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

TL;DR

This paper constructs proper moduli spaces for Calabi-Yau cones and Sasaki-Einstein manifolds with singularities, providing an algebraic approach that avoids previous invariants and uses local volume and stability techniques.

Contribution

It introduces a new algebraic construction of proper K-moduli of Q-Fano varieties and Calabi-Yau cones without relying on delta-invariants or Donaldson-Futaki invariants.

Findings

Constructed proper moduli spaces of K-polystable Q-Fano cones.

Provided an alternative algebraic proof of properness.

Utilized local normalized volume and higher Theta-stability methods.

Abstract

We construct proper moduli algebraic spaces of K-polystable $\mathbb{Q}$-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 $\mathbb{Q}$-Fano varieties. In contrast to the previous algebraic proof of its properness ([BHLLX, LXZ]), we do not use the $\delta$-invariants ([FO, BJ]) nor the $L^2$-normalized Donaldson-Futaki invariants. We use the local normalized volume of [Li] and the higher $\Theta$-stable reduction instead.