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$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism (complex conjugation!), so this is diffeomorphic to the blowup of $\Bbb{CP}^{2n+1}$ at one point.

On the other hand, $\Bbb{CP}^2 \# \Bbb{CP}^2$ does not even support an almost complex structure: Noether's formula demands that its first Chern class $c_1^2 = 2\chi + 3\sigma = 14$, but if $c_1 = ax_1 + bx_2$ (where $x_1, x_2$ generate $H^2$, $x_1^2 = x_2^2$ is the positive generator of $H^4$, and $x_1x_2 = 0$), then $c_1^2 = a^2 + b^2$, and you cannot write $14$ as a sum of two squares.

Using a higher-dimensional facsimile of the same proof, I wrote down a proof here that $\Bbb{CP}^4 \# \Bbb{CP}^4$ does not admit an almost complex structure. The computations using any similar argument would, no doubt, become absurd if I increased the dimension any further.

Can any $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ support an almost complex structure?

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    $\begingroup$ The original motivation for this question was another Math.SE question asking when, if $M$ and $N$ are complex manifolds, $M \# N$ supports a complex structure. For $4n+2$-manifolds, the answer is "always", and for $4$-manifolds, the answer is "never". This seems like a reasonable first place to find a pair of complex manifolds $M, N$ such that $M \# N$ is not complex. $\endgroup$ – mme Sep 2 '15 at 20:06
  • $\begingroup$ Pardon my ignorance, but I'm curious about the formal definition of the $\#$ operator... $\endgroup$ – Suvrit Sep 3 '15 at 13:14
  • $\begingroup$ @Suvrit: Sorry for not being explicit. I mean here the connected sum operation. You pick choices of embedding $D^n \hookrightarrow M$, $D^n \hookrightarrow N$, and glue the two manifolds together along these embedded discs; this does not end up depending on the choice of embedding. (One needs to be a little bit more careful to get a well-defined smooth structure on this, but this is not so hard). $\endgroup$ – mme Sep 3 '15 at 14:12
  • $\begingroup$ I am greatly appreciative of Alexsandar Milivojevic for pointing out that my intended argument that $M \# N$ supports an almost complex structure in singly even dimensions is false: I claimed that for arbitrary complex manifolds $M, N$, one may put a complex structure on $M \# \overline N$. This is not true, and $K3 \# \overline{K3}$ gives a counterexample, as for an almost complex 4-manifold $\chi + \sigma$ is divisible by 4. $\endgroup$ – mme Dec 7 '18 at 9:02

The $m$-fold connected sum $m\# {\mathbb{CP}}^{2n}$ admits an almost complex structure if and only if $m$ is odd, as we show in our recent preprint. (By the way, thanks to Mike for this interesting question, which motivated us to write the paper!)

Here's a brief summary of the proof's idea. Our main tool is a result by Sutherland resp. Thomas from the 60s which tells us when a stable almost complex structure is induced by an honest almost complex structure: this is the case iff its top Chern class equals the Euler class of the manifold.

As the connected sum of manifolds admitting a stable almost complex structure admits one as well, we certainly have stable almost complex structures on $m\# {\mathbb{CP}}^{2n}$ at our disposal, and we can understand the full set of all such structures by explicitly determining the kernel of the reduction map from complex to real K-theory. We then compute the top Chern class of all these structures: luckily for us, it turns out that in order to show the non-existence of almost complex structures for even $m$, it suffices to compute its value modulo 4 and compare it to the Euler characteristic of $m\# {\mathbb{CP}}^{2n}$. For odd $m$, we explicitly find a stable almost complex structure for which the criterion above is satisfied.

Edit: The paper Connected sums of almost complex manifolds from Huijun Yang generalizes our theorem to the following beautiful fact:

Let $M_i$ $i=1,\ldots,\alpha$ be $4n$-dimensional almost complex manifolds. Then the connected sum $$\#_{i=1}^\alpha M_i \#(\alpha-1) \mathbb C\mathbb P^{2n}$$ admits an almost complex structure. Moreover if $M$ is an almost complex manifold of dimension $2n$ then $M\#\overline{\mathbb C\mathbb P^n}$ admits an almost complex structure.

Thus the last statement means that the "blow-up" of almost complex manifolds is again almost complex in analogy to blow-ups of points in complex manifolds.

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    $\begingroup$ As is pointed out in the paper, the fact that $m\mathbb{CP}^{2n}$ does not admit an almost complex structure for $m$ even (in particular, $m = 2$) follows from a result of Hirzebruch which states that the Euler characteristic and signature of a closed almost complex manifold $M$ of dimension $4n$ satisfies $\chi(M) \equiv (-1)^n\sigma(M) \bmod 4$. This is proved using the Hirzebruch $\chi_y$ genus, see here for example. $\endgroup$ – Michael Albanese Sep 10 '18 at 14:17

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