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

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

2011-11-23

arxiv.org/abs/1111.5398v2

Mathematics

Algebraic Geometry

26 pages. The restriction that Picard number of the Fano variety equals one is removed by proving general facts on the Q-Fano

Scientific paper

For any flat projective family $(\mX,\mL)->C$ such that the generic fibre $\mX_\eta$ is a klt $Q$-Fano variety and $\mL|_{\mX_\eta}=-rK_{\eta}$, 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 the Donaldson-Futaki intersection numbers of the appearing models decrease. When the family is a test configuration of a fixed Fano variety $(X,-rK_X)$, this implies Tian's conjecture: given $X$ a Fano manifold, to test its K-(semi)stability, we only need to test on the special test configurations.

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