K-semistability is equivariant volume minimization

Abstract

This is a continuation to the paper [Li15a] in which a problem of minimizing normalized volumes over Q-Gorenstein klt singularities was proposed. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler-Einstein metrics on Fano varieties. In particular, we prove that for a Q-Fano variety V , the K-semistability of (V, −K V ) is equivalent to the condition that the normalized volume is minimized at the canonical valuation ord V among all C * -invariant valuations on the cone associated to any positive Cartier multiple of −K V . In this case, it's shown that ord V is the unique minimizer among all C * -invariant quasi-monomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over V .Definition 2.5 ([Ber15, Fuj15b]).1. Let (V, L)/C 1 be a normal semi-test configuration of (V, r −1 K −1 V ) and (V,L)/P 1 be its natural compactification. Let D (V,L) be the Qdivisor on V satisfying the following conditions: L) is a Z-divisor corresponding to the divisorial sheafL(r −1 KV /P 1 ). The Ding invariantDing invariant Ding(V, L) of (V, L)/C is defined as:3. V is called Ding semistable if Ding(V, L) ≥ 0 for any normal test configuration (V, L)/C of (V, −r −1 K V ).

Uniqueness of K-polystable degenerations of Fano varieties

“…The equivalence of the above definition with the original definitions was addressed in [Fuj19a,Fuj19b,Li17] and the arguments rely on the special degeneration theory of [LX14]. In Corollary 4.2, we will show that the wordy dreamy may be removed from Definition 2.4.2.…”

“…In the proof of Theorem 1.1, we will need a more general version of (6). The more general formula follows from the original argument in [Li17]. Let (X, ∆) be an n-dimensional log Fano pair and r ∈ Z >0 so that L := −r(K X +∆) is Cartier.…”

“…where A X,∆ (E) is the log discrepancy of E. This invariant was defined in [Fuj18] and the K-(semi)stability of (X, ∆) can be phrased in terms of the positivity of β X,∆ (E) [Fuj19a,Li17].…”

Volume minimization and obstructions to solving some problems in Kähler geometry

“…Odaka observed the decrease of the Donaldson-Futaki invariant along the minimal model program using the concavity of the volume functional. Li [38], [39] considered normalized volume functional on the space of valuations on Fano manifolds and characterized K-semistabilty in terms of volume minimization. Note that when a Sasakian manifold is the circle bundle of an ample line bundle L over M , then the Reeb vector field defines a valuation of the ring ⊕ ∞ k=0 H 0 (M, L k ).…”

K-semistable Fano manifolds with the smallest alpha invariant

“…This is an application of Birkar's answer to Tian's question [Bir16, Theorem 1.5], and Fujita-Li's criterion for K-semistability [Li15,Fuj16b]. It is natural to ask whether the same statement holds true for K-semistable Q-Fano varieties instead of manifolds.…”