11
$\begingroup$

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 $M$.

Improve this question
$\endgroup$
9
  • 1
    $\begingroup$ 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. $\endgroup$
    – Sasha
    Commented Jan 25, 2015 at 21:39
  • 2
    $\begingroup$ You want to consider symplectic forms up to scalar multiple at least. $\endgroup$
    – Michael Bächtold
    Commented Jan 25, 2015 at 21:55
  • $\begingroup$ @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. $\endgroup$
    – Nathaniel Bottman
    Commented Jan 25, 2015 at 22:34
  • $\begingroup$ @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. $\endgroup$
    – Sasha
    Commented Jan 26, 2015 at 9:44
  • 1
    $\begingroup$ 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. $\endgroup$
    – Thomas Kragh
    Commented Feb 6, 2015 at 6:57

1 Answer 1

Reset to default
2
$\begingroup$

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.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .