A minimizing valuation is quasi-monomial
Annals of Mathematics, 2019
111 citations recorded by the source corpus.
We prove a version of Jonsson-Mustaţǎ's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li's conjecture that a minimizer of the normalized volume function is always quasi-monomial. Applying our techniques to a family of klt singularities, we show that the volume of klt singularities is a constructible function. As a corollary, we prove that in a family of klt log Fano pairs, the K-semistable ones form a Zariski open set. Together with [Jia17], we conclude that all K-semistable klt Fano varieties with a fixed dimension and volume are parametrized by an Artin stack of finite type, which then admits a separated good moduli space by [BX18, ABHLX19], whose geometric points parametrize K-polystable klt Fano varieties.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/8ed2de06dbdc11c200356ab9d799e92a427e21e1
Citations in the corpus
- Openness of K-semistability for Fano varieties (2019, methodology): “We note that [Xu19] and the current paper have a similar strategy via the local-to-global principle.”
- Finite generation for valuations computing stability thresholds and applications to K-stability (2021, methodology): “By [BJ20, Proposition 4.8(ii)] and [Xu20, Theorem 1.1], the valuation v is quasi-monomial.”
- On positivity of the CM line bundle on K-moduli spaces (2019, methodology): “In fact, the boundedness result of [Jia17] can be replaced by the corresponding result in the log version (see [Che18,LLX18]) and other main technical results in [BX19,ABHLX19,BLX19,Xu19] are already established for log pairs.”
- Optimal destabilizing centers and equivariant K-stability (2020, background): “Combinedwith [8,43], this also gives an algebraic proof that a general Fano hypersurface is K-semistable (in fact K-stable if the degree is at least 3).”