ScalingStacks

An upper semicontinuous $L^{1}$-function $\phi$ on a coordinate ball is called plurisubharmonic (psh) if $\sqrt{-1}\partial\bar{\partial}\phi\geq 0$. The basic intuition is that regularity properties for psh functions in general dimensions are analogous to subharmonic functions on Riemann surfaces. This is captured by the basic version of the Skoda inequality:

The basic Skoda inequality immediately implies a global version. On a compact Kähler manifold $(X,\omega)$, we say an upper semicontinuous $L^{1}$-function $\phi\in PSH(X,\omega)$ if $\omega_{\phi}=\omega+\sqrt{-1}\partial\bar{\partial}\phi\geq 0$. This is the generalised notion of Kähler potentials.

Here we outline a method to estimate Kähler potentials, pioneered by Kolodziej, and further developed by [12] and [15][22]. Our exposition largely adapts [15][22][16], with special attention to the dependence of constants. Unlike in [15], we do not impose a volume normalisation.

Given an $n$-dimensional Kähler manifold $(X,\omega)$, for $\phi\in PSH(X,\omega)\cap L^{\infty}$, pluripotential theory allows one to make sense of the Monge-Ampère (MA) measure $\omega_{\phi}^{n}$, generalising the notion of volume forms. The basic problem is to estimate $\phi$ from a priori bounds on $\omega_{\phi}^{n}$. A key concept is the capacity of subsets $K\subset X$:

$$Cap_{\omega}(K)=\sup\{\int_{K}\omega_{u}^{n}|u\in PSH(X,\omega),0\leq u\leq 1\}.$$

We wish to sketch the main ideas behind a prototypical result:

The first ingredient is:

The second ingredient below contains the most substance:

The third ingredient is an elementary decay lemma:

Thm. 2.7 gives a criterion for two Kähler potentials to be close to each other.

This section is included for motivational purposes. On any compact complex manifold $X$ with a positive line bundle $L$, any fixed Kähler metric $\omega$ in the class $2\pi c_{1}(L)$ is the curvature form of a Hermitian metric $h$ on $L$. Consider the projective embedding $\iota_{k}:X\hookrightarrow\mathbb{P}(H^{0}(X,L^{k})^{*})$ for $k\gg 1$. The $L^{2}$ norms on sections induce Euclidean metrics on the vector spaces $H^{0}(X,L^{k})$, hence Fubini-Study metrics $\omega_{FS,k}$ on $\mathbb{P}(H^{0}(X,L^{k})^{*})$. A famous result of Tian says that $\omega$ is approximated by the algebraic metrics $k^{-1}\iota_{k}^{*}\omega_{FS,k}$ as $k\to\infty$; this idea has been much exploited in regularization theorems.

This construction is particularly transparent in the toric case, as explained in [13]. Let $(X,L)$ be an $n$-dimensional polarised toric manifold with moment polytope $P$, so a $T^{n}$-invariant basis $\{s_{m}\}$ of $H^{0}(X,L^{k})$ corresponds to $kP\cap\mathbb{Z}^{n}$, or equivalently $P\cap k^{-1}\mathbb{Z}^{m}$ after rescaling. The $L^{2}$-metric on $H^{0}(X,L^{k})$ is diagonal in the basis; i.e. the toric assumption reduces the unitary group acting on $H^{0}(X,L^{k})$ to its maximal torus. Concretely, let $\phi$ denote the torus invariant Kähler potential on $X\cap(\mathbb{C}^{*})^{n}$, equivalently thought as some convex function of $\vec{t}\in\mathbb{R}^{n}$ via the logarithm map $(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$. Then

$$I_{m}(k)=\left\lVert s_{m}\right\rVert_{L^{2}}^{2}=\int_{X}|s_{m}|^{2}d\text{Vol}=\text{const}\int_{\mathbb{R}^{n}}e^{-k(\phi-\vec{t}\cdot m)}d\vec{t},\quad m\in P\cap k^{-1}\mathbb{Z}^{n},$$

(4)

and the Fubini-Study potentials are

$$k^{-1}\iota_{k}^{*}\phi_{FS,k}=k^{-1}\log\left(\sum_{m\in P\cap k^{-1}\mathbb{Z}^{n}}I_{m}(k)^{-1}|s_{m}|^{2}\right).$$

(5)

Now the RHS of (4) is a Laplace type integral, and its dominant contribution comes from the neighbourhood of the point $\vec{t}_{0}$ where $\vec{t}\cdot m-\phi(\vec{t})$ is maximized among $\vec{t}\in\mathbb{R}^{n}$. The maximum is the value of the Legendre transform of $\phi$:

$$u(m)=\sup_{\vec{t}}(\vec{t}\cdot m-\phi(t)).$$

The steepest descent method yields the asymptote

$$k^{-1}\log I_{m}(k)=u(m)+O(k^{-1}\log k),\quad k\to\infty.$$

In the ‘continuum limit’ $k\to\infty$, the discrete sum $\sum_{m\in P\cap k^{-1}\mathbb{Z}^{n}}$ is replaced by an integral. Now the RHS of (5) is to leading order

$$k^{-1}\log\int_{P}e^{k(-u(m)+\vec{t}\cdot m)}dm,\quad\vec{t}\in\mathbb{R}^{n}.$$

This is another Laplace type integral, and its limit as $k\to\infty$ is the Legendre transform of $u$, which gives back the function $\phi$.

The moral is that in the presence of toric symmetry, algebraic approximation of Kähler metrics is related to Legendre transforms.

Extension theorems allow us to think extrinsically about Kähler currents on subvarieties in some ambient projective manifold.

Savin [33] 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.

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 [31].

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 [6][7][8] 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 $B_{1}\setminus\Sigma$.

• •

  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 $B_{1}$.

• •

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

Mooney [31] shows further that

• •

  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).

• •

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

A real $n$-dimensional submanifold $L$ of a compact Calabi-Yau n-fold $(X,\omega,J,\Omega)$ is called a special Lagrangian (SLag) with phase angle $\theta$ if

$$\omega|_{L}=0,\quad\text{Im}(e^{i\theta}\Omega)|_{L}=0.$$

(6)

They are special cases of calibrated submanifolds introduced by Harvey and Lawson [25], and in particular are minimal submanifolds. The classical result of McLean says that the deformation theory of SLags 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 SLag $T^{n}$-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 [41, 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. [41, 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 SLag 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 SLags. 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 SLag with respect to $(g_{k},\omega_{k},\Omega_{k})$ is

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

(7)

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 (7) 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 SLags indeed define a local SLag $T^{n}$-fibration on some open subset of $Y_{3r/2}$ containing $Y_{r}$.