## “monotone” versus “symplectic Fano”

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?

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