Grauert's Oka principle for vector bundles

Remarks

Grauert proved the theorem in 1958, for holomorphic principal bundles over Stein spaces whose structure group is a complex Lie group. Vector bundles of rank $r$ are the case of the group $GL_r(\C)$. For line bundles it is the Oka principle of 1939, which follows from the exponential sequence. For higher rank the group $GL_r(\C)$ is not commutative, and the proof needs a new, nonlinear method. Grauert deformed continuous transition functions to holomorphic ones by successive approximation over an exhaustion of $X$, using a Runge-type approximation for maps into complex Lie groups. Example: on $\C^n$, which is contractible, every topological complex vector bundle is trivial, so every holomorphic vector bundle on $\C^n$ is holomorphically trivial. The Stein hypothesis is needed. On the projective line, the rank $2$ bundle $\mathcal{O}(1) \oplus \mathcal{O}(-1)$ and the trivial bundle of rank $2$ are topologically isomorphic, since both have first Chern class $0$. They are not holomorphically isomorphic, since the first has a nonzero holomorphic section that vanishes at a point.

Going deeper

For a student

The theorem reduces the classification of holomorphic vector bundles on a Stein manifold to homotopy theory. Bundles of rank $r$ correspond to homotopy classes of maps from $X$ to the classifying space of $GL_r(\C)$. It is the starting point of modern Oka theory. This theory asks for which target manifolds $Y$ the same principle holds for holomorphic maps from Stein manifolds to $Y$; these targets are the Oka manifolds.