In Nakahara's GTaP chapter 8.2, he explains the almost complex structure on a real manifold of even dimension. I'm a bit confused by the dimension equation that immediately follows equation (8.27):
$$ \text{dim}_\mathbb C T_pM^+=\text{dim}_\mathbb C T_pM^-=\frac 12\text{dim}_\mathbb C T_pM^\mathbb C=\frac 12\text{dim}_\mathbb C M $$
$T_pM^+$ is the subspace of holomorphic vectors (those which transform as $J(Z)=iJ(Z)$ under the complex structure $J$).
The simplest example of a complex manifold is $M=\mathbb C$, and we have $\text{dim}_\mathbb C\mathbb C=1$. In this case, the expressions in this equation would have the value $\frac 12$.
How can the dimension of a vector space be half-integer? What am I mising?
I cannot find any indication that the complex dimension of $M$ must be even.
Let $A,B$ be vector spaces over $K$, then $$\operatorname{dim}(A\otimes_K B)=\operatorname{dim}A\cdot\operatorname{dim}B.$$ So, if $M$ is a $2n$-dimensional, then $\operatorname{dim}_\mathbb{R}T_pM=2n$. Now, the tangent space Nakahara is talking about is the complexified tangent space, $T_pM^\mathbb{C}:=T_pM\otimes_\mathbb{R}\mathbb{C}$. The real dimension of this space is $\operatorname{dim}_\mathbb{R} T_pM^\mathbb{C}=\operatorname{dim}_\mathbb{R}T_pM\cdot\operatorname{dim}_\mathbb{R}\mathbb{C}=4n$. Using $\operatorname{dim}_\mathbb{C}=\frac{1}{2}\operatorname{dim}_\mathbb{R}$, we see that $\operatorname{dim}_\mathbb{C}T_pM^\mathbb{C}=2\operatorname{dim}_\mathbb{C}M=2n.$ The last $1/2$ in your text is a typo.