Colloquium

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.