Timeline for Almost complex structures in Floer theory
Current License: CC BY-SA 2.5
3 events
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| Nov 15, 2010 at 15:03 | comment | added | Sam Lisi | Tim, what you say is correct, but can easily be worked around. For the linear analysis, you only need A to be asymptotically symmetric at the punctures, and it doesn't need to be with respect to the metric given by $\omega(\cdot, J\cdot)$. I can't find a reference right now, but I am certain I have seen the details worked out at least once in the tame case, essentially be deforming the metric near the intersections of the Lagrangians. (If you are willing to have tame except compatible in a neighbourhood of the Lagrangian intersection, then there is no need to do any work at all.) | |
| Nov 9, 2010 at 22:58 | comment | added | Tim Perutz | The linear analysis underlying a Floer theory typically involves (in the $L^2$ version) operators on $(d/ds)+A$ acting on maps $\mathbb{R}\to H$ for some Hilbert space $H$. Here $A$ should be a densely-defined symmetric operator on $H$. In Lagrangian Floer theory, this formulation arises when the inner product on $H$ is derived from the metric associated with a compatible $J$. In the tame setting one would presumably have to proceed differently. I'm not sure whether anyone has done this. | |
| Nov 9, 2010 at 21:30 | history | answered | Michael Hutchings | CC BY-SA 2.5 |