With this lengthy discussion involving the tangent cone in place, we return to the limit space $X_{\infty}$. Recall that we have a sequence of scalings $\sqrt{k_{\nu}}$. We consider embeddings $\chi_{\nu}:\tilde{U}\rightarrow X^{{\rm reg}}_{\infty}$. Given such a $\chi_{\nu}$ we write $J^{\nu}$ for the pull-back of the complex structure on $X_{\infty}^{{\rm reg}}$ and $g^{\nu}$ for the pull-back of $k_{\nu}$ times the metric.

This follows easily from the general assertions in Section 2.1 about convergence. We now fix this $k_{\nu}$ and define $k(p)=mk_{\nu}$ and $r(p)=\rho k(p)^{-1/2}$. We write $\chi_{k_{\nu}}=\chi$.

Let $X_{i}\in{\mathcal{K}}(n,C,V)$ be a sequence converging to $X_{\infty}$. We fix distance functions on $X_{\infty}\sqcup X_{i}$. We consider embeddings $\chi^{i}:\tilde{U}\rightarrow X_{i}$. Given such maps we write $g_{i},J_{i}$ for the pull backs of the metric and complex structure, $\Lambda_{i}$ for the pull-back of $L^{k_{\nu}}$ and $A_{i}$ for the pulled back connection.

Again this follows from our general discussion of convergence.

Fix $i$ large enough, as in Proposition 3.13. We apply Proposition 3.11 to find a $t$ such that the pull-back by $\mu_{t}$ of the data $tg_{i},J_{i},\Lambda_{i}^{t},A^{\otimes t}_{i}$ has Property (H) over $U$. Now write $k=tk_{\nu}$ so $k\leq k(p)$. We apply Proposition 2.4 to construct a holomorphic section $s$ of $L^{k}\rightarrow X_{i}$, with a fixed bound on the $L^{2,\sharp}$ norm and with $|s(x)|\geq 1/4$ at points $x$ with $d^{\sharp}(x,\chi_{i}(u_{*}))<(4K_{1})^{-1}$. Here we are writing $d^{\sharp}$ for the scaled metric, so in terms of the original metric the condition is $d(x,\chi_{i}(u_{*}))<k^{-1/2}(4K_{1})^{-1}$.

To finish, suppose $q\in X_{i}$ has $d(q,p)\leq r(p)$. By construction $r(p)\leq\rho k^{-1/2}$. Note also that if we set $p^{\prime}=\chi^{i}(t^{-1/2}z_{\rho})$. then

$$d(q,p^{\prime})\leq d(q,p)+d(p,\chi(t^{-1/2}z_{\rho}))+d(\chi(t^{-1/2}z_{\rho}),\chi^{(i)}(t^{-1/2}z_{\rho}))\leq 4\rho\sqrt{k}.$$

This means that $d^{\sharp}(q,p^{\prime})\leq 4\rho$ which is less than $(4K_{1})^{-1}$ by our choice of $\rho$.