Subsection 5.3 (00TX) · ScalingStacks

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5.3 Gromov-Hausdorff convergence

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

g_{\infty}=\begin{cases}\frac{1}{2}\sum_{i,j}\frac{\partial^{2}u_{\infty}}{\partial x_{i}\partial x_{j}}dx_{i}dx_{j},\quad\text{ on the face regions},\\ \frac{1}{2}\sum_{i,j}\frac{\partial^{2}u_{\infty,m}}{\partial x_{i}\partial x_{j}}dx_{i}dx_{j},\quad\text{ on the star regions}.\end{cases}

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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.

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Remark 5.9. We do not know if $\bar{\mathcal{R}}$ is homeomorphic to $\partial\Delta_{\lambda}^{\vee}\simeq S^{n}$ , as the regularity theory of the real MA equation on a singular affine manifold is not yet developed, and we know little about what can happen near singularities.

The goal of this section is to show

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Theorem 5.10. The subsequence of collapsing CY metrics $(X_{s},g_{CY,s})$ converges in the Gromov-Hausdorff sense to $(\bar{\mathcal{R}},g_{\infty})$ .

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Proposition 5.11. There is a uniform diameter bound

\text{diam}(X_{s},g_{CY,s})\leq C.

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Proof. This argument is essentially the same as [36, Thm 3.1]. We quote [36, Lem 3.2]:

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Lemma 5.12. Let $(M^{2n},g)$ be a closed Riemannian manifold with $Ric(g)\geq 0$ , let $p\in M$ and $1<R\leq diam(X,g)$ . Then $\frac{R-1}{4n}\leq\frac{\text{Vol}(B(p,2(R+1)))}{\text{Vol}(B(p,1))}$ .

Using Thm. 5.6, we can find inside the regular region of $X_{s}$ some geodesic ball $B_{g_{CY,s}}(p,r)$ of radius $r<1$ , occupying a nontrivial portion of the total volume:

\frac{\text{Vol}(B_{g_{CY,s}}(p,r)))}{\text{Vol}(X_{s})}\geq\epsilon>0,

with $\epsilon$ independent of $s$ . Now applying the Lemma to the rescaled CY metric $r^{-2}g_{CY,s}$ ,

\frac{\text{diam}(X_{s})-r}{4nr}\leq\frac{\text{Vol}(B_{g_{CY,s}}(p,2(\text{diam}(X_{s})+r)))}{\text{Vol}(B_{g_{CY,s}}(p,r))}\leq\frac{\text{Vol}(X_{s})}{\text{Vol}(B_{g_{CY,s}}(p,r))}\leq\epsilon^{-1},

so $\text{diam}(X_{s})\leq Cr\leq C$ as required. ∎

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Proof. (Thm. 5.10) By Thm 5.6 we already know the metric convergence over any properly contained open subset of $\mathcal{R}$ , which corresponds to a region $U_{s}\subset X_{s}$ , with nearly the full measure:

\text{Vol}(U_{s})>(1-\epsilon)\text{Vol}(X_{s}),

where $\epsilon$ can be chosen arbitrarily small. It now suffices to show any point $p\in X_{s}\setminus U_{s}$ is close to $U_{s}$ . For any $r>0$ such that the geodesic ball $B_{g_{CY,s}}(p,r)\subset X_{s}\setminus U_{s}$ , the Bishop-Gromov inequality implies

\left(\frac{r}{\text{diam}(X_{s})}\right)^{2n}\leq\frac{\text{Vol}(B_{g_{CY,s}}(p,r))}{\text{Vol}(X_{s})}\leq\frac{\text{Vol}(X_{s}\setminus U_{s})}{\text{Vol}(X_{s})}<\epsilon.

Taking the sup of all such $r$ ,

\text{dist}_{g_{CY,s}}(p,U_{s})\leq\epsilon^{1/2n}\text{diam}(X_{s})\leq C\epsilon^{1/2n},

which can be made arbitrarily small. ∎