ScalingStacks

Example 4.4 . [04D5]

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Example 4.4.

Consider any compact Lagrangian inside the unit ball of ℂn\mathbb{C}^{n}. By an easy maximum principle argument, during the flow LtL_{t} remains inside the shrinking ball {∑|zi|2≤1−2nt}\{\sum|z_{i}|^{2}\leq 1-2nt\}, so must develop a finite time singularity at some t≤12​nt\leq\frac{1}{2n}. From the Floer theoretic perspective, since such Lagrangians can always be displaced off itself by the Hamiltonian isotopy corresponding to translations in ℂn\mathbb{C}^{n}, its Floer cohomology is either obstructed or zero. As such, a compact Lagrangian supported in a small coordinate ball is invisible to the derived Fukaya category. From a different perspective, since such Lagrangians have zero homology class, they are excluded in the almost calibrated case.

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