Yum-Tong Siu, Harvard University: Hyperbolicity and holomorphic jet differentials

Yum-Tong Siu, Harvard University: Hyperbolicity and holomorphic jet differentials
For function theory the hyperbolicity problem seeks conditions for a compact complex manifold to admit no nonconstant holomorphic map from C. The corresponding problem in number theory is for the set of rational points to be finite or contained in some proper subvariety.
Nonexistence of nonconstant entire curves comes from sufficiently independent holomorphic jet differentials vanishing on some ample divisor. For a complex submanifold of an abelian variety such jet differentials are constructed from the position-forgetting map of the jet space of the submanifold to conclude hyperbolicity unless the submanifold is invariant under some linear translation. For complex hypersurfaces of the complex projective space, hyperbolicity for a generic hypersurface of sufficiently high degree follows from the method of vertical jet differentials on the universal complex hypersurface.
In this talk we will discuss the reduction of the lower bound for the degree of a generic n-dimensional hypersurface to be hyperbolic from the best known bound of the order of (n log n)n to polynomial order in n, possibly even to quadratic in n, by combining techniques from the abelian variety setting and the complex projective space setting. The analogy between hyperbolicity in function theory and in number theory will be considered. For abelian varieties differentiation in jet differentials in function theory is replaced by the difference map in number theory, but similar implementation for the hypersurface setting is not yet clear.