Uniqueness of birational structures on Inoue surfaces

Abstract

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

36 pages