I have two definitions of "Complex Tangent Space" over a Complex Manifold $M$ of (complex) dimension $n$. One of them is defining, over the real tangent space $T_p(M)$, the complex structure $J_p :T_p(M) \rightarrow T_p(M)$ given by
$$J_p \left(\frac {\partial}{\partial x_i}|_p \right) = \frac {\partial}{\partial y_i}|_p \quad ; \quad J_p \left(\frac {\partial}{\partial y_i}|_p \right) = -\frac {\partial}{\partial x_i}|_p$$
Where $\{ \partial/\partial x_i |_p, \partial/\partial y_i|_p\}_{i=1,...,n}$ is a (real) basis in $T_p(M)$ induced by a complex chart. It's easy to show that this, indeed, is a complex structure and does not depend on the choise of complex chart. This way, the tangent space is now a complex vector space of dimension $n$ and we can think of the "Complex Tangent Bundle" as a complex vector bundle over $M$ with fiber $T_p(M)$ (seen as a complex vector space) at each $p \in M$.
Another way to define the complex tangent space is by doing a "mimic" with the definition of the real tangent space. We define $H_p(M)$ as the space of pairs $(f,U)$, beeing $U \subseteq M$ open with $p \in U$ and $f :U \rightarrow \Bbb C$ holomorphic, quotient by the equivalence relation $$(f,U) \sim (g,V) \iff \exists \ W \subseteq U \cap V \ \ \mathrm{open} \ \mathrm{with} \ \ p\in W \ \ \mathrm{and} \ \ f|_W =g|_W$$ We define the holomorphic tangent space at $p\in M$ as the set of $\Bbb C$-linear functions $X_p :H_p(M) \rightarrow \Bbb C$ that satisfy the Leibniz Rule (we will write this space as $T_p^h (M)$). It can be shown that this is a complex vector space of dimension $n$. We will name "Holomorphic Tangent Bundle" the complex vector bundle over $M$ with fiber $T_p^h(M)$ at each $p \in M$. If $f : M \rightarrow N$ is an holomorphic function between complex manifolds, and $p \in M$, we define $\hat d f_p :T_p^h(M) \rightarrow T_p^h(N)$ by $$\hat d f_p (X_p) ([h]) := X_p(h \ \mathrm{o} \ f)$$
So, we defined the "Complex Tangent Bundle" $T(M)$ and the "Holomorphic Tangent Bundle" $T^h(M)$. My question is
- Can we find, for each complex manifold $M$, a natural isomorphism of complex vector bundles $F^M : T(M) \rightarrow T^h(M)$ such that, if $F_p^M :T_p(M) \rightarrow T_p^h(M)$ is the induced $\Bbb C$-linear isomorphism for each $p \in M$, then $\hat d f_p \ \mathrm{o} \ F_p^M = F_{f(p)}^N \ \mathrm{o} \ df_p$ for every holomorphic function $f : M \rightarrow N $ ? (I don't know how to draw conmutative diagrams)
If we can accomplish this, then I will be convinced that this two definitions are "Equivalent".