The Calabi conjecture and K-stability

TL;DR

This paper proves K-stability for certain Calabi-Yau and canonically polarized varieties with mild singularities, showing that some stable varieties are K-stable but not asymptotically stable, with implications for orbifold metrics.

Contribution

It provides an algebraic proof of K-stability for classes of varieties with singularities, clarifying the relationship between K-stability and asymptotic stability.

Findings

Stable varieties are K-stable despite not being asymptotically stable.

Counterexamples to the conjecture that K-stability implies asymptotic stability.

Orbifold varieties with Kahler-Einstein metrics are K-stable.

Abstract

We algebraically prove K-stability of polarized Calabi-Yau varieties and canonically polarized varieties with mild singularities. In particular, the} "stable varieties" introduced by Kollar-Shepherd-Barron and Alexeev, which form compact moduli space, are proven to be K-stable although it is well known that they are \textit{not} necessarily asymptotically (semi)stable. As a consequence, we have orbifold counterexamples, to the folklore conjecture "K-stability implies asymptotic stability". They have Kahler-Einstein (orbifold) metrics so the result of Donaldson does not hold for orbifolds.