Let $-CP^{2}$ denote the complex projective surface $CP^{2}$ with the reverse orientation. I have seen some results about the existence of symplectic structures on the connected sums $\#\_{l}CP^{2}\#\_{k}(-CP^{2})$ for **some** positive integers l,k. 


My question is whether there is a complete result which can decribe the existence of symplectic structures on the connected sums $\#\_{l}CP^{2}\#\_{k}(-CP^{2})$ for **all** positive integers l,k.



Does the smooth manifold $\#_{l}CP^{2}\#_{k}(-CP^{2})$ admit a symplectic structure?
For $l=1$, these are blowups of the projective plane, which are all Kaehler and hence symplectic. For $l>1$, these do not have symplectic structures. For if $l$ is even, then they don't even have almost complex structures [(cf. this MO thread)](https://mathoverflow.net/questions/59820/a-question-on-classification-of-almost-complex-structures-on-4-manifolds), which symplectic manifolds certainly do. If $l>1$ is odd, then their Seiberg-Witten invariants would vanish, since your manifolds decompose as a connected sum into pieces with positive $b\_2^+$. But a famous theorem of Taubes says that there is a non-vanishing Seiberg-Witten invariant.


Finally, if $l=0$, then $b\_2^+ = 0$, so your manifold cannot be symplectic, since the cohomology class of the symplectic form has positive square.


