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Hi. I have a question about the notion "symplectic Fano". Let $(M,\omega)$ be a symplectic manifold with a $\omega$-tamed almost complex structure $J$. According to "J-holomorphic curves and symplectic topology, by D.McDuff and D.Salamon, pp 388 - 389", $(M,\omega,J)$ is called "symplectic Fano" if for any $A \in H_2(M)$ which can be represented by a $J$-holomorphic curve, $\langle c_1(M), A \rangle$ is positive. (Note that $J$-curve that I meant is not just a rational curve. The genus of the curve can be positive.)

On the other hand, $(M,\omega,J)$ is called "monotone" if $[\omega] = \lambda c_1(M)$ in $H^2(M;\mathbb{R})$ for some positive $\lambda \in \mathbb{R}$.

It is obvious that if $M$ admits a monotone symplectic structure, then it is symplectic Fano. My questions is as follow.

Q : Does "symplectic Fano" imply the existence of a monotone symplectic structure?

Thank you in advance.

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Edited. For the definitions that you mention "Simplectic Fano" can be non-montone. For example, you can take a $4$-dimensional Kahler non-agebraic torus that does not have complex curves at all. Such a torus does not have complex curves at all and it is has no symplectic structure for which it is monotone. The same trick can be done with K3 surface

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  • $\begingroup$ 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. $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Jun 7, 2011 at 13:51
  • $\begingroup$ Mohammad, you are right I misread the question. But now it works :) $\endgroup$
    – Dmitri Panov
    Commented Jun 7, 2011 at 14:30
  • $\begingroup$ you were right, my examples were wrong. $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Jun 8, 2011 at 4:14
  • $\begingroup$ I deleted them. $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Jun 8, 2011 at 4:14
  • $\begingroup$ So the questions is if $\pm c_1$ is positive on any $J$_holomorphic curve, is itself a symplectic form? $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Jun 8, 2011 at 4:17

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