The ‘moduli continuity method’ permits an explicit algebraisation of the Gromov–Hausdorff compactification of Kähler–Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the ‘log setting’ to describe explicitly some compact moduli spaces of K‐polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval sufficiently close to one, by considering constructions arising from geometric invariant theory (GIT). More precisely, we discuss the cases of pairs given by cubic surfaces with anticanonical sections, and of projective space with non‐Fano hypersurfaces, and we show ampleness of the CM line bundle on their good moduli space (in the sense of Alper). Finally, we introduce a conjecture relating K‐stability (and degenerations) of log pairs formed by a fixed Fano variety and pluri‐anticanonical sections to certain natural GIT quotients.

中文翻译：

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模连续性方法在记录K稳定对中的应用

“模连续性方法”允许显式在一些基本示例中，对Fano流形上的Kähler-Einstein度量的Gromov-Hausdorff压缩进行了代数化。在本文中，我们将这种方法应用于“对数设置”中，以明确描述K-多稳态对数Fano对的紧凑模空间。通过考虑由几何不变理论（GIT）引起的构造，我们关注奇异角在足够接近一个间隔的情况下受到扰动的情况。更确切地说，我们讨论了具有反规范截面的三次曲面和具有非范诺超曲面的射影空间给定的对的情况，并且我们展示了CM线束在其良好的模空间上的充裕度（在Alper的意义上）。最后，