What are known examples of compact symplectic Fano manifolds, apart from those that come from algebraic geometry?
We define symplectic Fano manifold as a symplectic manifold $(M,w)$, such that $[c_1(M)]=[w]\in H^2(M)$. Note that $c_1(M)$ is canonically defined, because up to homothopy there is a unique almost complex structure on $M$ tamed by $w$. Such manifolds often called monotone, or strongly (strictly) montone (but I don't think that the terminology is fixed).
For an algebraic Fano manifold, defined over $\mathbb {C}$ there is a Kahler form in the class of $[c_1(M)]$, since $-K(M)$ is ample by definition. So, as Tim pointed out in his remark such a manifold is obviously a "symplectic Fano". I am aware of just one alternative construction that produces "symplectic Fanos", it is given in http://arxiv.org/abs/0905.3237 in section 5. It produces non-algebraic symplectic Fano manifolds strating from real dimension 12. The construction goes via non-compact coadjoint orbits. Is there any other construction in the literature?
Note, that in dimension 4 all "symplectic Fanos" come from Algebraic geometry (Gromov+Taubes+Mcduff), i.e. there are 10 examples, $\mathbb{C}P^1\times \mathbb{C}P^1$ and $\mathbb {C}P^2$ blown up in at most $8$ points.
