How to interpret the cotangent bundle of a complex manifold?
Let $X$ be a complex manifold. I am not sure what people mean when they talk about the cotangent bundle $T^*X$ of $X$. I have two interpretations:
- At each point $x\in X$, $T_x^*X$ is the complex vector space dual to the complex vector space $T_xX$, i.e. $T_x^*X$ is the space of all complex-linear maps $T_xX\to\Bbb C$.
- At each point $x\in X$, $T_x^*X$ is the dual space to the real vector space $T_xX$, i.e. the space of all real-linear maps $T_xX\to\Bbb R$.
Which one is the right interpretation?
Thinking: I was first thinking that there is a natural isomorphism between the two, but it doesn't seem like so. If I try to get a real isomorphism $$(T_xX)_{\Bbb R}^*\to (T_xX)^*_{\Bbb C},$$ where the first space are the real-linear maps $T_xX\to\Bbb R$ and the second one are the complex linear maps $T_xX\to\Bbb C$ (but viewed as a real vector space), then the isomorphism is always basis dependent.
Note that $$\dim_{\Bbb R}(T_xX)^*_{\Bbb R}=\dim_{\Bbb R} T_xX=2\dim_{\Bbb C}T_xX=2\dim_{\Bbb C}(T_xX)^*_{\Bbb C}=\dim_{\Bbb R}(T_xX)^*_{\Bbb C},$$ so the two spaces are indeed isomorphic. Although I cannot find a basis-independent isomorphism.
If $V$ is a vector space over $\mathbb{C}$, there is a natural $\mathbb{R}$-isomorphism $\varphi\colon V_{\mathbb{R}}^*\to V_{\mathbb{C}}^*$ defined by $$ \varphi(f)(v) \,=\, f(v) + i\,f(-iv). $$ for $f\in V_{\mathbb{R}}^*$ and $v\in V$, with inverse $\varphi^{-1}\colon V_{\mathbb{C}}^*\to V_{\mathbb{R}}^*$ defined by $$ \varphi^{-1}(g)(v) \,=\, \mathrm{Re}\bigl(g(v)\bigr). $$ Note that if $f\colon V \to \mathbb{R}$ is $\mathbb{R}$-linear, then $\varphi(f)$ is indeed $\mathbb{C}$-linear, since $$ \varphi(f)(iv) \,=\, f(iv) + i f(v) \,=\, if(v) - f(-iv) \,=\, i\bigl(f(v) + if(-iv)\bigr) \,=\, i\,\varphi(f)(v) $$ for any $v\in V$.