Special test configuration and K-stability of Fano varieties

For any flat projective family (X , L) C such that the generic fibre X is a klt Q-Fano variety and L|X Q -KX , 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.