Timeline for Is the Nijenhuis tensor an obstruction to the existence of non constant pseudo-holomorphic maps?
Current License: CC BY-SA 3.0
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| Jun 18, 2014 at 13:07 | comment | added | Ritwik | @Bryant and Chris: Actually Chris's comment was the real motivation for my question. In Symplectic Geometry the moduli space of J holomorphic maps from one Riemann Surface to a Symplectic Manifold is studied. I was wondering why a higher dimensional analogue of this is not studied; I think Chris's answer explains that. | |
| Jun 17, 2014 at 18:47 | comment | added | Robert Bryant | It's not completely clear what the OP had in mind. As I read it, he was asking whether having $J_N$ and $J_M$ be nonintegrable implied (in general) that all pseudoholomorphic maps from $N$ to $M$ must be constant. Is is really obvious that there does not exist a (probably very high dimensional and necessarily) non-integrable, almost complex manifold $(M,J_M)$ such that every almost complex $4$-manifold $(N,J_N)$ has a locally defined, nonconstant pseudoholomorphic map into $(M,J_M)$? I think that statement might not be as easy to actually prove as it is to believe. | |
| Jun 17, 2014 at 18:37 | comment | added | Chris Gerig | Ah I am implicitly fixing the target $(M,J_M)$. I speak of generic $(N,J_N)$ which is consistent with your observation. | |
| Jun 17, 2014 at 18:33 | comment | added | Robert Bryant | But remember that existence/nonexistence of a nonconstant pseudoholomorphic mapping depends on the target as well. As I observed above, the identity map is a nonconstant pseudoholomorphic map for any $(N,J_N)$. | |
| Jun 17, 2014 at 17:37 | history | answered | Chris Gerig | CC BY-SA 3.0 |