Wrapped Fukaya categories of Stein manifolds

Question

By the work of Abouzaid, we know that the wrapped Fukaya category of $T^\ast Q$ with $Q$ a closed smooth manifold is generated by a cotangent fiber. Basically, this is an application of Abouzaid's generation criterion for wrapped Fukaya categories of Liouville manifolds, which says that a full subcategory $\mathscr{B}\subset\mathscr{W}(X)$ of the wrapped Fukaya category of a Liouville manifold $X$ generates the wrapped Fukaya category $X$ if the image of the open-closed map

$\mathscr{OC}:HH_\ast(\mathscr{B},\mathscr{B})\rightarrow SH^\ast(X)$

contains the identity of $SH^\ast(X)$, where $SH^\ast(X)$ is the symplectic cohomology of the Liouville domain $X_0\subset X$, which is also an invariant of $X$ by Viterbo functoriality.

Since there is a Stein a structure on $T^\ast Q$, it's natural to ask which Lagrangians generate $\mathscr{W}(X)$ for a general Stein manifold $X$. However, it seems hard to apply Abouzaid's criterion for explicit candidates which are expected to generate $\mathscr{W}(X)$. The case for $T^\ast Q$ is relatively easier mainly because in this case, the geometric information of $\mathscr{W}(T^\ast Q)$ comes entirely from the loop space $\mathscr{L}X$. For example, someone considered Lagrangian sections of certain (local) Lagrangian fibrations on log Calabi-Yau surfaces or $T^\ast S^3$ in some papers concerning mirror symmetry, but we still don't know whether these Lagrangians generate $\mathscr{W}(X)$. (These considerations are of course motivated by $\mathbb{R}_+\subset\mathbb{C}^\ast$.)

On the other hand, for the study of homological mirror symmetry for affine varieties, the superpotential $W:X\rightarrow\mathbb{C}$ doesn't seem to provide a natural Lefschetz fibration on $X$ in dimensions $n\geq3$. For example, $W=z_1z_2z_3$ on $\mathbb{C}^3$ is not even a Morse-Bott fibration. Therefore, we need other techniques to find natural candidates to generate $\mathscr{W}(X)$. Recently, Auroux proposed in his IHES talk that for certain affine varieties (which I believe should be affine conic bundles), there is a single Lagrangian homeomorphic to $\mathbb{R}^n$ which is expected to generate the so-called fiberwise wrapped Fukaya category $\mathscr{F}(X,W)$. To my understanding, this category should be an $A_\infty$ subcategory of $\mathscr{W}(X)$.

Combining all these I come up with the following questions. For a Stein manifold $X$,

1. Will a single Lagrangian $L$ generate $\mathscr{W}(X)$?

2. Is it true that $L$ must be diffeomorphic to $\mathbb{R}^n$?

3. If we have counterexamples for the above two questions, how about restricting ourselves to the case when $X$ is an affine variety?

Answer 1

By now we understand quite well generators of such categories. The co-core disks for any relative skeleton of the sector associated to the superpotential will do. (The argument for this is geometric, and does not invoke Abouzaid's criterion.)

For the notion of relative skeleton, see e.g. section 1.3 of this.