The Little Picard Theorem says that a complex analytic function defined
everywhere on \(\bf C\), can miss at most one complex value. Its standard
proofs are all based on the fact that \(\bf C\) minus two points is
hyperbolic (in the sense of negative curvature as is the case of the unit
disk). The higher dimensional generalization of hyperbolicity, at least in
the birational context, is that of general type (almost everywhere negative
curvature). We will define the opposite notion to that of general type,
that of being special, and discuss our result that any object defined by
complex polynomials (a variety) \(X\) has a decomposition as a fiber space
over a base object of general type whose fibers are special. A conjectural
generalization of the Little Picard theorem would then be that there exist
an entire function with values in \(X\) not contained in any subvariety in
\(X\) if and only if \(X\) is special. We will conclude by our verification
of the conjecture for \(X\) that is of maximal albanese dimension, which is
the case for \(\bf C\) minus two points. This is joint work with Jorg
Winkelmann.
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Last modified: April 11 2016 - 18:14:43