# 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) $$begin{equation*}phi:overline{E}to E^* end{equation*}$$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 $$bar{partial_phi}$$?