I am reading the proof of Newlander-Nerenberg's theorem in the real analytic case, and there are some parts I don't understand, can someone help me please?
1) In the beginning, she said: " since everything is local, we may assume that X is an open set U of $\mathbb R^{2n}$". I think in the end, she proves that U is diffeomorphic to $\mathbb C^n$, and the operator I agrees with the complex structure on U. I don't understand why it's enough? do we need to also show that the open sets are also glued together by holomorphic functions?
2) It is not clear to me why the pullback of the holomorphic tangent bundle $T_{U_{\mathbb C}}$ along $U \rightarrow U_{\mathbb C}$ is the complexified tangent bundle $T_{{U,\mathbb R}} \otimes \mathbb C$
3) She defines $E_{\mathbb C}$ = eigenspace of the operator I on $T_{U_{\mathbb C}}$, and she says along U, $E_{\mathbb C}$ is the same as $T_U^{0,1}$. I don't understand why then $E_{\mathbb C}$ satisfies the condition for the Frobenius theorem, ie $[E_C, E_C] \subseteq E_C$. The argument is as follow: "the sections of $T_{U}^{0,1}$ are generated by vector fields $\chi + i I \chi$ where $\chi$ is a real vector field on U, and sections of $E_C$ on $U_C$ are generated by $\chi + iI\chi$ where $\chi$ is a real or complex vector field on $U_C$. Hence it follows immediately. " What is the relationship between vector fields on $U$ and $U_C$ here ? What is the definition of vector fields on $U_C$?
