We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable Fano varieties. Specifically, we prove that if the stability threshold of every K-unstable Fano variety is computed by a divisorial valuation, then such K-moduli spaces are proper. The argument relies on studying certain optimal destabilizing test configurations and constructing a \(\Theta \)-stratification on the moduli stack of Fano varieties.

中文翻译：

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关于 K-模空间的适当性和 Fano 变种的最优退化

我们建立了一种代数方法来证明 K-polystable Fano 变体模空间的适当性，并将问题简化为对 K-unstable Fano 变体不稳定的猜想。具体来说，我们证明如果每个 K 不稳定 Fano 变体的稳定性阈值都是通过除数估值计算的，那么这样的 K 模空间是合适的。该论点依赖于研究某些最佳不稳定测试配置并在 Fano 变体的模数堆栈上构建\(\Theta \) -分层。