dg.differential geometry – Is the complexification of the realization of a holomorphic bundle still holomorphic?

Let $mathcal{E}=(E,bar{partial}_E)$ be a holomorphic vector bundle over a complex manifold $X$ with rk$_mathbb{C}(E)=k$. Here $E$ is the underlying complex vector bundle and $bar{partial}_E$ is the integrable operator inducing the holomorphic structure on $E$. Now consider its realization $E_mathbb{R}$ as a real vector bundle such that rk$_mathbb{R}(E_mathbb{R})=2k$. Now take its complexification $(E_mathbb{R})^mathbb{C}$ and it becomes a complex vector bundle such that rk$_mathbb{C}big((E_mathbb{R})^mathbb{C}big)=2k$

Question 1

When is $(E_mathbb{R})^mathbb{C}$ a holomorphic bundle? Is there a canonical way to induce a “natural” holomorphic structure from the one of $mathcal{E}$?

Let’s try with an example: Take $mathcal{E}=(mathcal{T}_X, bar{partial})$ to be the holomorphic tangent bundle of $X$, then $(mathcal{T}_X)_mathbb{R}cong TX$ where $TX$ is the real tangent bundle of $X$. Now complexifying $(mathcal{T}_X)_mathbb{R}$ is equivalent to complexify $TX$. In particular we obtain $T^mathbb{C}Xcong T^{1,0}Xoplus T^{0,1}X$ where $T^{1,0}Xcongmathcal{T}_X$ is the holomorphic tangent bundle and $T^{0,1}X=overline{T^{1,0}X}$. We conclude that in general $big((mathcal{T}_X)_mathbb{R}big)^mathbb{C}cong T^mathbb{C}X$ is not holomorphic.

Let’s come back to $(E_mathbb{R})^mathbb{C}$. Inspired by the example I would say that $(E_mathbb{R})^mathbb{C}cong Eoplus overline{E}$. Now in general $overline{E}$ is not holomorphic since if ${g_{alphabeta}}$ are the holomorphic transition functions of $E$ then ${overline{g}_{alphabeta}}$ are the anti-holomorphic transition function of $overline{E}$.

Here comes the trouble!

If $mathcal{E}=(E,bar{partial}_E)$ is holomorphic then $mathcal{E}^*=(E^*, bar{partial}_{E^*})$ is holomorphic as well. Pick a hermitian metric $h$ on $E$, then it induces an isomorphism (as complex vector bundles) begin{equation*}phi:overline{E}to E^*
If I define $bar{partial}_{phi}:=phi^{-1}circbar{partial}_{E^*}circphi$ then it seems to me that it induces a holomorphic structure on $overline{E}$.

Question 2

Is the argument correct? If so, is there a way to describe the holomorphic transition functions induced by $bar{partial_phi}$?