Rationally simply connected varieties and pseudo algebraically closed fields

TL;DR

This paper proves Serre's Conjecture II for certain pseudo algebraically closed fields of cohomological degree 2, demonstrating their rational simple connectedness, $C_2$-property, and Period-Index equality in Brauer groups.

Contribution

It introduces a novel method using rational simple connectedness to establish Serre's Conjecture II and related properties for PAC fields of cohomological degree 2.

Findings

Proves Serre's Conjecture II for PAC fields of cohomological degree 2 in characteristic 0 or with roots of unity.

Shows such fields are $C_2$-fields, satisfying certain cohomological properties.

Establishes Period equals Index for Brauer groups of these fields.

Abstract

The cohomological dimension of a field is the largest degree with non-vanishing Galois cohomology. Serre's "Conjecture II" predicts that for every perfect field of cohomological dimension $2$, every torsor over the field for a semisimple, simply connected algebraic group is trivial. A field is perfect and "pseudo algebraically closed" (PAC) if every geometrically irreducible curve over the field has a rational point. These have cohomological dimension $1$. Every transcendence degree $1$ extension of such a field has cohomological degree $2$. We prove Serre's "Conjecture II" for such fields of cohomological degree $2$ provided either the field is of characteristic $0$ or the field contains primitive roots of unity for all orders $n$ prime to the characteristic. The method uses "rational simple connectedness" in an essential way. With the same method, we prove that such fields are $C_2$-fields, and we prove that "Period equals Index" for the Brauer groups of such fields. Finally, we use a similar method to reprove and extend a theorem of Fried-Jarden: every perfect PAC field of positive characteristic is $C_2$