Brody's lemma

Remarks

Robert Brody proved the theorem in 1978. Proof idea: if a compact $X$ is not hyperbolic, then there are holomorphic maps $f_k \colon \mathbb{D} \to X$ with $\norm{f_k'(0)} \to \infty$. The reparametrisation lemma recentres and rescales each $f_k$ at a point where $(1 - \abs{t}^2) \norm{f_k'(t)}$ is nearly maximal, which gives maps $g_k$ from discs of radii $R_k \to \infty$ with $\norm{g_k'(0)} = 1$ and $\norm{g_k'(t)} \le R_k^2 / (R_k^2 - \abs{t}^2)$. These maps are equicontinuous, and $X$ is compact, so by the Arzelà–Ascoli theorem a subsequence converges locally uniformly to the required $g$. Part 2 follows: a hyperbolic manifold has no entire curve because $k_\C \equiv 0$, and part 1 gives the converse. Example: a smooth plane curve of degree $d \ge 4$ has genus at least $3$, so it has no entire curve, and by the theorem it is hyperbolic. Compactness is necessary: there are noncompact complex manifolds that contain no entire curve but are not Kobayashi hyperbolic.

Going deeper

For a student

The lemma turns hyperbolicity, a metric property, into a statement about entire curves, which is much easier to test and to deform. It implies that hyperbolicity is an open condition in families of compact complex manifolds: a limit of entire curves in nearby fibres gives an entire curve in the limit fibre. It is the starting point of the strategy for the Kobayashi conjecture. There one shows that entire curves in a generic hypersurface must satisfy algebraic differential equations, and then that no such curves exist.