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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.

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Did you need to invoke Yau's work [on K-E metrics]? On a compact complex manifold with ample anticanonical bundle, you get an anti-canonical Kaehler form by rescaling a pullback of Fubini-Study. – Tim Perutz Jan 19 '10 at 2:12
    
Tim, you are absoulutely correct. No need to invoke it. I will correct the formulation, according to your suggestion. – Dmitri Jan 19 '10 at 2:53
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I don't know any examples pre Fine-Panov! McDuff asked me a related question: is there a monotone symplectic 2n-manifold whose minimal Chern number (on 2-spheres) exceeds n+1? Not an algebraic Fano (Mori, Annals 110, 1979). What about one with min Chern number n+1, not symplectomorphic to projective space? – Tim Perutz Jan 19 '10 at 3:30
    
Tim, many thanks! I never heard of this cool question of McDuff! – Dmitri Jan 19 '10 at 12:02

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