Special test configuration and K-stability of Fano varieties

Abstract

For any flat projective family (X , L) → C such that the generic fibre Xη is a klt Q-Fano variety and L|X η ∼ Q -KX 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, -KX ), 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. ContentsPart 1. Family of Fano Varieties CHI LI and CHENYANG XU Part 2. Application to KE Problem 218 7. Introduction: K-stability from Kähler-Einstein problem 218 8. Donaldson-Futaki invariant and K-stability 222 8.1.Intersection formula for the Donaldson-Futaki invariant 222 8.2.Normal test configuration 225 9. Proof of Theorem 7 226 9.1.Equivariant Semi-stable reduction 227 9.2.Proof of Tian's conjecture 227 References 228This article is motivated by studying Tian's conjecture, which says that to test K-(semi, poly)stability we only need to consider the test configurations whose special fibers are Q-Fano varieties.It consists of two parts.In the first part, inspired by Tian's conjecture, we obtain our main result on the existence of special degenerations of Fano varieties.In the second part, we apply the result from the first part to study K-stability of Fano varieties.In particular, we give an affirmative answer to Tian's conjecture.

Locations

• Annals of Mathematics - View - PDF