which boils down to

(dd^{c}\phi_{CY,J,t})^{n}=n!\det(D^{2}\phi_{0})\frac{|u_{J}|^{2}}{|u_{J}(E_{J})|^{2}}\prod_{i}\frac{1}{4\pi}\sqrt{-1}d\zeta_{i}\wedge d\bar{\zeta}_{i}.

Since $u_{J}$ has a Taylor expansion in $z_{0},\ldots z_{n}$ , we see that $\frac{|u_{J}|^{2}}{|u_{J}(E_{J})|^{2}}=1+f$ for some smooth function $f$ in $\zeta_{1},\ldots,\zeta_{n}$ with exponentially small $C^{k}$ -norm bound

\left\lVert f\right\rVert_{C^{k}(\text{Log}_{\mathcal{X}}^{-1}(W_{\delta}))}\lesssim_{k}\exp(-c(W_{\delta})|\log|t||)

for some exponent $c(W_{\delta})>0$ depending on $W_{\delta}$ .

We focus on balls in the local universal cover of $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$ with definite size in the $\zeta_{i}$ coordinates. For sufficiently small $t$ , then the volume relative error $f$ has arbitrarily small $C^{k}$ -norm bound, and Theorem 4.7 says the $C^{0}$ -norm of $\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}$ on the ball is also arbitrarily small. Thus we can apply Savin’s theorem 2.8, to deduce that $\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{k}}$ is arbitrarily small on shrinked balls. Since $W_{\delta}$ for varying $\delta$ give an exhaustion of the regular locus of $\phi_{0}$ , this shrinking can be compensated by starting with a larger $W_{\delta}$ , and we deduce the $C^{k}$ -convergence estimate as required. ∎

The geometric meaning is that inside the generic region, the CY metric $\omega_{CY,t}$ is $C^{\infty}$ -close to a semiflat metric:

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

In terms of the Riemannian metric tensors,

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

(11)

The name ‘semiflat’ means the metric restricted to the $T^{n}$ -fibres are flat Euclidean. The $T^{n}$ -fibres are precisely special Lagrangian in the model case

\begin{cases}\omega_{semiflat}=\frac{1}{4\pi|\log|t||^{2}}\sum_{1\leq i,j\leq n}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{j},\\ \Omega_{semiflat}=\text{const}\cdot\prod_{1}^{n}d\log z_{i},\end{cases}

or equivalently

\text{Log}_{\mathcal{X}}:(z_{1},\ldots z_{n})\mapsto\frac{1}{\log|t|}(\log|z_{1}|,\ldots\log|z_{n}|)

is a special Lagrangian fibration in the model case for some choice of the phase angle. Since $(\omega_{CY,t},\Omega_{t})$ is $C^{\infty}$ -close to the model case, standard perturbation theory allows one to perturb the $T^{n}$ -fibres into special Lagrangians with respect to $(\omega_{CY,t},\Omega_{t})$ in the generic region, to obtain a new special Lagrangian fibration. The details are carried out in [49], and more expositions can be found in [32].

Theorem 4.11.

(Special Lagrangian fibration on the generic region) For any given $0<\delta\ll 1$ , then for $t$ sufficiently small depending on $\delta$ , there is a special Lagrangian fibration on an open subset of $(X_{t},\omega_{CY,t},\Omega_{t})$ containing $W_{\delta}$ .

Consequently, assuming as always the comparison property between NA MA equation and real MA equation, then the special Lagrangian fibration exists on an open subset of arbitrarily large percentage of $X_{t}$ as $t\to 0$ , which is the main theorem of the paper.

Finally we make a few comments about the status of the Kontsevich-Soibelman/Gross-Wilson conjecture, which says that given a polarised algebraic maximally degenerate family of CY manifolds, whose holonomy groups are exactly $SU(n)$ , the Gromov-Hausdorff limit of the CY metrics $g_{CY,t}$ is the essential skeleton $Sk(X)$ equipped with a real Monge-Ampère metric on the regular locus, the singular locus has real codimension 2, and $Sk(X)$ is homeomorphic to $S^{n}$ .

What follows quickly from the metric asymptote (11) and [32] are the following facts, assuming the comparison property:

•

Over the regular locus of $\phi_{0}$ inside each $n$ -dimensional open faces $\text{Int}(\Delta_{J})$ , the metrics $g_{CY,t}$ converge in the Gromov-Hausdorff sense to a real MA metric as $t\to 0$ :

g_{CY,t}\to\frac{1}{2\pi}\sum_{1\leq i,j\leq n}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}dx_{i}\otimes dx_{j}.

This is immediate from the much stronger $C^{\infty}_{loc}$ metric asymptote (11).

•

There is a uniform diameter bound $\text{diam}(X_{t},g_{CY,t})\leq C$ [32, Prop. 5.11][34].
•

Any point in $X_{t}\setminus W_{\delta}$ is within $C\delta^{1/2n}$ -distance to a point on $W_{\delta}$ for sufficiently small $t$ . This follows from the Bishop-Gromov comparison argument in [32, section 5.3].
•

Consequently, the regular locus $W_{0}$ of $\phi_{0}$ inside the union of $n$ -dimensional open faces of $Sk(X)$ , is an open dense subset of any Gromov-Hausdorff limit space of $(X_{t},g_{CY,t})$ .

Remark 4.12.

Notice there is a gap between the above results and the Gromov-Hausdorff convergence to the real MA metric on $Sk(X)$ defined by the Hessian of $\phi_{0}$ , because the $n$ -dimensional open faces are disconnected, and therefore we cannot access the distance of two points on different faces. One needs further information on the $(n-1)$ -dimensional faces of $Sk(X)$ .

What remains to be resolved are the following questions, which seem to contain substantial difficulty:

•

Prove the comparison property.
•

Formulate a global notion of convex functions and the real MA equation on $Sk(X)$ , instead of just on the $n$ -dimensional open faces. Notice this is nontrivial because $Sk(X)$ only has a piecewise affine structure, not a global affine structure. See section 5.3 for some closely related discussions.
•

Develop a regularity theory for such real MA metrics, and prove/disprove that the singular locus has real codimension at least two. Notice this is false for real MA equations on the unit ball by a counterexample of Mooney [35], so if it is true then there has to be a global reason.
•

The regularity theory should also show that $Sk(X)$ equipped with the real MA metric has the same topology as $Sk(X)$ viewed as a simplicial complex. This is nontrivial because a priori the real MA equation can have singularities which contract lines to points, and the singular set may even be quite fractal, such as in Mooney’s example.
•

Prove an enhanced version of the comparison property between NA MA equation and real MA equation, which works globally on all faces of $Sk(X)$ , not just on the $n$ -dimensional open faces.
•

Extend the arguments in this paper over the global regular locus of $\phi_{0}$ , to show that the CY metrics converge smoothly there as well. Use this to identify the Gromov-Hausdorff limit of $(X_{t},g_{CY,t})$ with $Sk(X)$ equipped with the real MA metric defined by the Hessian of $\phi_{0}$ .
•

Show that $Sk(X)$ with the standard topology is homeomorphic to $S^{n}$ . This question does not refer to the metric, and is much studied in birational geometry [36][37]. This can be checked explicitly for many examples. In general, it is known that $Sk(X)$ is a ‘pseudomanifold’, its $\mathbb{Q}$ -homology groups agree with $S^{n}$ , and its fundamental group has trivial profinite completion, but the actual homeomorphism type is still elusive.