Local-global principles for homogeneous spaces over some two-dimensional geometric global fields

TL;DR

This paper investigates the failure of the local-global principle for certain homogeneous spaces over two-dimensional geometric global fields, introducing a new obstruction that improves upon the classical Brauer-Manin obstruction.

Contribution

It constructs a new torsor-based obstruction that explains failures not accounted for by the Brauer-Manin obstruction, and compares it with the descent obstruction.

Findings

The classical Brauer-Manin obstruction is insufficient in these cases.

A new torsor-based obstruction successfully explains some failures.

Comparison with descent obstruction reveals new insights.

Abstract

In this article, we study the obstructions to the local-global principle for homogeneous spaces with connected or abelian stabilizers over finite extensions of the field $\mathbb{C}((x,y))$ of Laurent series in two variables over the complex numbers and over function fields of curves over $\mathbb{C}((t))$. We give examples that prove that the usual Brauer-Manin obstruction is not enough to explain the failure of the local-global principle, and we then construct a variant of this obstruction using torsors under quasi-trivial tori which turns out to work. In the end of the article, we compare this new obstruction to the descent obstruction with respect to torsors under tori. For that purpose, we use a result on towers of torsors, that is of independent interest and therefore is proved in a separate appendix.