Lemma 3.2 . [05DJ]

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Lemma 3.2.

$\dim_{\mathbb{R}}\bar{S}=n$ , and there is a constant $\theta_{0}\in\mathbb{R}$ such that $\omega_{E}|_{\bar{S}}=0$ and ${\rm Im}e^{\sqrt{-1}\theta_{0}}\Omega_{E}|_{\bar{S}}=0.$ Moreover, $S$ is a special lagrangian submanifold of phase $\theta_{0}$ in $(X,\omega_{0},\Omega_{0})$ , i.e. $\dim_{\mathbb{R}}S=n$ ,

\omega_{0}|_{S}\equiv 0,\ \ {\rm and}\ \ {\rm Im}e^{\sqrt{-1}\theta_{0}}\Omega_{0}|_{S}=0.