According to a theorem of McDuff, if $(X,\omega)$ is a closed symplectic 4-manifold, which contains a symplectic sphere of self-intersection +1, then $X$ is symplectomorphic to $CP^2$ blown up a few times, with its standard symplectic form (the theorem contains results about other positive self-intersections as well). My question is whether there are similar results when $(X,\omega)$ contains symplectic surfaces of higher genus, for instance a symplectic torus. Although I think not, since we can already have curves of different genera repesenting different homology classes in $CP^2$. But I will appreciate if you can let me know of any other similar classification results.
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1$\begingroup$ What about the projective completion of a degree-$1$ complex line bundle over a genus-$g$ Riemann surface? This is a symplectic manifold that is not symplectomorphic to a blowing up of $\mathbb{C}P^2$, since the fundamental group is wrong. $\endgroup$– Jason StarrCommented Apr 21, 2015 at 0:31
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1$\begingroup$ You might want to have a look at Chris Wendl's notes From Ruled Surfaces to Planar Open Books. He reproves McDuff's result from a different viewpoint and states some generalizations, I think. $\endgroup$– Oldřich SpáčilCommented Apr 21, 2015 at 14:14
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