We have $p: E \to X$ an holomorphic vector bundle,where $X$ is a complex manifold and $f: Y \to X$ where $Y$ is anothet complex manifold. We can build up $f^* E$ pullback bundle on $Y$. Now,is it true that the sheaf of sections of $f^*E$ is isomorphic to the pullback sheaf of sections of $E$? (To be clear , $f$ induces naturally a morphism of the ringed spaces $(Y,\mathcal{O}_Y)$ and $(X,\mathcal{O}_X)$,so given $E$ identified with the $\mathcal{O}_X$ modules of its section, we can make $\mathcal{O}_Y \otimes_{f^{-1}\mathcal{O}_X} f^{-1}E$ pullack in the sheaf theoretic sense and for me it is not even clear why this is a locally free sheaf of the right rank).

  • $\begingroup$ As far as I know, this is true for the smooth case, but not for the holomorphic case. $\endgroup$ – Tim Apr 15 at 22:09

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