# The complex structure on $T\mathbb C^n$

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 $$\{\partial/\partial z_1\dots\partial/\partial z_n \}$$?

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.