Timeline for Unique Almost Complex Structure with a Two-Form
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 26, 2011 at 21:07 | answer | added | Claudio Gorodski | timeline score: 4 | |
| May 26, 2011 at 9:55 | comment | added | Gunnar Þór Magnússon | Oh, yes, and under these conditions $\omega$ will be a holomorphic symplectic form on $(M,T)$ - i.e. a non-degenerate holomorphic 2-form, which then trivializes the canonical bundle of $M$. | |
| May 26, 2011 at 9:53 | comment | added | Gunnar Þór Magnússon | This is how it's supposed to work: take a 4-manifold $M$ with the topology of a K3 surface, and a 2-form $\omega$ on $M$. Let $T = \Ker \omega \subset T_M$ be a subspace of the tangent bundle. Then $\omega \wedge \omega = 0$ and $\omega \wedge \overline \omega$ ensure that $T$ defines an almost complex structure on $M$ (i.e. $T$ is the space of $(0,1)$ or $(1,0)$ vectors). Then $d \omega = 0$ is supposed to be equivalent to integrability. -- Like I said, there are details to work out, and I haven't had time to do them yet. | |
| May 26, 2011 at 9:47 | comment | added | Gunnar Þór Magnússon | I've been meaning to work out the details of this for some time. It seems to be linked to K3 surfaces - there are semi-vague references to this fact in Buchdahl [springerlink.com/content/h2517445047r421r/] and Huybrechts [math.uni-bonn.de/people/huybrech/HKhabmod.ps]. Sorry I can't do better than that. | |
| May 26, 2011 at 7:22 | history | asked | Matt Fahrad | CC BY-SA 3.0 |