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Let $\mathscr{F}(X)$ be the exact Fukaya category of an exact symplectic manifold $(X^{2n},\omega)$, i.e. the objects in $\mathscr{F}(X)$ are all closed exact Lagrangian submanifolds with Maslov index 0. Consider a collection of Lagrangian submanifolds $L_1,\cdot\cdot\cdot,L_k\subset X$, and assume that $L_i\cong\mathbb{R}^n$. Since $H^1(L_i)=0$, $HF^\ast(L_i,L_j)$ is well-defined and $\mathbb{Z}$-graded for arbitrary $i$ and $j$. Denote by $\mathscr{F}^\bigstar(X)$ the $A_\infty$ category whoses objects consists of all the closed exact Lagrangian submanifolds with vanishing Maslov index together with the contractible Lagrangians $\{L_1,\cdot\cdot\cdot,L_k\}$ introduced above. My question is when $\{L_1,\cdot\cdot\cdot,L_k\}$ split-generates $D^\pi\mathscr{F}^\bigstar(X)$?

A well-known special case of my question is the work of Seidel, which concerns the question above in the case when $\{L_1,\cdot\cdot\cdot,L_k\}$ is a certain collection of Lefschetz thimbles. In the absence of a Lefschetz fibration, is there any concise criterion on the collection $\{L_1,\cdot\cdot\cdot,L_k\}$ to get the required split-generation result?

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  • $\begingroup$ Abouzaid gives a criterion in this paper: ams.org/mathscinet-getitem?mr=2737980. $\endgroup$ Dec 2, 2014 at 3:49
  • $\begingroup$ free version: arxiv.org/abs/1001.4593 $\endgroup$ Dec 2, 2014 at 3:50
  • $\begingroup$ @JohnPardon But that is for wrapped Fukaya category, and the manifold should be a Liouville domain to have well-defined symplectic cohomology. The case is slightly different from what I described above. $\endgroup$
    – YHBKJ
    Dec 2, 2014 at 3:54
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    $\begingroup$ What version of $HF^*(L_i,L_j)$ are you using here? $\endgroup$
    – Zack
    Dec 2, 2014 at 17:47
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    $\begingroup$ @YHBKJ How do you pick the perturbation? Usual Floer homology is not invariant under non-compactly supported Hamiltonian perturbations. $\endgroup$
    – Zack
    Dec 3, 2014 at 15:56

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