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

Question

This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here.

$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ 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?

Answer 1

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.