Families of rationally simply connected varieties over surfaces and   torsors for semisimple groups
  Under suitable hypotheses, we prove that a form of a projective homogeneous variety $G/P$ defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre's Conjecture II in Galois cohomology for function fields over an algebraically closed field. 
A. J. de Jong, Xuhua He, Jason Michael Starr