We begin with a simple observation.

Transforming the $C^{0}$-bound to the unscaled norms gives, for any holomorphic section of $L^{k}$:

$$\|s\|_{L^{\infty}}\leq K_{0}k^{n/2}\|s\|_{L^{2}}.$$

Write $\rho=\rho_{k,X}(x)$ so there is a section $s$ with $L^{2}$ norm $1$ and with $|s(x)|^{2}=\rho$. Then $s^{\mu}$ is a holomorphic section of $L^{k\mu}$ with

$$|s^{\mu}(x)|^{2}=\rho^{\mu}\ \ \ \|s^{\mu}\|^{2}_{L^{2}}\leq\|s\|^{2\mu-2}_{L^{\infty}}\|s\|^{2}_{L^{2}}\leq K^{2\mu-2}k^{n(\mu-1)},$$

from which the result follows.

We will use this several times below. In the context of our remarks in the Introduction, note that when $\mu$ is large this gives a rather poor estimate compared with what one would hope to be true, but it suffices for our purposes.

Here, as before, we assume we have fixed metrics on the $X_{i}\sqcup X_{\infty}$.

Proof of Proposition 3.3

We first use the compactness of $X_{\infty}$. The $r(p)/2$-balls centred at points $p$ cover $X_{\infty}$ so we can find a finite sub-cover by balls of radius $r(p_{\alpha})/2$ centred at points $p_{\alpha}\in X_{\infty}$. Let $r$ be the minimum of the $r(p_{\alpha})$. Let $i$ be large enough that for any $x\in X_{i}$ there is a point $x_{\infty}\in X_{\infty}$ with $d(x,x_{\infty})\leq r/4$. In addition suppose that $i\geq{\rm max}_{\alpha}i(p_{\alpha})$. Then $x_{\infty}$ lies in the $r(p_{\alpha})/2$ ball centred at $p_{\alpha}$ for some $\alpha$ and hence $d(x,p_{\alpha})<\frac{3}{4}r(p_{\alpha})$. Now Theorem 3.2 states that there are $k(p_{\alpha})$ and $b(p_{\alpha})$ such that for a suitable $k_{\alpha}\leq k(p_{\alpha})$ we $\rho_{k_{\alpha},X_{i}}(x)\geq b(p_{\alpha})$. Take $k_{X_{\infty}}$ to be the least integer such that each integer less than or equal to each $k(p_{\alpha})$ divides $k_{X_{\infty}}$. Then Lemma 3.1 implies that a positive lower bound on any $\underline{\rho}(k_{\alpha},X_{i})$ gives a positive lower bound on $\underline{\rho}(k_{X_{\infty}},X_{i})$ and the Lemma follows.

The same argument, using Lemma 3.1, shows that, given the statement of Lemma 3.4, there are for each integer $\mu\geq 1$ numbers $b_{\mu}>0$ (depending only on $X_{\infty}$)such that $\underline{\rho}(\mu k_{X_{\infty}},X_{i})\geq b^{2}_{\mu}$ once $i$ is sufficiently large. Now we prove Theorem 1.1 (assuming Theorem 3.2) by contradiction. If Theorem 1.1 is false then there are $X_{i,j}\in{\mathcal{K}}(n,C,V)$ such that $\underline{\rho}(X_{i,j},j!)$ tends to zero for fixed $j$ as $i\rightarrow\infty$. By Gromov’s Compactness theorem there is no loss in supposing that, for each fixed $j$, the $X_{i,j}$ converge to some limit $X_{j}$ as $i\rightarrow\infty$. Taking a subsequence $j(\nu)$ we can suppose also that the $X_{j(\nu)}$ converge to $X_{\infty}$. For large enough $\nu$ the integer $k_{X_{\infty}}$ divides $j(\nu)!$; say $j(\nu)!=m(\nu)k_{X_{\infty}}$. Now choose $i(\nu)$ so large that $X_{i(\nu),j(\nu)}$ converge to $X_{\infty}$ as $\nu\rightarrow\infty$ and also so that $\underline{\rho}(X_{i(\nu),j(\nu)},j(\nu)!)<b_{\mu(\nu)}$. This gives a contradiction.