The Chow–Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. We prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stability only for general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. We also present an application to the classification of Fano varieties. Additionally, our semi-positivity statements work in general for log-Fano pairs.

中文翻译：

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K 稳定 klt Fano 品种家族的 CM 系丛的正性

Chow-Mumford (CM) 线丛是基于任何 klt Fano 变体家族的函子线丛。据推测，它在 K-poly-stable klt Fano 品种的模空间上产生极化。证明 CM 线束的充足性归结为显示关于具有 K 半稳定/K 多稳定纤维的家庭的 CM 线束的半积极/积极陈述。我们证明了 K 半稳定情况下的必要半正性陈述，以及均匀 K 稳定情况下的必要正性陈述，包括在这两种情况下假设 K 稳定性仅适用于一般纤维的变体。我们的陈述适用于最一般的奇异情况（klt 奇异性），并且证明是代数的，除了通过概率论的中心极限定理计算实数序列的极限。我们还介绍了 Fano 品种分类的应用。此外，我们的半正性陈述一般适用于 log-Fano 对。