RIGID ANALYTIC PICARD THEOREMS By WILLIAM CHERRY and MIN RU
نویسنده
چکیده
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, which says there are no nonconstant rigid analytic maps from the affine line to nonsingular projective curves of positive genus, and of Cherry’s result 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. 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 nonnegative 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 nonsingular 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.
منابع مشابه
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, ...
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* It is my pleasure to acknowledge the generous financial support provided by Academia Sinica and National Central University which made these lectures possible. These lectures are largely based on Min Ru's concise exposition of the subject in [Ru].
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