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 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.

16 pages, no figure. v2: minor revision, v3: title changed, re-editted (the latter half moved to arXiv:0807.1716 v4)

References in corpus (14)

• A simply connected surface of general type with p_g=0 and K^2=2
• K-stability of constant scalar curvature polarization
• Log canonical singularities and complete moduli of stable pairs
• A stronger concept of K-stability
• Moduli spaces $M_{g,n}(W)$ for surfaces
• An example of asymptotically Chow unstable manifolds with constant scalar curvature
• Stable varieties with a twist
• Examples of non-symmetric Kähler-Einstein toric Fano manifolds
• Quotients by finite equivalence relations
• Semi log resolution
• Two examples of surfaces with normal crossing singularities
• On Vorontsov's theorem on K3 surfaces
• Constructions of singular rational surfaces of Picard number one with ample canonical divisor
• K3 surfaces with non-symplectic automorphisms of prime order

Cited by in corpus (2)

• Alpha invariant and K-stability of Q-Fano varieties
• On the limit of spectral measures associated to a test configuration