Special test configuration and K-stability of Fano varieties

Chi Li, Chenyang Xu

Research output: Contribution to journalArticlepeer-review

75 Scopus citations

Abstract

For any flat projective family (X;L) C such that the generic bre Xη is a klt Q-Fano variety and L| ~Q -K, 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 conguration of a xed Fano variety (X,-KX), 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.

Original languageEnglish (US)
Pages (from-to)197-232
Number of pages36
JournalAnnals of Mathematics
Volume180
Issue number1
DOIs
StatePublished - 2014
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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