Special test configuration and K-stability of Fano varieties

Abstract

For any flat projective family $(\mathcal{X},\mathcal{L})\rightarrow C$ such that the generic fibre $\mathcal{X}_\eta$ is a klt $\mathbb{Q}$-Fano variety and $\mathcal{L}|_{\mathcal{X}_\eta}\sim_{\mathbb{Q}}-K_{X_{\eta}}$, 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 $\mathbb{Q}$-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.