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.