Existence of valuations with smallest normalized volume
By H. Blum. Compositio Mathematica, 2016
70 citations recorded by the source corpus.
Li introduced the normalized volume of a valuation due to its relation to K-semistability. He conjectured that over a Kawamata log terminal (klt) singularity there exists a valuation with smallest normalized volume. We prove this conjecture and give an explicit example to show that such a valuation need not be divisorial.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/1cf4f5122134c6d8b07e132753ade0eec48ecac1
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Uniqueness of the minimizer of the normalized volume function (2020, methodology): “been a lot of progress on the solution of different parts of the Stable Degeneration Conjecture in [Blu18,Li17,LX18,LX16,Xu20]. In particular, it has been known that a minimizing valuation exists (see [Blu18]) and it is always quasi-monomial (see [Xu20]). 1.1. Main Theorems. In this paper, we aim to ”
- A Guided Tour to Normalized Volume (2018, methodology): “o algebrize the work in [DS17] without invoking the metric, but it is also of a completely local nature. Since then, the investigation on this local stability theory points to different directions. In [Blu18], the existence of a minimum (opposed to only infimum) which was conjectured in [Li18a] is affir”
- A non-Archimedean approach to K-stability (2018, background): “The cone point is a klt singularity, and there has been much recent activity on the study of general klt singularities [Blu16, LL16, LX17, BL18], but we shall not discuss this further here.”
- K-stability of cubic threefolds (2017, background): “e the volume of a klt singularity x∈ Xto be vol(cx,X) = defn minvol(cv) for all v∈ ValX,x. We remark that in [Li18] it was show the infimum of the right hand side exists as a positive number; later in [Blu18], the existence of a minimum is confirmed. There are two other different ways to characterize t”
- Algebraicity of the metric tangent cones and equivariant K-stability (2018, background): “ck ·ordS k →v0 and vol(ordc S k ) →vol(cv) as k →+∞. Moreover, If v0 is divisorial, then v0 is given by a K-semistable Kolla´r component. In the above theorem, when v0 is a divisorial minimizer, then [Blu18] also shows it yields a Koll´ar component. In the case of K-semistable log Fano cones, the ap”
- Thresholds, valuations, and K-stability (2017, methodology): “of a valuation vcentered at a closed point on Xas vol(cv) := A(v)nvol(v) when A(v) <1, and vol(cv) := 1 when A(v) = 1. This is a homogeneous function of degree 0 on Val X. The rst author proved in [Blu18a] that for any closed point ˘2X, the normalized volume function restricted to valuations cent”
- K-stability of Fano varieties: an algebro-geometric approach (2020, background): “y many works, and by now only one part remains to be open (see Conjecture 4.21). In [Li17,LL19,LX20], the cone singularity case is thoroughly investigated. The existence of the minimizer is proved in [Blu18], then in [Xu20a] it is shown that they are all quasi-monomial and in [XZ20b] the uniqueness ”