11

Something I have wondered idly about from time to time is:

If $(M,\omega), (M',\omega')$ are symplectic manifolds, and you "know what the Lagrangians $L \subset M$ resp. $L' \subset M'$ are," can you determine whether $M$ and $M'$ are symplectomorphic?

The sort of thing I have in mind is this corresponding statement for coherent sheaves:

Theorem (Gabriel). Suppose $X$ and $Y$ are smooth projective varieties. If there exists an equivalence $\mathrm{\bf Coh}(X) \simeq \mathrm{\bf Coh}(Y)$, then $X$ and $Y$ are isomorphic.

(That's proven in Huybrecht's book on FM transforms using Orlov's theorem and skyscraper sheaves.)

One issue is, what's the right way to formalize "know all $L \subset M$"? (Maybe you mean some version of the Fukaya category up to $A_\infty$-equivalence?) A bigger issue is, how the heck do you prove it?? For certain $M$, there's an analogue of skyscraper sheaves coming from mirror symmetry, but I don't know how to ape the argument beyond that point; for other kinds of $M$, I know how to get some bits of information from the Fukaya category. But it would be awfully nice to have an argument for fairly general $M$, e.g. all compact $M$.

Nathaniel Bottman
  • 1,569
  • 1
    If you replace $Coh(X)$ by $D^b(Coh(X))$ in Gabriel's Theroem it will no longer be true. So, it is really important to keep track of the t-structure. So, to expect something similar on the symplectic side one probably needs a t-structure in the Fukaya category (or some other structure to replace it) to recover the symplectic manifold. – Sasha Jan 25 '15 at 21:39
  • 2
    You want to consider symplectic forms up to scalar multiple at least. – Michael Bächtold Jan 25 '15 at 21:55
  • @Sasha That's true, though if you add the assumption that $X$ has ample canonical or anticanonical bundle and assume that the equivalence of derived categories is exact, then it's still true. – Nathaniel Bottman Jan 25 '15 at 22:34
  • @Nate Bottman: Fukaya category is miror dual of the derived category of a CY variety. The mirror of varieties with ample or antiample canonical class is given by an appropriate LG model, so this is a different question. – Sasha Jan 26 '15 at 09:44
  • @Sasha Not exactly true. A Landau-Ginzburg model $(X,W)$ can be identified with a lower dimensional variety $H$, which is roughly the critical locus of $W$. On $H$, one can still consider the wrapped Fukaya category or $D^b(H)$. This is in some sense because of the Knorrer periodicity $D^b(X,W)\cong D^b(H)$. The Knorrer periodicity has its analogue on the symplectic side, see for example, Ivan Smith's "pencil of quadrics" paper. – YHBKJ Jan 26 '15 at 11:32
  • @YHBKJ: Yes, it may be a wrapped Fukaya category instead of LG model. But the point is that with a usual Fukaya category it would be strange to expect some kind of reconstruction. – Sasha Jan 26 '15 at 22:18
  • Note that the $A_\infty$ categories of cotangent bundles of homotopy spheres of the same dimension are equivalent - yet Abouzaid proved that some of them are not symplectomorphic, and if the nearby Lagrangian conjecture is true many more examples of this sort exists (allthough I doubt that it is true, but I still think there are many more examples of this). – Thomas Kragh Feb 05 '15 at 20:33
  • @ThomasKragh Good point! Are you using Nadler--Zaslow for the quasiequivalence of Fukaya categories? – Nathaniel Bottman Feb 05 '15 at 23:00
  • 1
    Both Nadler-Zaslow and Abouzaid-(Fukaya-Seidel-Smith) kind of says that the wrapped Fukaya category is equivalent to a derived category of sheaves of complexes with locally constant homology... or something like that.. which implies it. Abouzaid just states it slightly differently in terms of modules over the fiber, which is the based loop space of the base. – Thomas Kragh Feb 06 '15 at 06:57

1 Answers1

2

I'd like to add two more examples.

The first one is in the negative direction. Let $\pi:E^{2n}\rightarrow\mathbb{C}$ be an exact Lefschetz fibration, by the work of Seidel one can associate its Fukaya category $\mathscr{F}(\pi)$, which can be realized as a directed $A_\infty$ algebra or a diagonal bimodule. Denote by $\mathscr{F}(\pi)^\vee$ its dual bimodule. Seidel constructed the following bimodule map

$\beta:\mathscr{F}(\pi)^\vee[-n]\rightarrow\mathscr{F}(\pi)$

which describes the natural transformation of degree $n$ from the Serre functor to the identity. It's an observation due to Maydanskiy that there are examples of exact Lefschetz fibrations with quasi-isomorphic $\mathscr{F}(\pi)$, but with different bimodule maps $\beta$. This shows that one should include $\beta$ as an additional piece of information to distinguish exact symplectic manifolds which admit Lefschetz fibrations.

The second example is in the positive direction. Weiwei Wu proved the following:

The special isogenous tori are symplectomorphic if and only if they have equivalent derived Fukaya categories.

A special isogenous torus, by definition, is obtained by taking products of tori which are finite group quotients of a standard product torus. The proof is a combination of the result of Abouzaid-Smith on homological mirror symmetry for standard $T^{2n}$ and finite group actions on Fukaya categories which generalize the case of a $\mathbb{Z}/2$-action considered by Seidel in the definition of $\mathscr{F}(\pi)$.

This result is also expected to be true for symplectic tori equipped with linear symplectic forms.

Addendum Of course you may also think of monotone Fukaya categories, and that suggests you to take into consideration the scaling of the symplectic form. The reconstruction is most likely to be possible in the case when there is a well-defined mirror functor, which in turn means that the SYZ picture of mirror symmetry should somehow be compatible with homological mirror symmetry. Up to my knowledge, this is true only for $T^{2n}$, $(\mathbb{C}^\ast)^n$, Kodaira-Thurston manifold, or things like that.

YHBKJ
  • 3,157