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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