A cornerstone of Kähler geometry is Yau’s celebrated proof of the Calabi conjecture, which implies

We now give a very sketchy account of (the modern view on) Yau’s proof of Theorem 4.1. Fix a background Kähler metric $\omega$ in the given class, and the task is to find a Kähler potential $\phi$ solving the complex Monge-Ampère equation for any given density function $e^{f}$ satisfying the cohomological constraint $\int_{X}e^{f}\omega^{n}=\int_{X}\omega^{n}$,

$$\omega_{\phi}^{n}=(\omega+dd^{c}\phi)^{n}=e^{f}\omega^{n},\quad\int_{X}\phi\omega^{n}=0,$$

(9)

In particular for $e^{f}\omega^{n}=\text{const}\cdot\Omega\wedge\overline{\Omega}$ one obtains the Calabi-Yau metric. The uniqueness follows from a simple integration by parts argument.

The existence proof follows the continuity method, namely to deform through the space of Kähler metrics as $f$ deforms from zero to the desired function. Viewing (9) as defining a map from the potential to the density function, we need to show the map is submersive and proper. The submersion property is a consequence of standard Hodge theory on the Laplacian. To show properness one needs a priori estimates on $\phi$ depending only on $X$ and bounds on $f$, and for that matter we are free to impose any order of regularity on $f$.

The first step is to bound $\left\lVert\phi\right\rVert_{C^{0}}$ (known as ‘$C^{0}$-estimate’), using a technique called Moser iteration. We rewrite the equation as

$$(1-e^{f})\omega^{n}=-dd^{c}\phi\wedge(\omega^{n-1}+\omega^{n-1}\wedge\omega_{\phi}+\ldots+\omega_{\phi}^{n-1}).$$

For any $p>1$, multiply the equation by $\phi|\phi|^{p-2}$ and integrate by parts,

$$\int_{X}|\nabla|\phi|^{p/2}|_{\omega}^{2}\omega^{n}\leq\frac{C(n,\left\lVert f\right\rVert_{L^{\infty}})p^{2}}{p-1}\int_{X}|\phi|^{p-1}\omega^{n}.$$

The key is that a weaker norm on RHS controls a stronger norm on LHS. This is reverse to the direction of the Poincaré inequality

$$\int_{X}u^{2}\omega^{n}\leq C\int_{X}|du|_{\omega}^{2}\omega^{n},\quad\int_{X}u=0,$$

and the Sobolev inequality

$$\int_{X}|u|^{\frac{2n}{n-1}}\omega^{n}\leq C(\int_{X}|du|^{2}_{\omega}\omega^{n}+\int_{X}|u|^{2}\omega^{n}).$$

The combined force is that $\left\lVert\phi\right\rVert_{L^{2}_{1}}\leq C$, and one can inductively bound $\left\lVert\phi\right\rVert_{L^{p}}$ for large $p$, which turns out to be uniform in $p$. Taking the limit $p\to\infty$, this gives a bound $\left\lVert\phi\right\rVert_{L^{\infty}}\leq C(X,\left\lVert f\right\rVert_{L^{\infty}})$.

The second step is to bound $\left\lVert dd^{c}\phi\right\rVert_{C^{0}}$ (known as $C^{1,\bar{1}}$-estimate), i.e. to prove a uniform equivalence between $\omega$ and $\omega_{\phi}$. The starting point is a differential inequality by local calculations (in the analyst’s convention of Laplacian)

$$\Delta_{\omega_{\phi}}\log\Tr_{\omega}\omega_{\phi}\geq-C\Tr_{\omega_{\phi}}\omega-C.$$

Subtracting a large enough multiply of the identity

$$\Delta_{\omega_{\phi}}\phi=n-\Tr_{\omega_{\phi}}\omega,$$

we get a differential inequality

$$\Delta_{\omega_{\phi}}(\log\Tr_{\omega}\omega_{\phi}-C\phi)\geq\Tr_{\omega_{\phi}}\omega-C^{\prime}.$$

Using the a priori bound on $\phi$, an application of maximum principle then shows $\Tr_{\omega}\omega_{\phi}\leq C$.

The third step is to get higher order estimates. Standard elliptic theory implies it is sufficient to have a $C^{2,\alpha}$ bound on $\phi$. Evans-Krylov theory bridged the gap between $C^{1,\bar{1}}$-bound and $C^{2,\alpha}$-bound (cf. [66] Chapter 2, Section 4 for details). This argument is of local nature, and crucially uses that $\log\det(\partial_{i}\partial_{\bar{j}}\phi)$ is a concave function of the matrix $(\partial_{i}\partial_{\bar{j}}\phi)$, and the main tool is a Harnack inequality. The geometric insight is that higher order regularity is a manifestation of the local Euclidean nature of Kähler manifolds.

Yau’s proof strategy is highly influential and permeates the vast majority of works on CY metrics. The brief sketch above, however, highlights two reasons why it is difficult to adapt to the setting of SYZ conjecture:

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  The CY metrics undergoing large complex structure limit are highly degenerate, to the extent that the Sobolev constant becomes too big, so that the Moser iteration technique does not give useful $C^{0}$-estimate on the potential.

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  The $C^{1,\bar{1}}$ bound has global nature, so in order to obtain useful estimates on the Calabi-Yau metric only in the generic region, the design of the maximum principle must hold globally. This is hard on highly degenerate manifolds, where Riemannian curvature cannot be globally uniformly bounded.

For these reasons, our approach to the SYZ conjecture has significant departure from Yau’s proof. The potential estimates use complex pluripotential theory instead, and the metric estimate in the generic region uses a deep theorem of Savin in elliptic PDE theory, bypassing Yau’s estimates.

Complex pluripotential theory concerns the study of weak notions of Kähler potentials, and their wider implications on algebraic geometry and PDEs. They can be viewed as the generalization of potential theory on Riemann surfaces to several complex variables. Some common themes include:

• •

  The weak compactness theory for the space of potentials. This is suited for variational methods in Kähler geometry.

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  The relations between potential theory and algebraic geometry, via Hodge theory, $\bar{\partial}$-operator, intersection theory, etc. This provides transcendental methods to birational geometry.

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  The weak formulation of the complex Monge-Ampère equation, and a priori potential estimates under weak assumptions on the volume density, or in the presence of singularities for the ambient complex variety. This is useful for studying singularity formation of canonical metrics.

Unlike Yau’s proof of the Calabi conjecture, whose techniques are typical of elliptic PDEs, complex pluripotential theory is more akin to complex analysis, and frequently provides stronger results. To build up the intuition, we will first review the standard versions in the literature, before stating our uniform versions, which are foundational to our approach to the SYZ conjecture.

A basic problem in pluripotential theory is to estimate Kähler potentials. Kolodziej pioneered a method to achieve the following effects. The basic versions of his theorems work on a fixed ambient Kähler manifold $(Y,\omega)$, and deal with Kähler potentials $\phi\in PSH(Y,\omega)$ normalized to $\sup_{Y}\phi=0$.

• •

  (‘Potential estimate’) If the volume density of $\omega_{\phi}^{n}$ has some integrability control such as an $L^{p}$-bound

  $$\int_{Y}|\frac{\omega_{\phi}^{n}}{\omega^{n}}|^{p}\leq C,\quad p>1,$$

  (10)

  then $\phi$ has an $L^{\infty}$ bound depending only on $(X,\omega),n,p,C$ [45][20][24]. In fact this can be improved to a $C^{\alpha}$ Hölder bound on $\phi$ [47]. (Notice $p=1$ would not suffice, as the $L^{1}$-bound $\int_{Y}\omega_{\phi}^{n}\leq\int_{Y}\omega^{n}$ is automatic).

• •

  (‘$L^{1}$-stability estimates’) Suppose $\phi,\psi\in PSH(Y,\omega)$ are both subject to the $L^{p}$ volume density integrability control (10), and the normalization $\sup_{Y}\phi=\sup_{Y}\psi=0$. If

  1. 1.

     Either $\phi,\psi$ are close together in the $L^{1}$-sense $\int_{Y}|\psi-\phi|\omega^{n}\ll 1$, (‘$L^{1}$-potential stability’)
     

  2. 2.

     Or the volume densities of $\omega_{\phi}^{n}$ and $\omega_{\psi}^{n}$ are close together in the $L^{1}$-sense, namely the total variation $\int_{Y}|\omega_{\phi}^{n}-\omega_{\psi}^{n}|\ll 1$, (‘$L^{1}$-volume stability’)

  Then $|\phi-\psi|$ is small in the $L^{\infty}$ sense with quantitative estimates [46].

In our applications, we need to work with a family of Kähler manifolds, and the estimates need to be uniform under very severe complex structure degenerations, and allowing the total volume to collapse to zero. Some subtleties are:

• •

  The Calabi-Yau volume measure is very different from the volume form of a Fubini-Study metric (cf. section 3.1), so the idea of volume density with respect to a Fubini-Study style ambient metric is no longer appropriate. The replacement of (10) turns out to be a Skoda type estimate (11).

• •

  Unlike $L^{\infty}$ bounds, the Hölder norm depends strongly on the choice of the ambient metric, which specifies a choice of distance function. We think it is highly non-obvious how to make a semi-explicit choice uniformly in the family, and therefore we do not attempt to generalize Hölder estimates.

• •

  A technical problem in our applications involving the comparison of two potentials, only one potential has volume density control. Thus unlike the $L^{1}$-stability estimates above, our version treats the two potentials asymmetrically.

Our analogue of the potential estimate is

The strength of this result is that it still applies when the complex/Kähler structures are highly degenerate, as it distills the dependence on $(Y,\omega)$ to only 3 constants $n,\alpha,A$. The relations to the more standard version above can be explained as follows:

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  For a fixed ambient Kähler metric, by the Hölder inequality and the standard Skoda inequality Theorem 4.3, under the density integrability assumption (10), then for $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, and some small enough $\beta>0$,

  $$\int_{Y}e^{-\beta u}\frac{\omega_{\phi}^{n}}{\text{Vol}(Y)}\leq\frac{1}{\text{Vol}(Y)}\left(\int_{Y}|\frac{\omega_{\phi}^{n}}{\omega^{n}}|^{p}\omega^{n}\right)^{1/p}\left(\int_{Y}e^{-p^{\prime}\beta u}\omega^{n}\right)^{1/p^{\prime}}\leq\text{const}.$$

  This means the Skoda type estimate (11) is a weaker assumption than (10), even in the standard setting.

• •

  The first part of the conclusion recovers Kolodziej’s potential estimate.

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  The second part of the conclusion is about comparing the two potentials $\phi$ versus $-t_{0}$. The condition $\frac{\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(Y)}<(2B)^{-2n}$ means $\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}$ is small, which by the Hölder inequality and the density integrability (10) follows from the smallness of $\int_{\phi\leq-t_{0}}\omega^{n}$. This should be viewed as one half of the $L^{1}$-potential stability condition $\int_{Y}|\phi+t_{0}|\omega^{n}\ll 1.$ The conclusion for the lower bound on $\min\phi$, should be viewed as one half of a smallness bound on the $C^{0}$-norm of $\phi+t_{0}$, namely the two potentials $\phi$ and $-t_{0}$ are close together in $C^{0}$-norm.

• •

  To generalize to the cases with $\psi$ not necessarily constant, it suffices to replace $\omega$ by $\omega_{\psi}$, and $\phi$ by $\phi-\psi$ in Theorem 4.5.

Combining Thm. 4.4 with Thm. 4.5, we immediately obtain

Our adaption of the $L^{1}$-volume stability estimate is

Theorem 4.7 should be viewed as a one-sided version of $L^{1}$-volume stability estimate. The goal is to compare the two potentials $\phi$ and $0$, without loss of generality. The Skoda type estimate is the weakened version of the volume density integrability assumption (10) as before. Assume for the moment that this holds for both measures $d\mu$ and $d\nu$. After adjusting $\phi$ by an additive constant, we might as well assume $\mu(E_{0})\approx\nu(E_{0})\approx\frac{1}{2}$, noticing that the total variation between the two measures is small by assumption. Then we can reverse the role of $\omega_{\phi}$ and $\omega$, to deduce a two-sided smallness bound on $\phi$, which is the content of the $L^{1}$-volume stability estimate.

Savin [65] proved that for a large class of second order elliptic equations satisfying certain structural conditions, any viscosity solution $C^{0}$-close to a given smooth solution has interior $C^{2,\gamma}$-bound. In particular this applies to complex MA equation. Combined with the Schauder estimate,

Savin’s theorem has fully nonlinear nature, because the perturbative machinery only applies once the solution has a priori $C^{2}$ bound. His proof has two main parts: first he shows a Harnack inequality by a nontrivial application of Aleksandrov-Bakelman-Pucci estimates, and then uses a compactness argument to prove $C^{2,\gamma}$ estimate, similar to De Giorgi’s almost flatness theorem for minimal surfaces [21].

There is an extensive literature on the local regularity theory for the real Monge-Ampère equation, largely due to the Caffarelli school. All results surveyed here can be found in [61].

Any convex function on an open set $v:\Omega\subset\mathbb{R}^{n}\to\mathbb{R}$ has an associated Borel measure called the Monge-Ampère measure, defined by

$$MA(v)(E)=|\partial v(E)|,$$

where $|\partial v(E)|$ denotes the Lebesgue measure of the image of the subgradient map on $E\subset\Omega$. Given a Borel measure $\mu$, a solution to $MA(v)=\mu$ is called an Aleksandrov solution to $\det(D^{2}v)=\mu;$ if $v\in C^{2}$, this is the classical real Monge-Ampère equation. We shall assume a two-sided density bound

$$\det(D^{2}v)=f\text{ in }B_{1},\quad 0<\Lambda_{1}\leq f\leq\Lambda_{2}.$$

Let $B_{1}\setminus\Sigma$ be the set of strictly convex points of $v$, namely there is a supporting hyperplane touching the graph of $v$ only at one point. Then Caffarelli [7][8][9] shows

• •

  If $f\in C^{\gamma}(B_{1})$, then $v\in C^{2,\gamma}_{loc}(B_{1}\setminus\Sigma)$. Then by Schauder theory, if $f$ is smooth, then $v$ is smooth in $B_{1}\setminus\Sigma$.

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  If $L$ is a supporting affine linear function to $v$, such that the convex set $\{v=L\}$ is not a point. Then $\{v=L\}$ has no extremal point in the interior of $B_{1}$.

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  The above affine linear set $\{v=L\}$ has dimension $k<n/2$.

Mooney [61] shows further that

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  The singular set $\Sigma$ has $(n-1)$-Hausdorff measure zero. Consequently $B_{1}\setminus\Sigma$ is path connected (because a generic path joining two given points does not intersect a subset of zero $(n-1)$-Hausdorff measure).

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  The solution $v\in W^{2,1}_{loc}(B_{1})$ even if $\Sigma$ is nonempty.

The classical result of McLean says that the deformation theory of special Lagrangians with phase $\theta$ is unobstructed, and the first order deformation space is isomorphic to $H^{1}(L,\mathbb{R})$. Thus if $L$ is diffeomorphic to $T^{n}$, then the deformation space is $n$-dimensional, compatible with the SYZ conjecture that $X$ admits a special Lagrangian $T^{n}$-fibration. A sufficient condition to construct Slag fibrations, under the very strong hypothesis of collapsing metric with locally bounded sectional curvature, is obtained by Zhang [80, Thm 1.1].

The essence of Zhang’s result is a standard application of the implicit function theorem, and we shall summarize the key points (cf. [80, section 4] for more details). Denote $Y_{r}=T^{n}\times B(0,r)\subset T^{n}_{x_{i}}\times\mathbb{R}^{n}_{y_{i}}\simeq T^{*}T^{n}$, where $r\gg 1$ is fixed. The trivial example of a special Lagrangian fibration is the following: the CY structure is the flat model

$$g=\sum(dx_{i}^{2}+dy_{i}^{2}),\quad\omega=\sum dx_{i}\wedge dy_{i},\quad\Omega=\bigwedge(dx_{j}+\sqrt{-1}dy_{j}),$$

and the Slag fibration is just the projection to the $\mathbb{R}^{n}_{y_{i}}$ factor, namely the tori $T^{n}\times\{y\}$ are special Lagrangians. Zhang considers a family of CY structures $(g_{k},\omega_{k},\Omega_{k})$ converging to $(g,\omega,\Omega)$ in the $C^{\infty}$-sense on $Y_{2r}$ (which follows from his bounded sectional curvature assumptions by elliptic bootstrap), such that $\omega_{k}\in[\omega]\in H^{2}(Y_{2r},\mathbb{R})$. Small deformations of the standard $T^{n}$ fibres can be represented as graphs on $T^{n}$: for $y\in\mathbb{R}^{n}$ and a 1-form $\sigma$ on $T^{n}$ orthogonal to the harmonic 1-forms $dx_{1},\ldots dx_{n}$, write

$$L(y,\sigma)=\text{Graph}(x\mapsto y+\sigma(x))\subset T^{*}T^{n}.$$

The condition for $L(y,\sigma)$ to be a special Lagrangian with respect to $(g_{k},\omega_{k},\Omega_{k})$ is

$$\omega_{k}|_{L(y,\sigma)}=0,\quad\text{Im}(e^{\sqrt{-1}\theta_{k}}\Omega_{k})|_{L(y,\sigma)}=0,$$

(12)

where $\theta_{k}$ are chosen so that $\int_{T^{n}}e^{\sqrt{-1}\theta_{k}}\Omega_{k}>0$. Zhang shows by perturbation arguments that for each $y\in B(0,\frac{3r}{2})$ and $k\geq k_{0}\gg 1$, there is a unique $\sigma=\sigma_{k,y}$ such that $L(y,\sigma_{k,y})$ solves (12) with small norm bound $\left\lVert\sigma_{k,y}\right\rVert<\delta\ll 1$. He then uses another implicit function argument to show that these special Lagrangians indeed define a local special Lagrangian $T^{n}$-fibration on some open subset of $Y_{3r/2}$ containing $Y_{r}$.