It is clear to me that a complex manifold is orientable. However is there a canonical choice of one of the two possible orientation or is it just a matter of convention? And, if it is a convention, is there a universally accepted one ?
It seems to me that on a manifold of complex dimension $n$ using the complex structure $J$, both the choices \begin{equation} \{\partial_{x_1}, \ldots,\partial_{x_n},J\partial_{x_1}, \ldots,J\partial_{x_n}\} \end{equation} and \begin{equation} \{\partial_{x_1},J\partial_{x_1}, \ldots,\partial_{x_n},J\partial_{x_n}\} \end{equation} are natural, but they result in opposite orientation for $n$ even. Similarly, given a Kahler form $\Omega$, wedging it with itself $n$ times gives an orientation form, however defining $\Omega(X,Y)=g(X,JY)$ or $\Omega(X,Y)=g(JX,Y)$ seems to be a matter of convention, and the resulting orientation forms are opposite for $n$ odd.
Finally, as a special case of the question above, what is the standard orientation of $\mathbb{C}^{n}$ as a complex manifold?