Dimension of the complex projective space $\mathbb{C}\mathbb{P}^n$ as almost complex manifold

Question

I have already shown that the complex projective space $\mathbb{C}\mathbb{P}^n$ is a complex manifold by checking the required properties of the transition maps. Since every complex manifold is an almost complex manifold, $\mathbb{C}\mathbb{P}^n$ is also an almost complex manifold. I wonder what is its dimension. I have proved that almost complex manifolds always have even dimension. But how can I determine the dimension of the complex projective space $\mathbb{C}\mathbb{P}^n$ as an almost complex manifold? And how can I prove this answer? It would be great if someone could help me with this proof. Thanks in advance for your help!

Answer 1

An almost complex manifold is a real smooth manifold $M$ with a bundle map $$ J: TM\rightarrow TM $$ such that $J^2=-id$. When you ask what is the dimension of $\mathbb{CP}^n$ as an almost complex manifold, what you're asking is what is its dimension as a real manifold. If $M$ is a complex manifold of complex dimension $n$, then the dimension of $M$ as a real manifold is $2n$. In other words, $$ \dim_{\mathbb{R}}M=2\dim_{\mathbb{C}}M $$ Since $\dim_{\mathbb{C}}\mathbb{CP}^n=n$, then $\dim_{\mathbb{R}}\mathbb{CP}^n=2n$.