Are complex projective spaces orientable?

Every complex manifold is orientable, as every complex vector space as a canonical orientation as a real space. Namely if $V$, is a complex vector space, and $B= (u_1,...,u_n)$ is a base (over C), then $B^*=(u_1,...,u_n, iu_1,...iu_n)$ is a base over $\bf R$. Note that if $B'$ is another base over $C$ and $l$ the unique linear map $C$ map such that $lB=B'$, $lB^*=B'^*$, and the determinant of $l$, viwed as a $R$ linear map is the squre of the modulus of the determinant of $l$, hence positive. Thus the orientation given by $B$ is the same than that given by $B'$, and complex linear maps preserves this orientation.

1) An orientation on a topological manifold $X^n$ is a choice of generator $X_p$ of each local homology group $H_n(X, X\setminus, p, \mathbb{Z}) = \mathbb{Z}$ compatible with the inclusions $(X, X\setminus U) \to (X, X\setminus p)$ for neighborhoods $U$ of $p$. For a smooth, rather than merely topological, manifold, unraveling the definition above gives the usual definition of orientability in terms of the triviality of $\det \bigwedge^n T^*X $.

In this particular case, an easy way of proving that $\mathbb{CP}^n$ is orientable is noting that $\pi_1 \mathbb{CP}^n = 0$. That follows immediately from the cellular approximation theorem, for example, or the fibration $S^1 \to S^{2n+1} \to \mathbb{CP}^n$, where $S^1\subset \mathbb{C}$ acts on $S^{2n+1}\subset \mathbb{C}^{n+1}$ by $z.(w_0, \dots, w_n) = (zw_0, \dots, zw_n)$.

2) More generally, the result holds for any almost-complex manifold; that is, any real manifold $X$ with a map $J:TX \to TX$ such that $J_p:T_p X \to T_p X$ with $J^2 = -1$. (Any complex manifold is also almost-complex; take $J$ to be multiplication by $i$ on each fiber.) To orient $X$, take a complex basis $v_1, \dots, v_n$ of $T_p X$, and let $v_1, \dots, v_n, Jv_1, \dots, Jv_n$ be its corresponding orientation as a real vector space. Alternatively, for a smooth complex manifold, consider the top form $\omega = dz_1 \wedge d\overline{z}_1 \wedge \dots dz_n \wedge d\overline{z}_n$ for a local coordinate chart $(z_1, \dots, z_n)$.