# 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).

Proof: To prove the theorem, it suffices to check that any holomorphically varying isomorphism ${f_b:F\rightarrow F}$ must be constant (here ${b\in B}$ is the varying parameter). This will imply that the gluing maps are constant.

Next, it suffices to check this for small disks ${{\mathbb D}\in B}$, so consider a holomorphic map ${f:{\mathbb D}\times F\rightarrow F}$, such that ${f(0,-):F\rightarrow F}$ is a biholomorphism. By applying this same biholomorphism to both sides, we can assume ${f(0,-)=id:F\rightarrow F}$, so it suffices to check ${f(-,p):{\mathbb D}\rightarrow F}$ is constant, for every ${p\in F}$ (knowing ${f(0,p)=p}$). Finally, observe that we can define ${f_k:{\mathbb D}\times F\rightarrow F}$ by ${f_k(x,p)=f(x,f_{k-1}(x,p))}$, with ${f_1=f}$, in other words iterating ${f}$.

Two facts will finish the proof. The first is a property of hyperbolic manifolds: for a local chart ${U}$ around ${p\in F}$ and any natural ${N}$, there exists a constant ${C_N}$ such that ${|\partial^N \phi(0)| for any holomorphic map ${\phi:{\mathbb D}\rightarrow F}$. The second is that the map ${f_k(-,p):{\mathbb D}\rightarrow F}$ has growing (in ${k}$) derivative.

Indeed, suppose in local coordinates

$\displaystyle f(z,w_1,w_2,\ldots,w_n)= w_1 + w_2 + \ldots + w_n + c_i z^i\cdot p_i(w_1,\ldots,w_n) + \cdots$

Then in local coordinates, we have

$\displaystyle f_k(z,w_1,w_2,\ldots,w_n)= w_1 + w_2 + \ldots + w_n + k\cdot c_i z^i \cdot p_i(w_1,\ldots,w_n) + \cdots$

Here, the index ${i}$ is the first non-zero term in the ${z}$-expansion. This implies that the ${z}$-derivative vanishes to all orders, thus the map must be constant. $\Box$