Subsection 5.1 (00TV) · ScalingStacks

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5.1 Limiting real MA metric

We work in the context of section 4.7, and use the notations therein. We shall extract some subsequential limit of local potentials for the CY metric $\omega_{CY,s}$ , and check that up to a constant it solves the real MA equation on $\partial\Delta_{\lambda}^{\vee}\setminus Sing$ according to Def. 3.29 (cf. also section 2.6).

Since the convex functions $u_{CY,s}$ on $N_{\mathbb{R}}$ produced by double Legendre transform have uniform Lipschitz bounds (25), by the Arzela-Ascoli theorem we can take a subsequential limit as $s\to\infty$ , such that $u_{CY,s}\to u_{\infty}$ in $C^{0}_{loc}$ -topology. Later we will sometimes suppress mentioning the subsequence for brevity. In particular $u_{\infty}$ is convex and admissible. We can also pass Cor. 4.15 to the limit, to see that 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 $u_{\infty,m}=u_{\infty}-m$ is constant upon translation in the $w$ -direction. In particular in such regions the $C^{0}_{loc}$ convergence improves to $\left\lVert u_{CY,s,m}-u_{\infty,m}\right\rVert_{C^{0}}\to 0$ .

By construction $\psi_{CY,s,m}=u_{CY,s,m}\circ\text{Log}_{s}$ , and $u_{CY,s,0}=u_{CY,s}\circ\text{Log}_{s}.$ Thus the stability estimate Cor. 4.26 implies that

•

Inside $U^{s,*}_{w}\subset X_{s}$ , for $\langle m,w\rangle=1$ , the local potentials satisfy

$|\varphi_{CY,s,m}-u_{\infty,m}\circ\text{Log}_{s}|\to 0.$
•

Inside $U^{s,face}_{w}$ , the local potentials satisfy

$|\varphi_{CY,s,0}-u_{\infty}\circ\text{Log}_{s}|\to 0.$

The rest of this section is devoted to proving

00SK

Theorem 5.1. On $\partial\Delta_{\lambda}^{\vee}\setminus Sing$ , the locally convex function $u_{\infty}$ solves the real MA equation in the sense of Def. 3.29 up to a scaling constant:

MA(u_{\infty})=\frac{a_{\infty}}{\pi^{n}n!}d\mu_{\infty},

(31)

where $d\mu_{\infty}$ is the Lebesgue measure on $\partial\Delta_{\lambda}^{\vee}$ , and the constant $a_{\infty}$ is defined by (21).

The intuitive idea is to pass the complex MA equation to some weak limit. The main problem is that the sequence $\varphi_{CY,s}$ live on different manifolds, so we need more effective estimates to pass to the limit.

00SL

Lemma 5.2. Let $u$ be a bounded convex function on the square $\{|x_{i}|<1\}\subset\mathbb{R}^{n}$ . Via the rescaled log map $s^{-1}\text{Log}:(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$ , the function $u$ pulls back to a psh function on $\{|\log|z_{i}||<s\}$ . Then the real MA measure of $u$ is related to the pushforward of the complex MA measure of $u\circ s^{-1}\text{Log}$ by

MA(u)=\frac{s^{n}}{\pi^{n}n!}(s^{-1}\text{Log})_{*}(\sqrt{-1}\partial\bar{\partial}(u\circ s^{-1}\text{Log}))^{n}.

00SM

Proof. If $u$ is smooth, then

MA(u)(K)=\int_{K}\det(D^{2}u)dx_{1}\ldots dx_{n}=\frac{s^{n}}{\pi^{n}n!}\int_{(s^{-1}\text{Log})^{-1}(K)}(\sqrt{-1}\partial\bar{\partial}(u\circ s^{-1}\text{Log}))^{n}.

Since $u\in C^{0}$ , and both the real and complex MA operators are weakly continuous with respect to $C^{0}$ -limits, this equality passes to general $u$ . ∎

00SN

Lemma 5.3. (Chern-Levine type estimate) Let $u$ be a psh function on the annulus region $U=\{|\log|z_{i}||<s,\forall i\}\subset(\mathbb{C}^{*})^{n}$ , with $\left\lVert u\right\rVert_{L^{\infty}}\lesssim 1$ . Then

•

On the shrinked set $E=\{|\log|z_{i}||<s/2\}$ the measure

$\int_{E}(\sqrt{-1}\partial\bar{\partial}u)^{n}\leq Cs^{-n}.$

•

Let $u+v$ is another psh function, with $\left\lVert v\right\rVert_{L^{\infty}}\ll 1$ . Let $f$ be any compactly supported function on the square $\{|x_{i}|<1\}\subset\mathbb{R}^{n}$ . Then

\int f\{(\sqrt{-1}\partial\bar{\partial}(u+v))^{n}-(\sqrt{-1}\partial\bar{\partial}u)^{n}\}\leq Cs^{-n}\left\lVert f\right\rVert_{C^{2}}\left\lVert v\right\rVert_{L^{\infty}}.

00SP

Proof. Let $\chi$ be a compactly supported nonnegative smooth function on the square $\{|x_{i}|<1\}\subset\mathbb{R}^{n}$ , equal to one on $\{|x_{i}|\leq 1/2\}$ . We identify $\chi$ with $\chi\circ s^{-1}\text{Log}$ , and denote $\omega_{std}=\sqrt{-1}\sum d\log z_{i}\wedge d\overline{\log z_{i}}$ . Then

-Cs^{-2}\omega_{std}\leq\sqrt{-1}\partial\bar{\partial}\chi\leq Cs^{-2}\omega_{std}.

The basic obervation is that if $T$ is a positive current of bidegree $(n-1,n-1)$ , then by integration by part,

\begin{split}&\int_{E}\sqrt{-1}\partial\bar{\partial}u\wedge T\leq\int_{\text{supp}(\chi)}\chi\sqrt{-1}\partial\bar{\partial}u\wedge T\\ &=\int_{\text{supp}(\chi)}u\sqrt{-1}\partial\bar{\partial}\chi\wedge T\leq Cs^{-2}\int_{\text{supp}(\chi)}\omega_{std}\wedge T.\end{split}

Iterating this argument to lower the power of $\sqrt{-1}\partial\bar{\partial}u$ ,

\int_{E}(\sqrt{-1}\partial\bar{\partial}u)^{n}\leq Cs^{-2n}\int_{U}\omega_{std}^{n}\leq Cs^{-n}.

The second statement is proved similarly by removing $\sqrt{-1}\partial\bar{\partial}v$ factors iteratively. ∎

00SQ

Proof. (Thm. 5.1) There are two subcases: the interior of the top dimensional faces of $\partial\Delta_{\lambda}^{\vee}$ , and the star of the vertices $\text{Star}(w)$ . Since the arguments are almost the same we focus on the latter.

On the interior of $\text{Star}(w)$ , we have local affine coordinates $x^{m_{1}},\ldots x^{m_{n}}$ , related to the holomorphic $\mathbb{C}^{*}$ -coordinates $z^{m_{1}},\ldots z^{m_{n}}$ by $x^{m_{i}}=s^{-1}\log|z^{m_{i}}|$ . The star type region $U^{s,*}_{w}\subset X_{s}$ can be viewed as a subset of $(\mathbb{C}^{*})^{n}$ , so we use the rescaled map $s^{-1}\text{Log}:(\mathbb{C}^{*})^{n}_{z^{m_{i}}}\to\mathbb{R}^{n}_{x^{m_{i}}}$ to pullback the function $u_{\infty,m}$ on $\text{Star}(w)$ . On the other hand, $X_{s}\cap(\mathbb{C}^{*})^{n+1}$ maps into $N_{\mathbb{R}}$ via $\text{Log}_{s}$ , so we can also pullback $u_{\infty,m}$ via $\text{Log}_{s}$ . These two pullbacks differ by at most $Cs^{-1}$ using Cor. 4.15. We also write $\phi=\varphi_{CY,s,m}$ .

Take a local test function $f\in C^{2}_{c}$ supported in the interior of $\text{Star}(w)$ , then $f$ is identified as a local function on $U^{s,*}_{w}\subset X_{s}$ via $s^{-1}\text{Log}$ . By the Chern-Levine type estimate above,

\begin{split}&s^{n}\int f\{(\sqrt{-1}\partial\bar{\partial}\phi)^{n}-(\sqrt{-1}\partial\bar{\partial}(u_{\infty,m}\circ s^{-1}\text{Log}))^{n}\}\\ &\leq C\left\lVert\phi-u_{\infty,m}\circ s^{-1}\text{Log}\right\rVert_{L^{\infty}}\left\lVert f\right\rVert_{C^{2}}\to 0,\end{split}

as $s\to+\infty$ . By the Calabi-Yau condition (20) and Prop. 3.14,

s^{n}(\sqrt{-1}\partial\bar{\partial}\phi)^{n}=a_{s}d\mu_{s}=a_{\infty}d\mu_{s}(1+o(1)),\quad s\to+\infty.

Pushing forward via $s^{-1}\text{Log}$ , and applying Lemma 5.2,

\pi^{n}n!\int fMA(u_{\infty})=\lim_{s\to\infty}\int fa_{s}(s^{-1}\text{Log})_{*}d\mu_{s}=a_{\infty}\int fd\mu_{\infty}.

Since this holds for every $f\in C^{2}_{c}$ , on the interior of this top dimensional face we obtain the measure equality (31). ∎