Timeline for Is there a quaternionic analogue of Kodaira's embedding theorem?
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9 events
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| Aug 5, 2017 at 10:05 | comment | added | Malkoun | Many natural constructions that were inspired by a family of Wolf spaces associated to a family of classical groups seem to extend to more general Quaternion-Kähler manifolds (usually via twistor theory). I am trying to turn this philosophy into some more concrete conjecture. But perhaps it is only good as a philosophy, rather than a concrete conjecture (for instance, the conjectured link between integrability and twistor theory for instance, which is a good philosophy, when turned into a conjecture, seems to have some exceptions, so perhaps it is better to leave my statement as a philosophy). | |
| Aug 5, 2017 at 9:57 | comment | added | Malkoun | Prof. Bryant, I understand. I did say that I was off on a wrong track with the notion of quaternionic embedding. But I wonder if my vague intuition which is explained (vaguely) in my last comment can be formalized somehow (not using quaternionic embeddings). | |
| Aug 5, 2017 at 9:55 | comment | added | Robert Bryant | @Malkoun: Even other Wolf spaces won't help that much: The more general fact (also easy to prove) is that any quaternionic submanifold of a quaternion-Kähler manifold (whatever the scalar curvature) must be totally geodesic. In particular, there is at most a finite dimensional space of quaternionic submanifolds of any given quaternion-Kähler manifold, and most quaternion-Kähler manifolds don't have any nontrivial quaternionic submanifolds. It's a very rigid category. | |
| Aug 5, 2017 at 9:39 | comment | added | Malkoun | What I mean is that many constructions that work for $\mathbb{H}P^m$, and are "natural" enough, seem to extend to Quaternion-Kähler manifolds in general (with maybe some restriction on the scalar curvature). Perhaps $\mathbb{H}P^m$ can be replaced by a family of Wolf spaces associated to a family of classical groups. I am trying to formalize this vague intuition. I have been unsuccessful so far. | |
| Aug 5, 2017 at 9:35 | comment | added | Malkoun | I was off on a wrong track again... My intuition is that $\mathbb{H}P^m$ are kind of "universal" among QK manifolds (maybe restricted to have positive scalar curvature). Maybe similar to how $SL(2,\mathbb{C})$'s are "universal" in the theory of complex semisimple Lie groups, in some sense. Perhaps the only way to formalize this intuition is via the LeBrun-Salamon conjecture. | |
| Aug 5, 2017 at 7:25 | comment | added | W. Cadegan-Schlieper | Heck, Quaternion-Kaehler manifolds are scarce: I don't think we know of any other than the Wolf manifolds... | |
| Aug 5, 2017 at 7:00 | comment | added | Malkoun | Ah ok. Thank you Prof. Bryant. It looks like quaternionic maps are scarse. | |
| Aug 5, 2017 at 2:46 | comment | added | Robert Bryant | The answer is 'no', in general. In fact, as is easy to show, the only 'quaternionic' submanifolds of $\mathbb{HP}^n$ are the 'linear' submanifolds $\mathbb{HP}^m\subset\mathbb{HP}^n$. | |
| Aug 4, 2017 at 22:59 | history | asked | Malkoun | CC BY-SA 3.0 |