ScalingStacks

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

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.

Once we know the subsequential limit $u_{\infty}$ satisfies the real MA equation, then by the local regularity theory surveyed in section 2.6,

We now proceed to a very explicit coordinate version of higher order estimates for the local CY potentials, by transferring regularity from the real MA equation to the complex MA equation.

Let $x\in\mathcal{R}$, then $u_{\infty}$ (resp. the appropriate $u_{\infty,m})$ has $C^{k,\gamma}$-bound on some coordinate ball $B(x,2r(x))\subset\mathcal{R}$ contained in a shrinked face (resp. $\text{Star}(w)$). For clarity we focus on the face case. The radius $r(x)$ and the $C^{k,\gamma}$-bound depend on the choice of $x$, but are uniform for $x$ in any fixed compact subset of $\mathcal{R}$. We identify $u_{\infty}$ with its pullback to $(s^{-1}\text{Log})^{-1}(B(x,r(x)))\subset U^{s,face}_{w}\subset X_{s}$.

The local CY potential $\varphi_{CY,s,0}$ on $(s^{-1}\text{Log})^{-1}(B(x,2r(x)))$ satisfies

along the subsequence. We may regard $(s^{-1}\text{Log})^{-1}(B(x,2r(x)))$ as an open subset of $(\mathbb{C}^{*})^{n}$. On the universal cover of $(\mathbb{C}^{*})^{n}$, we use the natural coordinates $s^{-1}\log z^{m_{i}}$ for $i=1,\ldots n$.

Now $\varphi_{CY,s,0}$ satisfies the complex MA equation (cf. (20)(14))

By the holomorphic volume form formula (12),

where the $o(1)$ term in fact has exponentially small $C^{\infty}$ bounds in $s^{-1}\log z^{m_{i}}$ coordinates; the higher order bound uses that $\Omega_{s}$ is holomorphic. On the other hand by the calculation in section 5.1, the pullback of $u_{\infty}$ satisfies

To summarize, the deviation of RHS is negligible and the deviation between $\varphi_{CY,s,0}$ and $u_{\infty}\circ s^{-1}\text{Log}$ is small in $C^{0}$-norm. Applying Savin’s Thm. 2.14,

The intuition is that in the generic regular locus in the toric part of $X_{s}$, the local CY potentials converge in some $C^{\infty}_{loc}$ sense.

Next we discuss CY metrics in $(s^{-1}\text{Log})^{-1}(B(x,r(x)))\subset U_{s,K}$.

• •

  In the face type region case, up to $C^{\infty}$-small error in the $s^{-1}\log z^{m_{i}}$ coordinates,

  $$\begin{split}&\omega_{CY,s}=\sqrt{-1}\partial\bar{\partial}\varphi_{CY,s,0}\\ \approx&\sqrt{-1}\partial\bar{\partial}u_{\infty}\circ s^{-1}\text{Log}=\frac{1}{4}\frac{\partial^{2}u_{\infty}}{\partial x^{m_{i}}\partial x^{m_{j}}}\sqrt{-1}s^{-1}d\log z^{m_{i}}\wedge s^{-1}d\overline{\log z^{m_{j}}},\end{split}$$

  hence the CY metrics $g_{CY,s}$ is up to $C^{\infty}$-small error

  $$g_{CY,s}\approx\text{Re}\{\frac{1}{2}\frac{\partial^{2}u_{\infty}}{\partial x^{m_{i}}\partial x^{m_{j}}}s^{-1}d\log z^{m_{i}}\otimes s^{-1}d\overline{\log z^{m_{j}}}\}.$$

  (32)

• •

  Likewise in the star type region case, up to $C^{\infty}$ small error in the $s^{-1}\log z^{m_{i}}$ coordinates,

  $$\begin{cases}\omega_{CY,s}\approx\frac{1}{4}\frac{\partial^{2}u_{\infty,m}}{\partial x^{m_{i}}\partial x^{m_{j}}}\sqrt{-1}s^{-1}d\log z^{m_{i}}\wedge s^{-1}d\overline{\log z^{m_{j}}},\\ g_{CY,s}\approx\text{Re}\{\frac{1}{2}\frac{\partial^{2}u_{\infty,m}}{\partial x^{m_{i}}\partial x^{m_{j}}}s^{-1}d\log z^{m_{i}}\otimes s^{-1}d\overline{\log z^{m_{j}}}\}.\end{cases}$$

  (33)

Notice in such local $(\mathbb{C}^{*})^{n}$ coordinates, the rescaled log map $s^{-1}\text{Log}$ gives a local $T^{n}$-fibration. The metric associated to $\sqrt{-1}\partial\bar{\partial}(u_{\infty}\circ s^{-1}\text{Log})$ is a semiflat metric, namely a $T^{n}$-invariant metric which is flat when restricted to any $T^{n}$-fibre. Thus (32)(33) assert that the Calabi-Yau metrics $g_{CY,s}$ are $C^{\infty}$-approximated by semiflat metrics in the regular regions.

On the regular locus $\mathcal{R}\subset\partial\Delta_{\lambda}^{\vee}$ we have a well defined real MA metric,

Notice the definitions are compatible on overlapping regions. Let $(\bar{\mathcal{R}},g_{\infty})$ be the metric completion. The metric asymptotes (32)(33) say that in some $C^{\infty}_{loc}$ sense the collapsing CY metrics $g_{CY,s}$ converge to the metric $g_{\infty}$ on $\mathcal{R}$, and we know $\mathcal{R}$ is path connected because its complement has zero $\mathcal{H}^{n-1}$-measure.

The goal of this section is to show

In the setting of section 5.2, the very strong regularity bounds in the generic region leads to the existence of special Lagrangian $T^{n}$-fibrations thereon.