Lemma 3.15 . [0430]

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Lemma 3.15.

The region $\{\ell\gtrsim A^{1/2}\}\subset M^{+}_{\nu}$ is covered by subregions of diameter $\sim\ell$ where the Kähler ansatz is approximated by suitably gauge fixed flat models $g_{\text{flat}}$ , with metric deviation estimate

\left\lVert\tilde{g}^{(1)}-g_{\text{flat}}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CA^{-3/4}\nu,\quad\left\lVert\tilde{\Omega}^{(1)}-\Omega_{\text{flat}}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CA^{-3/4}\nu.