So the real tangent space of $\mathbb C^n$ at the point $p$ is given by $T_p\mathbb C^n=Span_\mathbb R\{\partial/\partial x_1,\partial/\partial y_1, \dots \partial/\partial x_n,\partial/\partial y_n\}$ where $z_j=x_j+iy_j$.
And if we complexify: $T_p\mathbb C^n\otimes \mathbb C=\{\partial/\partial z_1\dots\partial/\partial z_n \}\oplus\{\partial/\partial \bar z_1\dots\partial/\partial\bar z_n \}$ where $\partial/\partial z_j=\frac12(\partial/\partial x_j-i\partial/\partial y_j)$ and $\partial/\partial \bar z_j=\frac12(\partial/\partial x_j+i\partial/\partial y_j)$.
Now there is a complex structure $J$ on $T_p\mathbb C^n\otimes \mathbb C$ given by $J(\partial/\partial z_j)=i\partial/\partial z_j$ and $\partial/\partial \bar z_j=-i\partial/\partial \bar z_j$.
My questions: What is $J_{T_p\mathbb C^n}$? Does $T_p\mathbb C^n$ have a compex structure? Is $T_p\mathbb C^n$ isomorphic a (complex) vector space to $\{\partial/\partial z_1\dots\partial/\partial z_n \}$?
Best Answer
Yes, $T_p\mathbb{C}^n$ has a complex structure $J$ given by $$J\frac {\partial}{\partial x^i}=\frac{\partial}{\partial y^i},\quad J\frac{\partial}{\partial y^i}=-\frac{\partial}{\partial x^i}\;.$$ Holomorphicity is defined in means of the above $J$; a differentiable map $$u:\Omega\subset\mathbb{C}^n\to\mathbb{C}^m$$ is holomorphic if and only its differential at any point commutes with the complex structures on both sides, that is, $$du\circ J^{\mathbb{C}^n}=J^{\mathbb{C}^m}\circ du.$$ (This is just a somewhat more compact way to write the Cauchy-Riemann equations). This complex structure extends linearly to $T_p\mathbb{C}^n\otimes\mathbb{C}$, and then, for example, $$\begin{align}J\frac{\partial}{\partial z^i}&=\frac {1}{2}J\left(\frac{\partial}{\partial x^i}-\sqrt{-1}\frac{\partial}{\partial y^i}\right)\\&=\frac{1}{2}\left(\frac{\partial}{\partial y^i}+\sqrt{-1}\frac{\partial}{\partial x^i}\right)\\&=\sqrt{-1}\frac{\partial}{\partial z^i}\;.\end{align}$$
The complexified space $T_p\mathbb{C}^n\otimes\mathbb{C}$ decomposes as the direct sum $$\begin{align}T_p\mathbb{C}^n\otimes\mathbb{C}&=\mathrm{span}\left(\frac{\partial}{\partial z^1},\ldots,\frac{\partial}{\partial z^n}\right)\bigoplus\mathrm{span}\left(\frac{\partial}{\partial \overline{z}^1},\ldots,\frac{\partial}{\partial \overline{z}^n}\right)\\&=:T_p^{1,0}\mathbb{C}^n\oplus T_p^{0,1}\mathbb{C}^n.\end{align}$$ Also, the real tangent space $T_p\mathbb{C}^n$ is naturally embedded in its complexification by $v\mapsto v\otimes1.$ Then, every $v\in T_p\mathbb{C}^n$ can be uniquely expressed as the sum of its holomorphic and anti-holomorphic parts; $$v=v^{1,0}+v^{0,1}.$$ For example, $$\frac{\partial}{\partial x^i}=\frac{\partial}{\partial z^i}+\frac{\partial}{\partial \overline{z}^i}.$$ Finally, the map $$\pi:T_p\mathbb{C}^n\to T_p^{1,0}\mathbb{C}^n,\quad v\mapsto v^{1,0},$$ is an isomorphism of complex vector spaces. It may be worth pointing out that $T_p^{1,0}\mathbb{C}^n$ and $T_p^{0,1}\mathbb{C}^n$ are the $\sqrt{-1}$ and $-\sqrt{-1}$ eigenspaces of $J$, respectively. In other words, $T_p^{1,0}\mathbb{C}^n\subset T_p\mathbb{C}^n\otimes\mathbb{C}$ is the subspace on which both complex structures, $J$ and multiplication by $\sqrt{-1},$ coincide. This means that for a holomorphic function $$f:\Omega\subset\mathbb{C}^n\to\mathbb{C}$$ the spaces $T_p\mathbb{C}^n$ and $T_p^{1,0}\mathbb{C}^n$ coincide as spaces of derivations; if $f$ is holomorphic, then we have $$X(f)=X^{1,0}(f)$$ for any (real) vector field.