Rigid Analytic Picard Theorems

We prove a geometric logarithmic derivative lemma for rigid analytic mappings to algebraic varieties in characteristic zero. We use the lemma to give a new and simpler proof (at least in characteristic zero) of Berkovich’s little Picard theorem [Ber, Theorem 4.5.1], which says there are no nonconstant rigid analytic maps from the affine line to non-singular projective curves of positive genus, and of Cherry’s result [Ch 1] that there are no nonconstant rigid analytic maps from the affine line to Abelian varieties. Furthermore, we use the lemma to prove new theorems of little and big Picard type for dominant mappings, in close analogy with Griffiths and King [GK]. For the little Picard type theorem, we prove that if X is a smooth projective variety with a simple normal crossings divisor D such that (X, D) has non-negative logarithmic Kodaira dimension, then there are no dominant rigid analytic maps f from Am to X \ D. For the big Picard type theorem, we prove that if Y is a non-singular rigid analytic space, E is an effective simple normal crossings divisor on Y, and if X is a smooth projective variety with a simple normal crossings divisor D such that (X, D) is of log-general type, then any dominant rigid analytic map f : Y \ E → X \D extends to an analytic map from Y to X.