ScalingStacks

The Strominger-Yau-Zaslow (SYZ) conjecture [34] is the following: given a family of $n$-dimensional polarised Calabi-Yau (CY) manifolds $(X_{s},g_{s},J_{s},\omega_{s},\Omega_{s})$ of holonomy $SU(n)$ degenerating to the large complex structure limit, then

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  After suitable scaling, the metric spaces $(X_{s},g_{s})$ converge in the Gromov-Hausdorff sense to a singular affine manifold $B$ homeomorphic to $S^{n}$. The limiting metric $g_{\infty}$ is a real Monge-Ampère metric on the smooth locus. (This part is also known as the Kontsevich-Soibelman conjecture [29][28].)

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  Near the degenerating limit, the manifold $X_{s}$ admits a special Lagrangian $T^{n}$ fibration over the base $B$ with some singular fibres. The diameters of the fibres are much smaller compared to $\text{diam}(B)$. In the generic region on $X_{s}$, which covers most of the measure on $X_{s}$, the metric $g_{s}$ is a small perturbation of a semiflat metric, meaning that the $T^{n}$ fibres are almost flat.

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  Mirror manifolds should be constructed as another $T^{n}$ fibration over the same base $B$, by fibrewise replacing the $T^{n}$ fibres with the dual tori.

An early achievement is Gross and Wilson’s gluing construction [21] of degenerating CY metrics on K3 surfaces with elliptic fibrations, which becomes a special Lagrangian $T^{2}$-fibration after hyperkähler rotation. In this setting the metric is known semi-explicitly. The same period brought forth many insights concerning topological [19], combinatorial [23][24], and differential geometric [39] aspects of the SYZ conjecture, until Joyce [27] discovered through his study of special Lagrangian singularities that the SYZ fibration map cannot be naïvely expected to be smooth, indicating the difficulty of the metric problem.

Later research on the SYZ conjecture gradually shifted focus from its metric geometric roots, in favour of softer approaches based on algebraic or symplectic methods, taking the original SYZ conjecture mainly as an inspiration. This has led to spectacular progress in the mathematical understanding of mirror symmetry, described in the excellent survey [18].

In the metric vein, the SYZ conjecture fits into the more general question of understanding how CY metrics degenerate as the complex and Kähler structures vary. The main dichotomy is whether the family of metrics are noncollapsed, meaning there is a uniform lower bound on the volume once the diameter is normalised to one. In the noncollapsing case much is known: for example, a polarised family of noncollapsed CY manifolds can only degenerate to normal CY varieties with klt singularities, and the notion of metric convergence agrees with the algebro-geometric notion of flat limit [14].

The collapsing case is widely open. Tosatti et al. made substantial progress on describing collapsing metrics associated with holomorphic fibrations [37][20], in particular generalising much of [21] to hyperkähler manifolds with holomorphic Lagrangian fibrations. Recently there are many efforts to describe the degenerating CY metrics in special cases, notably for K3 surfaces [17][26][32], and higher dimensional generalisations [35].

The metric SYZ conjecture resisted most attempts because the large complex structure limit is a very severe degeneration mechanism. An interesting program of Boucksom et al. [4][3] proposes that in the case of polarised algebraic degenerations the underlying Calabi-Yau manifolds converge naturally into a non-archimedean (NA) space, and the CY metrics should converge in a potential theoretic sense to their NA analogue. Their greatest achievements so far is to define and solve the NA Monge-Ampère (MA) equation, building on heavy machinery from birational geometry. To make contact with the SYZ conjecture, it would still remain to compare the non-archimedean MA equation with the real MA equation, prove the potential theoretic convergence, and improve it to the metric convergence. Notwithstanding these difficulties, this program has the promise to prove the SYZ conjecture in great generality.

The viewpoint of this paper is much more concrete. We focus on the Fermat family of projective hypersurfaces of any dimension $n$, approaching the large complex structure limit:

The most striking aspect of our results is

We also summarize informally the other results in this paper:

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  (cf. section 5.3) The subsequence of CY metrics converge in the Gromov-Hausdorff sense to the metric completion of a smooth real MA metric on an open dense subset $\mathcal{R}\subset\partial\Delta_{\lambda}^{\vee}$, where $\partial\Delta_{\lambda}^{\vee}$ denotes the boundary of a certain $(n+1)$-dimensional simplex $\Delta_{\lambda}^{\vee}$ in $\mathbb{R}^{n+1}$ arising naturally from tropical geometry, and $\partial\Delta_{\lambda}^{\vee}\setminus\mathcal{R}$ has zero $(n-1)$-Hausdorff measure.

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  (cf. Prop. 5.11) The diameters of the subsequence of CY metrics are uniformly bounded.

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  (cf. section 5.2) In the generic region of $X_{s}$ for $s\gg 1$, the CY metrics are $C^{\infty}_{loc}$ close to a sequence of semiflat metrics. In particular the sectional curvature in the generic region is uniformly bounded.

A basic feature of the complex geometry of CY hypersurfaces near the large complex structure limit, is that in generic regions the local structure is a large annulus region in $(\mathbb{C}^{*})^{n}$, equipped with a holomorphic volume form which modulo a scale factor is very close to $d\log z_{1}\wedge\ldots d\log z_{n}$. An elementary observation is that plurisubharmonic (psh) functions are intimately related to convex functions:

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  Let $\phi$ be psh on an annulus $\{1<|z_{j}|<\Lambda\}\subset(\mathbb{C}^{*})^{n}$, then the fibrewise average function

  $$\bar{\phi}(x_{1},\ldots x_{n})=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{T^{n}}\phi(e^{x_{1}+i\theta_{1}},\ldots e^{x_{n}+i\theta_{n}})d\theta_{1}\ldots d\theta_{n}$$

  is convex.

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  Let $u$ be a convex function on $\{0<x_{j}<\log\Lambda\}$, then the pullback of $u$ to $\{1<|z_{j}|<\Lambda\}\subset(\mathbb{C}^{*})^{n}$ via the logarithm map is psh, and $u$ solves the real MA equation $\det(D^{2}u)=\text{const}$ iff its pullback solves the complex MA equation $\det(\frac{\partial^{2}u}{\partial\log z_{i}\partial\overline{\log z_{j}}})=\text{const}$.

Our strategy is to show that in the highly collapsed regime $s\gg 1$, the local Kähler potentials are $C^{0}$-approximated by convex functions, whose regularity properties can be then transferred back to the local Kähler potentials at least in the generic region. In effect, this implies in the generic region the Calabi-Yau metrics are collapsing with uniformly bounded sectional curvature; then the existence of the special Lagrangian fibration in the generic region is a simple perturbation argument. Keeping in mind that the local complex structure is an annulus in $(\mathbb{C}^{*})^{n}$, the special Lagrangian fibration is just a small $C^{\infty}$-perturbation of the logarithm map $(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$.

The essential problem is to obtain uniform estimates on the CY metrics as $s\to\infty$. Our techniques differ very significantly from Yau’s proof of the Calabi conjecture. Our Kähler potential estimates are largely based on Kolodziej’s method in pluripotential theory, which has the advantage of robustness even in collapsing settings. The technical core of our contribution is to produce a regularisation of the Calabi-Yau potential, and prove an improved version of the global Skoda inequality, which for large $s$ forces the potential to be very close to its regularisation. As convexity is built into the construction of the regularisation, this furnishes a bridge between holomorphic and convex geometry, and one can start to transfer the a priori much better regularity from the convex world into the holomorphic world near the collapsing limit $s\to\infty$. Our higher order estimates exploit the local regularity theory of real MA equations, and a result of Savin from nonlinear PDE theory.

The structure of the paper is as follows. We survey the rather extensive analytical backgrounds in section 2. The complex geometry of the degenerating hypersurfaces is discussed in section 3, with particular emphasis on its interplay with tropical geometry. We estimate the Calabi-Yau potentials in section 4; in particular we prove the Skoda type estimates, the uniform $L^{\infty}$ bound, and the $C^{0}$-approximation by the convex regularisations. In section 5, we use uniform Lipschitz bounds on the regularisation to extract a subsequential limit, and show that this defines a real MA metric. We then use the local regularity theory of real MA metrics to show the higher order estimates on the CY local potentials, and prove the existence of the special Lagrangian fibration.

We now discuss some directions of future research.

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  It seems highly plausible that the SYZ conjecture on generic regions will hold also on many other degenerating CY manifolds, or at least CY hypersurfaces. In fact the only reason we restrict to the Fermat case is to utilize the large discrete symmetry group to give a relatively simple proof of a technical extension property for locally convex functions, which seems likely to generalise to other contexts.

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  One would like to study the existence, uniqueness, and regularity of the real MA equation on compact polyhedral sets, which are covered by charts whose transition functions are only piecewise linear but not smooth in general; the SYZ conjecture predicts the solutions to such real MA equations should arise as possible limits of the collapsing CY metrics. This question may be parallel to the non-archimedean MA approach taken up in [5]. At present according to the author’s knowledge, it is not clear how to define the real MA equation globally on such sets, and in fact we do not even have an established notion of local convexity.

  Such questions on the real MA equations have direct bearings on improving our main theorem. For instance, if one can establish uniquenss, then there is no need to pass to subsequences in all of our results. If one can establish sufficient regularity, then it may be possible to prove the Gromov-Hausdorff limit is homeomorphic to $\partial\Delta_{\lambda}^{\vee}\simeq S^{n}$.

  The problem to set up the real MA equation is quite subtle. On a piecewise linear manifold the notion of a convex function is dependent on charts, and so does the real MA operator. To set up an invariant notion of the real MA equation, it is necessary to make branch cuts to charts. The location of such cuts seems to depend on some gradient condition on the convex function in question, and is hard to predict in the absence of symmetry. Thus the global real MA equation on polyhedral sets has the feature of a free boundary problem.

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  The a priori estimate approach in this paper says very little about the CY metrics in regions with high curvature concentration. In the case of CY 3-folds, the author [30] recently constructed the 3-dimensional analogues of the Ooguri-Vafa metric, which are conjectured to be the universal metric models for the neighbourhood of the most singular fibres in a generic SYZ fibration. A program to tackle the 3-fold case of the SYZ conjecture based on gluing ideas is outlined in [30], which has the ultimate aim to give a global description of the metric, and to produce a special Lagrangian fibration globally. This gluing approach requires very refined information on the singularities of the real MA equation, which is still far from what we can establish by a priori estimate considerations.