# Are holomorphic structures on complex vector bundles unique up to isomorphism?

I'm looking for an example of the following situation:

$$M$$ is a complex manifold, $$E_1, E_2$$ are non-isomorphic holomorphic vector bundles on $$M$$, but $$E_1$$ is isomorphic to $$E_2$$ as a complex smooth vector bundle on $$M$$.

Does such a thing exist, or are holomorphic structures on complex vector bundles unique?

• See this question which asks about the case where the underlying smooth bundle is trivial. Aug 30, 2022 at 10:46
• You've got a good (+1) answer, but in case a concrete example helps, if $E$ is an elliptic curve and we fix a point $p_0$ of $E$, then for each $p$ in $E$ the divisor $[p_0] - [p]$ defines a topologically trivial holomorphic line bundle. This bundle is trivial if and only if $p = p_0$, however, since a non-trivial section of a trivial bundle is non-vanishing. Sep 5, 2022 at 15:35

First, note that $$\mathcal{C}^{\infty}$$ is a soft sheaf, hence acyclic.
It follows that $$H^1(M,(\mathcal{C}^{\infty})^{\times}) \cong H^2(M,\mathbb{Z})$$ for any smooth manifold $$M$$. In particular, if $$M$$ is compact, there are only countably many isomorphism classes of smooth line bundles.
Now assume that $$M$$ is a compact Riemann surface of nonzero genus. Then its Jacobian $$J$$ parametrizes a subset of isomorphic classes of line bundles on $$M$$, and $$J$$ has uncountably many points – so some of them correspond to line bundles that are smoothly isomorphic.