Let $f:M\to N$ be a holomorphic map between two oreintable complex manifolds. Given a volume form $dx_1\wedge dy_1\wedge…dx_n\wedge dy_n=(\frac{i}{2})^ndz_1\wedge d\overline{z_1}…dz_n\wedge d\overline{z_n}$ on $Y$, then $f^*(dx_1\wedge dy_1\wedge…dx_n\wedge dy_n)=f^*(\frac{i}{2})^ndz_1\wedge d\overline{z_1}…dz_n\wedge d\overline{z_n}=(\frac{i}{2})^ndf_1\wedge d\overline{f_1}…df_n\wedge d\overline{f_n}$.
Note $df_i\wedge d\overline{f_i}=(\sum_j\frac{\partial f_i}{\partial w_j}dw_j)\wedge (\sum_j\frac{\overline{f_i}}{\overline{w_j}}d\overline{w_j})$.
It's easy to see when $n=1$, we have $f^*dx\wedge dy=f^*\frac{i}{2}dz\wedge d\bar{z}=\frac{i}{2}|f'(w)|^2dw\wedge d\bar{w}=|f'(w)|^2dx'\wedge dy'$, which is a volume form on $X$. But I can see it's true for $n>1$. So I wonder if a holomorphic map preserves orientation for any dimension?
Best Answer
This is really just a linear algebra question: restricting our attention to the tangent space at a single point, the question is whether an invertible complex-linear map $\mathbb{C}^n\to\mathbb{C}^n$ (the derivative of $f$ at our point) always has positive determinant when considered as a map $\mathbb{R}^{2n}\to\mathbb{R}^{2n}$. The answer is yes. Here is one way to prove it. Since the determinant is a continuous function, it suffices to show that $GL_n(\mathbb{C})$ is connected, since the identity obviously has positive determinant. There are many ways to show $GL_n(\mathbb{C})$ is connected; for instance, by Jordan normal form it suffices to show the set of invertible upper triangular matrices is connected and that is easy since it is just a product of copies of $\mathbb{C}\setminus\{0\}$ (for the diagonal entries) and $\mathbb{C}$ (for the entries above the diagonal).
(Note that applying this result to the transition functions of the complex charts on a complex manifold, this shows that every complex manifold is orientable and has a canonical orientation (the one given by its complex charts, identifying $\mathbb{C}^n$ with $\mathbb{R}^{2n}$ in a fixed way). This then says that any holomorphic immersion between complex manifolds of the same dimension is orientation-preserving if you give both of them the canonical orientation. Of course, if you do not require any compatibility between the orientation and the complex structure then you cannot conclude a holomorphic map is orientation-preserving, since you could just reverse the orientation on the domain or codomain.)