On properness of K-moduli spaces and optimal degenerations of Fano varieties
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Author(s)
Blum, Harold
Halpern-Leistner, Daniel
Liu, Yuchen
Xu, Chenyang
Date Issued
July 28, 2021
Publisher
Springer International Publishing
Citation
Selecta Mathematica. 2021 Jul 28;27(4):73
Version
Author's final manuscript
Abstract
Abstract
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
$$\Theta $$
Θ
-stratification on the moduli stack of Fano varieties.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00029-021-00694-7
