Thresholds, valuations, and K-stability

Abstract

Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter generalizes a notion by Fujita and Odaka, and can be used to characterize when a Q-Fano variety is K-semistable or uniformly K-stable. It can also be used to generalize volume bounds due to Fujita and Liu. The two thresholds can be written as infima of certain functionals on the space of valuations on X. When L is ample, we prove that these infima are attained. In the toric case, toric valuations acheive these infima, and we obtain simple expressions for the two thresholds in terms of the moment polytope of L.arXiv:1706.04548v1 [math.AG] 14 Jun 2017for all positive integers p, q. This, together with the Noetherianity of X, implies that {J (X, (c/p) · a p )} p∈N has a unique maximal element that is called the c-th asymptotic multiplier ideal and denoted by J (X, c · a • ). Note that J (X, c · a • ) = J (X, (c/p) · a p ) for all p divisible enough.Asymptotic multiplier ideals also satisfy a subadditivity property. See [Laz04, Theorem 11.2.3] for the case when X is smooth.Corollary 5.6. Let a • be a graded sequence of ideals on X. If m ∈ N * and c ∈ Q * + , thenNext we give a containment relation for the multiplier ideal associated to the graded sequence of valuation ideals. The result appears in [ELS03] in the case when v is divisorial.Proof. It is an immediate consequence of the valuative criterion for membership in the multiplier ideal [BdFFU15, Theorem1.2] that J (X, c · a • (v)) ⊂ a cv(a•(v))−A(v) (v).Since v(a • (v)) = 1 (see [Blu16b, Lemma 3.5]), the proof is complete.

Uniqueness of K-polystable degenerations of Fano varieties

“…Then in a series of papers [Fuj16, Fuj19a, Fuj18] of K. Fujita, all divisorial valuations were studied and an invariant β was defined for each divisorial valuation. After [Li17], it became more natural to extend the setup to all valuations over the log Fano variety rather than only considering divisorial valuations (see also [LX16,BJ17]). Moreover, a characterization of K-stability notions in terms of the sign of β-invariant for divisorial valuations was proved in [Li17,Fuj19a] and lead to another characterization by the δ-invariant in [FO18,BJ17].…”

“…After [Li17], it became more natural to extend the setup to all valuations over the log Fano variety rather than only considering divisorial valuations (see also [LX16,BJ17]). Moreover, a characterization of K-stability notions in terms of the sign of β-invariant for divisorial valuations was proved in [Li17,Fuj19a] and lead to another characterization by the δ-invariant in [FO18,BJ17]. These interpretations of K-stability using valuations have made it easier to apply techniques from birational geometry, especially the Minimal Model Program, to the study of K-stability.…”

“…In Section 2, we will have a short discussion on the above materials. More precisely, we will provide information on the language of valuations and filtrations following [BHJ17,Fuj19b,Li17] and the invariants β and δ associated to them following [Fuj19b,FO18,BJ17]. We also discuss the normalized volume function from [Li18] and its relation with the K-stability of Fano varieties (see [Li17,LX16]).…”

“…Next, is the δ-invariant of (X, ∆), which, as defined in [FO18], measures log canonical thresholds of a certain classes of anti-log canonical divisors of (X, ∆). It is shown in [BJ17] that…”

Some Recent Developments in Kähler Geometry and Exceptional Holonomy

“…• Fujita and Odaka [43] defined a numerical invariant δ(X) of a Fano manifold and conjectured that uniform K-stability is equivalent to δ(X) > 1, which was later confirmed by Blum and Jonsson [5]. This invariant δ is obtained from the theory of the log canonical threshold (lct) (which in this context is a numerical invariant measuring how singular a divisor is).…”