Uniqueness of birational structures on Inoue surfaces

TL;DR

This paper proves the uniqueness of the birational structure on Inoue surfaces, extending previous results about affine and projective structures, and establishing that the natural affine structure is the sole birational structure.

Contribution

It demonstrates that the natural birational structure on Inoue surfaces is unique, generalizing earlier results about affine and projective structures on complex surfaces.

Findings

The natural $( ext{Aff}_2( ext{C}), ext{C}^2)$-structure on Inoue surfaces is unique.

This structure is the only $( ext{Bir}( ext{P}^2), ext{P}^2( ext{C}))$-structure on these surfaces.

The result extends Klingler's theorem from projective to birational structures.

Abstract

We prove that the natural $(\operatorname{Aff} _2(\mathbf{C}),\mathbf{C}^2)$-structure on an Inoue surface is the unique $(\operatorname{Bir}(\mathbb{P}^2),\mathbb{P}^2(\mathbf{C}))$-structure, generalizing a result of Bruno Klingler which asserts that the natural $(\operatorname{Aff}_2(\mathbf{C}),\mathbf{C}^2)$-structure is the unique $(\operatorname{PGL} _3(\mathbf{C}),\mathbb{P}^2(\mathbf{C}))$-structure.