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.