Subsection 4.5 (00BG) · ScalingStacks

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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’.

00AL

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 $i=1,2,\ldots,n$ .

00AM

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].

00AN

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 $(X_{t},g_{CY,t})$ .

00AP

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.