# Holomorphic line bundle and complex line bundle

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.

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.
• 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? Aug 23, 2022 at 6:45
• 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. Aug 23, 2022 at 6:47