holomorphic line bundle over germ of complex space

Question

I wonder if holomorphic line bundle over zero of holomorphic function(over $\mathbb{C}^n$) is trivial?(We can assume it to be a manifold if necessary)

I Know there is a principle that, for Stein manifolds, we can reduce the classification of holomorphic bundle to the classification of topological bundle. But I have no idea to obtain any information about the topological data.

Thank you for your answer!

*Some backgrounds:

We say a space is complex if it is locally the zero set of holomorphic function over $\mathbb{C}^n$. We define the germ of space $(T, t_0)$ as the equivalent class of those spaces $(S, s_0)$ such that there exist two isomorphic neighborhood $U$ of $s_0$ and $V$ of $t_0$.

The book I read state something like: holomorphic line bundles over germ of complex space are holomorphically trivial.

The book doesn't give a explicitly definition about bundle over germ of space, so I understand the statement above as: holomorphic line bundles over zero of holomorphic function is holomorphic trivial.

Answer 1

1. Every real-analytic variety is homeomorphic to the geometric realization of a simplicial complex, see for instance

Łojasiewicz, Stanisław, Triangulation of semi-analytic sets, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 18, 449-474 (1964). ZBL0128.17101.

In particular, if $A\subset {\mathbb C}^n$ is a complex-analytic subvariety, then every point $p\in A$ has a contractible neighborhood in $A$.

2. Now, to your question: Suppose that $V\to A$ is a holomorphic vector bundle. By the definition (I am taking here the weakest definition possible), this bundle is locally trivial as a smooth vector bundle. Since $A$ is locally a Stein space, you can now quote Oka-Grauert principle,

Grauert, Hans, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135, 263-273 (1958). ZBL0081.07401.

This principle implies that if a holomorphic vector bundle over a Stein space is smoothly trivial, then it is holomorphically trivial. Thus, holomorphic vector bundles over analytic varieties are locally holomorphically trivial.