We conclude this section by quantifying the failure of $\bm{\underline{\omega}}_{\epsilon}=(\omega^{1}_{\epsilon},\omega^{2}_{\epsilon},\omega^{3}_{\epsilon})$ to define a hyperkähler structure. Since by construction $d\omega^{i}_{\epsilon}=0$ for all $i=1,2,3$, we only have to check that $\bm{\underline{\omega}}_{\epsilon}$ is a definite triple and estimate the difference between the associated intersection matrix and the identity.

In the regions $\rho_{j}\leq\epsilon^{\frac{2}{5}}$, $\rho_{i}\leq\epsilon^{\frac{2}{5}}$ and when $\rho_{i},\rho_{j}\geq 2\epsilon^{\frac{2}{5}}$ for all $i,j$ the triple $\bm{\underline{\omega}}_{\epsilon}$ defines a genuine hyperkähler structure. In the transition regions $\epsilon^{\frac{2}{5}}\leq\rho_{j}\leq 2\epsilon^{\frac{2}{5}}$ and $\epsilon^{\frac{2}{5}}\leq\rho_{i}\leq 2\epsilon^{\frac{2}{5}}$ we have, respectively,

$$\bm{\underline{\omega}}_{\epsilon}-\bm{\underline{\omega}}_{q_{j},\epsilon}=O(\epsilon^{2-\frac{1}{5}}),\qquad\bm{\underline{\omega}}_{\epsilon}-\bm{\underline{\omega}}_{p_{i},\epsilon}=O(\epsilon^{2-\frac{1}{5}}).$$

Since $\bm{\underline{\omega}}_{q_{j},\epsilon}$ and $\bm{\underline{\omega}}_{p_{i},\epsilon}$ are hyperkähler triples, we conclude that $\bm{\underline{\omega}}_{\epsilon}$ is a definite triple for $\epsilon$ sufficiently small.

Let $\mu_{\epsilon}$, $g_{\epsilon}$ and $Q_{\epsilon}$ be the volume form, metric and intersection matrix associated to the definite triple $\bm{\underline{\omega}}_{\epsilon}$ as in Section 2. Using Lemma 5.4, (5.5), (5.6) and the definition of $\bm{\underline{\omega}}_{\epsilon}$ we calculate that

(5.8)

$$|Q_{\epsilon}-\text{id}|\leq C\epsilon^{2-\frac{1}{5}}$$

in every transition region $\epsilon^{\frac{2}{5}}\leq\rho_{j}\leq 2\epsilon^{\frac{2}{5}}$, $j=1,\dots,8$, and $\epsilon^{\frac{2}{5}}\leq\rho_{i}\leq 2\epsilon^{\frac{2}{5}}$, $i=1,\dots,n$. Outside the transition regions $Q_{\epsilon}\equiv\text{id}$.