Stable degeneration conjecture for klt singularities

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Stable degeneration conjecture for klt singularities

   Let $x\in (X=\operatorname{Spec}(R),D)$ be an arbitrary klt singularity. A valuation $v$ minimizing the normalized volume is considered, and write

   $$R_0=\operatorname{gr}_v(R).$$

   Let $\xi_v$ be the induced Reeb vector. Stable degeneration conjecture. There is a unique minimizer $v$ up to rescaling; moreover, $v$ is quasi-monomial, $R_0$ is finitely generated, and the induced degeneration

   $$(X_0=\operatorname{Spec}(R_0),D_0,\xi_v)$$

   is a K-semistable Fano cone singularity. This conjecture is the proposed structure theorem for minimizers of normalized volume and is presented as a central open problem, although important special cases are known.

   source: Chi Li, Yuchen Liu and Chenyang Xu, “A Guided Tour to Normalized Volume”, arXiv:1806.07112 (2019).

2. The stable degeneration conjecture for klt singularities

   Let $Y$ be a klt singularity, let $x\in Y={\rm Spec}(R)$ be a closed point, and let $v$ be a minimizer of the normalized volume function $\widehat{\rm vol}_{Y,x}$ on the non-archimedean link of $(Y,x)$. Let $R_0:={\rm gr}_vR$.

   Stable degeneration conjecture. The associated graded ring $R_0$ is finitely generated.

   The source identifies this as the main remaining part of the stable degeneration conjecture after existence and uniqueness, up to rescaling, of a quasi-monomial normalized-volume minimizer. Finite generation is expected to produce the stable degeneration associated with the singularity.

   source: Chenyang Xu, “K-stability of Fano varieties: an algebro-geometric approach”, arXiv:2011.10477 (2020).