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dg.differential geometry – Is the complexification of the conclusion of a holomorphic bundle quiet holomorphic? Answer

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dg.differential geometry – Is the complexification of the conclusion of a holomorphic bundle quiet holomorphic?

Let $mathcal{E}=(E,barrier{partial}_E)$ breathe a holomorphic vector bundle over a complicated manifold $X$ with rk$_mathbb{C}(E)=okay$. Here $E$ is the underlying complicated vector bundle and $barrier{partial}_E$ is the integrable operator inducing the holomorphic construction on $E$. Now deem its realization $E_mathbb{R}$ as a actual vector bundle such that rk$_mathbb{R}(E_mathbb{R})=2k$. Now take its complexification $(E_mathbb{R})^mathbb{C}$ and it turns into a complicated vector bundle such that rk$_mathbb{C}huge((E_mathbb{R})^mathbb{C}huge)=2k$

Question 1

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

Let’s attempt with an instance: Take $mathcal{E}=(mathcal{T}_X, barrier{partial})$ to breathe the holomorphic tangent bundle of $X$, then $(mathcal{T}_X)_mathbb{R}cong TX$ the place $TX$ is the actual tangent bundle of $X$. Now complexifying $(mathcal{T}_X)_mathbb{R}$ is equal to complexify $TX$. In explicit we secure $T^mathbb{C}Xcong T^{1,0}Xoplus T^{0,1}X$ the place $T^{1,0}Xcongmathcal{T}_X$ is the holomorphic tangent bundle and $T^{0,1}X=overline{T^{1,0}X}$. We resolve that in common $huge((mathcal{T}_X)_mathbb{R}huge)^mathbb{C}cong T^mathbb{C}X$ shouldn’t be holomorphic.

Let’s come advocate to $(E_mathbb{R})^mathbb{C}$. Inspired by the instance I’d say that $(E_mathbb{R})^mathbb{C}cong Eoplus overline{E}$. Now in common $overline{E}$ shouldn’t be holomorphic since if ${g_{alphabeta}}$ are the holomorphic transition features of $E$ then ${overline{g}_{alphabeta}}$ are the anti-holomorphic transition duty of $overline{E}$.

Here comes the ail!

If $mathcal{E}=(E,barrier{partial}_E)$ is holomorphic then $mathcal{E}^*=(E^*, barrier{partial}_{E^*})$ is holomorphic as nicely. Pick a hermitian metric $h$ on $E$, then it induces an isomorphism (as complicated vector bundles) commence{equation*}phi:overline{E}to E^*
aim{equation*}
If I outline $barrier{partial}_{phi}:=phi^{-1}circbar{partial}_{E^*}circphi$ then it appears to me that it induces a holomorphic construction on $overline{E}$.

Question 2

Is the dispute rectify? If so, is there a route to narrate the holomorphic transition features induced by $barrier{partial_phi}$?

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