Rational points of rationally simply connected varieties over global function fields

TL;DR

This paper proves that certain algebraic varieties over global function fields have rational points if they deform to rationally simply connected varieties, leading to new proofs of major conjectures in the field.

Contribution

It establishes a link between deformation to rationally simply connected varieties and the existence of rational points over global function fields, providing new proofs of key theorems.

Findings

Proves existence of rational points under deformation conditions

Provides uniform proofs of Period-Index Theorem over these fields

Confirms cases of Serre's Conjecture II and Lang's property

Abstract

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.