Timeline for Cohomology theory for symplectic manifolds
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 10, 2011 at 18:31 | comment | added | R.S. | Just a comment about the flat section: from a physical point of view, the use of a polarisation is very natural and once you include this on the scheme that you describe above one gets a "natural sheafification" of L. In 'On the definition of Quantization' (Colloques Internationaux C.N.R.S., No. 237 – Géométrie symplectique et physique mathématique.) Bertram Kostant defines a hermitian structure on the cohomology groups exploring a transverse polarisation. I guess that if you consider the polarisation (and the paper) on your picture you could get a cup product interpretation. | |
| Jul 9, 2011 at 23:12 | comment | added | Chris Schommer-Pries | It's true that you can twist ordinary cohomology by a line bundle with flat connection. That's one equivalent way to understand $H^*(M, L)$. If memory serves you can twist 2-periodic cohomology (i.e. add all the even/odd parts together) by a line bundle with a connection which is not necessarily flat. Maybe your $(L,s)$ pairs represent cycles in this twisted cohomology? | |
| Jul 9, 2011 at 22:34 | comment | added | Paul | If I recall correctly, in the special case when $M$ is the character variety of $SU(2)$ representations of a surface groups and the $L_i$ come from a Heegaard splitting of a 3-manifold this pairing was considered in some notes of Dennis Johnson, perhaps also by Kevin Walker in his extension of Casson's invariant. But I dont think the issue of whether this comes from an intersection pairing on some cohomology theory is considered. | |
| Jul 9, 2011 at 21:36 | history | edited | John Pardon | CC BY-SA 3.0 |
added 46 characters in body
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| Jul 9, 2011 at 20:49 | comment | added | Mariano Suárez-Álvarez | Well, then your «doesn't make sense» is more of a «does not help at all»... That was my point :) | |
| Jul 9, 2011 at 20:41 | comment | added | John Pardon | There is no natural good sheafification of $\mathcal L$. If I use smooth sections of $\mathcal L$, then the higher cohomology vanishes since it's a fine sheaf. To get twisted (co)homology in the context of sheaf cohomology, one needs the sheaf of locally constant sections of your bundle. I don't have a flat connection on $\mathcal L$, there's no notion of a "locally constant" section of $\mathcal L$. | |
| Jul 9, 2011 at 20:28 | comment | added | Mariano Suárez-Álvarez | Why doesn't $H^\*(X,\mathcal L)$ make sense? Look at $\mathcal L$ as a sheaf... | |
| Jul 9, 2011 at 19:36 | answer | added | Eigenbunny | timeline score: 4 | |
| Jul 9, 2011 at 17:06 | history | asked | John Pardon | CC BY-SA 3.0 |