Special test configuration and K-stability of Fanco varities

For any flat projective family (X,L) -> C such that the generic fibre X-eta is a klt Q-Fanco variety and L vertical bar(X eta) similar to(Q) -K-X eta , we use the techniques from the minimal model program (MMP) to modify family. The end product is a family such that every fiber is a kit Q-Fanco variety. Moreover, we can prove that the Donaldson-Futski invariants of the appearing models decrease. When the family is a test configuration of a fixed Fanco variety (X,-K-X), this implies Tian's conjecture: given X a Fanco manifold, to test its K-(semi, poly)stability, we only need to test on the special test configurations.