Holomorphic bundles with hyperbolic fibers

This brief note was inspired by this MathOverflow post. A definition of a Kobayashi hyperbolic manifold is available here. Informally, it says that one can’t holomorphically map arbitrarily large complex unit disks into the complex manifold. In a certain sense “most” complex manifolds are Kobayashi-hyperbolic. Below is the proof of the following theorem of Royden, proved in this paper.

Theorem 1 Suppose {E\rightarrow B} is a holomorphic fibration, with fiber a Kobayashi-hyperbolic manifold {F}. Then the fibration is locally flat, i.e. the gluing maps on overlapping local charts must be (locally) constant.

Remark 1 The requirement that the fibration be holomorphic means that locally it is holomorphically isomorphic to {U\times F}. For example, if {{\mathbb B}=\{(z,w)\in {\mathbb C}^2 \rvert |z|^2+|w|^2<1\}} is the unit ball in {{\mathbb C}^2}, the natural projection {{\mathbb B}\rightarrow {\mathbb D}} to the unit disk in {{\mathbb C}} is not a holomorphically trivial fibration. One can see this using the maximum principle (see the MO post for details).

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