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 δ-invariants ([FO, BJ]) nor the L²-normalized Donaldson-Futaki invariants. We use the local normalized volume of [Li] and the higher Θ-stable reduction instead.