Non-archimedean Hyperbolicity

A complex manifold X is said to be hyperbolic (in the sense of Brody) if every analytic map from the complex plane C to X is constant. From Picard’s “little” theorem, an entire function missing more than two values must be constant. It is equivalent to say that P \ {0, 1,∞} is hyperbolic. Picard’s theorem also show that a Riemann surface of genus one omitting one point and Riemann surfaces of genus at least 2 are hyperbolic. The higher dimensional case is a much more difficult problem. It was proved by Siu and Yeung [20] that an abelian variety omitting a very ample divisor is hyperbolic. A deep conjecture is that complex manifolds of general type are hyperbolic. It was also conjectured by Kobayashi [16] and Zaidenberg [22] that the complements of “generic” hypersurfaces in P with degree at least 2n + 1 are hyperbolic. There have been many results related to this conjecture. We will only mention a few here. This conjecture was verified by Green [14] in the case of 2n + 1 hyperplanes in general position. More generally, Babets [4], Eremenko-Sodin [12] and Ru [18] independently showed that P \{ 2n+ 1 hypersurfaces in general position} is hyperbolic. When n = 2, the conjecture is correct for the case of four generic curves (cf.[10]). For the case of three generic curves C1, C2, C3, Dethloff, Schmacher, and Wong ([10], [11]) showed that P \ ∪i=1Ci is hyperbolic if degCi ≥ 2 for i = 1, 2, 3. When one of the Ci is a line they show that any holomorphic map f : C→ P \ ∪i=1Ci is algebraically degenerate if, up to enumeration, d1 = 1, d2 ≥ 3 and d3 ≥ 4. Similar questions can be asked in the case of non-archimedean ground fields. Let K be an algebraically closed field of arbitrary characteristic, complete with respect to a non-archimedean absolute value | |. A variety X over K is said to be K-hyperbolic if every analytic map from K to X is constant. In contrast to the situation over the complex numbers, it is much easier to study hyperbolic problems over non-archimedean ground fields. For example, an non-archimedean entire function with no zero is constant, i.e., P \ {0,∞} is hyperbolic. The