# Complex bundles and (quasi)-complex structures of manifolds

Could you help me with some hint or reference for the following questions?

I'm reviewing the Milnor-Stasheff for references.

1. Is there some $$(2k)$$-manifold with stably quasi-complex structure, such that the structure is not complex?

2. Is there some $$(2k+1)$$-manifold that is stably quasi-complex?

3. Is there any embedding of $$\mathbb{CP}^4$$ in $$\mathbb{R}^{11}$$?

4. For which values of $$n$$ does it have that $$\mathbb{CP}^n$$ has structure Spin?

Where,

a) Let $$E \rightarrow M$$ real vector bundle of range $$2n$$, a quasi-complex structure in the bundle is a function $$J: E \rightarrow E$$ which is $$\mathbb{R}$$ - linear in each fiber and such that $$J \circ J = -Id$$

b) Stably quasi-complex structure for $$M$$ means that there exists a trivial vector bundle $$\eta: \mathbb{R}^k \rightarrow M$$ such that $$\eta \oplus \tau M$$ (the Whitney sum of bundles) has a complex structure for $$M$$ (with $$\tau M$$ the tangent bundle of $$M$$).

First of all, Thanks

• For $4.$ you can use Theorem 4.5 to calculate the Chern classes of $\mathbb{C}P^n$. The theorem is stated in terms of the SW classes of $\mathbb{R}P^n$, but the same proof works. Then you should know the relationship between $c_1$ and $w_2$, and what this implies for spin structure. Apr 25, 2019 at 9:21
• By 'quasi-complex' do you mean 'almost complex'? Apr 25, 2019 at 9:22
• I do not know the definition of almost-complex, but I leave you the definition of quasi-complex in the publication, thanks Apr 26, 2019 at 21:42
• This is what is usually called an almost complex structure, en.wikipedia.org/wiki/Almost_complex_manifold . Milnor and Stasheff also use this term in the 1974 printing. Apr 27, 2019 at 9:00
• For 1., spheres are stably parallelizable, so an even dimension sphere is stably almost complex. However the only spheres which are almost complex are $S^2$ and $S^6$. Of these $S^2$ is complex, and as far as I understand it is still undecided as to whether $S^6$ is complex or not. Apr 27, 2019 at 12:40