Reductivity of the automorphism group of K-polystable Fano varieties

@article{Alper2019ReductivityOT,
  title={Reductivity of the automorphism group of K-polystable Fano varieties},
  author={Jarod Alper and Harold Blum and Daniel Halpern-Leistner and Xu Chen},
  journal={arXiv: Algebraic Geometry},
  year={2019}
}
We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $\Theta$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space. 
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