Special test configurations and $K$-stability of Fano varieties

TL;DR

This paper uses the minimal model program to modify families of Fano varieties, showing that special test configurations suffice to determine K-stability, thereby confirming Tian's conjecture.

Contribution

It introduces a MMP-based method to simplify test configurations for Fano varieties, proving that only special configurations need testing for K-stability.

Findings

Modified families have fibers that are all klt Q-Fano varieties.

Donaldson-Futaki invariants decrease under the modifications.

Tian's conjecture on K-stability testing is confirmed for Fano manifolds.

Abstract

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