ScalingStacks

Proposition 4.6 . [05E2]

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Proposition 4.6.

For k≫1k\gg 1, there is an open set Y2​r⊃Wk⊃YrY_{2r}\supset W_{k}\supset Y_{r} such that (Wk,ωk,Ωk)(W_{k},\omega_{k},\Omega_{k}) admits a equivariant special lagrangian fibration fk:Wk⟶Bkf_{k}:W_{k}\longrightarrow B_{k} of phase θk\theta_{k} over Bk⊂ℝnB_{k}\subset\mathbb{R}^{n}, i.e. there is a Γ\Gamma-action on BkB_{k}, fkf_{k} is a Γ\Gamma-equivariant map, and fkf_{k} is a special lagrangian fibration of phase θk\theta_{k}, i.e.

ωk|fk−1​(b)≡0,Im​e−1​θk​Ωk|fk−1​(b)≡0,\omega_{k}|_{f_{k}^{-1}(b)}\equiv 0,\ \ \ {\rm Im}e^{\sqrt{-1}\theta_{k}}\Omega_{k}|_{f_{k}^{-1}(b)}\equiv 0,

for any b∈Bkb\in B_{k}.

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