On properness of K-moduli spaces and optimal degenerations of Fano varieties

Harold Blum ^[1] ; Daniel Halpern-Leistner ^[2] ; Yuchen Liu ^[3] ; Chenyang Xu ^[4]
1. [1] Stony Brook University
  
  Stony Brook University
  
  Town of Brookhaven, Estados Unidos
2. [2] Cornell University
  
  Cornell University
  
  City of Ithaca, Estados Unidos
3. [3] Northwestern University
  
  Northwestern University
  
  Township of Evanston, Estados Unidos
4. [4] Princeton University; MIT, USA; Beijing International Center for Mathematical Research, China
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Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 4, 2021
Idioma: inglés
DOI: 10.1007/s00029-021-00694-7
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Resumen
- We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable Fano varieties. Specifically, we prove that if the stability threshold of every K-unstable Fano variety is computed by a divisorial valuation, then such K-moduli spaces are proper. The argument relies on studying certain optimal destabilizing test configurations and constructing a Θ-stratification on the moduli stack of Fano varieties.