TITLE: Does the smooth manifold $\#_{l}CP^{2}\#_{k}(-CP^{2})$ admit a symplectic structure?
QUESTION [9 upvotes]: 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.

REPLY [11 votes]: 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), 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.