We work in the rescaled metric. Given $\epsilon>0$ we can choose a maximal set of points $x_{i}$ in $X$ such that the distance between any two is at least $\epsilon$. Then the $2\epsilon$ balls with these centres cover $X$ and the $\epsilon/2$ balls are disjoint. Consider the evaluation map

$${\rm ev}:H^{0}(X;L^{k})\rightarrow\bigoplus L^{k}_{x_{i}}.$$

We first show that if $\epsilon$ is sufficiently small then this map is injective. For if it is not injective there is a holomorphic section $s$ with $L^{2,\sharp}$ norm $1$ vanishing at all the $x_{i}$. Since the $2\epsilon$ balls cover we get $\|s\|_{L^{\infty}}\leq 2K_{1}\epsilon$. This gives a contradiction to $\|s\|_{L^{2\sharp}}=1$ if $\epsilon$ is small enough. On the other hand since the $\epsilon/2$ balls are disjoint the non-collapsing condition gives an upper bound on the number of the points $x_{i}$ which completes the proof.

In fact the estimate one gets by this argument is

$$N_{k}+1=\frac{2^{4n}K_{1}^{2n}V^{n+1}n!}{c\pi^{n}}k^{n^{2}},$$

which is very poor compared with the asymptotics we know that ${\rm dim}H^{0}(X,L^{k})\sim(2\pi)^{-n}Vk^{n}$ for a fixed $X$, as $k\rightarrow\infty$.

For our purposes there is no loss of generality in supposing that the $k_{0}$ of Theorem 1.1 is $1$. Then the sections of $L^{k}$ define a regular map of $X$ for all $k$. Suppose we choose isometric embeddings

$$\phi_{k}:H^{0}(X;L^{k})^{*}\rightarrow\mbox{${\mathbb{C}}$}^{N_{k}+1},$$

using the $L^{2}$ norm on the left hand side and the fixed standard Hermitian form on the right. Then we get projective varieties

$$V(X,\phi_{k})\subset\mbox{${\mathbb{C}}$}\mbox{${\mathbb{P}}$}^{N_{k}},$$

and holomorphic maps

$$T_{k}:X\rightarrow V(X,\phi_{k}).$$

Of course $T_{k}$ depend on the choice of $\phi_{k}$ which is arbitrary, but any two choices differ by the action of the unitary group $U(N_{k}+1)$. The fact that this group is compact will mean that in the end the choice of $\phi_{k}$ will not be important. Soon we will reduce to the case when $T$ is generically 1-1 but we do not need to assume that yet, so $T_{k}$ could map to a variety of dimension less than $n$ or be a multiple cover of an $n$-dimensional variety. In any case we get, by straightforward arguments, a fixed upper bound on the degree of $V(X,\phi_{k})$ (depending on $k,n,V$).

By standard general principles there is a system of morphisms of projective varieties, for integer $\lambda$,

$$f_{\lambda}:V(X,\phi_{\lambda k})\rightarrow V(X,\phi_{k}),$$

with $f_{\lambda\mu}=f_{\lambda}\circ f_{\mu}$ and $f_{\lambda}T_{\lambda k}=T_{k}.$

Now we bring in the crucial lower bound provided by Theorem 1.

Here we are referring to the “operator norm” of the derivative, regarded as a map from the tangent space of $X$ at a point, with the given metric $g$, to the tangent space of $\mbox{${\mathbb{C}}$}\mbox{${\mathbb{P}}$}^{N_{1}}$ with the standard Fubini-Study metric.

The proof of the Lemma comes directly from the definitions. Given a point $x\in X$ we can choose an orthonormal basis of sections $s_{0},s_{1},\dots,s_{N}$ with $s_{i}(x)=0$ for $i>0$ and $|s_{0}(x)|=B\geq b$. There is no loss of generality in supposing that $\phi_{1}$ maps the dual basis to the first $N+1$ basis vectors in $\mbox{${\mathbb{C}}$}^{N_{1}+1}$. Fix a unitary isomorphism of the fibre $L_{x}$ with ${\mathbb{C}}$. Then the derivative of each $s_{i}$, for $i>0$ can be regarded as an element of the cotangent space of $X$ at $x$. Identifying the tangent space of $\mbox{${\mathbb{C}}$}\mbox{${\mathbb{P}}$}^{N_{1}}$ at $(1,0,\dots,0)$ with $\mbox{${\mathbb{C}}$}^{N_{1}}$ in the standard way, the derivative of $T$ at $x$ is represented by

$$B^{-1}(\partial s_{1},\dots,\partial s_{N},\dots,0),$$

and the lemma follows.

Using Lemma 3.1 we get similar universal bounds on the derivatives of all maps $T_{k}$, for suitable constants which we do not need to keep track of.

Now suppose that $X_{i}$ is a sequence in ${\mathcal{K}}(n,c,v)$ with Gromov-Hausdorff limit a polarised limit space $X_{\infty}$. For each fixed $k$ we choose $\phi_{k,i}$ so we have a sequence of projective varieties $V(X_{i},\phi_{k,i})$ of bounded dimension and degree. By standard results we can, choosing a subsequence suppose that for each $k$ these converge in the algebro-geometric sense to a limit $W_{k}$. (More precisely, we can suppose that for each $k$ the $V(X_{i},\phi_{k,i})$ have fixed degree and dimension and converge as points in the Chow variety parametrising algebraic cycles of that type. Then we take $W_{k}$ to be the corresponding algebraic set.) It follows easily from the compactness of $U(N_{k}+1)$ that $W_{k}$ is independent, up to projective unitary transformations, of the choice of maps $\phi_{k,i}$.

More precisely what we mean is that we suppose we have fixed metrics on the $X_{i}\sqcup X_{\infty}$ then for all $\epsilon>0$ we can find $\delta>0$ so that the distance in the projective space between $T_{k}(y),T_{k}(y)$ is less than $\epsilon$ if $d(x,y)<\delta$.

The proof of the Lemma is very easy using the equicontinuity of the maps $T_{k}$ on the $X_{i}$. The limit map $T_{k}$ on $X_{\infty}$ is unique up to unitary transformations preserving $W_{k}$ and the possible existence of such maps is the only reason that we may need to pass to a subsequence.

In the next subsection we will collect some further analytical results which will give a much clearer view of the situation. Then we return to discuss the relation between $X_{\infty}$ and the $W_{k}$ further in subsection 4.3.

Recall that we have a uniform $C^{0}$ estimate (Prop. 2.1) for holomorphic sections of $L^{k}\rightarrow X$, for any $X$ in ${\mathcal{K}}(n,c,V)$. We will now extend this to a polarised limit space $X_{\infty}$.

Here of course we are writing $L^{k}\rightarrow X^{{\rm reg}}_{\infty}$ for the limiting line bundle and we are defining the $L^{2,\sharp}$ norm with the rescaled metric.

We prove the Lemma by contradiction. The argument is very similar to our main construction in Section 3. Suppose there is a holomorphic section $s$ with $\|s\|_{L^{2,\sharp}}=1,\|s\|_{L^{\infty}}=B$ and there is a point $p\in X^{{\rm reg}}_{\infty}$ with $|s(p)|=K_{0}+\lambda$ for some $\lambda>0$. Choose a neighbourhood $D$ of $p$ which lies inside $X_{\infty}$. There is some constant $C$ so that an estimate like that in (H3) of Property (H) holds. The singular set in $X_{\infty}$ has Hausdorff codimension strictly bigger than $2$ so by the argument of Prop. 3.5 we can construct a cut-off function $\beta$ equal to $1$ over $D$ and with $\|\nabla\beta\|_{L^{2}}$ as small as we like. In particular we can make this much smaller than $\lambda B^{-1}C^{-1}$. When $i$ is large we can choose maps $\chi_{i}$ from a neighbourhood of the support of $\beta$ into $X_{i}$ and lifts $\hat{\chi}_{i}$ so that the structures match up as closely as we please. Transport $\beta s$ by these maps to a section of $L^{k}\rightarrow X_{i}$ and adjust to get a holomorphic section $s_{i}$ just as in Section 3. Then when $i$ is large enough we see that $s_{i}$ contradicts Prop. 2.1, by arguments just like those in Section 3.

Now we define $H^{0}(X_{\infty},L^{k})$ to be the space of bounded holomorphic sections over the regular part. Let $S\subset\mbox{${\mathbb{R}}$}$ be the set

$$S=\{0,1,1/2,1/3,1/4,\dots 1/i\dots\},$$

and for integers $j$ let $S_{j}\subset S$ be the subset $\{0,j^{-1},(j+1)^{-1}\dots\}$. We are regarding $S$ as a topological space, so any sequence tending to zero would do equally well. Let

$${\mathcal{X}}=\bigsqcup_{i=1,2,\dots,\infty}X_{i}.$$

Thus there is a map of sets $\pi:{\mathcal{X}}\rightarrow S$ which takes $X_{i}$ to $i^{-1}$ for $i=1,\dots\infty$. The distance functions on $X_{i}\sqcup X_{\infty}$ define a natural topology on ${\mathcal{X}}$ such that $\pi$ is continuous.

Now let

$${\mathcal{H}}=\bigsqcup_{i=1,2,\dots,\infty}H^{0}(X_{i},L^{k}),$$

taking the above definition in the case $i=\infty$. There is an obvious map of sets $\varpi:{\mathcal{H}}\rightarrow S$. We put a topology on ${\mathcal{H}}$ by saying that sections are close if they are close when compared by maps $\chi_{i},\hat{\chi}_{i}$, as above.

(Note that this is for fixed $k$: for different values of $k$ one might a priori have to take different values of $j$.)

The proof uses much the same construction as in Lemma 4.6. The content of the statement is that, for large enough $i$, we can define linear isomorphisms

$$Q_{i}:H^{0}(X_{\infty},L^{k})\rightarrow H^{0}(X_{i},L^{k})$$

such that $Q_{i}(s)$ tends to $s$ as $i\rightarrow\infty$, in the sense above. We choose a family of compactly supported cut-off functions $\beta_{i}$ on $X^{{\rm reg}}_{\infty}$ with the following properties.

• •

  The compact sets $\beta_{i}^{-1}(1)$ give an exhaustion of $X_{\infty}^{{\rm reg}}$;

• •

  The support of $\beta_{i}$ is contained in the domain of a map $\chi_{i}$ under which the structures compare with a small error $\eta_{i}$ with $\eta_{i}\rightarrow 0$ as $i\rightarrow\infty$;

• •

  $\|\nabla\beta_{i}\|_{L^{2,\sharp}}\rightarrow 0$ as $i\rightarrow\infty$. In particular $\|\nabla\beta_{i}\|_{L^{2,\sharp}}$ can be taken very small compared with $K_{0}^{-1}$.

Then for any holomorphic section $s\in H^{0}(X_{\infty},L^{k})$ we transport $\beta_{i}s$ to $X_{i}$ using $\chi_{i}$ and project to get an element $Q_{i}(s)\in H^{0}(X_{i},L^{k})$ in the familiar way. Our standard argument shows that $Q_{i}(s)$ can be made as close as we please to $s$ by taking $i$ large. In particular this shows that $Q_{i}$ is injective, for large $i$. (Note that the point of establishing Lemma 4.4 first is that the bounds we require on $\|\nabla\beta_{i}\|$ do not depend on $s$, but only on $K_{0}$.) To prove surjectivity we argue by contradiction. If $Q_{i}$ is not surjective we can find $s_{i}\in H^{0}(X_{i},L^{k})$ of $L^{2,\sharp}$ norm $1$ and $L^{2,\sharp}$-orthogonal to the image of $Q_{i}$. Passing to a subsequence and taking a limit as $i\rightarrow\infty$ we get a section $s_{\infty}\in H^{0}(X_{\infty},L^{k})$. The $C^{0}$ estimate shows that $s_{\infty}$ has $L^{2,\sharp}$ norm $1$ and we easily get a contradiction to the fact that $s_{i}$ is orthogonal to $Q_{i}(s_{\infty})$ for all $i$.

Our reason for formulating things in this way is that it is natural to consider families $\pi:{\mathcal{X}}\rightarrow B$ over a general base. Here we want the fibres of $\pi$ to be either smooth manifolds in ${\mathcal{K}}(n,c,V)$ or polarised Gromov-Hausdorff limits of such, and we want the topology on ${\mathcal{X}}$ to be compatible with the Gromov-Hausdorff distance in an obvious way. It is not hard to set up the definitions and the proof of Lemma 4.5 shows that, if $B$ is connected, there is a “direct image” which is a vector bundle over $B$. However there does not seem much point in developing the theory in detail since in the end, after we have proved Theorem 1.2, this construction can be obtained from the standard algebraic geometry direct image.

We now turn to the problem of separating points.

The proof is a small extension of our main argument in Section 3. By a compactness argument, it suffices to find a $k$ which works for a fixed pair of distinct points $p_{1},p_{2}$. We choose a sequence $X_{i}$ from ${\mathcal{K}}(n,c,V)$ converging to $X_{\infty}$. We can find a sequence $k_{\nu}\rightarrow\infty$ such that rescaling $X_{\infty}$ by $\sqrt{k_{\nu}}$ at each of the points we get convergence to tangent cones $C(Y_{1}),C(Y_{2})$ and we construct $U_{1},U_{2}$ etc. in each case. We then choose $k$ so that we get maps $\chi_{s}:U_{s}\rightarrow X_{i}$ as in Section 3. Clearly we can also suppose that $\chi_{1}(U_{1}),\chi_{2}(U_{2})$ are disjoint. (For this we will need to take $\sqrt{k}$ large compared with $\rho^{-1}$.) Then we get holomorphic sections $s_{1},s_{2}$ of $L^{k}\rightarrow X_{i}$ with fixed $L^{2,\sharp}$ norm and such that $|s_{i}|\geq 1/2$ say at points close to $p_{i}$. Consider the section $s_{1}$ at points in $X_{i}$ close to the image of $\chi_{2}$. Recall that $s_{1}=\sigma_{1}-\tau_{1}$ where $\sigma_{1}$ vanishes on the image of $\chi_{2}$ and the $L^{2,\sharp}$ norm of $\tau_{1}$ can be made as we please by our original choice of parameters. Let $u_{*}\in D\subset U_{2}$ be the base point. Since $\tau_{1}$ is holomorphic over $\chi_{2}(D)$ the size of $\tau_{1}(\chi_{2}(u_{*}))$ can be controlled by the $L^{2}$ norm of $\tau_{1}$ over $\chi_{2}(D)$. Thus by a suitable choice of original parameters (depending only on knowledge of $Y_{1},Y_{2}$) we can arrange that $|s_{1}(x)|=|\tau_{1}(x)|\leq 1/100$, say, for points $x$ close to $\chi_{2}(u_{*})$. Taking the limit as $i\rightarrow\infty$ we get sections $s_{1},s_{2}\in H^{0}(X_{\infty},L^{k})$ with $|s_{i}(p_{i})|\geq 1/2$ and $|s_{i}(p_{j})|\leq 1/100$ for $i\neq j$.

This is another straightforward application of the Hörmander technique.

We can go back to the discussion of (4.1) and state things in a much clearer way. For a given $k$ we can suppose that all the spaces $H^{0}(X_{i},L^{k})$ have the same dimension and identify them with $H^{0}(X_{\infty},L^{k})$ as in Lemma 4.5. In the usual way, the sections in $H^{0}(X_{\infty},L^{k})$ define a holomorphic map from $X^{{\rm reg}}_{\infty}$ to $\mbox{${\mathbb{P}}$}(H^{0}(X_{\infty},L^{k})^{*})$. We fix a basis in $H^{0}(X_{\infty},L^{k})$ so that we can say that we map to $\mbox{${\mathbb{C}}$}\mbox{${\mathbb{P}}$}^{N}$. The same argument as in lemma 4.2 gives a bound on the derivative of this map so it has a unique continuous extension to $X_{\infty}$. Pulling back the hyperplane bundle by this map (in the case $k=1$) defines an extension of the line bundle $L$ to $X_{\infty}$ (at this stage, as a topological bundle). Theorem (1.1) implies that the original metric on $L$ is uniformly equivalent to the metric pulled back from the hyperplane bundle. The convergence of the maps $T_{k}:X_{i}\rightarrow\mbox{${\mathbb{C}}$}\mbox{${\mathbb{P}}$}^{N}$ to $T_{k}:X_{\infty}\rightarrow\mbox{${\mathbb{C}}$}\mbox{${\mathbb{P}}$}^{N}$ over the regular part is completely clear because of the way we chose our identifications of $H^{0}(X_{i},L^{k}),H^{0}(X_{\infty},L^{k})$. The algebraic set $W_{k}$ is the image $T_{k}(X_{\infty})$. It is also clear that we have a system of morphisms $f_{\lambda}:W_{k\lambda}\rightarrow W_{k}$ such that $f_{\lambda\mu}=f_{\lambda}\circ f_{\mu}$ and

$$T_{k}=f_{\lambda}\circ T_{k\lambda}:X_{\infty}\rightarrow W_{k}.$$

(The morphism $f_{\lambda}$ can be viewed as induced by the linear map which is the transpose of $s^{\lambda}(H^{0}(X_{\infty},L^{k}))\rightarrow H^{0}(X_{\infty},L^{\lambda k})$ composed with the inverse of the Veronese map.)

Suppose we have any collection of sets $W_{k}$, for integers $k\geq 1$, and maps $f_{\lambda}:W_{k\lambda}\rightarrow W_{k}$ with $f_{\lambda\mu}=f_{\lambda}\circ f_{\mu}$. Then we can form the limit set $W_{\leftarrow}\subset\Pi_{k}W_{k}$ given by sequences $(w_{1},w_{2},\dots)$ such that $f_{\lambda}(w_{k\lambda})=w_{k}$ for all $k,\lambda$. If we have another set $X_{\infty}$ and maps $T_{k}:X_{\infty}\rightarrow W_{k}$ compatible with the $f_{\lambda}$ then we get an induced map from $X_{\infty}$ to $W_{\leftarrow}$. In our situation, Proposition 4.6 implies that this map is a bijection so what we know at this stage is that we can recover the Gromov-Haussdorf limit algebro-geometrically (at least as a set) in this way.

It is interesting to compare this with [10], [11] where the first-named author made a different attack on the same kind of problem. This attack was made in the absence of Theorem 1.1, and the cost of that absence was that one got a system like the $f_{\lambda}$ but only of rational maps (or “web of descendants” in the language of [10]). The core of the problem was that, without something like Theorem 1.1, one does not know that the $W_{k}$ are irreducible. This difficulty is also explained by Tian in [23]. The construction of [10] should probably best be thought of as an attempt to define the Gromov-Hausdorff limit as a “limit” of algebraic sets or schemes (in the sense of ${\rm lim}\leftarrow$) in this fashion. (From a more algebraic point of view the limiting process we conceive of here is related to considering rings that are not finitely generated.) But, having now Theorem 1.1, we can take a simpler and more direct path (in the context of manifolds satisfying the hypotheses (1.1),(1.2)). However it seem likely that related ideas on the algebraic side may play a role in the future in the study of constant scalar curvature Kähler metrics (lacking (1.1), (1.2)). In this direction, see the recent work of Szekelyhidi [20].

We will now restrict attention to the case when $X_{\infty}$ is the limit of Kähler-Einstein manifolds $X_{i}\in{\mathcal{K}}(n,c,V)$ with Ricci curvature +1, -1/2 or 0. We suppose that $L=K_{X}^{-1}$ or $K_{X}^{2}$ in the first and second situations and in the third situation we suppose that the manifolds are Calabi-Yau, so we have fixed holomorphic $n$ forms $\Theta_{i}$ over $X_{i}$ with $\Theta_{i}\wedge\overline{\Theta}_{i}$ the volume form. For brevity we just call this “the Kähler-Einstein case”.