Geodesic
Special test configuration and K-stability of Fano varieties
Chi Li1 ; Chenyang Xu2
1 Princeton University, Princeton, NJ
2 Beijing International Center of Mathematics Research, Beijing 100871, China
Annals of mathematics, Tome 180 (2014) no. 1, pp. 197-232.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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.
DOI : 10.4007/annals.2014.180.1.4

Chi Li 1 ; Chenyang Xu 2

1 Princeton University, Princeton, NJ
2 Beijing International Center of Mathematics Research, Beijing 100871, China 
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Chi Li; Chenyang Xu. Special test configuration  and K-stability of Fano varieties. Annals of mathematics, Tome 180 (2014) no. 1, pp. 197-232. doi : 10.4007/annals.2014.180.1.4. http://geodesic.mathdoc.fr/articles/10.4007/annals.2014.180.1.4/

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