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Given a Riemann surface $S$, e.g. $\mathbb{P}^1(\mathbb{C})$, with complex conjugation on the coordinates and a holomorphic vector bundle $E$ over $S$.

The complex conjugation $f$ is not holomorphic, but $C^{\infty}$. So we can look at $f^{*}E$ as a smooth real vector bundle. Since $f$ is an involution $E$ and $f^{*}E$ are isomorphic as smooth real bundles (this is most likely wrong). But somehow there should be more structure on $f^{*}E$. I mean $f$ is antiholomorphic, that should be useful somehow.

Is there a way to define a (canonical?) complex structure on $f^{*}E$, such that for any point $p$ we have an antilinear isomorphism $E_{p} \rightarrow f^{*}E_{p}=E_{\overline{p}}f^{*}E_{p}=E_{\overline{p}}$?

I think we don't get a linear iso, since the complex conjugation itself is only antilinear.

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Given a Riemann surface $S$, e.g. $\mathbb{P}^1(\mathbb{C})$, with complex conjugation on the coordinates and a holomorphic vector bundle $E$ over $S$.

The complex conjugation $f$ is not holomorphic, but $C^{\infty}$. So we can look at $f^{*}E$ as a smooth real vector bundle. Since $f$ is an involution $E$ and $f^{*}E$ are isomorphic as smooth real bundles (this is most certainly likely wrong). But somehow there should be more structure on $f^{*}E$. I mean $f$ is antiholomorphic, that should be useful somehow.

Is there a way to define a (canonical?) complex structure on $f^{*}E$, such that for any point $p$ we have an antilinear isomorphism $E_{p} \rightarrow f^{*}E_{p}=E_{\overline{p}}? I think we don't get a linear iso, since the complex conjugation itself is only antilinear. 2 added 41 characters in body Given a Riemann surface$S$, e.g.$\mathbb{P}^1(\mathbb{C})$, with complex conjugation on the coordinates and a holomorphic vector bundle$E$over$S$. The complex conjugation$f$is not holomorphic, but$C^{\infty}$. So we can look at$f^{*}E$as a smooth real vector bundle. Since$f$is an involution$E$and$f^{*}E$are isomorphic as smooth real bundles (this is most certainly wrong). But somehow there should be more structure on$f^{*}E$. I mean$f$is antiholomorphic, that should be useful somehow. Is there a way to define a (canonical?) complex structure on$f^{*}E$, such that for any point$p$we have an antilinear isomorphism$E_{p} \rightarrow f^{*}E_{p}=E_{\overline{p}}?

I think we don't get a linear iso, since the complex conjugation itself is only antilinear.

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