Rational points of rationally simply connected varieties over global function fields

Starr, Jason; Xu, Chenyang

Annales Henri Lebesgue, Volume 3 (2020), pp. 1399-1417.

For a complex projective manifold that is rationally connected, resp. rationally simply connected, every finite subset is 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 has a rational point if it deforms to a rationally simply connected variety in characteristic $0$ with vanishing elementary obstruction. 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.

Si une variété projective complexe est rationnellement connexe, chaque ensemble fini de points est contenu dans une courbe rationnelle ; si elle est rationnellement simplement connexe, les espaces paramétrant ces courbes rationnelles sont eux-mêmes rationnellement connexes. Nous montrons qu’un schéma projectif sur un corps global de caractéristique non nulle possède un point rationnel s’il se déforme en une variété rationnellement simplement connexe de caractéristique zéro dont l’obstruction élémentaire s’évanouit. Pour de tels corps, on obtient ainsi des preuves uniformes du théorème période-indice, du cas quasi-déployé de la « Conjecture II » de Serre, et de la propriété $C_2$ de Lang.

Classification: 14G05, 14M22, 14D06
Keywords: rationally simply connected varieties, rational points, degenerations

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