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 $X$,

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

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

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

disjointcocords which generates (in fact this might work in general), but it is a counter example to 2. since one can prove that a connected Lagrangian does not generate (I believe). $\endgroup$ – Thomas Kragh Sep 22 '14 at 7:43