The Calabi Conjecture and K-stability

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 Kollár–Shepherd-Barron [“Threefolds and deformation of surface singularities.” Inventiones Mathematicae 91 (1988): 299–338] and Alexeev [“Moduli spaces Mg, n(W) for surfaces.” Proceeding of “Higher-dimensional complex varieties (Trento, 1994)”, de Gruyter, Berlin, 1996, 1–22], which form compact moduli space, are proven to be K-stable although it is well known that they are not necessarily asymptotically (semi) stable. As a consequence, we have orbifold counterexamples to the folklore conjecture “K-stability implies asymptotic stability”. They have Kähler–Einstein (orbifold) metrics, so the result of Donaldson [“Scalar curvature and projective embeddings. I.” Journal of Differential Geometry 59, no. 3 (2001): 479–522] does not hold for orbifolds.