Compact moduli of Calabi-Yau cones and Sasaki-Einstein spaces

We construct proper moduli algebraic spaces of K-polystable Q-Fano cones (a. k. a. Calabi-Yau cones) or equivalently their links i. e. , Sasaki-Einstein manifolds with singularities. As a byproduct, it gives alternative algebraic construction of proper K-moduli of Q-Fano varieties. In contrast to the previous algebraic proof of its properness (BHLLX, LXZ), we do not use the -invariants (FO, BJ) nor the L²-normalized Donaldson-Futaki invariants. We use the local normalized volume of Li and the higher -stable reduction instead.