I'm currently reading chapter 1 of Mei Chi's book, and I have a few questions on 1.3.
For any point $p\in \mathbb{C}^n$, the tangent space $T_p(\mathbb{C}^n)$ is spanned by
$$ \Big(\frac{\partial}{\partial x_1}\Big)_p,\Big(\frac{\partial}{\partial y_1}\Big)_p,\dots, \Big(\frac{\partial}{\partial x_n}\Big)_p, \Big(\frac{\partial}{\partial y_n}\Big)_p $$
Define an $\mathbb{R}$-linear map from $T_p(\mathbb{C}^n)$ onto itself by
$$ J\Big(\frac{\partial}{\partial x_i}\Big)_p=\Big(\frac{\partial}{\partial y_i}\Big)_p, J\Big(\frac{\partial}{\partial y_i}\Big)_p=-\Big(\frac{\partial}{\partial x_i}\Big)_p $$
$J$ is called the complex structure on $\mathbb{C}^n$. The complex structure $J$ induces a natural splitting of the complexified tangent space $\mathbb{C}T_p(\mathbb{C}^n)=T_p(\mathbb{C}^n)\otimes_{\mathbb{R}}\mathbb{C}$. First we extend $J$ to the whole complexified tangent space by $J(x\otimes \alpha)=(J(x)\otimes\alpha)$ It follows that J is a $\mathbb{C}$ linear map from $\mathbb{C}T_p(\mathbb{C}^n)$ onto itself with $J^2=-1$. QUESTIONS:
- I've not had a course in differential geometry. The tangent space is spanned by vectors like $\Big(\frac{\partial}{\partial x_1}\Big)_p$. What is this operator acting on? The partial derivative w.r.t $x_1$ of what? $\mathbb{C}^n$ is a complex manifold trivially, so the identity map?
- In an attempt to understand a complex structure, I have read the first couple of pages of Heybrechts' book "Complex Geometry". I for the most part understand the ideas. Do we complexify the real vector space $T_p(\mathbb{C}^n)$ by using the complex structure $J$ to "build" the space $\mathbb{C}T_p(\mathbb{C}^n)$? Heybrechts' book extends a real v.s. to a complex v.s. by $(a+bi)v=av+bJ(v)$. Is that what $J(x\otimes \alpha)=(J(x)\otimes\alpha)$ is referring to? Does someone know of a resource online that could elaborate on this process?
Any answers or references to good resources would be appreciated!
Best Answer
For the second question, complexification of a real vector space is canonical, it is given by $V\otimes_{\mathbb R} \mathbb C$. The complex structure $J$ is an additional piece of information, it equips the original real vector space a structure of complex vector space. In this case the complexification has twice the complex dimension and it naturally decomposes into two eigenspaces with respect to the complex linear extension of $J$, i.e. the $J(x\otimes \alpha) =J(x) \otimes\alpha$ that you mentioned.
You can use basis to think about this. Originally the real vector space $T_p(\mathbb C^n)$ has real basis $\{\frac{\partial}{\partial x_i},\frac{\partial}{\partial y_i}\} _{i=1}^n$. The complexified tangent space $T_p(\mathbb C^n)_{\mathbb C}$ has the same set of vectors as basis, but it is now a complex vector space. With the additional $J$ naturally defined, $T_p(\mathbb C^n)$ now has the structure of complex vector space with basis $\{\frac{\partial}{\partial x_i} \} _{i=1}^n$, note that now $\frac{\partial}{\partial y_i}$ is in the line containing $\frac{\partial}{\partial x_i}$. Finally the decomposition of complexified tangent space can be realized by a change of basis, instead of using $x_i, y_i$, we take $\{\frac{\partial}{\partial z_k}, \frac{\partial}{\partial\bar z_k}\} _{k=1}^n$, where $\frac{\partial}{\partial z_k} =\frac 12(\frac{\partial}{\partial x_k} - i\frac{\partial}{\partial y_k}) $ and $\frac{\partial}{\partial\bar z_k} =\frac12(\frac{\partial}{\partial x_k}+i\frac{\partial}{\partial y_k}) $. You can check that they are eigenvectors of the complexified operator $J$