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Let $M$ be a complex manifold, with $\mathcal{O}$ be sheaf of holomorphic function and $\mathcal{A}$ be sheaf of smooth function.

Therefore there is canonical inclusion from $$\mathcal{O}^\times \to \mathcal{A}^\times$$

Then it will induce a map between :

$$H^1(X,\mathcal{O}^\times) \to H^1(X,\mathcal{A}^\times)$$

By the Cech resolution we know $H^1(X,\mathcal{O}^\times)$ isomorphic to group of holomorphic line bundle, and $H^1(X,\mathcal{A}^\times)$ isomorphic to group of smooth line bundle. I was confused how to map a holomorphic bundle (with complex rank 1) to smooth line bundle (with real rank 1). This map should maps holomorphic line bundle to complex line bundle , however complex line bundle is real rank 2.

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1 Answer 1

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In order to have an inclusion $\mathcal{O}\to\mathcal{A}$, you have to take $\mathcal{A}$ to be the sheaf of complex valued smooth functions. Then $\mathcal{A}^\times$ is the sheaf of smooth functions with values in $GL(1,\mathbb C)$ and hence $H^1(X,\mathcal{A}^\times)$ classifies complex line bundles.

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  • $\begingroup$ thank you , Does this complex line bundle has something to do with complex structure on $X$ ? $H^1(X,\mathcal{A}^\times) $ know nothing about complex structure of $X$ correct? $\endgroup$
    – yi li
    Aug 23, 2022 at 6:45
  • $\begingroup$ There is no relation to the complex structure on $X$, one just looks at bundles whose fibers are 1-dimensional complex vector spaces and whose transition functions are complex linear in each fiber. $\endgroup$ Aug 23, 2022 at 6:47
  • $\begingroup$ clear now thank you. $\endgroup$
    – yi li
    Aug 23, 2022 at 6:48

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