AI 中文总结
本文证明了正亏格光滑复射影曲线上，GL(n,C) 和 SL(n,C) 的 Deligne--Hitchin 扭曲空间及其稳定轨迹（亏格≥2时）均无局部有限型复概形代数化。
AI 中文摘要
本文证明了在正亏格的平滑复射影曲线上，对于结构群 $\mathrm{GL}(n,\mathbb{C})\\ (n\geq1)$ 和 $\mathrm{SL}(n,\mathbb{C})\\ (n\geq2)$，Deligne--Hitchin 扭曲空间不存在由局部有限型复概形给出的代数化。当亏格至少为 2 时，该结论同样适用于其稳定轨迹，从而也适用于相应的 Hitchin 扭曲空间。
英文摘要
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. Nevertheless, the Deligne--Hitchin twistor spaces admit $\mathbb{R}_{\mathrm{alg}}$-definable complex analytic structures when the rank or the genus is one.
Comments12 pages. Added Section 4 on definable structures when the rank or the genus is one. Comments are welcome!
