Special test configurations and $K$-stability of Fano varieties

Abstract

For any flat projective family $(\mX,\mL)\rightarrow C$ such that the generic fibre $\mX_η$ is a klt Q-Fano variety and $\mL|_{\mX_η}\sim_{Q}-K_{X_η}$, we use the techniques from the minimal model program (MMP) to modify the total family. The end product is a family such that every fiber is a klt Q-Fano variety. Moreover, we can prove that the Donaldson-Futaki invariants of the appearing models decrease. When the family is a test configuration of a fixed Fano variety $(X,-K_X)$, this implies Tian's conjecture: given $X$ a Fano manifold, to test its K-(semi, poly)stability, we only need to test on the special test configurations.

v3: Final version. To appear Annals of Mathematics. v2: 26 pages. The restriction that Picard number of the Fano variety equals one is removed by proving general facts on the Q-Fano degeneration of a (punctured) family of Q-Fano varieties. So a full version of Tian's conjecture is proved. The calculation on the decreasing of Donaldson-Futaki intersection number is streamlined

References in corpus (4)

• K-stability and Kähler-Einstein metrics
• Moduli of varieties of general type
• CM Stability and the Generalized Futaki Invariant I
• Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 π

Cited by in corpus (21)

• K-stability and Kähler-Einstein metrics
• Finiteness of algebraic fundamental groups
• K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
• Algebraicity of the Metric Tangent Cones and Equivariant K-stability
• On the moduli of Kahler-Einstein Fano manifolds
• Conical Kahler-Einstein metric revisited
• Compact Moduli Spaces of Del Pezzo Surfaces and Kähler-Einstein metrics
• Kahler-Einstein metrics and stability
• A note on the definition of K-stability
• Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds
• Extremal Kähler metrics
• The dual complex of singularities
• Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
• Testing log K-stability by blowing up formalism
• Torus equivariant K-stability
• On the limit of spectral measures associated to a test configuration
• K-stability implies CM-stability
• Alpha invariants and K-stability for general polarisations of Fano varieties
• On parametrization, optimization and triviality of test configurations
• Canonical Kahler metrics and Arithmetics -- Generalising Faltings heights
• Separatedness of moduli of K-stable varieties