Special test configuration and K-stability of Fano varieties

For any flat projective family (X;L) C such that the generic bre X_η is a klt Q-Fano variety and L|_Xη ~Q -K_Xη, 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 conguration of a xed 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.