I just started learning about Fukaya categories and got slightly confused by the following question. It looks like the statement that a collection of objects generate Fukaya category is stronger than the statement that these objects split generate it. Am I right? There is a criterion due to Mohammed Abouzaid for split generation, but there is no criterion for generation. Am I right?
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2 Answers
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Yes, generation is stronger than split-generation, because you don't need to split off summands after forming iterated mapping cones with degree shifts. The simplest example I have in mind is when $M$ is the symplectic 4-manifold obtained by blowing up the origin of $\mathbb{C}^2$. By choosing the symplectic form carefully one can define its Fukaya category $\mathcal{F}(M)$ whose objects are closed monotone Lagrangian submanifolds equipped with Spin structures or gradings if necessary.
Assuming $\mathcal{F}(M)$ is defined over $\mathbb{C}$, then its non-zero eigensummand is split-generated by a monotone Lagrangian torus $T$ lying inside the unit sphere bundle of $\mathcal{O}(-1)\rightarrow\mathbb{P}^1$. But it's easy to see $T$ does not generate $D^\pi\mathcal{F}_\lambda(M)$, where $\lambda\neq0$.
However, since there is a Lefschetz fibration $\pi:M\rightarrow\mathbb{C}$ obtained by starting from the trivial projection $\mathbb{C}^2\rightarrow\mathbb{C}$ and then attaching the exceptional curve to the fiber above the origin of $\mathbb{C}$, we can enlarge the monotone Fukaya category $\mathcal{F}(M)$ by including also the Lefschetz thimbles of $\pi$. This gives us the Fukaya category $\mathcal{F}(\pi)$. In this case, $\pi$ has only one critical value and the thimbles of $\pi$ are all Lagrangian isotopic. Take any thimble $\Delta\subset M$, then by a theorem of Seidel, we know that $\Delta$ generates $D\mathcal{F}(\pi)$.
In fact, we know in this case that $\Delta$ can be identified with a non-trivial idempotent of $T$.
In general, knowing split-generation is good enough in order to compute Fukaya categories. If there are some additional restrictions on your triangulated category, say it admits a bounded $t$-structure, then split-generation implies generation. So I'm not sure whether a generation criterion instead of a split-generation criterion would lead to some important consequences.
Abouzaid's work is for wrapped Fukaya categories rather than Fukaya category of closed Lagrangians. The most important example for his theory, namely the wrapped Fukaya category of $T^\ast Q$ for any closed manifold $Q$, can actually be generated by a cotangent fiber.
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I will try to answer the second question.
In the most recent paper of Lazarev (Geometric and algebraic presentations of Weinstein domains), it is shown that Abouzaid's criterion can actually imply generation instead of split generation, as long as the manifold $M$ is Weinstein and the Lagrangians considered in the subcategory generate $H^n(M) = H_n(M, \partial_\infty M)$.
Roughly speaking, split generation plus surjection on the Grothendieck group $K_0(\mathcal{W}(M))$ will imply generation. What Lazarev showed was a surjection from $H^n(M)$ to $K_0(\mathcal{W}(M))$. This depends on the structral results of the wrapped Fukaya category (and to check it can be well-defined one needs to consider the relations between the $n$-cells in $C^n(M)$).
