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.
Best Answer
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