Timeline for "monotone" versus "symplectic Fano"
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 8, 2011 at 15:41 | comment | added | Mohammad Farajzadeh-Tehrani | Exactly, an example like what you mentioned at the end is the answer. | |
| Jun 8, 2011 at 8:30 | comment | added | Dmitri Panov | Mohammad, I understand this question as follows: Suppose $c_1$ is positive on all $J$-holomorphic curves (for some $J$), is it true then that we have a symplectic form $\omega$ on M such that $\omega=\lambdaC_1(M)$? It could be possible, to get further examples using discrepancy with Kleiman's theorem, for example if you find a Kahler manifold such that $(C_1(M),\Sigma)>0$ for all complex curves $\Sigma$ but $C_1(M)^{1/2 dim M}=0$. | |
| Jun 8, 2011 at 4:26 | comment | added | Mohammad Farajzadeh-Tehrani | Or saying better, does it have a representative which is a symplectic form. This question is related to Kleiman's criteria for ampleness on varieties. Kleiman's theorem says (under some mild assumptions): If $w^k$ is positive on any sub variety of dimension $k$ then $w$ is a kahler form. But here we only have the condition above for curves and not symplectic manifolds of any dimension. | |
| Jun 8, 2011 at 4:17 | comment | added | Mohammad Farajzadeh-Tehrani | So the questions is if $\pm c_1$ is positive on any $J$_holomorphic curve, is itself a symplectic form? | |
| Jun 8, 2011 at 4:14 | comment | added | Mohammad Farajzadeh-Tehrani | I deleted them. | |
| Jun 8, 2011 at 4:14 | comment | added | Mohammad Farajzadeh-Tehrani | you were right, my examples were wrong. | |
| Jun 7, 2011 at 14:30 | comment | added | Dmitri Panov | Mohammad, you are right I misread the question. But now it works :) | |
| Jun 7, 2011 at 14:30 | history | undeleted | Dmitri Panov | ||
| Jun 7, 2011 at 14:29 | history | edited | Dmitri Panov | CC BY-SA 3.0 |
deleted 50 characters in body
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| Jun 7, 2011 at 14:18 | history | deleted | Dmitri Panov | ||
| Jun 7, 2011 at 13:51 | comment | added | Mohammad Farajzadeh-Tehrani | Yes the word Fano is not correct but what he really wants is: $w(A)$'s all have the same sign over classes that can be represented by J-holomorphic curves. | |
| Jun 7, 2011 at 13:42 | history | answered | Dmitri Panov | CC BY-SA 3.0 |