Section 2 (00AY) · ScalingStacks

2.1 Uniform Skoda inequality

Given a Kähler manifold $(Y,\omega)$ , an upper semicontinuous $L^{1}_{loc}$ function $\phi\in PSH(Y,\omega)$ , if $\omega_{\phi}=\omega+dd^{c}\phi\geq 0$ . A Skoda type inequality captures the apriori regularity of such functions. The following uniform version is the main result in the author’s companion paper [33].

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Theorem 2.1. (Uniform Skoda estimate) Given a polarised algebraic degeneration family of Calabi-Yau manifolds $\pi:X\to S\setminus\{0\}$ as in the Introduction. Let $\omega_{FS}$ be a fixed Fubini-Study metric on $(X,c_{1}(L))$ induced by a projective embedding via the sections of a high power of $L$ , and use $\omega_{FS,t}=\frac{1}{|\log|t||}\omega_{FS}|_{X_{t}}$ to define a family of background metrics on $X_{t}$ in the class $\frac{1}{|\log|t||}c_{1}(L)$ . Then there are uniform positive constants $\alpha,A$ independent of $t$ for $0<|t|\ll 1$ , such that for the normalised Calabi-Yau measures $d\mu_{t}$ ,

\int_{X_{t}}e^{-\alpha u}d\mu_{t}\leq A,\quad\forall u\in PSH(X_{t},\omega_{FS,t})\text{ with }\sup_{X_{t}}u=0.

2.2 Kolodziej’s estimate on pluripotentials

Given an $n$ -dimensional Kähler manifold $(Y,\omega)$ , for $\phi\in PSH(Y,\omega)\cap L^{\infty}$ , pluripotential theory allows one to make sense of the Monge-Ampère (MA) measure $\omega_{\phi}^{n}$ , generalising the notion of volume forms. A basic problem is to estimate $\phi$ from a priori bounds on $\omega_{\phi}^{n}$ . A prototypical result is (cf. [32, section 2.2] for an exposition based on [15][16]):

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Theorem 2.2. Let $(Y,\omega)$ be a compact Kähler manifold, and $\phi\in PSH(Y,\omega)\cap C^{0}$ , such that $\omega_{\phi}^{n}$ is an absolutely continuous measure. Assume there are positive constants $\alpha,A$ , such that the Skoda type estimate holds with respect to $\omega_{\phi}^{n}$ :

\int_{Y}e^{-\alpha u}\frac{\omega_{\phi}^{n}}{\text{Vol}(Y)}\leq A,\quad\forall u\in PSH(Y,\omega)\text{ with }\sup_{Y}u=0.

(3)

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For fixed $n,\alpha,A$ , there is number $B(n,\alpha,A)$ , such that if $\frac{\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(Y)}<(2B)^{-2n}$ for some $t_{0}$ , then $\min\phi\geq-t_{0}-4B(\frac{\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(Y)})^{1/2n}$ .
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If $\sup_{Y}\phi=0$ , then $\left\lVert\phi\right\rVert_{C^{0}}\leq C(n,\alpha,A)$ .

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 $n,\alpha,A$ . A minor variant gives a criterion for two Kähler potentials to be close to each other.

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Corollary 2.3. (Stability estimate) Let $(Y,\omega)$ be a compact Kähler manifold, and $\phi,\psi\in PSH(Y,\omega)\cap C^{0}$ , such that $\omega_{\psi}^{n}$ is absolutely continuous. Assume $\left\lVert\phi\right\rVert_{C^{0}}\leq A^{\prime}$ and the Skoda type estimate (3). Then there is a number $B(n,A,A^{\prime},\alpha)$ , such that if $\frac{\int_{\psi-\phi\leq-t_{0}}\omega_{\psi}^{n}}{\text{Vol}(Y)}<(2B)^{-2n}$ for some $t_{0}$ , then

\min(\psi-\phi)\geq-t_{0}-4B\left(\frac{\int_{\psi-\phi\leq-t_{0}}\omega_{\psi}^{n}}{\text{Vol}(Y)}\right)^{1/2n}.

.

2.3 $L^{1}$ -stability estimate

In complex pluripotential theory, an $L^{1}$ -stability estimate is an assertion about the $C^{0}$ -closeness of two Kähler potentials given that their volume densities are close in $L^{1}$ . We now adapt an argument of Kolodziej [30] to prove a uniform version which allows the complex structure to be highly degenerate and the volume to collapse. Our formulation also brings out the asymmetrical role of the two Kähler potentials; in fact one of them is set to zero. We do not pursue optimality.

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Lemma 2.4. (Comparison principle)[30, Thm. 2.1] If $u$ and $v$ are $\omega$ -psh on $Y$ , then on $\Omega=\{u<v\}$ , we have

\int_{\Omega}\omega_{u}^{n}\geq\int_{\Omega}\omega_{v}^{n}.

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Lemma 2.5. (Concavity of $det^{1/n}$ ) On an open domain, suppose $u,v$ are continuous $\omega$ -psh functions, with

\omega_{u}^{n}=f\omega^{n},\quad\omega_{v}^{n}=g\omega^{n}

for $f,g\in L^{\infty}$ . Then for $0<s<1$ , we have $\omega_{su+(1-s)v}^{n}\geq(sf^{1/n}+(1-s)g^{1/n})^{n}\omega^{n}$ .

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Proof. In the smooth case this is a pointwise inequality expressing the concavity of $A\mapsto\det^{1/n}A$ on the set of Hermitian matrices. In general one shows this by an approximation argument [30, Lemma 1.2]. ∎

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Theorem 2.6. (Uniform $L^{1}$ -stability) Let $(Y,\omega)$ be a compact Kähler manifold, and $\phi\in PSH(Y,\omega)\cap C^{0}$ , satisfying the complex MA equations

\frac{\omega^{n}}{\text{Vol}(Y)}=d\mu,\quad\frac{\omega_{\phi}^{n}}{\text{Vol}(Y)}=d\nu

for probability measures $d\mu$ and $d\nu$ . Assume

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There is a Skoda estimate

$\int_{Y}e^{-\alpha u}d\mu\leq A,\quad\forall u\in PSH(Y,\omega)\text{ with }\sup_{Y}u=0.$
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The complement of $E_{0}=\{\phi>0\}$ has a mass lower bound

$\int_{E_{0}^{c}}d\mu\geq\lambda>0.$
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( $L^{1}$ -stability assumption) The total variation $\int_{Y}|d\mu-d\nu|\leq s^{2n+3}<1$ .
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$\phi$ is smooth away from a (possibly empty) closed subset $S$ with $d\mu$ -measure zero. Globally $\left\lVert\phi\right\rVert_{C^{0}}\leq A^{\prime}$ .

Then for $0<s<s_{0}(\lambda,n,\alpha,A,A^{\prime})\ll 1$ , there is a uniform estimate

\sup_{Y}\phi\leq C(\lambda,n,\alpha,A,A^{\prime})s.

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Proof. We can reduce to the case with $d\mu(E_{0})\geq\frac{1}{2}$ by shifting $\phi$ by a constant. We construct an auxiliary continuous $\omega$ -psh function $\rho$ by solving the complex MA equation with $L^{\infty}$ -density [15]

\frac{\omega_{\rho}^{n}}{\text{Vol}(Y)}=\frac{1}{d\mu(E_{0})}d\mu\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule height=0.55974pt,depth=0.0pt,width=5.59721pt}E_{0},\quad\sup_{Y}\rho=0.

where $d\mu\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule height=0.55974pt,depth=0.0pt,width=5.59721pt}E_{0}(F)=d\mu(E\cap F)$ is the restricted measure. By Theorem 2.2 we have $\left\lVert\rho\right\rVert_{C^{0}}\leq C(n,\alpha,A)$ , since the RHS measure satisfies a Skoda estimate. We choose $a=\max(C(n,\alpha,A),A^{\prime})$ , so that

-a\leq\rho\leq 0,\quad\phi\geq-a,

implying the set inclusion

E^{\prime}=\{\phi>2as+s\max\phi\}\subset E=\{(1-s)\phi+s\rho-as>0\}\subset E_{0}.

Our next goal is to show $d\mu(E^{\prime})$ is small.

Let $G=\{1-s^{2}\geq\frac{d\nu}{d\mu}\}\subset Y\setminus S$ . On the open set $E_{0}\setminus(S\cup G)$ , by the concavity Lemma 2.5,

\frac{\omega_{s\rho+(1-s)\phi}^{n}}{\text{Vol}(Y)}\geq\left(sd\mu(E_{0})^{-1/n}+(1-s)(1-s^{2})^{1/n}\right)^{n}d\mu.

Now $d\mu(E_{0})^{-1/n}\geq(1-\lambda)^{-1/n}$ by assumption. Choose $0<q<n\{(1-\lambda)^{-1/n}-1\}$ , so for $0<s\ll 1$ depending on $\lambda,n$ , by Taylor expansion in $s$ ,

\left(sd\mu(E_{0})^{-1/n}+(1-s)(1-s^{2})^{1/n}\right)^{n}\geq 1+qs.

Combining this with the comparison principle Lemma 2.4, and the assumption $d\mu(S)=0$ ,

(1+qs)\int_{E\setminus G}d\mu\leq\int_{E}\frac{\omega_{s\rho+(1-s)\phi}^{n}}{\text{Vol}(Y)}\leq\int_{E}\frac{\omega^{n}}{\text{Vol}(Y)}=\int_{E}d\mu.

On the other hand, by the definition of $G$ and the $L^{1}$ -stability assumption,

d\mu(G)\leq s^{-2}\int_{G}(d\mu-d\nu)\leq s^{-2}\int_{Y}|d\mu-d\nu|\leq s^{2n+1},

hence

(1+qs)\int_{E\setminus G}d\mu\leq\int_{E\setminus G}d\mu+s^{2n+1}.

We conclude $\int_{E\setminus G}d\mu\leq q^{-1}s^{2n}$ , so

\int_{E^{\prime}}d\mu\leq\int_{E}d\mu\leq q^{-1}s^{2n}+s^{2n+1}\leq(q^{-1}+1)s^{2n}.

We now apply the stability estimate Cor. 2.3 to compare the potentials $\phi$ and $0$ , to see for $s$ sufficiently small depending on $n,A,A^{\prime},\alpha,\lambda$ ,

\min_{Y}(-\phi)\geq-2as-s\max_{Y}\phi-4B(\int_{E^{\prime}}d\mu)^{1/2n}\geq-2as-s\max_{Y}\phi-4B(1+q^{-1})^{1/2n}s,

whence $\max_{Y}\phi\leq C(n,A,A^{\prime},\alpha,\lambda)s$ as required. ∎

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Remark 2.7. It is not clear to the author why in Kolodziej’s original argument [30, Lemma 1.2] applies in the proof of [30, Thm. 4.1], as $E_{0}\setminus G$ is not an open domain if one considers general $L^{p}$ -densities.

2.4 Savin’s small perturbation theorem

Savin [40] 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,

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Theorem 2.8. Fix $k\geq 2$ and $0<\gamma<1$ . On the unit ball, let $v$ be a given smooth solution to the complex Monge-Ampère equation $(dd^{c}v)^{n}=1$ . Then there are constants $0<\kappa\ll 1$ and $C$ depending on $n,k,\gamma,\left\lVert v\right\rVert_{C^{k,\gamma}}$ , such that if

(dd^{c}(u+v))^{n}=1+f,\quad\left\lVert f\right\rVert_{C^{k-2,\gamma}}<\kappa,

and $\left\lVert u\right\rVert_{C^{0}}<\kappa$ , then $\left\lVert u\right\rVert_{C^{k,\gamma}(B_{1/2})}\leq C\kappa$ .

2.5 Regularity theory for real Monge-Ampère

There is an extensive literature on the local regularity theory for the real Monge-Ampère equation, largely due to the Caffarelli school. The author thanks C. Mooney for bringing some of these results to his attention. All results surveyed here can be found in [35].

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_{\mathbb{R}}(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 [8][9][10] shows

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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 [35] 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.

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Remark 2.9. A classical counterexample of Pogorelov shows that for $n=3$ , the singular set $\Sigma$ can contain a line segment. This is generalised by Caffarelli [10], who for any $k<n/2$ constructs examples where $f$ is smooth but $\Sigma$ contains a $k$ -plane. A surprising example of Mooney [35] shows that the Hausdorff dimension of $\Sigma$ can be larger than $n-1-\epsilon$ for any small $\epsilon$ . This means the local regularity theory surveyed above is essentially optimal.

2 Analytic backgrounds

2.1 Uniform Skoda inequality

2.2 Kolodziej’s estimate on pluripotentials

2.3 L1L^{1}-stability estimate

2.4 Savin’s small perturbation theorem

2.5 Regularity theory for real Monge-Ampère

2.3 $L^{1}$ -stability estimate