Non-algebraicity of Deligne--Hitchin twistor spaces

In this paper, we prove that on a smooth complex projective curve of positive genus, the Deligne–Hitchin twistor spaces for the structure groups $\mathrm{G}\mathrm{L}(n,\mathbb{C})(n\ge 1)$ and $\mathrm{S}\mathrm{L}(n,\mathbb{C})(n\ge 2)$ 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.