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].