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Remark 3.14 . [04BT]

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Remark 3.14.

In the above argument, once we achieved C​F0​(L1′,L)=0CF^{0}(L_{1}^{\prime},L)=0, there is a different way to proceed. We reinterpret the distinguished triangle L1′→L→L′′→L1′​[1]L_{1}^{\prime}\to L\to L^{\prime\prime}\to L_{1}^{\prime}[1] as an isomorphism in Db​F​u​k​(X)D^{b}Fuk(X) between LL and a twisted complex built from L1′∪L′′L_{1}^{\prime}\cup L^{\prime\prime}. This would give rise to a bordism current constructed from the universal family of holomorphic curves, with ∂𝒞=L−L1′−L′′\partial\mathcal{C}=L-L_{1}^{\prime}-L^{\prime\prime}. However, C​F0​(L1′,L)=0CF^{0}(L_{1}^{\prime},L)=0 implies that no holomorphic curve contributing to 𝒞\mathcal{C} passes from L1′L_{1}^{\prime} to LL in the clockwise direction of ∂Σ\partial\Sigma. The twisted complex structure on L1′∪L′′L_{1}^{\prime}\cup L^{\prime\prime} forbids the passage from L1′L_{1}^{\prime} to L′′L^{\prime\prime} in the clockwise direction of ∂Σ\partial\Sigma. Thus if the holomorphic curve has any boundary portion on L1′L_{1}^{\prime}, its entire boundary would lie on L1′L_{1}^{\prime}, which cannot happen in the almost calibrated setting.

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