This is crucial, as we indicated above, but the proof is easy. The set $X^{{\rm reg}}_{\infty}$ is dense in $X_{\infty}$ so its image is dense in $W_{k}$. Thus we can choose a point $x_{0}\in X^{{\rm reg}}_{\infty}$ so that $T_{k}(x_{0})$ lies in a unique component $U$ of $W_{k}$. Suppose there is a point $w$ in $W_{k}$ which is not in $U$. Then we can find a polynomial $P$ of degree $\lambda$ say so that $P$ vanishes on $U$ but not at $w$. Regarding $P$ as a section of a line bundle we can suppose $|P(w)|=1$. Now $P$ also defines holomorphic sections $\sigma_{i}$ of $L^{\lambda k}$ over $X_{i}$ for each $i$ (including $i=\infty$) which satisfy a fixed $L^{\infty}$ bound (because of the equivalence of the metrics on the line bundle). By construction the section $\sigma_{\infty}$ vanishes in a neighbourhood of $x_{0}$ and so by analytic continuation and the fact that the regular set is dense and connected it vanishes identically. It follows from the $L^{\infty}$ bound on $\sigma_{i}$, the general estimate of (2.1) and convergence on compact subsets of the regular set that $\|\sigma_{i}\|_{L^{\infty}}$ tends to $0$ as $i\rightarrow\infty$. But this contradicts the fact that $|P(w)|=1$ (again using the equivalence of the two metrics on $L^{k}$).

(Notice that in this proof we do use the fact that $X_{\infty}^{{\rm reg}}$ has an analytic, not just $C^{2,\alpha}$, structure.)

Recall that we have compatible maps $T_{k}:X_{\infty}\rightarrow W_{k}$ and $f_{\lambda}:W_{\lambda k}\rightarrow W_{k}$. Proposition (4.8) implies that the $T_{k}$ asymptotically separate points, in the sense that the induced map from $X_{\infty}$ to $\lim_{\leftarrow}W_{k}$ is injective. What we want to show now is that in fact there is some fixed $k$ for which this is true.

First we can plainly use Proposition 4.6 to arrange that $T_{k}$ is generically 1-1, i.e. so that the fibre $T_{k}^{-1}(w)$ is a single point for a generic $w\in W_{k}$. As usual we may as well suppose that this happens for $k=1$ and hence for all $k$. Thus all maps $f_{\lambda}:W_{\lambda k}\rightarrow W_{k}$ are also generically $1-1$. Our main theorem 1.1 and the first derivative estimate imply that there is a number $r>0$ so that for any $X\in{\mathcal{K}}(n,c,V)$ and any point $x\in X$ there is a holomorphic section of $L$ which does not vanish on the ball of radius $r$ about $x$. The argument extends easily to the limit space $X_{\infty}$ and $H^{0}(X_{\infty},L)$. Choose $k$ in accordance with Proposition 4.6 taking $\rho=r/2$ say. Thus if $p_{1},p_{2}$ are two points in the same fibre $F=T_{k}^{-1}(w)$ of $T_{k}:X_{\infty}\rightarrow W_{k}$ the distance between them is less $r/2$. In other words the fibre $F$ is contained in the $r/2$ ball about $p_{1}$, so there is a section $s\in H^{0}(X_{\infty},L)$ of $L$ which does not vanish on $F$. By construction, $F$ maps by $T_{k\lambda}$ onto $F_{\lambda}=f_{\lambda}^{-1}(w)$ for any $f_{\lambda}:W_{\lambda k}\rightarrow W_{k}$. The section $s^{\lambda k}\in H^{0}(X_{\infty},L^{\lambda k})$ defines one component of $T_{k\lambda}$ so the fact that $s$ does not vanish on $F$ implies that $F_{\lambda}$ lies in the corresponding affine subspace. Since $F_{\lambda}$ is a compact algebraic set it must be finite. Thus all maps $f_{\lambda}:W_{k\lambda}\rightarrow W_{k}$ have finite fibres. Let $N(w)$ be the number of local irreducible components of $W_{k}$ at $w$. Since $f_{\lambda}$ is generically 1-1 the number of points in $f_{\lambda}^{-1}(w)$ is at most $N(w)$. It follows then the number of points in $T_{k}^{-1}(w)$ is also finite, and in fact bounded by $N(w)$.

As usual we may as well suppose that the value of $k$ in the previous Lemma is $1$. Thus $T_{1}:X_{\infty}\rightarrow W_{1}\subset\mbox{${\mathbb{C}}$}\mbox{${\mathbb{P}}$}^{N_{1}}$ has finite fibres. For any given point $w_{1}\in W_{1}$ we can find a $k$ such that $T_{1}^{-1}(w_{1})$ is mapped injectively to $W_{k}$ by $T_{k}$. It is clear then there is a decomposition of $W_{1}$ into a finite number of quasi-projective subvarieties $Z_{\alpha}$ such that $T_{1}^{-1}(Z_{\alpha})$ is a disjoint union of a number $n_{\alpha}$ of copies of $Z_{\alpha}$. Pick points $z_{\alpha}\in Z_{\alpha}$. If for some $\alpha$ some $T_{k}$ separates the points $T_{1}^{-1}(z_{\alpha})$ then it is clear that $T_{k}$ separates points in $T_{1}^{-1}(z)$ for generic $z\in Z_{\alpha}$. Now the Proposition follows from a simple induction argument, using induction on the maximal dimension of a $Z_{\alpha}$ with $n_{\alpha}>1$ and the number of components $Z_{\alpha}$ with this maximal dimension.

We have now achieved our main goal—the central statement in Theorem 1.2. We have a continuous bijection $T_{k}:X_{\infty}\rightarrow W_{k}$ which is a homeomorphism, since the spaces are compact. As usual we may as well suppose that this $k$ is $1$, so all $T_{k}$ are homeomorphisms.

Recall that we denote the differential geometric singular set, the complement of $X_{\infty}^{{\rm reg}}$ by $\Sigma$. Let $S_{k}\subset W_{k}$ denote the algebro-geometric singular set.

Of course it is equivalent to say that $T_{k}$ maps $X_{\infty}^{{\rm reg}}$ to smooth points of $W_{k}$. The proof is similar to that of the previous Lemma. It follows from Proposition 4.7 that for any given compact subset $K\subset X_{\infty}^{{\rm reg}}$ we can choose $k$ so that $T_{k}$ maps $K$ into the smooth points of $W_{k}$. On the other hand the singular set $S_{1}$ has a finite number of irreducible components. If there is a component which meets $T_{1}(X_{\infty}^{{\rm reg}})$ we choose one of maximal dimension, say $V$. Thus there is a point $x\in X_{\infty}^{{\rm reg}}$ with $T_{1}(x)\in V$. We apply Proposition 4.7 with $K=\{x\}$ to find a $k$ such that $T_{k}(x)$ lies in the smooth set of $W_{k}$. Then it is clear that the number of irreducible components of $S_{k}$ is strictly less than for $S_{1}$, and the proof is completed by induction.

As usual we can suppose that the $k$ in Lemma 4.11 is 1. In the next subsection we will show that, at least for Kähler-Einstein limits, the singular sets match up but we do not need to use this fact.

Suppose $W_{1}$ is not normal. Let $\nu:\hat{W}_{1}\rightarrow W_{1}$ be the normalisation. Thus $\nu$ is a bijection outside the singular set $S_{1}$ of $W_{1}$. It is a general fact that the pull back ${\mathcal{L}}=\nu^{*}({\mathcal{O}}(1))$ is an ample line bundle on $\hat{W}_{1}$, so we can choose $k$ such that sections of ${\mathcal{L}}^{k}$ define a projective embedding of $\hat{W}_{1}$ in ${\mathbb{P}}$ say. The map $T_{1}:X_{\infty}^{{\rm reg}}\rightarrow W_{1}$ maps into the smooth part and so lifts to $\hat{T}_{1}:X_{\infty}^{{\rm reg}}\rightarrow\hat{W}_{1}$. Clearly the pull back of ${\mathcal{L}}$ to $X_{\infty}^{reg}$ by this map is identified with our polarising bundle $L$. Moreover, Theorem 1 implies that the metrics on the bundle agree up to a bounded factor. So the sections of ${\mathcal{L}}^{k}$ over $\hat{W}_{1}$ define bounded sections of $L^{k}$ over $X_{\infty}^{reg}$ that is, elements of $H^{0}(X_{\infty},L^{k})$. Write $U\subset H^{0}(X_{\infty},L^{k})$ for the image of this map from $H^{0}(\hat{W}_{1},{\mathcal{L}}^{k})$. These sections define a map $\alpha$ from $X_{\infty}^{{\rm reg}}$ to ${\mathbb{P}}$ and the definitions mean that this is just the composite of $\hat{T}$ with the above projective embedding of $\hat{W}_{1}$. The subspace $U$ contains the kth. powers of sections in $H^{0}(X_{\infty},L)$ which uniformly generate the fibres, so we have a first derivative estimate on the map $\alpha$. Hence $\alpha$ extends to a Lipschitz map, which we also call $\alpha$, from $X_{\infty}$ to ${\mathbb{P}}$ with image $\hat{W}_{1}$. Let $Z$ be the intersection of smooth part of $\hat{W}_{1}$ with $\alpha(\Sigma)$. The Lipschitz bound implies that the Hausdorff dimension of $Z$ is at most $2n-4$ and it follows that any local holomorphic function defined on the complement of $Z$ extends holomorphically over $Z$ [18]. This means that $H^{0}(X_{\infty},L^{k})$ can be identified with bounded holomorphic sections of the hyperplane bundle over the smooth part of $\hat{W}_{1}$. But it is a basic general fact about a normal variety that its structure sheaf can be defined by bounded holomorphic functions on the smooth part. So the subspace $U$ is in fact the whole of $H^{0}(X_{\infty},L^{k})$. Thus $\alpha$ is exactly $T_{k}$ and $W_{k}$ is $\hat{W}_{1}$, and hence normal.

To complete the story we have

This follows from the same argument as above.

We have now almost completed the proof of Theorem 1.2. For any given polarised limit space $X_{\infty}$ we can choose a $k$ so that $H^{0}(X_{\infty},L^{k})$ represents $X_{\infty}$ as a normal variety and if $X_{i}$ is a sequence converging to $X_{i}$ in the Gromov-Hausdorff sense we can choose a convergent sequence of embeddings. (Notice that the only reason for passing to a subsequence in the statement of Theorem 1.2 is that we can have different polarisations on the same Riemannian limit space.) The last point is to show that there is a single $k_{1}$ which works for all $X_{\infty}$. But this follows from Gromov compactness and the easy fact that if $k$ has the desired property for $X_{\infty}$ it does also for all limit spaces sufficiently close to $X_{\infty}$, in the Gromov-Hausdorff sense.

To spell out a little more the consequences of Theorem 1.2, observe that now that we are considering embeddings the degree of $W$ is determined by $k_{1}$ and $V$. So (for theoretical purposes) we can operate in a fixed quasi projective Chow variety ${\mathcal{T}}$ parameterising normal $n$-dimensional subvarieties of the given degree in a suitable large projective space $\mbox{${\mathbb{C}}$}\mbox{${\mathbb{P}}$}^{N}$. “Algebro-geometric convergence” of $X_{i}$ to $X_{\infty}$ means convergence in ${\mathcal{T}}$. There is a universal variety ${\mathcal{U}}\rightarrow{\mathcal{T}}$ and by general facts ([14], Theorem 9.11) this is a flat family. So we see that if $X_{i}$ converge to $X_{\infty}$ in the Gromov-Hausdorff sense then $X_{i}$ and $W=X_{\infty}$ can be realised as fibres in a flat family. So, for example, the Hilbert polynomials of $X_{i}$ and $W=X_{\infty}$ are the same.

There are different ways of going about the proofs of Theorem 1.2. We mention one elegant alternative, based on a result from the thesis of Chi Li [15], Prop. 7. This in turn depends upon results of Siu and Skoda. For $X$ in ${\mathcal{K}}(n,c,V)$ let $R_{X}$ be the graded ring

$$R_{X}=\bigoplus_{k}H^{0}(X;L^{k}).$$

Then from standard theory we know that $R_{X}$ is finitely generated and $X={\rm Proj}(R_{X})$. Assuming the lower bound in Theorem 1.1, Li proves an effective form of finite generation in the sense that if $\sigma_{i}$ is an orthonormal basis in the finite dimensional space $\bigoplus_{j=0}^{(n+2)k_{0}}H^{0}(X,L^{j})$ then the $\sigma_{i}$ generate $R_{X}$ and for each $k$ there is a number $B_{k}$ such that any element of $L^{2}$ norm $1$ in $H^{0}(X,L^{k})$ can be expressed as a polynomial in the $\sigma_{i}$ with co-efficients bounded by $B_{k}$. It follows easily that for a polarised limit space $X_{\infty}$ the graded ring

$$R_{X_{\infty}}=\bigoplus_{k}H^{0}(X_{\infty};L^{k})$$

is finitely generated. Then we can immediately define the algebraic variety $W$ as ${\rm Proj}(R_{X_{\infty}})$. Of course there is still some work to do in checking the properties of $W$.