Separatedness of moduli of K-stable varieties

Given a one parameter flat family of polarized algebraic varieties, we show that any K-stable limit is unique. In particular, moduli spaces of K-stable polarized varieties are automatically Hausdorff when they exist. We also give a characterization of K-stable limits in terms of the CM line bundle, and some applications to moduli. Our methods work for arbitrary projective schemes in any characteristic.