# Complexified Tangent Spaces

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:

1. 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?
2. 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!

• For the first question, $(\frac{\partial}{\partial x_1})_p$ acts on smooth functions $f: U \to \mathbb{R}$, where $U$ is an open neighborhood of $p$ in $\mathbb{C}^n \cong \mathbb{R}^{2n}$. The action is given by $f \mapsto \frac{\partial f}{\partial x_1}\big|_p$ Commented Feb 4, 2020 at 1:40

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 $$J$$