Rational simple connectedness and weak approximation -- Jason Starr

This is joint work with A. J. de Jong. Rational simple connectedness is a property of projective, complex manifolds that is to simple connectedness as rational connectedness is to path connectedness. I will explain the basic ideas and show that positivity conditions on the first two Chern classes imply rational simple connectedness (at least if the cohomology ring is simple). In particular, smooth, low degree complete intersections are rationally simply connected. Together with an argument of Hassett's, this proves "weak approximation" for low degree complete intersections over the function field of a curve: power series solutions are arbitrarily approximable by polynomial solutions.