Timeline for Can a symplectic manifold be recovered from its Lagrangians?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 18, 2015 at 12:51 | answer | added | YHBKJ | timeline score: 2 | |
| Feb 6, 2015 at 6:57 | comment | added | Thomas Kragh | 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. | |
| Feb 5, 2015 at 23:00 | comment | added | Nathaniel Bottman | @ThomasKragh Good point! Are you using Nadler--Zaslow for the quasiequivalence of Fukaya categories? | |
| Feb 5, 2015 at 20:33 | comment | added | Thomas Kragh | 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). | |
| Jan 26, 2015 at 22:18 | comment | added | Sasha | @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. | |
| Jan 26, 2015 at 11:32 | comment | added | YHBKJ | @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. | |
| Jan 26, 2015 at 9:44 | comment | added | Sasha | @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. | |
| Jan 25, 2015 at 22:34 | comment | added | Nathaniel Bottman | @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. | |
| Jan 25, 2015 at 21:55 | comment | added | Michael Bächtold | You want to consider symplectic forms up to scalar multiple at least. | |
| Jan 25, 2015 at 21:39 | comment | added | Sasha | 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. | |
| Jan 25, 2015 at 20:49 | history | asked | Nathaniel Bottman | CC BY-SA 3.0 |