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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.

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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.

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  • $\begingroup$ "The cohomology class of the symplectic form has positive square" : I wonder what is your definition of a symplectic strucuture. For me, a symplectic form remains symplectic if one changes the orientation. In particular, the answer should be positive for $k=1$ and all $l$. $\endgroup$
    – abx
    Commented Dec 12, 2013 at 14:44
  • $\begingroup$ @abx But a symplectic form canonically defines an orientation of the underlying manifold. If you had already chosen a preferred orientation of your manifold, it makes sense to look for symplectic forms inducing the same orientation. There are no symplectic forms on $CP^{2}$ which would induce an orientation different from the standard one. $\endgroup$
    – Oldřich Spáčil
    Commented Dec 12, 2013 at 15:48
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    $\begingroup$ I agree, but then the question should be slightly modified (… a symplectic structure compatible with the orientation). $\endgroup$
    – abx
    Commented Dec 12, 2013 at 15:53
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    $\begingroup$ Thank a lot for a clear answer! Moreover, I'd like to know how about considering all the smooth manifolds homeomorphic to $\#_{l}CP^{2}\#_{k}(-CP^{2})$ for $l$ is odd. $\endgroup$
    – user44052
    Commented Dec 13, 2013 at 8:13
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    $\begingroup$ That's a much more complicated answer; there are some that have symplectic structures, some that don't. By the argument I outlined above, you should assume that your manifold is irreducible, ie not a connected sum of manifolds with $b_2^+ >0$. There is a very large literature on the subject, starting with [Szabó, Zoltán, Simply-connected irreducible 4-manifolds with no symplectic structures. Invent. Math. 132 (1998), no. 3, 457–466.] I would also look at [R. Fintushel and R. Stern, Knots, links, and 4-manifolds, Invent. Math. 134 (1998), 363–400.] $\endgroup$
    – Danny Ruberman
    Commented Dec 13, 2013 at 14:49

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