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)|<C_N} 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

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