4 Estimates on the Kähler potential

4.1 Harnack inequality

Consider a general possibly singular Kähler potential $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$ on $X_{s}$ , normalised to $\sup_{X_{s}}\varphi=0$ . We think of $\varphi$ equivalently as a collection of local potentials $\{\varphi_{0},\varphi_{m}\}$ as in section 3.3. In the region $U_{w}^{s}\subset X_{s}$ , we can find $m\in\Delta_{\mathbb{Z}}$ with $\langle m,w\rangle=1$ and $\mathbb{C}^{*}$ -coordinates $z^{m_{1}},\ldots z^{m_{n}}$ as in section 3.1. Recall $d\mu_{s}$ is the normalised canonical measure induced by the holomorphic volume form.

Notation.

Denote $X^{toric}_{s}$ as the union of all the toric regions $U_{w,\delta}^{s}$ for various choices of $m$ and $w$ . It is tacitly understood that slightly shrinked domains correspond to a slightly larger choice of $\delta$ , and we shall abusively use the same notation for shrinked domains.

Proposition 4.1.

(Harnack type inequality) Suppose $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$ with $\sup_{X_{s}}\varphi=0$ . Then the average integral

\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_{X^{toric}_{s}}|\varphi|d\mu_{s}\leq C.

Proof.

(cf. proof of Prop. 3.1 in [2]) Consider the local potentials $\phi=\varphi_{m}$ on various coordinate charts in section 3.1, both of the toric type and of the boundary type. The charts can be chosen so that the Lebesgue measures thereof are uniformly equivalent to $d\mu_{s}$ up to a scaling factor. We have $|\phi-\varphi|\leq C$ uniformly on charts. Suppose a coordinate ball $B(p,3R)$ is contained in (the universal cover of) the local chart. Since $\phi$ is psh and $\phi-C\leq 0$ , for $z\in B(p,R)$ ,

\phi(y)-C\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_{B(y,2R)}(\phi-C)\lesssim\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_{B(p,R)}(\phi-C),

hence

\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_{B(p,R)}|\varphi|\lesssim 1+\inf_{B(p,R)}(-\varphi).

To deduce the global version of the Harnack type inequality we need a transitivity property, namely we can connect the chart containing the maximum point of $\varphi$ to any of the toric charts in $X_{s}^{toric}$ via a chain of $O(1)$ number of charts, such that $\inf_{B(p,R)}|\varphi|$ on charts increase by only $O(1)$ in each step. This last fact is because we can choose the chains of successive charts $B(p_{i},5R_{i})$ such that the measure of the overlap occupies a nontrivial portion of the previous chart:

|B(p_{i},R_{i})\cap B(p_{i+1},R_{i+1})|\gtrsim\frac{1}{10}|B(p_{i},R_{i})|,

which would force

\begin{split}\inf_{B(p_{i+1},R_{i+1})}|\varphi|\leq&\inf_{B(p_{i+1},R_{i+1})\cap B(p_{i},R_{i})}|\varphi|\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_{B(p_{i+1},R_{i+1})\cap B(p_{i},R_{i})}|\varphi|\\ \lesssim&\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_{B(p_{i},R_{i})}|\varphi|\lesssim 1+\inf_{B(p_{i},R_{i})}|\varphi|.\end{split}

∎

Remark 4.2.

Notice this transitivity argument allows us to move from boundary type charts into toric charts, but not conversely, because the measure is much larger on toric charts.

4.2 Local potentials: convexity

We continue with a general $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$ normalised to $\sup_{X_{s}}\varphi=0$ , whose local potentials are $\{\varphi_{0},\varphi_{m}\}$ . A simple obeservation is:

Lemma 4.3.

Let $\Phi$ be any psh function on the open subset of $\{1<|\zeta_{i}|<\Lambda,i=1,\ldots n\}\subset(\mathbb{C}^{*})^{n}$ . Then the $T^{n}$ -invariant function

\bar{\Phi}(\log|\zeta_{1}|,\ldots,\log|\zeta_{n}|)=\frac{1}{(2\pi)^{n}}\int_{T^{n}}\Phi(|\zeta_{1}|e^{i\theta_{1}},\ldots|\zeta_{n}|e^{i\theta_{n}})d\theta_{1}\ldots d\theta_{n}

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

Proof.

Since the $T^{n}$ -action on $(\mathbb{C}^{*})^{n}$ is holomorphic, $\Phi(\zeta_{1}e^{i\theta_{1}},\ldots\zeta_{n}e^{i\theta_{n}})$ is psh in $\zeta$ for any choice of $\theta_{i}$ , so the average function $\bar{\Phi}$ is also psh. Any $T^{n}$ -invariant psh function must be convex in the log coordinates, because of the formula

\sqrt{-1}\partial\bar{\partial}\bar{\Phi}=\frac{1}{4}\sum\frac{\partial^{2}\bar{\Phi}}{\partial x_{i}\partial x_{j}}\sqrt{-1}d\log\zeta_{i}\wedge d\overline{\log\zeta_{j}}\geq 0.

∎

In the region $U_{w}^{s}\subset X_{s}$ , we can find $m\in\Delta_{\mathbb{Z}}$ with $\langle m,w\rangle=1$ and $\mathbb{C}^{*}$ -coordinates $z^{m_{1}},\ldots z^{m_{n}}$ as in section 3.1, and consider the local potential $\phi=\varphi_{m}$ . Denote $x^{m_{i}}=\frac{\log|z^{m_{i}}|}{s}$ . We produce the local average function

\bar{\phi}(x^{m_{1}},\ldots x^{m_{n}})=\frac{1}{(2\pi)^{n}}\int_{T^{n}}\phi(|z^{m_{1}}|e^{i\theta_{1}},\ldots|z^{m_{n}}|e^{i\theta_{n}})d\theta_{1}\ldots d\theta_{n}.

(22)

Proposition 4.4.

In the chart $U^{s}_{w}$ the average function $\bar{\phi}$ is convex, and on the shrinked chart $U^{s}_{w,\delta}$ it has a Lipschitz bound:

|\bar{\phi}|\leq C,\quad|\bar{\phi}(x)-\bar{\phi}(x^{\prime})|\leq C|x-x^{\prime}|.

(23)

Proof.

By Lemma 4.3, $\bar{\phi}$ is convex, and by Prop. 4.1 it has an $L^{1}$ bound in the $x^{m_{i}}$ coordinates:

\int|\bar{\phi}|dx^{m_{1}}\ldots dx^{m_{n}}\leq C.

Clearly $\bar{\phi}$ is also bounded above, so for the argument we may pretend $\bar{\phi}\leq 0$ upon shifting by a bounded constant.

We claim $\bar{\phi}(x)$ is bounded from below for $x$ in a shrinked interior region. The ball $B(x,2r)$ is contained in the coordinate chart, with $r$ bounded below by a positive constant. For $y$ in the annulus $B(x,2r)\setminus B(x,r)$ , we have $2\bar{\phi}(\frac{x+y}{2})\leq\bar{\phi}(x)+\bar{\phi}(y)$ , so upon integration

\int|\bar{\phi}|\gtrsim\int 2|\bar{\phi}(\frac{x+y}{2})|dy\geq\int|\bar{\phi}(x)|+|\bar{\phi}(y)|dy,

which bounds $|\bar{\phi}(x)|$ . Thus on a slightly shrinked $x$ -domain the oscillation is bounded:

\text{osc }\bar{\phi}=(\sup-\inf)\bar{\phi}\leq C,

and the Lipschitz bound follows again by convexity. ∎

Remark 4.5.

We discuss some intuition about log scales. Let $P\in X_{s}$ lie in $U_{w,\delta}^{s}$ , then a log scale $|z^{m_{i}}|\sim|z^{m_{i}}(P)|$ around $P$ refers to the subregion

\{\frac{1}{2}|z^{m_{i}}(P)|\lesssim|z^{m_{i}}|\lesssim 2|z^{m_{i}}(P)|,\quad 1\leq i\leq n\}.

Now $\log|z^{m_{i}}|$ vary by order $O(s)$ within $U_{w,\delta}^{s}$ , so there are an enormous number of log scales. The long range behaviour of $X_{s}$ is similar to $(\mathbb{C}^{*})^{n}$ , with half of the dimensions compactified into $T^{n}$ . On the other hand, over one log scale $X_{s}$ behaves qualitatively like the unit disc in $\mathbb{C}^{n}$ . The concept of local oscillation of a function refers to the oscillation within one log scale. In particular the Lipschitz bound (23) implies a local oscillation bound

\text{osc}_{|z^{m_{i}}|\sim|z^{m_{i}}(P)|}\bar{\phi}\leq Cs^{-1}.

4.3 Local potentials: plurisubharmonicity

The following lemma is a special case of the principle that for a subharmonic function, the standard mean value inequality has interesting strengthenings if there is more information about microscopic averages.

Lemma 4.6.

Let $\Phi$ be a subharmonic function on $B_{2}^{n}\times T^{k}=B_{2}\times\mathbb{R}^{k}/\epsilon\mathbb{Z}^{k}$ equipped with the Euclidean metric $g=\sum_{1}^{n}dx_{i}^{2}+\sum_{1}^{k}dy_{j}^{2}$ , where $0<\epsilon\ll 1$ . Let $v$ be the averaging function of $\Phi$ over the $T^{k}$ fibres. Assume $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int|\Phi|\lesssim 1$ and a Lipschitz bound $\text{Lip}(v)\lesssim 1$ , then on $B_{1}\times T^{k}$ we have $\Phi\leq v+C\epsilon^{1/2}$ .

Proof.

(courtesy of W. Feldman) By passing to the universal cover $B_{2}\times\mathbb{R}^{k}$ , the standard mean value inequality implies

\sup_{B_{3/2}\times T^{k}}\Phi\lesssim\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|\Phi|\lesssim 1.

Let $p\in B_{1}\times T^{k}$ , which lifts to a point $p$ in $B_{1}\times\mathbb{R}^{k}$ . Consider the Euclidean ball $B_{g}(p,\epsilon R)\subset B_{3/2}\times\mathbb{R}^{k}$ , where $R\gg 1$ is a parameter to be chosen. Then by the mean value inequality,

\Phi(p)\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_{B_{g}(p,\epsilon R)}\Phi.

Define the subset $E\subset B_{g}(\epsilon R)$ as the union of all interior lattice cubes, then

B_{g}(p,\epsilon R)\setminus E\subset B_{g}(p,\epsilon R)\setminus B_{g}(p,\epsilon(R-C)),

and by the lattice periodicity of $\Phi$ we have $\int_{E}\Phi=\int_{E}v$ . By partitioning the integral $\int_{B_{g}(p,\epsilon)}\Phi$ into the contributions from $E$ and $B_{g}(p,\epsilon R)\setminus E$ ,

\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_{B_{g}(p,\epsilon R)}\Phi\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_{B_{g}(p,\epsilon R)}v+CR^{-1}\sup_{B_{g}(p,\epsilon R)}(\Phi-v)\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_{B_{g}(p,\epsilon R)}v+CR^{-1}.

By the Lipschitz bound of $v$ , the RHS is bounded above by

v(p)+\text{Lip}(v)\epsilon R+CR^{-1}\leq v(p)+C(\epsilon R+R^{-1}).

Choosing $R=\epsilon^{-1/2}$ gives $\Phi(p)\leq v(p)+C\epsilon^{1/2}$ . ∎

Back to the setting of Prop. 4.4,

Corollary 4.7.

(Local potential upper bound) On $U^{s}_{w,\delta}$ , then $\phi-\bar{\phi}\leq Cs^{-1/2}.$

Proof.

The psh property of $\phi$ implies subharmonicity. By the Harnack inequality in Prop. 4.1 the average $L^{1}$ -integral is bounded, and by Prop. 4.4 there is a Lipschitz bound on the local average function $\bar{\phi}$ . ∎

Corollary 4.8.

(Local $L^{1}$ -oscillation bound) Over one log scale inside $U_{w,\delta}^{s}$ ,

\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_{|z^{m_{i}}|\sim|z^{m_{i}}(P)|}|\phi-\bar{\phi}|d\mu_{s}\leq Cs^{-1/2}.

Proof.

Recall the local oscillation of $\bar{\phi}$ in one log scale is $O(s^{-1})$ . Since the local sup of $\phi$ differs from the local average of $\phi$ by $O(s^{-1/2})$ , the local $L^{1}$ -oscillation is likewise bounded by $O(s^{-1/2})$ . ∎

Remark 4.9.

The $s$ -dependence is probably not optimal.

We now seek a local $L^{1}$ -oscillation bound on the charts of boundary type $U_{P}$ (cf. Remark 3.10). The idea is that any chart of boundary type overlaps with some chart of toric type in an annulus region, where the $L^{1}$ -oscillation bound is already known. It would be enough to transfer the $L^{1}$ -oscillation bound from the annulus to the deep interior of the chart.

Lemma 4.10.

Let $\Phi$ be a psh function on the $\{|z_{i}|\leq 4,\forall i\}\subset\mathbb{C}^{n}$ . Then

\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int_{B_{1}}|\Phi|\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int_{\{1<|z_{i}|<4,\forall i\}}|\Phi|.

Proof.

We induct on dimension. For $n=1$ , the unit ball is already enclosed by an annulus, so $\sup_{B(1)}\Phi$ is bounded above, and the mean value property applied to all balls $B(p,2)$ with $1<|p|\leq 2$ gives a lower bound on $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(1)}\Phi$ . Thus the $L^{1}$ -bound in $B(1)$ is clear.

For general $n$ , notice by induction we can bound for each $i\leq n$ ,

\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_{\{1<|z_{i}|<4,|z_{j}|<4,\forall j\neq i\}}|\Phi|\lesssim\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_{\{1<|z_{j}|<4,\forall j\}}|\Phi|,

so $\Phi$ is controlled in $L^{1}$ on an annulus enclosing $B(1)$ , and we can bound $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(1)}|\Phi|$ similar to the $n=1$ case. ∎

Corollary 4.11.

(Local $L^{1}$ -oscillation bound II) In the chart of boundary type $U_{P}$ , the local potential $\phi$ satisfies

\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_{U_{P}}|\phi-\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_{U_{P}}\phi|d\mu_{s}\leq Cs^{-1/2}.

4.4 Locally convex function

In section 4.2 we produced a collection of local average functions $\bar{\phi}=\bar{\phi}_{m,w}$ on $U_{w}^{s}$ corresponding to various choices of $w$ and $m$ with $\langle m,w\rangle=1$ . But the local coordinates $x^{m_{1}},\ldots x^{m_{n}}$ are naturally interpreted also as coordinates on $\partial\Delta_{\lambda}^{\vee}$ (cf. section 3.2), so $\bar{\phi}_{m,w}$ can be alternatively viewed as a collection of convex functions on the charts $U_{w}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$ of $\partial\Delta_{\lambda}^{\vee}$ . (Notice these local functions are defined without the need to shrink the domain to $U^{\infty}_{w,\delta}$ ).

The intuition is that up to $C^{0}$ -small error, the differences of these local functions agree with the cocycle $\{m-m^{\prime}\}$ , or equivalently, up to some $C^{0}$ -small fuzziness $\bar{\phi}_{m,w}+\langle m,x\rangle$ glue to a locally convex function on $\partial\Delta_{\lambda}^{\vee}$ in the sense of Definition 3.22. The more precise statement is

Lemma 4.12.

On overlapping charts of $\partial\Delta_{\lambda}^{\vee}$ ,

|\bar{\phi}_{m,w}-\bar{\phi}_{m^{\prime},w^{\prime}}+(m-m^{\prime})|\leq Cs^{-1/2}.

Proof.

Since we know the local $L^{1}$ -oscillation estimate holds in every local region, in a log scale in $U^{s}_{w}$ , not necessarily in the shrinked region $U^{s}_{w,\delta}$ ,

\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_{|z^{m_{i}}|\sim|z^{m_{i}}(P)|}|\varphi_{m}-\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\varphi_{m}|d\mu_{s}\leq Cs^{-1/2}.

Since $\bar{\phi}_{m,w}$ is convex, a local $L^{1}$ -bound implies a local $L^{\infty}$ -bound in a slightly shrinked region, so in the log scale,

|\bar{\phi}_{m,w}-\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\varphi_{m}|\leq Cs^{-1/2}.

Likewise for $\bar{\phi}_{m^{\prime},w^{\prime}}$ . By definition the local potentials differ by

\varphi_{m}-\varphi_{m^{\prime}}=\langle m^{\prime}-m,\text{Log}_{s}(z)\rangle

Notice that for a given point $P$ on $\partial\Delta_{\lambda}^{\vee}$ , the log scales on $U^{s}_{w}$ and $U^{s}_{w^{\prime}}$ around $P$ have a nontrivial percentage of overlapping measure. Thus

\begin{split}|\bar{\phi}_{m,w}-\bar{\phi}_{m^{\prime},w^{\prime}}+(m-m^{\prime})|&\lesssim s^{-1/2}+|\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\varphi_{m}-\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\varphi_{m^{\prime}}+(m-m^{\prime})|\\ &\lesssim s^{-1/2}+\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_{overlap}|\varphi_{m}-\varphi_{m^{\prime}}+(m-m^{\prime})|\\ &\lesssim s^{-1/2}.\end{split}

∎

Remark 4.13.

The tropical version $U^{\infty}_{w}$ of $U^{s}_{w}$ is in general larger than $U^{\infty}_{w}\cap\partial\Delta_{\lambda}^{\vee}$ ; it typically contains also some subset stretching to infinity along the $w$ -direction. If we regard $\bar{\phi}_{m,w}$ as local functions on $\mathcal{A}_{\lambda}^{\infty}$ instead of $\partial\Delta_{\lambda}^{\vee}$ , then there is a delicate issue. The Lemma above does not imply that $\bar{\phi}_{m,w}+\langle m,x\rangle$ for various choices of $m,w$ glue approximately on overlapping regions far from $\partial\Delta_{\lambda}^{\vee}$ . The problem is that such overlapping regions have too small measure, which breaks down the proof.

4.5 Legendre transform, extension, regularisation

We restrict to the Fermat case, and consider a general $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$ with $\sup_{X_{s}}\varphi=0$ , invariant under the symmetric group permuting the monomials $Z_{0}^{n+2},\ldots,Z_{n+1}^{n+2}$ . The goal of this section is to canonically patch together the local convex functions in section 4.4 approximately to produce a convex admissible function on $N_{\mathbb{R}}=\mathbb{R}^{n+1}$ . We will then induce a potential $\psi\in PSH(X_{s},s^{-1}\omega_{FS})\cap C^{0}$ which is a regularisation of $\varphi$ in the sense that it enjoys better a priori bounds than $\varphi$ .

Proposition 4.14.

There is an admissible convex function $u$ on $N_{\mathbb{R}}$ , such that on $U_{w}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$ ,

|u-(\bar{\phi}_{m,w}+m)|\leq Cs^{-1/2}.

(24)

Proof.

The idea is to regard $\bar{\phi}_{m,w}+\langle m,x\rangle$ as approximately defining a locally convex function on $\partial\Delta_{\lambda}^{\vee}$ in the sense of Def. 3.22, and then the problem is essentially to prove an effective version of the extension property (cf. Prop. 3.27). We will outline the main modifications.

We will produce $u$ by mimicking the Legendre duality construction in Prop. 3.19. For $p\in\Delta$ , define

u^{*}(p)=\sup_{x\in\partial\Delta_{\lambda}^{\vee}}\{\langle x,p\rangle-(\bar{\phi}_{m,w}+\langle m,x\rangle)\},

where it is tacitly understood that $\bar{\phi}_{m,w}+\langle m,x\rangle$ is defined only over $\partial\Delta_{\lambda}^{\vee}\cap U^{\infty}_{w}$ , and the sup is taken over all choices of $m,w$ whenever $\bar{\phi}_{m,w}$ is defined. Since $\bar{\phi}_{m,w}$ are uniformly bounded on $\partial\Delta_{\lambda}^{\vee}$ , we see $\left\lVert u^{*}\right\rVert_{C^{0}(\Delta)}\leq C$ . We then define a convex function $u$ on $N_{\mathbb{R}}$ by another Legendre transform

u(x)=\sup_{p\in\Delta}\{\langle p,x\rangle-u^{*}(p)\},

which is admissible because $u^{*}$ is bounded. By the same reasoning in Prop. 3.19, on $\partial\Delta_{\lambda}^{\vee}\cap U^{\infty}_{w}$ ,

u(x)\leq\bar{\phi}_{m,w}+\langle m,x\rangle+Cs^{-1/2}.

We are only left to show

u(x)\geq\bar{\phi}_{m,w}+\langle m,x\rangle-Cs^{-1/2},

which amounts to showing that there exists $p\in\Delta$ , such that for any $y\in\partial\Delta_{\lambda}^{\vee}$ ,

\bar{\phi}_{m^{\prime},w^{\prime}}(y)+\langle m^{\prime},y\rangle\geq\bar{\phi}_{m,w}(x)+\langle m,x\rangle+\langle p,y-x\rangle-Cs^{-1/2}.

Notice our setting enjoys the discrete symmetry. This last step is the effective version of Prop. 3.27, and the proof is basically the same. ∎

By construction $u$ has a number of additional properties:

Corollary 4.15.

The canonical extension $u$ satisfies an a priori Lipschitz bound

\begin{cases}|u-\max_{m}\langle m,x\rangle|\leq C,\quad\forall x\in N_{\mathbb{R}},\\ |u(x)-u(x^{\prime})|\leq C|x-x^{\prime}|,\quad\forall x,x^{\prime}\in N_{\mathbb{R}}.\end{cases}

(25)

Morever, in the region $\text{Star}(w)+\mathbb{R}_{\geq 0}w\subset N_{\mathbb{R}}$ , for any $m$ with $\langle m,w\rangle=1$ , the function $u_{m}=u-m$ is constant upon translation in the $w$ -direction.

Proof.

The first inequality is because the Legendre transform $u^{*}(p)$ is bounded on $\Delta$ as in the above proof, and the second is because $\nabla u\in\Delta$ . The morever statement is essentially identical to Cor. 3.28. ∎

By a small variant of Prop. 3.16, when we pullback the admissible convex functions $u$ via $\text{Log}_{s}$ , we obtain a torus invariant Kähler current on $(\mathbb{P}_{\Delta},s^{-1}[\Delta])$ with continuous local potentials. In details, we write $\psi_{0}=u\circ\text{Log}_{s}$ , and define

\begin{cases}\psi=\psi_{0}-\frac{(n+2)}{2s}\log(\sum_{m^{\prime}\in vertex(\Delta)}e^{\frac{2}{n+2}\langle m^{\prime},\text{Log}(z)\rangle}),\\ \psi_{m}=\psi_{0}-\langle m,\text{Log}_{s}(z)\rangle=u_{m}\circ\text{Log}_{s}.\end{cases}

(26)

By construction $\psi\in PSH(\mathbb{P}_{\Delta},s^{-1}\omega_{FS})\cap C^{0}$ , and $\psi_{0},\psi_{m}$ are the local potentials of $\psi$ (cf. (19)). By Cor. 4.15, $\left\lVert\psi\right\rVert_{C^{0}}\leq C$ , and $\psi$ inherits the Lipschitz bound from $u$ . By a slight abuse of notation, the restriction to $X_{s}$ will still be denoted as $\psi\in PSH(X_{s},s^{-1}\omega_{FS})\cap C^{0}$ . We think of $\psi$ as a regularisation of $\varphi$ .

Remark 4.16.

As explained in section 2.3, on toric manifolds the Legendre transform arises from a limiting version of approximation by algebraic metrics, which in turn is a more standard way to regularise an arbitrary Kähler potential. Now $X_{s}$ is not a toric manifold, but the toric symmetry holds approximately in generic regions, which motivates us to take the Legendre transform as a replacement of algebraic regularisation.

We now specify some subregions on $X_{s}^{toric}$ with coordinate descriptions. These are intimately related to $\partial\Delta_{\lambda}^{\vee}\setminus Sing$ , which is covered by the stars of the vertices and the interior of the top dimensional faces (cf. section 3.5).

Notation.

(Star type regions on $X_{s}$ ) On the region $U^{s}_{w}\subset X_{s}$ , recall the coodinates $z^{m_{i}}$ and regard $x^{m_{i}}=s^{-1}\log|z^{m_{i}}|$ as local coordinates also on $U^{\infty}_{w}\cap\partial\Delta_{\lambda}^{\vee}$ . Let $U^{s,*}_{w}\subset U^{s}_{w,\delta}$ be the subset where the $x^{m_{i}}$ coordinates correspond to points in $\text{Star}(w)\subset\partial\Delta_{\lambda}^{\vee}$ . The tropical analogue of $U^{s,*}_{w}$ is $(\text{Star}(w)+\mathbb{R}_{\geq 0}w)\cap A_{\lambda}^{\infty}$ .

Notation.

(Face type regions on $X_{s}$ ) Consider a slightly shrinked subset of the interior of a given top dimensional face of $\partial\Delta_{\lambda}^{\vee}$ . This can be regarded as a subset of $U_{w,\delta}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$ , where we regard $x^{m_{i}}=s^{-1}\log|z^{m_{i}}|$ as local affine coordinates. Let $U_{w}^{s,face}\subset U_{w}^{s}$ be the subset where the $x^{m_{i}}$ coordinates correspond to points in this shrinked face. The tropical analogue of $U_{w}^{s,face}\subset U_{w}^{s}$ is the shrinked face.

The intuition is that when $z\in X_{s}$ has $\text{Log}_{s}$ image close to $\partial\Delta_{\lambda}^{\vee}$ , or if this image approaches infinity in specific directions, then $\varphi-\psi$ is bounded above by a very small number:

Proposition 4.17.

(Local potential upper bound)

•

Inside $U^{s,*}_{w}\subset X_{s}$ , for $\langle m,w\rangle=1$ , the local potentials satisfy $\varphi_{m}-\psi_{m}\leq Cs^{-1/2},$ or equivalently $\varphi-\psi\leq Cs^{-1/2}$ .
•

Inside $U^{s,face}_{w}$ , the local potentials satisfies $\varphi_{0}-\psi_{0}\leq Cs^{-1/2}$ , or equivalently $\varphi-\psi\leq Cs^{-1/2}$ .

Proof.

In the star type region case, by Cor. 4.7, we have the upper bound $\varphi_{m}-\bar{\phi}_{m,w}\leq Cs^{-1/2}$ . By Prop. 4.14 and Cor. 4.15, in $U^{s,*}_{w}$ we can replace $\bar{\phi}_{m,w}$ by $\psi_{m}$ up to an error bounded by $Cs^{-1/2}$ , hence the claim. The face type region follows the same argument, without the translational invariance statement of Cor. 4.15. ∎

4.6 Improved Skoda inequality

Recall the local $L^{1}$ -oscillation bounds in both toric and boundary type regions, from Cor. 4.8 and 4.11. Consequently,

Lemma 4.18.

(Local Skoda estimate) Consider any $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$ normalised to $\sup_{X_{s}}\varphi=0$ . There are uniform positive constants $\alpha$ , $C$ , such that the local potentials $\phi$ satisfy

•

In a log scale in the toric region,

\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int_{|z^{m_{i}}|\sim|z^{m_{i}}(P)|}e^{-\alpha\sqrt{s}(\phi-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-6.57559pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-4.84631pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-3.71837pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-3.36836pt}}\!\int\phi)}d\mu_{s}\leq C.

•

In a boundary type chart,

\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int_{U_{P}}e^{-\alpha\sqrt{s}(\phi-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-6.57559pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-4.84631pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-3.71837pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-3.36836pt}}\!\int\phi)}d\mu_{s}\leq C.

Proof.

Apply the standard Skoda inequality (cf. Thm 2.1) to the rescaled function $s^{1/2}(\phi-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int\phi)$ . ∎

Remark 4.19.

The local average $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int\phi$ can be replaced by the local supremum using the mean value inequality.

Corollary 4.20.

(global Skoda estimate) Consider any $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$ normalised to $\sup_{X_{s}}\varphi=0$ . There are uniform positive constants $\alpha$ , $C$ , such that

\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_{X_{s}}e^{-\alpha\varphi}d\mu_{s}\leq C.

(27)

Proof.

By the local Skoda estimate and the Remark above, for both a log scale in the toric region, and a boundary type chart, the local average

\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 e^{\alpha\sqrt{s}(-\varphi+\sup_{loc}\varphi)}d\mu_{s}\leq C,

(28)

so in particular $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int e^{\alpha(-\varphi+\sup_{loc}\varphi)}\leq C.$ But we have already achieved a $C^{0}$ -bound on local average functions, and in particular a lower bound on local suprema. Thus

\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 e^{-\alpha\varphi}d\mu_{s}\leq Ce^{-\alpha\sup_{loc}\varphi}\leq C,

or equivalently $\int_{loc}e^{-\alpha\varphi}d\mu_{s}\leq C\int_{loc}d\mu_{s}$ for local integrals. To pass from this to the global Skoda estimate, we need to take a large collection of log scales and boundary type charts and sum over the estimates:

\int_{X_{s}}e^{-\alpha\varphi}d\mu_{s}\leq C\sum\int_{loc}d\mu_{s}.

The only problem is to ensure that the local charts can be chosen without substantially overcounting the measure. For points on $X_{s}$ whose $\text{Log}_{s}$ image is at $O(s^{-1})$ Euclidean distance to $\partial\Delta_{\lambda}^{\vee}$ , it is easy to choose the charts so that each point is contained in $O(1)$ number of charts. Away from $\partial\Delta_{\lambda}^{\vee}$ , the points deep inside the boundary type charts in general do not have this local finiteness property, but this is compensated by the fact that the measure $d\mu_{s}$ decays exponentially away from $\partial\Delta_{\lambda}^{\vee}$ (cf. (16)). The conclusion is that

\sum\int_{loc}d\mu_{s}\leq C\int_{X_{s}}d\mu_{s},

whence the global Skoda estimate. ∎

We now specialize to the Fermat case, and consider $\varphi\in PSH(X,s^{-1}\omega_{FS})$ normalised to $\sup_{X_{s}}\varphi=0$ with discrete symmetry, as in section 4.5. The regularisation of $\varphi$ produced via Legendre transform is denoted as $\psi$ .

Theorem 4.21.

(Improved Skoda estimate) In the Fermat case above, there are uniform constants $\alpha$ , $C$ , such that

\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int_{X_{s}}e^{-\alpha\sqrt{s}(\varphi-\psi)}d\mu_{s}\leq C.

(29)

Proof.

On either a log scale in the toric region, or a boundary type chart, we have by the local $L^{1}$ -oscillation estimate and the mean value inequality that

|\sup_{loc}\varphi_{m}-\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_{loc}\varphi_{m}|\leq Cs^{-1/2},\quad|\sup_{loc}\psi_{m}-\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_{loc}\psi_{m}|\leq Cs^{-1/2}.

Notice also the local averages of $\varphi_{m}$ and $\psi_{m}$ differ by $O(s^{-1/2})$ , so

|\sup_{loc}\varphi_{m}-\sup_{loc}\psi_{m}|\leq Cs^{-1/2},\quad|\sup_{loc}\varphi-\sup_{loc}\psi|\leq Cs^{-1/2}.

Combined with (28),

\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_{loc}e^{-\alpha\sqrt{s}(\varphi-\psi)}d\mu_{s}\leq C.

The summation argument as in the global Skoda estimate proves the claim. ∎

Remark 4.22.

This means $\phi-\psi$ can only fail to be bounded below by $Cs^{-1/2}$ on a set with exponentially small probability measure. Notice we have not yet used the complex MA equation.

4.7 $L^{\infty}$ and stability estimates for CY potentials

We finally impose the Calabi-Yau condition, and consider the CY potential $\varphi=\varphi_{CY,s}$ normalised to $\sup_{X_{s}}\varphi=0$ , solving (20):

\omega_{CY,s}^{n}=(s^{-1}\omega_{FS}+\sqrt{-1}\partial\bar{\partial}\varphi)^{n}=a_{s}s^{-n}d\mu_{s}.

Theorem 4.23.

( $L^{\infty}$ -estimate) The Calabi-Yau potential $\varphi_{CY,s}$ satisfies the uniform $L^{\infty}$ -estimate $\left\lVert\varphi_{CY,s}\right\rVert_{L^{\infty}}\leq C$ .

Proof.

We apply Kolodziej’s estimate in Thm 2.7. The Skoda type inequality (2) is verified in Cor. 4.20, hence the $L^{\infty}$ estimate. ∎

We now specialize to the Fermat case. Clearly $\varphi$ is invariant under the discrete symmetry of the hypersurface. Recall the regularisation is denoted as $\psi=\psi_{CY,s}$ , coming from the double Legendre transform construction $u=u_{CY,s}$ (cf. section 4.5). The local potentials of $\varphi_{CY,s}$ and $\psi_{CY,s}$ are denoted $\varphi_{m}=\varphi_{CY,s,m}$ and $\psi_{m}=\psi_{CY,s,m}$ according to the same convention as (19).

Theorem 4.24.

In the Fermat case, there is a uniform stability estimate

\varphi_{CY,s}-\psi_{CY,s}\geq-Cs^{-1/2}\log s.

(30)

Proof.

We apply Cor. 2.12. The Skoda estimate is verified in Cor. 4.20. The improved Skoda estimate Thm. 4.21 implies an exponential volume decay:

\frac{\int_{\varphi-\psi\leq-t}\omega_{\phi}^{n}}{\text{Vol}(X_{s})}\leq Ce^{-\alpha t\sqrt{s}},

hence there exists $c\gg 1$ , such that for $t_{0}=cs^{-1/2}\log s$ ,

\left(\frac{\int_{\varphi-\psi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(X_{s})}\right)^{1/2n}\leq Ce^{-\alpha t_{0}\sqrt{s}/2n}=Ce^{-\alpha c\log s/2n}\leq Cs^{-1/2}.

Thm 2.7 then implies $\varphi-\psi\geq-Cs^{-1/2}\log s$ as required. ∎

Remark 4.25.

In the theorems above only an upper bound on the volume measure is actually needed. The intuition is that the Skoda inequality is already so close to an $L^{\infty}$ estimate, that a very tiny amount of extra assumptions are needed to conclude $L^{\infty}$ -estimate.

Combining this with the upper bound from Prop. 4.17,

Corollary 4.26.

In the Fermat case, there is a uniform $C^{0}$ -stability estimate:

•

Inside $U^{s,*}_{w}\subset X_{s}$ , for $\langle m,w\rangle=1$ , the local potentials satisfy $|\varphi_{CY,s,m}-\psi_{CY,s,m}|\leq Cs^{-1/2}\log s,$ or equivalently $|\varphi_{CY,s}-\psi_{CY,s}|\leq Cs^{-1/2}\log s$ .
•

Inside $U^{s,face}_{w}$ , the local potentials satisfy $|\varphi_{CY,s,0}-\psi_{CY,s,0}|\leq Cs^{-1/2}\log s$ , or equivalently $|\varphi_{CY,s}-\psi_{CY,s}|\leq Cs^{-1/2}\log s$ .

The point is that in the generic region of $X_{s}$ the Calabi-Yau local potentials are $C^{0}$ -approximated by their regularisations, which build in convexity by construction, and therefore have a priori Lipschitz bounds.

4 Estimates on the Kähler potential [00TL]