ScalingStacks

Conjecture 3.34 . [04BX]

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Conjecture 3.34.

Assume further that the Kähler metric on XX is Calabi-Yau. Suppose LL is an almost calibrated exact Lagrangian brane in Db​F​u​k​(X)D^{b}Fuk(X), with Harder-Narasimhan decomposition (4)

0=ℰ0→ℰ1→…→ℰN=L,0=\mathcal{E}_{0}\to\mathcal{E}_{1}\to\ldots\to\mathcal{E}_{N}=L,

fitting into the distinguished triangles

ℰi−1→ℰi→Li→ℰi−1​[1],\mathcal{E}_{i-1}\to\mathcal{E}_{i}\to L_{i}\to\mathcal{E}_{i-1}[1],

such that Li∈𝒫⁡(ϕi)L_{i}\in\mathcal{P}(\phi_{i}) with ϕ1>…>ϕN\phi_{1}>\ldots>\phi_{N}. We have θ^i=πϕi=arg∫LiΩ\hat{\theta}_{i}=\pi\phi_{i}=\arg\int_{L_{i}}\Omega. Then the phase angle inequality (35) and the volume lower bound (36) hold.

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