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


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 \(\Theta \)-stratification on the moduli stack of Fano varieties.

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    This definition differs from that in [4, 22] by a sign convention to conform to the convention in the K-stability literature that non-negativity of the Futaki invariant corresponds to semistability.

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    The current arXiv version of [22] states a weaker version of Theorem 8.3 that applies to real valued numerical invariants, and also includes the condition of uniqueness of HN filtrations. The theorem we discuss here appears in an update of [22] that is not yet available on the arXiv.


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The authors thank Jarod Alper and Jochen Heinloth for helpful conversations. HB was partially supported by NSF Grant DMS-1803102. DHL was partially supported by a Simons Foundation Collaboration grant and NSF CAREER Grant DMS-1945478. YL was partially supported by NSF Grant DMS-2001317. CX was partially supported by NSF Grant DMS-1901849.

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Correspondence to Harold Blum.

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Blum, H., Halpern-Leistner, D., Liu, Y. et al. On properness of K-moduli spaces and optimal degenerations of Fano varieties. Sel. Math. New Ser. 27, 73 (2021).

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  • Fano varieties
  • K-stability
  • Moduli

Mathematics Subject Classification

  • 14D20
  • 14J45