Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

By A. J. Jong, Xuhua He, J. Starr. 2008

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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.

Subfields: Geometry and Topology, Mathematical Physics

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