Cfrgtkky

Let $M$ and $N$ be real manifolds of dimension $n$ which happen to admit complex structures (so that necessarily $n=2k$ and both are orientable). Then does their connected sum $M# N$ also admit a complex structure?

This is true for $n=2$, because every oriented $2$-dimensional topological manifold admits a complex structure. However, this operation doesn't seem to be compatible with the information given by the structure; in particular we would want $Msqcup N$ to at least be almost-complex cobordant to $M# N$, or that these two manifolds should have the same Chern numbers. We have
$$c_1[Msqcup N]=langle c_1(tau Msqcuptau N),[Msqcup N]rangle = langle c_1(tau M),[M]rangle + langle c_1(tau N),[N]rangle = c_1[M]+c_1[N] $$
But then recall that the first Chern class of a complex rank 1 bundle $V$ is the Euler class of its realification, and so the first Chern number of a complex 1-manifold is its Euler characteristic. Thus
$$c_1[M# N] = chi(M#N)=chi(M)+chi(N)-2 neq c_1[M] + c_1[N] $$
So it seems like it's possible that the fact that complex $1$-manifolds are closed under connected sum could be a coincidence.

In higher dimensions there are no complex structures on $S^{2k}$ (except MAYBE $S^6$). But, in the positive direction, there ARE these complex Calabi-Echkmann manifolds homeomorphic to $S^{2k+1}times S^{2l+1}$. So does anyone know if, for example, the connected sum of $g$ copies of $S^5times S^5$ admits a complex structure for any $g>1$?

EDIT: I am greatly appreciative of Aleksandar Milivojevic for pointing out the first paragraph of my original answer is false, and therefore does not give a proof of the following two paragraphs. (For instance, $K3 # overline{K3}$ supports no complex structure: Aleksandar kindly explained to me that this has signature zero but Euler characteristic $46$, and $chi + sigma$ is always divisible by $4$ on an almost complex 4-manifold.

I leave these here for posterity, separated from the correct content by lines. It remains an interesting question whether or not $M # N$ can support a complex structure when $M$ and $N$ are singly-even dimensional complex manifolds. There is no obstruction coming from the nonexistence of an almost complex structure, so it is not totally impossible.

Let $M$ and $N$ be complex manifolds. Because we have a local form (there is a neighborhood of a point isomorphic to the the complex unit disc) you can verify that $M # overline N$ has a natural complex structure - just line up the copies of $Bbb C^n setminus {0}$ you're gluing together so the almost complex structures match up. Then the Nijenhuis tensor still vanishes.

So, if $N$ supports a complex structure, does $overline N$? This would mean that $M # N$ supports a complex structure. If $text{dim}(N) = 4n+2$, then this is true: if $J$ is your (integrable almost) complex structure, then the complex conjugate $overline J$ gives a complex structure on $overline N$. The reason this doesn't work in dimension $4n$ is because $overline J$ induces the same orientation as $J$!

So I wouldn't quite call it a fluke. It's true for every $4n+2$-dimensional complex manifold, not just $2$-dimensional ones. But it's not true in every dimension.

For 4-manifolds, it is a theorem of Wu that $M$ admits an almost complex structure with $c_1(J) = c in H^2(M;Bbb Z)$ if and only if $c$ reduces to $w_2$ mod 2, and $c^2 = 3sigma + 2chi$. You can prove using this criterion that a most two of $M, N$, and $M # N$ can admit almost complex structures; in particular, $Bbb{CP}^2 # Bbb{CP}^2$ cannot admit an almost complex structure.

I don't know about higher dimensions $4n$. It might be wise to think about almost complex structures instead of complex ones first, because those are easier to work with and there are no known high dimensional (complex dimension $> 2$) examples of almost complex manifolds that do not admit a complex structure.

$X = Bbb{CP}^4 #Bbb{CP}^4$ does not support an almost complex structure. I suspect this is true for $Bbb{CP}^{2n} # Bbb{CP}^{2n}$ for any $n>0$; I can't prove it. The general idea is to obtain a "higher-dimensional" Wu theorem on which manifolds admit complex structures in terms of various characteristic classes of your manifold. I think this is possible in all dimensions, but probably becomes absurd very quickly. The 8-dimensional version is here. This is what we will use.

Preliminaries: the cohomology ring of $X$ is the direct sum of two copies of the cohomology ring of $Bbb{CP}^4$, identifying the unit and volume form in each copy. Denote the two generators of $H^2(X;Bbb Z)$ as $x_1, x_2$.

1) Wu classes. Because we know precisely what the cohomology ring is, it's easy to calculate this. In particular, $Sq^2(x_1^3) = x_1^4$, and $Sq^2(x_2^3) = x_2^4$, so $v_2 = x_1+x_2$. Similarly we see that $Sq^4(x_1^2) = x_1^4, Sq^4(x_2^2)=x_2^4$, so $v_4 = x_1^2+x_2^2$. Trivially $v_6 = 0$.

2) Stiefel-Whitney classes. Because the total SW-class $w$ satisfies $Sq(v)$, where $Sq$ is the total Steenrod square and $v$ is the total Wu class, we obtain $w_2 = v_2 = x_1+x_2$, $w_4 = v_2^2+v_4 = 0$, $w_6 = Sq^2(v_4) = 0$.

3) Pontryagin classes. We can give $X$ a cell structure such that its 4-skeleton is $Bbb{CP}^2 vee Bbb{CP}^2$. Because $X_4 hookrightarrow X$ induces an isomorphism on $H^4$, the Pontryagin class is determined by the restriction of the classifying map $X to BSO(8)$ to $X_4$. We can choose this classifying map to be of the form $f vee f$, where $f$ is the restriction of the classifying map of $Bbb{CP}^4$ to its 4-skeleton. Because we know $p_1(Bbb{CP}^4) = 5x_1^2$, we see that $$p_1(X) = 5x_1^2+5x_2^2.$$
Now the Hirzebruch signature theorem tells us that $p_2 = 20x_1^4$.

Now let's plug into the theorem. Our cohomology classes $u, v$ would have to be of the form $u = (2k+1)x_1 + (2ell+1)x_2$, and $v = 2mx_1^3 + 2nx_2^3$, where $m,n$ have the same parity (by b, c).

Then because $chi(M) = 16$, d) takes the form $$128 = 80 + 32(km+ell n) + 16(m+n) - (2k+1)^4 - (2ell+1)^4 + 10(2k+1)^2 + 10(2ell+1)^2 - 50.$$

Simplifying we get $$80 = 32(km + ell n - k^3 - ell^3 + k^2 + ell^2) + 16(k+ell - k^4 - ell^4 +m +n)$$
Reducing mod 32 and remembering that $m+n$ is even we obtain $$16 equiv 16(k+ell - k^4- ell^4) mod 32$$ But $k+ell$ is odd iff $k^4 + ell^4$ is odd, so the right side is $0 mod 32$. This is a contradiction, as desired.

That this was so much work suggests that this is, uh, the wrong approach to prove that $Bbb{CP}^{2n} # Bbb{CP}^{2n}$ never supports a complex structure. Maybe someone more gifted with obstruction theory than I can prove this.