  A complex projective manifold is rationally connected, resp. rationally
simply connected, if finite subsets are connected by a rational curve, resp.
the spaces parameterizing these connecting rational curves are themselves
rationally connected. We prove that a projective scheme over a global function
field with vanishing "elementary obstruction" has a rational point if it
deforms to a rationally simply connected variety in characteristic 0. This
gives new, uniform proofs over these fields of the Period-Index Theorem, the
quasi-split case of Serre's "Conjecture II", and Lang's $C_2$ property.
