Timeline for A Poisson Geometry Version of the Fukaya Category
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 26, 2010 at 19:47 | vote | accept | Daniel Pomerleano | ||
| Mar 24, 2010 at 1:38 | comment | added | Thomas Nevins | Ah, ok...I was secretly (ignorantly!) imagining that boundary conditions in the Fukaya category should work similarly to what I'd imagine one would want to do in D-module--land...thanks for the explanation! | |
| Mar 23, 2010 at 15:02 | comment | added | Tim Perutz | @Thomas. Nadler's picture also fits into a general class of good boundary conditions involving Lagrangians with Legendrian boundary on a convex contact boundary. I don't see that having a Poisson compactification can be a meaningful source of intuition about holomorphic curves (rather than about e.g. constructible sheaves) because we know nothing at all about holomorphic curves in Poisson manifolds. For instance, Dan's punctured plane has a degenerate symplectic form; it's not clear that viewing it as Poisson is relevant. | |
| Mar 23, 2010 at 3:56 | comment | added | Thomas Nevins | I'm far from knowledgeable about Fukaya categories, but I had the vague impression that a game akin to what Dan is asking about is happening also in the cotangent bundle case: one would rather work with a compact manifold, but the natural compactification is only Poisson. So instead one tries morally to reproduce the noncompact symplectic manifold by working on the compact manifold but with boundary conditions. [I thought in Nadler's stuff, Hamiltonian isotopies have boundary conditions at infinity?] A boundary condition at the origin seems implicit in Dan's discussion of the plane? | |
| Mar 22, 2010 at 18:51 | comment | added | Tim Perutz | I'd say that if you can prove a compactness theorem, then you have a chance. Non-compact symplectic leaves look like trouble to me... | |
| Mar 22, 2010 at 18:40 | comment | added | Daniel Pomerleano | Thanks Tim. I was hoping that it could be enough to say the complex structure is something like Poisson semicompatible replacing the usual positivity with a semipositivity condition on the degenerate locus. It was your last point about quantization which lead me to think about this in the first place actually. I was trying to understand what conditions one could put on a Poisson structure so that one could have a chance to define Gromov Witten invariants. | |
| Mar 22, 2010 at 18:24 | history | edited | Tim Perutz | CC BY-SA 2.5 |
More focused comments on Poisson topology.
|
| Mar 22, 2010 at 18:18 | history | answered | Tim Perutz | CC BY-SA 2.5 |