Positivity of the CM line bundle for families of K-stable klt Fano varieties

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.