Non-algebraicity of Deligne--Hitchin twistor spaces

Source abstract

In this paper, we prove that on a smooth complex projective curve of positive genus, the Deligne--Hitchin twistor spaces for the structure groups $\mathrm{GL}(n,\mathbb{C})\ (n\geq1)$ and $\mathrm{SL}(n,\mathbb{C})\ (n\geq2)$ admit no algebraization by a complex scheme locally of finite type. When the genus is at least two, the same holds for their stable loci and hence for the corresponding Hitchin twistor spaces.