Timeline for Almost complex structure gluing
Current License: CC BY-SA 3.0
6 events
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| Jun 20, 2015 at 13:31 | comment | added | Grigory Solomadin | So the answer $A\cup_{M} B$ is $\mathbb{P}(\nu\oplus\overline{\underline{\mathbb{C}}}),$ because one can cut out $D(\nu)$ from it using "affine chart fibration" given by zero section in trivial summand of $\nu\oplus\overline{\underline{\mathbb{C}}}$ and get $\mathbb{P}(\nu).$ $-A\cup_{M} B$ isn't corectly defined. The desired formula is $[\tilde{X}]-[X]=-\mathbb{P}(\nu\oplus\overline{\underline{\mathbb{C}}}).$ It can be proven by applying the aforementioned lemma from Totaro's paper to $A,B$ and $C=-X\setminus A$ (prop,; actually I'd like to choose signs $A,B,C$ as +1,-1,-1 corr.). That's it. | |
| May 20, 2015 at 22:11 | comment | added | Grigory Solomadin | Sorry, last passage is wrong. The idea is to see what is the image of difference bundle in the corresponding K-group of the projectivisation. | |
| May 20, 2015 at 18:09 | comment | added | Grigory Solomadin | @quinque I believe that here it is not important (I've presented the corresponding differential manifold). Moreover you can check lemma 2.1 from arxiv.org/abs/math/0003240 . The question is actually about stably complex structure on the projective bundle. I see two possible ways of determining it. Either through the comparison of Chern numbers of $\tilde{X}$ and $X,$ or with help of K-theory (one can understand stably complex structure as an element of reduced K-group). Actually the complex structure is $\sum_{i=0}^{k}(-1)^{k}\lambda^{i}[E-E^{-1}],$ where $E$ is st. comp. structure on bundle | |
| May 19, 2015 at 23:05 | comment | added | quinque | If you glue to smooth manifolds along their boundary, you do not have canonical structure of smooth manifold on this topological space. If you glue two complex curves then you do have canonical smooth structure due to conformal welding. But is there conformal welding for higher dimensions? | |
| May 16, 2015 at 22:13 | comment | added | Michael Albanese | Crossposted on MSE. | |
| May 16, 2015 at 21:36 | history | asked | Grigory Solomadin | CC BY-SA 3.0 |