This subsection is a rapid summary of many formidable results. We will not attempt to give detailed references, but refer to the surveys [4] [5]and the references therein.

Recall that if $Z,W$ are two compact metric spaces then the Gromov-Hausdorff distance $d_{GH}(Z,W)$ is the infimum of numbers $\delta$ such that there is a metric on $Z\sqcup W$ extending the given metrics on the components and such that each of $Z,W$ is $\delta$-dense. The starting point of the theory is Gromov’s Theorem that a sequence of compact $m$-dimensional Riemannian manifolds $(M_{i},g_{i})$ with bounded diameter and with Ricci curvature bounded below has a Gromov-Hausforff convergent subsequence with some limit $M_{\infty}$, which is a compact metric space. In our situation, with a sequence in ${\mathcal{K}}(n,c,V)$ the diameter bound follows from the non-collapsing condition (1.2). Passing to this subsequence, we can fix metrics on the disjoint unions $M_{i}\sqcup M_{\infty}$ such that $M_{i},M_{\infty}$ are $\delta_{i}$ dense, where $\delta_{i}\rightarrow 0$.

Now suppose that, as in our situation, the Ricci tensors of $M_{i}$ satisfy fixed upper and lower bounds. Suppose also that a non-collapsing condition (1.2) holds and the volumes are bounded below. Then there is a connected, open, dense subset $M^{{\rm reg}}_{\infty}\subset M_{\infty}$ which is an $m$-dimensional $C^{2,\alpha}$-manifold and has a $C^{1,\alpha}$ Riemannian metric $g_{\infty}$ (for all Hölder exponents $\alpha<1$.) This is compatible with the metric space structure in the following sense. For any compact $K\subset M_{{\rm reg}}$ we can find a number $s>0$ such that if $x_{1},x_{2}$ are points of $K$ with $d(x_{1},x_{2})\leq s$ then $d(x_{1},x_{2})$ is the infimum of the length of paths in $M_{{\rm reg}}$ between $x_{1},x_{2}$. Moreover the convergence on this subset is $C^{1,\alpha}$, in the following sense. Given any number $\delta>0$ and compact subset $K\subset M_{{\rm reg}}$ we can find for large enough $i$ open embeddings $\chi_{i}$ of an open neighbourhood of $K$ into $M_{i}$ such that:

1. (1)

   The pull-backs by the $\chi_{i}$ of the $g_{i}$ converge in $C^{1,\alpha}$ over $K$ to $g_{\infty}$.

2. (2)

   $d(x,\chi_{i}(x))\leq\delta$ for all $x\in K$.

The second item here refers to the chosen metric on $M_{i}\sqcup M_{\infty}$.

The volume form of the Riemannian metric on the dense set $M_{\infty}^{{\rm reg}}$ defines a measure on $M_{\infty}$ and the volume of $M_{\infty}$ is the limit of the volumes of the $M_{i}$. The Hausdorff dimension of the singular set $\Sigma=M_{\infty}\setminus M_{\infty}^{{\rm reg}}$ does not exceed $m-2$.

There is a variant of the theory in which one considers spaces with base points and convergence over bounded distance from the base points. In particular we can take a point $p\in M_{\infty}$ and any sequence $R_{i}\rightarrow\infty$ and then consider the sequence of based metric spaces given by scaling $M_{\infty}$ by a factor $R_{i}$. The compactness theorem implies that, passing to a subsequence, we get convergence and a fundamental result is that, under our hypotheses, the limit of such is a metric cone $C(Y)$—a tangent cone of $M_{\infty}$ at $p$. Here $Y$ is a metric space which contains a dense open subset $Y^{{\rm reg}}$ which is a smooth $(m-1)$-dimensional Einstein manifold, with Ricci curvature equal to $(m-2)$, and the metric and Riemannian structures on $Y^{{\rm reg}}$ are related in a similar way to that above. Likewise for the natural measure on $Y$. The singular set $\Sigma_{Y}=Y\setminus Y^{{\rm reg}}$ has Hausdorff dimension at most $m-3$. The cone $C(Y)$ has a smooth Ricci-flat metric outside the singular set $\{O\}\cup C(\Sigma_{Y})$ (where $O$ is the vertex of the cone) and the convergence of the rescaled metrics is $C^{1,\alpha}$ in the same sense as before. An important numerical invariant of this situation is the volume ratio

(2.1)

$$\kappa=\frac{\text{Vol}(Y)}{\text{Vol}(S^{m-1})}.$$

The Bishop inequality implies that $\kappa\leq 1$ and if a noncollapsing bound like (1.2) holds for the original manifolds $M_{i}$ we have $\kappa\geq c$.

All of the preceding discussion is in the general Riemannian context. Suppose now that our manifolds are $X_{i},g_{i}$ and we have additional structures $J_{i},L_{i},A_{i}$ as in Section 1. We define a polarised limit space to be a metric limit $X_{\infty},g_{\infty}$ as above, together with extra data as follows

• •

  A $C^{1,\alpha}$ complex structure $J_{\infty}$ on the regular set with respect to which the metric is Kähler with $2$-form $\omega_{\infty}$.

• •

  A $C^{2,\alpha}$ line bundle $L_{\infty}$ over the regular set;

• •

  A $C^{1,\alpha}$ connection $A_{\infty}$ on $L_{\infty}$ with curvature $-i\omega_{\infty}$.

(Notice that the integrability theorem for complex structures extends to the $C^{1,\alpha}$ situation [16]. Thus in fact we could say that $X^{{\rm reg}}_{\infty}$ and $L$ have smooth structures while the convergence is in $C^{1,\alpha}$. But this is largely irrelevant for our purposes.)

We define convergence of a sequence $(X_{i},J_{i},L_{i},A_{i})$ to such a polarised limit by requiring that for compact $K\subset X_{\infty}^{reg}$ we have maps $\chi_{i}$ as before but in addition so that the pulled back complex structures $\chi_{i}^{*}(J_{i})$ converge to $J_{\infty}$ (in $C^{1,\alpha}$) and we have bundle isomorphisms $\hat{\chi}_{i}:L_{\infty}\rightarrow\chi^{*}_{i}(L_{i})$ with respect to which the connections converge to $A_{\infty}$ in $C^{1,\alpha}$. It is straightforward to extend the compactness theorem to this polarised situation, using the fact that $J_{i}$ is a covariant-constant tensor with respect to the $C^{,\alpha}$ Levi-Civita connection of $g_{i}$.

Likewise, the regular part of a tangent cone $C(Y)$ at a point in $X_{\infty}$ has a smooth, Ricci-flat, Kähler metric which is induced from a Sasaki-Einstein structure on $Y^{{\rm reg}}$. In particular the Kähler form on the smooth part can be written as $\frac{i}{2}\partial\overline{\partial}|z|^{2}$, where $|z|$ denotes the distance to the vertex of the cone.

A significant difference in the Kähler case is that the singular sets (both in $X_{\infty}$ and in $Y$) are known to have Hausdorff codimension at least $4$. (This is conjectured but not established in the real case.) In particular, which will be crucial for us, the codimension is strictly greater than $2$.

In the case of primary interest—Kähler-Einstein metrics–we obtain $C^{\infty}$ convergence on compact subsets of the regular sets. (This is also part of the standard literature.) In addition, if we consider the “Fano case”, when $L_{i}$ is the anticanonical bundle of $X_{i}$, then the limit line bundle is just the anticanonical bundle of the regular set in $X_{\infty}$. The reader may well prefer to restrict to this case. More generally, if we just assume (in addition to (1.1), (1.2)) that the metrics have constant scalar curvature, then one can still establish this $C^{\infty}$ convergence, for example using the results of Chen and Weber [6].

We begin by recalling that, under our hypotheses, there is a uniform Sobolev inequality

(2.2)

$$\|f\|_{L^{2n/(n-1)}}\leq C_{1}\|\nabla f\|_{L^{2}}+C_{2}\|f\|_{L^{2}},$$

for functions $f$ on a manifold $X$ in the class ${\mathcal{K}}(n,c,V)$, where $C_{1},C_{2}$ depend only on $n,c,V$ [8]. Here of course we are referring to norms defined by the metric $g$. When working with the line bundle $L^{k}$ it will be convenient to use the norms defined by the rescaled metrics $kg$ (for integers $k\geq 1$). Thus lengths are scaled by $\sqrt{k}$ and volumes by $k^{n}$. We will use the notation $L^{2,\sharp}$ etc. to denote norms defined by these rescaled metrics. Then the scaling weight gives

(2.3)

$$\|f\|_{L^{2n/(n-1),\sharp}}\leq C_{1}\|\nabla f\|_{L^{2,\sharp}}+C_{2}k^{-1/2}\|f\|_{L^{2,\sharp}}.$$

So the scaling only helps in the Sobolev inequality. Of course the Ricci tensor ${\rm Ric}^{\sharp}$ of the rescaled metric is bounded between $-1/(2k)$ and $1/k$.

In the second item $\Delta_{\overline{\partial}}=\overline{\partial}^{*}\overline{\partial}+\overline{\partial}\overline{\partial}^{*}$, with adjoints defined using the rescaled metric, and the statement is that for all $\phi$

(2.4)

$$\langle\Delta_{\overline{\partial}}^{-1}\phi,\phi\rangle_{\sharp}\leq 2\|\phi\|^{2}_{L^{2,\sharp}}$$

This Proposition summarises results which are well-known to workers in the field and which all hinge on various formulae of Bochner-Weitzenbock type. We use the rescaled metrics throughout the discussion. First on $C^{\infty}$ sections $s$ of $L^{k}$ we have

$$\nabla^{*}\nabla s=2\overline{\partial}^{*}\overline{\partial}s+s,$$

so when $s$ is holomorphic $\nabla^{*}\nabla s=s$ which implies that

(2.5)

$$\Delta|s|\leq|s|,$$

where the lack of differentiability of $|s|$ at the zero set is handled in a standard way. (Note that we use the “geometers convention” for the sign of the Laplacian in this paper.) Now the bound on the $L^{\infty}$ norm follows from the Moser iteration argument applied to this differential inequality, using the uniform Sobolev inequality (see [22]).

The first derivative bound is obtained in a similar way. Changing notation slightly, for a holomorphic section $s$ with $\overline{\partial}s=0$ we write $\nabla s=\partial s$ where

$$\partial:\Omega^{p,q}(L^{k})\rightarrow\Omega^{p+1,q}(L^{k})$$

is defined using the connection. Since $\partial^{2}=0$ we have

$$\Delta_{\partial}\partial s=\partial\Delta_{\partial}s,$$

where $\Delta_{\partial}=\partial^{*}\partial+\partial\partial^{*}$. Then for a holomorphic section $s$ , $\Delta_{\partial}s=\nabla^{*}\nabla s=s$ and

$$\Delta_{\partial}(\partial s)=\partial s.$$

Now the Bochner-Weitzenbock formula comparing $\Delta_{\partial}$ and $\nabla^{*}\nabla$ on $\Omega^{1,0}(L^{k})$ has the form

$$\Delta_{\partial}=\nabla^{*}\nabla-1+{\rm Ric}^{\sharp},$$

so

$$(\nabla^{*}\nabla(\partial s),\partial s)\leq\frac{5}{2}|\partial s|^{2}.$$

It follows that

$$\Delta|\partial s|\leq\frac{5}{2}|\partial s|,$$

and the Moser argument applies as before. Notice that, with some labour, the constants $K_{0},K_{1}$ could be computed explicitly in terms of $n,c,V$.

For the second item in the Proposition we need a Bochner-Weizenbock formula on $\Omega^{0,1}(L^{k})$ i.e. sections of the bundle $\overline{T}^{*}\otimes L^{k}$. We decompose the covariant derivative on this bundle into (0,1) and (1,0) parts: $\nabla=\nabla^{\prime}+\nabla^{\prime\prime}$. Then the formula we want is

(2.6)

$$\Delta_{\overline{\partial}}=(\nabla^{\prime\prime})^{*}\nabla^{\prime\prime}+{\rm Ric}^{\sharp}+1$$

Given this we have, in the operator sense, $\Delta_{\overline{\partial}}\geq 1/2$ since ${\rm Ric}^{\sharp}\geq-1/2$ from which the invertibility and bound on the inverse follow immediately. An efficient way to derive (2.6) is to make the identification

$$\Omega^{0,1}(L^{k})=\Omega^{n,1}(K^{*}\otimes L^{k}),$$

under which $\nabla^{\prime\prime}$ becomes identified with

$$\partial^{*}:\Omega^{n,1}(K_{X}^{*}\otimes L^{k})\rightarrow\Omega^{n-1,1}(K_{X}^{*}\otimes L^{k}).$$

The formula (2.6) then becomes a special case of the Kodaira-Nakano formula ([13] p.154), using the fact that the Ricci form is the curvature of $K_{X}^{*}$.

With this background in place we move on to recall a version of the “Hörmander” construction of holomorphic sections. Suppose we have the following data

• •

  A (non-compact) manifold $U$, a base point $u_{*}\in U$ and an open neighbourhood $D\subset\subset U$ of $u_{0}$.

• •

  A $C^{\infty}$ Hermitian line bundle $\Lambda\rightarrow U$.

• •

  A complex structure $J$ and Kähler metric $g$ on $U$ with Kähler form $\Omega$.

• •

  A connection $A$ on $\Lambda$ having curvature $-i\Omega$.

We use this connection to define a $\overline{\partial}$-operator on sections of $\Lambda$, and hence a holomorphic structure.

We define a “Property (H)” which this data might have. Fix any $p>2n$.

Property (H):

There is a number $C>0$ and a compactly supported section $\sigma$ of $\Lambda\rightarrow U$ such that the following hold.

H1: $\|\sigma\|_{L^{2}}<(2\pi)^{n/2};$

H2: $|\sigma(u_{*})|>3/4;$

H3: For any smooth section $\tau$ of $\Lambda$ over a neighbourhood of $\overline{D}$ we have

$$|\tau(u_{*})|\leq C\left(\|\overline{\partial}\tau\|_{L^{p}(D)}+\|\tau\|_{L^{2}(D)}\right);$$

H4: $\|\overline{\partial}\sigma\|_{L^{2}}<\min(1/(8\sqrt{2}C),(2\pi)^{n/2}/10\sqrt{2});$

H5: $\|\overline{\partial}\sigma\|_{L^{p}(D)}<1/(8C).$

Many of the specific numbers here are arbitrary but it is convenient to fix some definite numbers.

We have

Notice first that for any choice of data there is some constant $C$ for which the bound in (H3) holds. This follows from the elliptic estimate

(2.7)

$$\|\tau\|_{L^{p}_{1}(D_{0})}\leq C_{3}\left(\|\overline{\partial}\tau\|_{L^{p}(D)}+\|\tau\|_{L^{2}(D_{0})}\right),$$

and the Sobolev inequality

$$|\tau(u_{*})|\leq C_{4}\|\tau\|_{L^{p}_{1}(D_{0})}.$$

Here $D_{0}\subset D$ is some interior domain containing $u_{*}$. We can write the $\overline{\partial}$-operator on functions for a perturbed complex structure as $\overline{\partial}+\mu\partial$ where $\mu$ is a “Beltrami differential”. Similarly if the variation of the connection is given by a $1$-form $a$ then the perturbed $\overline{\partial}$-operator on sections can be written as

$$(\overline{\partial}+\mu\partial)+(a^{\prime\prime}+\mu a^{\prime}),$$

where $a=a^{\prime}+a^{\prime\prime}$ is the decomposition into type. It follows that if $\mu$ and $a$ are small in $C^{0}$ then the perturbation of the $\overline{\partial}$ operator is small in the $L^{p}_{1}\rightarrow L^{p}$ operator norm and it is then clear that the inequality in the third item holds for the perturbed operator, with a slightly larger constant $C$. For the perturbed structure we use the same section $\sigma$, so the first and second item is automatic. Then it is also clear that, for sufficiently small perturbations, the bounds in the fourth and fifth item (with a slightly larger constant $C$) are also preserved, since we impose strict inequality.

For a connection $A$ on a line bundle $L$ write $A^{\otimes k}$ for the induced connection on $L^{k}$. The following proposition—basically well-known—will provide the core of our proof of Theorem 1.

To prove this we transport the section $\sigma$ using the maps $\chi,\hat{\chi}$ and regard it as a smooth section of $L^{k}$ over $X$, extending by zero. The norms we considered over $U$ match up with the $\sharp$-norms over $X$. We write $s=\sigma-\tau$ where $\tau=\overline{\partial}^{*}\Delta_{\overline{\partial}}^{-1}\overline{\partial}\sigma$. By simple Hodge Theory we have $\overline{\partial}s=0$. Now

$$\|\tau\|_{L^{2,\sharp}}=\langle\Delta_{\overline{\partial}}^{-1}\overline{\partial}\sigma,\overline{\partial}\ \overline{\partial}^{*}\Delta_{\overline{\partial}}^{-1}\overline{\partial}\sigma\rangle=\langle\Delta_{\overline{\partial}}^{-1}\overline{\partial}\sigma,\overline{\partial}\sigma\rangle,$$

since $\overline{\partial}\ \overline{\partial}\sigma=0$. Thus

(2.8)

$$\|\tau\|_{L^{2,\sharp}}\leq\sqrt{2}\|\overline{\partial}\sigma\|_{L^{2,\sharp}}\leq\min((8C)^{-1},\frac{1}{10}(2\pi)^{n/2}).$$

Hence in particular

$$\|s\|_{L^{2,\sharp}}\leq\|\sigma\|_{L^{2,\sharp}}+\|\tau\|_{L^{2,\sharp}}\leq\frac{11}{10}2\pi^{n/2}.$$

Now work over the image $\chi(D)$. Applying item (H3) to the section $\tau$ and using (H4), (H5) we get $|\tau(\chi(u_{*}))|\leq 1/4$, so $|s(\chi(u_{*}))|\geq 1/2$. By the derivative bound, $|s|$ exceeds $1/4$ at points a distance less than $(4K_{1})^{-1}$ from $\chi(u_{*})$.

To sum up we have the following.

This is just a direct combination of Lemma 2.2 and Proposition 2.3. (Here we use the notation $\|\ \|_{U}$ to indicate the $C^{0}$-norm over $U$.)

To illustrate this, take the case when $U$ is the ball of radius $R>2$ in $\mbox{${\mathbb{C}}$}^{n}$ with the standard flat metric and standard Kähler form $\Omega_{0}$. Let $\Lambda$ be the trivial holomorphic line bundle with metric $\exp(-|z|^{2}/2)$ so the trivialising section, $\sigma_{0}$ say, has norm $\exp(-|z|^{2}/4)$ and the induced connection $A_{0}$ has curvature $-i\Omega_{0}$ as required. Let $u_{*}$ be the origin and $D$ be the unit ball. Let $\beta_{R}$ be a standard cut-off function of $|z|$, equal to $1$ when $|z|\leq R/2$ and vanishing when $|z|\geq 9R/10$. Define $\sigma=\beta_{R}\sigma_{0}$. Then we have $\overline{\partial}\sigma=(\overline{\partial}\beta_{0})\sigma_{0}$. The $L^{2}$ norm of $\sigma$ is slightly less than $(2\pi)^{n}$ and $|\sigma(0)|=1$. The section $\sigma$ is holomorphic over $D$, so we get (H4) and there certainly is some constant $C$ as in item (H3) of Property (H), independent of $R$. It is clear that, because of the exponential decay, we can fix $R$ so that item (H5) is satisfied. So we have a set of data satisfying Property (H). Now let $x$ be a point in some X in ${\mathcal{K}}(n,c,V)$. Since the ball is simply connected $U(1)$ connections over it are determined up to isomorphism by their curvature tensors. It is then clear that, when $k$ is sufficiently large, we can find a map $\chi$ with $\chi(0)=x$ and such that the pull back of $kg_{X},J_{X},A_{X}^{\otimes k}$ differs by an arbitrarily small amount from the model $g_{0},J_{0},A_{0}$. Then we construct a holomorphic section of $L^{k}\rightarrow X$, of controlled $L^{2}$ norm and of a definite positive size on a definite neighbourhood of $x$.