aa r X i v : . [ m a t h . DG ] D ec PLURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS
MINGCHEN XIA
Abstract.
Given a compact polarized manifold (
X, L ), we introduce two new stability thresholds in termsof singularity types of global quasi-plurisubharmonic functions on X . We prove that in the Fano setting,the new invariants can effectively detect the K-stability of X . We study some functionals of geodesic raysin the space of Kähler potentials by means of the corresponding test curves. In particular, we introduce anew entropy functional of quasi-plurisubharmonic functions and relate the radial entropy functional to thisnew entropy functional. Contents
1. Introduction 12. Setup 43. Preliminaries 104. The theory of test curves 135. Intersection theory of b-divisors 186. Radial functionals in terms of Legendre transforms 207. Variational approach on Berkovich spaces 258. Second order expansion of Donaldson’s L -functionals 279. Stability thresholds 2810. Further problems 30Appendix A. Integrals and summations 32References 351. Introduction
Let X be a complex projective manifold of dimension n . Let L be an ample line bundle on X . Fix aKähler form ω ∈ c ( L ). Let V = ( L n ).A central problem in Kähler geometry is to give conditions for the existence of canonical metrics inthe Kähler class [ ω ]. The celebrated Yau–Tian–Donaldson conjecture asserts that the existence of Kähler–Einstein metrics (or more generally cscK metrics) is equivalent to certain algebro-geometric stability condi-tions. Classically, dating back to the work of Donaldson ([Don02]), the algebro-geometric stability notion,known as K-stability has been defined in terms of test configurations. Later on, a stronger condition knownas uniform K-stability is also introduced and studied ([BHJ16], [BHJ17]). It is known that the equivariantversion of uniform K-stability gives a characterization of the existence of Kähler–Einstein metrics, whicheven generalizes to the log Fano setting, see [Li19] and references therein.On the other hand, more recently a valuative approach to K-stability is introduced in [Fuj19], [FO18],[BJ20], which we briefly recall. The δ -invariant of L is defined as(1.1) δ ( L ) := inf E A X ( E ) S L ( E ) , where E runs over the set of prime divisors over X , A X ( E ) denotes the log discrepancy of E and S L ( E )is the expected order of vanishing of L along E . It is well-known that uniform twisted K-stability (resp.twisted K-semistability) is equivalent to δ ( L ) > δ ( L ) ≥ Date : December 23, 2020.
In this paper, we introduce a different stability threshold in terms of the singularity types of quasi-plurisubharmonic functions on X :(1.2) δ pp := inf [ ψ ] R ∞−∞ Ent([ ψ + τ ]) d τnV − R ∞−∞ (cid:16)R X ω ∧ ω n − ψ + τ − R X ω nψ + τ (cid:17) d τ , where [ ψ ] runs over the set of singularity types of quasi-psh functions with some non-zero Lelong numbers on X , ψ + • is a test curve associated to ψ . Recall that a test curve is the Legendre transform of a geodesic ray asin [RWN14]. The test curve ψ + • is the maximal extension of the test curve corresponding to deformation tothe normal cone (see Section 2.6). The quantity Ent[ • ] is an invariant of the singularity types of quasi-pshfunctions (see Definition 2.11). To the best of the author’s knowledge, this invariant has never been studiedin the literature. Observe that the quotient in (1.2) depends only on the singularity type of ψ . Moreover, itis easy to see that the quotient in (1.2) does not change under the rescaling ψ → cψ for c ∈ R > , hence onecould restrict ψ to run only in the set of ω -psh functions.Now we state the main theorem. Theorem 1.1.
Assume that X is Fano and L = − K X . Then δ pp ≥ δ . When δ ≤ , δ = δ pp . We recall that in the Fano setting, when δ ≤ δ is also equal to the greatest Ricci lower bound, seeSection 10.1 for details. We remark that although our theorem concerns only Kähler geometry, our proofrelies essentially on the non-Archimedean tools developed by Boucksom–Jonsson, as we recall later.As a corollary, Corollary 1.2.
Assume that X is Fano and L = − K X . Then(1) δ pp ≥ iff X is K-semistable.(2) δ pp > iff X is uniformly K-stable. Corollary 1.2 integrates into the program of characterizing K-stability in terms of some more explicitdata dating back to [RT07]. In [RT07], Ross–Thomas introduced the notion of slope stability in termsof test configurations associated to deformation to the normal cone, which gives a necessary condition forthe K-stability. Later on, this theory was extended in [Oda13] using flag ideals, in [Wit12], [Szé15] usingfiltrations. In [Fuj19], Fujita gives a characterization of K-stability in terms of all divisorial valuations. Ourresult gives a different characterization in terms of psh singularity types. Our approach can also be seenas a generalization of those of [RT07] in the sense that our definition of δ pp is based on a generalizationof deformation to the normal cone. We also notice that very recently in [DL20b], Dervan–Legendre havepartially extended Fujita’s work to general polarizations and studied the valuative stability.When the stability threshold is less than 1, it is interesting to understand its minimizers. We propose thefollowing conjecture: Conjecture 1.1.
When δ pp ≤ , there is always a minimizer of δ pp .When X is a Fano manifold, L = − K X and when δ < , the pluricomplex Green function G in the senseof [MT19] is a minimizer of δ pp . The first part in the Fano case follows from our proof of the main theorem. For general polarization, itseems difficult.There are some similar results for δ in the more general log Fano variety setting: there is always a quasi-monomial valuation that computes δ ([BLX19]). In the smooth Fano setting, it is even known that there isa divisor computing δ ([DS20], see [BLZ19, Theorem 6.7] for details). The same is conjectured to be true inlog Fano setting ([BLZ19, Conjecture 1.2], [BHLLX20, Conjecture 1.1]).The δ pp -invariant is closely related to δ in the following manner: take an extractable (Definition 2.7) divisor E . One can prove that in this case, there is always an ω -psh function ψ with analytic singularities such thaton a suitable birational model π : Y → X , the singularities of π ∗ ψ are just the hyperplane singularity along E . Recall that E induces a test configuration ( X , L ). Now one can make explicit computations to expressvarious functionals of ( X , L ) in terms of ψ • , the result turns out to be of the form of (1.2). More precisely,we prove that for a test curve induced by a general (semi-ample) test configuration, the non-Archimedeanentropy and ˜ J NA -functionals (the latter is more frequently denoted by I NA − J NA in the literature) areboth integrals of some corresponding functionals of psh singularities along the test curve (see Theorem 7.4,Corollary 7.5, Corollary 6.8). Conversely, given an ω -psh function ψ with analytic singularities, we can LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 3 always take a log resolution so that ψ has singularities along a snc Q -divisor D = P i a i D i . This divisorthen induces a higher rank valuation ( a − i ord D i ) i of C ( X ).We also define a slightly different stability threshold:(1.3) δ ′ := inf ψ ( K Y/X · ( − div Y ψ ) n − ) + n ( G n − ( L, div Y ψ ) · red div Y ψ ) n R (cid:16)R X ω ∧ ω n − τψ − R X ω nτψ (cid:17) d τ , where [ ψ ] runs over the set of singularity types of unbounded ω -psh functions with analytic singularities, π : Y → X is a log resolution of ψ , G n − is a polynomial defined by Equation (A.1) in the appendix, red ofa divisor D is the divisor with the same support as D but with all non-zero coefficients of D set to 1. Notethat the quotient in (1.3) is not invariant under the rescaling ψ cψ ( c ∈ Q > ), hence δ ′ is an invariant of ω -psh functions on X , not an invariant of all quasi-psh functions on X as δ pp is. We also prove that Theorem 1.3.
We always have δ ′ ≥ δ . In general, we do not expect them to be equal even if δ ≤
1. The interesting point is that the invariant δ ′ puts strong constrains on the possible singularities of ω -psh functions. Strategy of the proof
In the discussion, we fix a maximal geodesic ray in the sense of [BBJ15] and its Legendre transform ψ = ˆ ℓ .As discussed above, we need to express various energy functionals of ℓ in terms of ψ • .The part for ˜ J follows from the strategy introduced in [RWN14] and further developed in [DX20]. Seethe proof of Theorem 6.7 for details.The corresponding result for the entropy functional is the main new feature in this paper. As the variationof the entropy functional is not easily controlled, we try to tackle the non-Archimedean counterpart of theentropy at first, namely the non-Archimedean entropy:Ent NA ( φ ) := 1 V Z X an A X MA( φ ) , φ ∈ E , NA . In [Li20], Li showed that Ent NA ( φ ) is dominated by the slope at infinity of the usual entropy functionalEnt along ℓ , where φ is the non-Archimedean potential induced by ℓ . In [DX20], we have expressed thenon-Archimedean Monge–Ampère energy in terms of the test curves. Since the non-Archimedean Monge–Ampère energy is nothing but the primitive function of the Chambert-Loir measure MA( φ ), we get a fortiori a good understanding of MA( φ ). We make use of this description to compute the non-Archimedean entropyfunctional. The result turns out to be an integral of the variation of volumes along ψ • .Recall the potentials ψ τ are all I -model in the sense of [DX20]. In order to compute the variation ofvolumes of an I -model potential, we need to express this volume algebraically. We prove that the volumeof an I -model potential can be realized as certain (movable) intersection number of a b-divisor (in thesense of Shokurov) associated to the singularities of the potential, if the intersection number is properlydefined (see Theorem 5.2). Now we can carry out a purely algebraic computation to get a formula for thenon-Archimedean entropy (See Theorem 7.4, Corollary 7.5).Now it comes to Theorem 1.1 and Theorem 1.3. That these new invariants dominate δ is an easy con-sequence of the formulae of ˜ J and Ent NA . We simply embed the set of ω -psh functions into the set ofmaximal geodesic rays using the deformation to the normal cone like construction. This kind of embeddingwas already studied in [DDNL19b]. For the equality δ = δ pp when δ ≤
1. We make use of the results from[BLZ19], which says that when δ ≤ δ can be computed by a divisor E . Then [BCHM10] allows us toextract the divisor. We can therefore construct an ω -psh ψ whose singularities are exactly given by E . Then ψ minimizes δ pp as well and we conclude that δ = δ pp .We remark that although we have carried out our computations only on smooth complex varieties, it iseasy to generalize most results to normal Kähler varieties. However, due to the lack of a comprehensivereference on the pluripotential theory in this generality, we decide to limit ourselves to the smooth settingin order to keep the present paper at a readable length.Finally, let us explain the philosophy behind this paper. We regard the global pluripotential theory ofsingular metrics on a compact Kähler manifold as a differential version of the theory of geodesic rays in thespace of Kähler potentials. One of the justifications is given by Theorem 6.5 (namely, [DX20, Theorem 1.1]).Similarly, by the computations in this paper and in [DX20], a number of radial invariants of geodesic rays are MINGCHEN XIA in fact an integral along the corresponding test curves of some corresponding invariants defined by quasi-pshfunctions. See Table 1 for more examples.Classically, the K-stability of a polarized manifold is detected by test configurations, valuations, filtrationsand non-Archimedean potentials, which can all be embedded in the space of geodesic rays. By our philosophy,there should be a pluripotential-theoretic counterpart, which leads to the present paper. Note that ourprevious work [DX20] already followed this philosophy.
Organization of the paper
In Section 2, we present a few results necessary for understanding the definition of δ pp .In Section 3 and Section 4, we recall some basic notions in Kähler geometry and pluripotential theory.In Section 5, we recall the notion of Shokurov’s b-divisors and apply it to define the entropy of qpshsingularities.In Section 6 and Section 7, we express several functionals on the space of geodesic rays in terms of thecorresponding test curves.In Section 8, based on the assumption that Donaldson’s L -functionals have second order asymptotic ex-pansion (Conjecture 8.1), we get a partial refinement of the results obtained in Section 7 (see Corollary 8.3).In Section 9, we relate the new δ -invariants to the classical δ -invariant.In Section 10, we propose several further problems.We collect a few messy computations in Appendix A. Conventions
In this paper, all Monge–Ampère type operators are taken in the non-pluripolar sense (see [BEGZ10]).The functional ˜ J defined in (3.1) is usually written as I − J in the literature. Our definition of test curves in Definition 4.1 corresponds to maximal test curves in the literature (see [RWN14] for example). For aquasi-psh function ϕ , we write I ( ϕ ) for Nadel’s multiplier ideal sheaf of ϕ . The d c operator is normalizedso that dd c = i π ∂ ¯ ∂ . The definition of a birational model in Definition 2.3 requires that the model besmooth, hence stronger than the usual definition. When ω is a Kähler form, we adopt the convention that ω −∞ = ω + dd c ( −∞ ) = 0. A snc (strictly normal crossing) divisor is always assumed to be effective. By avaluation of a field, we refer to real valuations unless otherwise specified. We adopt the additive conventionfor valuations.We do not distinguish a holomorphic line bundle and the corresponding invertible sheaf in the analytic category. We use interchangeably additive and multiplicative notations for tensor products of invertiblesheaves.We make use of the results of [BHJ19] in an essential way. We only refer to the latest version on arXiv[BHJ16] with errata instead of the journal version. Acknowledgements
The author would like to thank Robert Berman, Chen Jiang, Kewei Zhang for discussions and TamásDarvas, Sébastien Boucksom, Mattias Jonsson for remarks and suggestions on the draft version of the paper.Part of the paper is used as assignments of the course
GFOK035 Academic Writing at Chalmers TekniskaHögskola. The author would like to thank his classmates Mohammad Farsi and Víctor López Juan for their(non-mathematical) suggestions. 2.
Setup
In this section, we present a minimal amount of preliminaries necessary to understand the definition ofthe new delta invariant (1.2).2.1.
Log resolution of analytic singularities.
Let X be a projective manifold of dimension n . Let L bea big and semi-ample line bundle on X . Let h be a smooth non-negatively curved Hermitian metric on L .Let ω = c ( L, h ). Let PSH(
X, ω ) denote the set of all ω -psh functions on X , namely the set of usc functions ϕ : X → [ −∞ , ∞ ) such that ω + dd c ϕ ≥ Definition 2.1.
A potential ϕ ∈ PSH(
X, ω ) is said to have analytic singularities if for each x ∈ X , there isa neighbourhood U x ⊆ X of x in the Euclidean topology, such that on U x , ϕ = c log N x X j =1 | f j | + ψ , LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 5 where c ∈ Q ≥ , f j are analytic functions on U x , N x ∈ Z > is an integer depending on x , ψ ∈ C ∞ ( U x ). Remark . Note that the definition of analytic singularities in Definition 2.1 is weaker than that in [MM07,Definition 2.3.9].
Definition 2.2.
Let D be an effective snc R -divisor on X . Let D = P i a i D i with D i being prime divisorsand a i ∈ R > . We say that ϕ ∈ PSH(
X, ω ) has analytic singularities along D if locally (in the Euclideantopology), ϕ = X i a i log | s i | h + ψ , where s i is a local section of L that defines D i , ψ is a smooth function.Note that a potential with analytic singularities along a snc Q -divisor has analytic singularities in thesense of Definition 2.1. Definition 2.3. A birational model of X is a projective birational morphism π : Y → X from a smooth projective variety Y to X . Definition 2.4.
Let ϕ ∈ PSH(
X, ω ) be a potential with analytic singularities. Then there is a bira-tional model π : Y → X of X , such that π ∗ ϕ has analytic singularities along a snc Q -divisor (see [MM07,Lemma 2.3.19]). We call any such π a log resolution of ϕ .2.2. I -model potentials. Let X be a compact Kähler manifold of dimension n . Let L be an ample linebundle. Let ω ∈ c ( L ) be a Kähler form. For any quasi-psh function ϕ on X , let I ( ϕ ) denote Nadel’smultiplier ideal sheaf of ϕ , namely, the coherent ideal sheaf on X locally generated by holomorphic functions f such that R | f | e − ϕ ω n < ∞ .The concept of I -model potential is developed in [DX20]. Definition 2.5.
Let ϕ ∈ PSH(
X, ω ). A quasi-equisingular approximation of ϕ is a sequence ϕ j of potentialsin PSH( X, ω ) with analytic singularities, such that(1) ϕ j converges to ϕ in L .(2) The singularity types of ϕ j are decreasing.(3) For any δ > k >
0, we can find j = j ( δ, k ) >
0, so that for j ≥ j , I ((1 + δ ) kϕ j ) ⊆ I ( kϕ ) . Recall that quasi-equisingular approximations always exist ([Cao14, Lemma 3.2], [DPS01, Theorem 2.2.1]).Recall that a potential ϕ ∈ PSH(
X, ω ) is said to be I -model if ϕ = P [ ϕ ] I := sup* { ψ ∈ PSH(
X, ω ) : ψ ≤ , ψ an ≤ ϕ an } . We use the notation PSH
Model I ( X, ω ) to denote the set of I -model potentials in PSH( X, ω ). Theorem 2.1 ([DX20, Theorem 1.4]) . Let ϕ ∈ PSH
Model ( X, ω ) , R X ω nϕ > . Then the following areequivalent:(1) ϕ ∈ PSH
Model I ( X, ω ) .(2) ϕ = sup* { ψ ∈ PSH(
X, ω ) : ψ ≤ , I ( kψ ) ⊆ I ( kϕ ) for any k ∈ Z > } . (3) lim k →∞ n ! k n h ( X, K X ⊗ L k ⊗ I ( kϕ )) = Z X ω nϕ . (4) For one (or equivalently any) quasi-equisingular approximation ϕ j of ϕ , lim j →∞ Z X ω nϕ j = Z X ω nϕ . Here and in the whole paper, products like ω nϕ are taken in the non-pluripolar sense, see [BEGZ10].For the definition of model potentials, we refer to [DDNL18b]. The set of model potentials in PSH( X, ω ) isdenoted by PSH
Model ( X, ω ). Recall that a model potential with analytic singularities is I -model ([Bon98]). MINGCHEN XIA
Let X div Q denote the set of all Q -divisorial geometric valuation on X . Namely, elements of X div Q are c ord E , where c ∈ Q > , E is a prime divisor over X (i.e. a prime divisor on a birational model of X ). Let ψ ∈ PSH(
X, ω ), recall that ψ an is a function X div Q defined as follows: let v ∈ X div Q , then set(2.1) − v ( ψ ) = ψ an ( v ) := − lim k →∞ k v ( I ( kψ )) . The function ψ an can be extended to the Berkovich analytification X an (see Section 3.3) of X in the obviousway. Lemma 2.2.
Let ϕ ∈ PSH(
X, ω ) . Let ϕ j be a quasi-equisingular approximation of ϕ j . Then ϕ j, an → ϕ an pointwisely on X div Q as j → ∞ .Proof. It suffices to prove that for any prime divisor E over X , ϕ j, an (ord E ) → ϕ an (ord E ). Fix k ∈ Z > , δ ∈ Q > , take j >
0, so that when j > j , I ((1 + δ ) kϕ j ) ⊆ I ( kϕ ). When j > j , we get1 k ord E ( I ( kϕ )) ≤ k ord E ( I ((1 + δ ) kϕ j )) . By Fekete’s lemma, − ϕ j, an (ord E ) = sup k ∈ Z > k ord E ( I ( kϕ j )) . So 1 k ord E ( I ( kϕ )) ≤ (1 + δ )( − ϕ j, an (ord E )) . Take sup with respect to k ∈ Z > , we get − ϕ an (ord E ) ≤ (1 + δ )( − ϕ j, an (ord E )) . Let δ → ϕ an (ord E ) ≥ lim j →∞ ϕ j, an (ord E ) . The converse is trivial. (cid:3)
Singularity divisors.
Let X be a compact Kähler manifold of dimension n . Let L be a semi-ampleline bundle with a smooth non-negatively curved Hermitian metric h . Let ω = c ( L, h ). Definition 2.6.
Let ψ ∈ PSH(
X, ω ). Let π : Y → X be a birational morphism from a normal Q -factorialprojective variety. Define the singularity divisor of ψ on Y asdiv Y ψ := X E ν E ( ψ ) E , where E runs over the set of prime divisors on Y , ν E ( ψ ) is the generic Lelong number of π ∗ ψ along E . Notethat this is a countable sum by Siu’s semi-continuity theorem.Let D be an effective R -divisor on Y . We say that the singularities of ψ are determined on Y by D if forany birational model Π : Z → Y , div Z ψ = Π ∗ D .We can regard div Y ψ as the divisorial part of Siu’s decomposition of dd c π ∗ ψ .As a consequence of resolution of singularities, any potential with analytic singularity admits a modelwhere its singularities are determined ([MM07, Lemma 2.3.19]). Definition 2.7.
Let E be a prime divisor over X . An extraction of E is a proper birational morphism π : Y → X from a normal Q -factorial variety Y , such that E is a prime divisor on Y and that − E is π -ample.If there is an extraction of E , we call E an extractable divisor.Observe that when X is Fano, an extractable divisor E is dreamy in the sense that the doubly gradedalgebra(2.2) M m ∈ Z ≥ M p ∈ Z H ( Y, − mπ ∗ K X − pE )is finitely generated. LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 7
In general, when the log discrepancy of E is well-behaved, one can run a suitable MMP to extract E . See[BCHM10, Corollary 1.4.3], [Kol13, Section 1.4] for details.Assume that L is ample. Let F be an extractable divisor. Let π : Y → X be an extraction of F . We cantake A ∈ Q > large enough, so that Aπ ∗ L − F is semi-ample. In particular, take B large enough, so that B ( Aπ ∗ L − F ) is base-point free. Take a basis s , . . . , s N of H ( X, B ( Aπ ∗ L − F )). Let ψ = 1 AB log max i =1 ,...,N | s j | h AB . Then the singularities of ψ are determined on Y by A − F (see Definition 2.6).2.4. Quasi-analytic singularities.
Let X be a projective manifold of dimension n . Let L be a big andsemi-ample line bundle on X . Let h be a smooth non-negatively curved Hermitian metric on L . Let ω = c ( L, h ). Definition 2.8.
We say a potential ϕ ∈ PSH(
X, ω ) has quasi-analytic singularities if there is a birationalmodel π : Y → X , a snc R -divisor D on Y , such that the singularities of ψ are determined on Y by D (seeDefinition 2.6). In this case, we also say that ϕ has quasi-analytic singularities along D . Lemma 2.3.
Let ϕ ∈ PSH(
X, ω ) be a potential with quasi-analytic singularities along a snc Q -divisor D ona birational model π : Y → X , then I ( kπ ∗ ϕ ) = O Y ( − ⌊ kD ⌋ ) for any k ∈ Q > .Proof. Without loss of generality, we take k = 1. Recall that we have assumed that the model is projective.Take a sufficiently ample line bundle H on Y , so that H − D is semi-ample and H − π ∗ L is ample. Takea m ∈ Z > so that m ( H − D ) is globally generated. Fix a smooth positively curved metric h on H . Let ω ′ := c ( H, h ), we may assume that ω ′ > π ∗ ω . Take a basis s , . . . , s N of H ( Y, m ( H − D )). Let ψ = 1 m log max i =1 ,...,N | s i | h m . Then we know that I ( ψ ) = O Y ( − ⌊ D ⌋ ) . But we know that π ∗ ϕ ∼ I ψ as ω ′ -psh functions, so we conclude. (cid:3) Remark . We rephrase the proof of Lemma 2.3 in fancier terms: LetPSH
Model ( X ) := lim −→ ω PSH
Model ( X, ω ) , where ω runs over all Kähler forms on X , when ω ≤ ω ′ , the map PSH Model ( X, ω ) → PSH
Model ( X, ω ′ ) is givenby the P ω ′ [ • ]. We take the filtered colimit in the category of sets. We define a class [ ϕ ] ∈ PSH
Model ( X ) to beanalytic if some representative is analytic. Now the proof of Lemma 2.3 says that when ϕ ∈ PSH
Model ( X, ω )is quasi-analytic, the class [ ϕ ] ∈ PSH
Model ( X ) is analytic. Lemma 2.4.
Let ϕ ∈ PSH(
X, ω ) be a potential with quasi-analytic singularities along a snc R -divisor D on X , then L − D is nef. If moreover R X ω nϕ > , then L − D is big and nef.Proof. Consider the positive current ω ϕ − [ D ] in c ( L − D ). Take a quasi-equisingular approximation h j of ω ϕ − [ D ] ([Cao14]). The Lelong number condition and the fact that h j has analytic singularities show thatits local potential is in fact bounded. Hence L − D is nef. Now the assumption R X ω nϕ > L − D ) n >
0, hence L − D is big ([DP04]). (cid:3) Non-archimedean envelopes.
Let X be a compact Kähler manifold of dimension n . Let L be anample line bundle with a smooth strictly positively curved metric h . Let ω = c ( L, h ).Let v = ( v , . . . , v m ) be a valuation of C ( X ) with value in R m . We assume for simplicity that each v i isdivisorial. MINGCHEN XIA
Definition 2.9.
Let a = ( a , . . . , a m ) ∈ R m ≥ . Define a potential in PSH( X, ω ) ∪ {−∞} :(2.3) ψ v ≥ a := sup* { ψ ∈ PSH(
X, ω ) : ψ ≤ , v i ( ψ ) ≥ a i for i = 1 , . . . , m } . We also define(2.4) ψ ′ v ≥ a := sup* k ∈ Z > k sup* (cid:26) log | s | h k : s ∈ H ( X, L k ) , sup X | s | h k ≤ , v i ( s ) ≥ ka i for i = 1 , . . . , m (cid:27) . Observe that ψ v ≥ a itself is a candidate in the sup in (2.3), provided that ψ v ≥ a = −∞ . Obviously, ψ v ≥ a is either I -model or −∞ . Lemma 2.5.
Assume that ψ v ≥ a has positive mass, then P [ ψ ′ v ≥ a ] I = ψ v ≥ a .Proof. That P [ ψ ′ v ≥ a ] I ≤ ψ v ≥ a is trivial, we prove the converse. Write ψ = ψ v ≥ a . It suffices to show that forany small enough ǫ >
0, any fixed divisorial valuation v of C ( X ), we can construct a section s ∈ H ( X, L k )for some large k , so that k − v i ( s ) ≥ v i ( ψ ) for all i and k − v ( s ) ≤ v ( ψ ) + ǫ . We may assume that a i > i . Let b i = v i ( ψ ). Then b i ≥ a i . Let b = v ( ψ ).By [DDNL19b, Lemma 4.4], we can construct a potential ψ ′ , more singular than ψ , such that v i ( ψ ′ ) > b i , v ( ψ ′ ) ≤ v ( ψ ) + ǫ . Take a small enough δ ∈ Q > , such that (1 + δ ) − v i ( ψ ′ ) > b i for all i . Regarding ψ ′ as a metric on (1 + δ ) L and applying [Dem12, Corollary 13.23] and its proof, we finda sequence of sections s k ∈ H ( X, L k ) for some sequence k increasing to ∞ , such that1 + δk [div s k ] → δω + ω ψ ′ , δk v i ( s k ) → v i ( ψ ′ ) . Thus for k large enough, 1 k v ( s k ) ≤ b + ǫ , k v i ( s k ) > b i . (cid:3) As a particular case, let v i = c i ord F i be a Q -divisorial valuation, a i ∈ R ( i = 1 , . . . , m ). We have ψ ′ v ≥ a = sup* k ∈ Z > sufficiently divisible k sup* ( log | s | h k : s ∈ H ( X, kL − m X i =1 ka i c − i F i ) , sup X | s | h k ≤ ) . Let D be a snc Q -divisor on X with finitely many irreducible components. Write D = P i a i D i . We define(2.5) ψ ≥ D := ψ (ord Di ) ≥ ( a i ) . Extended deformation to the normal cone.
Let ψ ∈ PSH(
X, ω ) be a potential with analyticsingularities. Let π : Y → X be a log resolution of ψ . Let Psef( ψ ) be the pseudo-effective threshold of ψ .Namely, Psef( ψ ) = sup { t ≥ π ∗ L − t div Y ψ is pseudo-effective } . We define a test curve ψ + • as follows (see Section 4 for the general theory of test curves): ψ + τ := , τ ≤ ,ψ ≥ τ div Y ψ , τ ∈ < τ ≤ Psef( ψ ) , −∞ , τ > Psef( ψ ) . We call this construction the extended deformation to the normal cone with respect to ψ . See Lemma 2.6for an explanation of this terminology.This construction can be realized geometrically. Definition 2.10.
Let ψ ∈ PSH(
X, Cω ) ( C ∈ Z > ) be a potential with analytic singularities. Let π : Y → X be a log resolution. Let A > A div Y ψ is integral. We say ψ is dreamy if thedouble-graded ring R ( X, L, ψ ) := M k ∈ Z ≥ M s ∈ Z ≥ H ( Y, kACL − sA div Y ψ )is finitely generated. LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 9
Note that ψ is dreamy or not does not depend on the choice of A .Assume that ψ ∈ PSH(
X, ω ) is dreamy and L is ample Let ( X , L ) be the relative proj of M k ∈ Z ≥ M s ∈ Z ≥ t − s H ( Y, kAL − sA div Y ψ )over C . Then ( X , L ) is a test configuration of ( X, L A ). It follows from Lemma 2.5 that the correspondingtest curve is just ψ + .More generally, let ψ ∈ PSH(
X, ω ). We define Psef( ψ ) as the sup of t ≥
0, such that on each birationalmodel π : Y → X , π ∗ L − t div Y ψ is pseudo-effective. This definition coincides with the previous one when ψ has analytic singularities.We define the corresponding test curve ψ + • as follows: When τ ≤
0, set ψ + τ = 0. When 0 < τ < Psef( ψ ),we define ψ + τ = lim Y ψ ≥ τ div Y ψ , where the limit is a limit of decreasing net taken over all birational models π : Y → X . Note that the limitis I -model by [DX20, Lemma 2.20] ∗ . Define ψ +Psef( ψ ) = lim τ → Psef( ψ ) − ψ + τ and ψ + τ = −∞ if τ > Psef( ψ ).2.7. Generalized deformation to the normal cone.
Let ψ ∈ PSH(
X, ω ) be a model potential withanalytic singularities. We define a test curve (see Section 4 for the precise definition) ψ • by ψ τ := , τ ≤ − ,P [(1 + τ ) ψ ] , τ ∈ ( − , , −∞ , τ > . The test curve ψ • is a truncated version of ψ + • : Lemma 2.6.
When τ ∈ [0 , , ψ + τ = ψ τ − . The test curve ψ • and its associated geodesic ray were studied in [Dar17a] and [DDNL19b].The following result due to Darvas ([Dar17b]) characterizes the geodesic ray induced by ψ • . Proposition 2.7.
Let ψ ∈ PSH
Model ( X, ω ) , then ˇ ψ t ( t ≥ ) is the increasing limit of ℓ kt , where ( ℓ kt ) t ∈ [0 , − E (max {− k,ψ } )] is the geodesic from to max {− k, ψ } . Assume that ψ has analytic singularities along a Z -divisor div X ψ on X and that L − div X ψ is semi-ample.In this case, let X = Bl div X ψ ×{ } X × C be the deformation to the normal cone. Let E be the exceptionaldivisor. Let Π : X → X × C be the natural map and let p : X × C → X be the natural projection. Let L = Π ∗ p ∗ L ⊗ O X ( −E ). Then we have a test configuration ( X , L ) of ( X, L ). By Example 4.1, the test curveinduced by the filtration of this test configuration is exactly ψ • . Note that ψ • is induced by the filtration inExample 4.3.We make the following observation: Lemma 2.8.
Let X be the deformation to the normal cone of X with respect to a Z -divisor mD with D aprime divisor. Then X is regular in codimension .Proof. After localization we are reduced to blowing-up ( x m , y ) on C , which is a quotient singularity, henceregular in codimension 1. (cid:3) ∗ Strictly speaking, [DX20, Lemma 2.20] only deals with decreasing sequences, but the proof works for decreasing nets aswell.
Entropy and delta invariant.
Let X be a compact Kähler manifold of dimension n . Let L be anample line bundle. Let ω ∈ c ( L ) be a Kähler form.We recall that for an R -Weil divisor D = P i a i D i with a i = 0, D i prime and pairwise distinct, red D := P i D i . Definition 2.11.
Let ψ ∈ PSH(
X, ω ). We define the entropy of [ ψ ] asEnt([ ψ ]) := nV lim Y (cid:16) h π ∗ L − div Y ψ i n − · ( K Y/X + red div Y ψ ) (cid:17) ∈ [0 , ∞ ] , where π : Y → X runs over all birational models on X . Here the product h•i is the movable intersection inthe sense of [BFJ09], [Bou02]. We formally set Ent([ −∞ ]) = 0.We observe that Ent([ ψ ]) depends only on the I -singularity type of ψ . To the best of the author’sknowledge, this invariant has never been defined in the literature. Remark . The condition ψ ∈ PSH(
X, ω ) is not essential. We can define the same quantity for anyquasi-psh function.We can now define our new delta invariant:
Definition 2.12.
We define the pluripotential-theoretic δ -invariant as δ pp = inf ψ R ∞−∞ Ent([ ψ + τ ]) d τnV − R ∞−∞ (cid:16)R X ω ∧ ω n − ψ + τ − R X ω nψ + τ (cid:17) d τ , where V = ( L n ), ψ runs over the set of ω -psh functions with some non-zero Lelong number on X . Thequotient depends only on the I -singularity type of ψ .Similarly, Definition 2.13.
We define the δ ′ -invariant of ( X, L ) as δ ′ := inf ψ ( K Y/X · ( − div Y ψ ) n − ) + n ( G n − ( L, div Y ψ ) · red div Y ψ ) n R (cid:16)R X ω ∧ ω n − τψ − R X ω nτψ (cid:17) d τ , where π : Y → X is a log resolution of ψ . Here ψ runs over the set of unbounded ω -psh functions withanalytic singularities. The quotient depends only on the singularity type of ψ .3. Preliminaries
Let X be a compact Kähler manifold of dimension n . Let ω be a Kähler form on X .3.1. Archimedean functionals.
In this section, we recall the definitions of several functionals in Kählergeometry. For the definition of E = E ( X, ω ), we refer to [Dar19] and references there in. We write E ∞ ( X, ω )for the set of bounded potentials in PSH(
X, ω ).Define V = V ω := R X ω n . Let E : E → R denote the Monge–Ampère energy functional: E ( ϕ ) = 1 V n X j =0 Z X ϕ ω jϕ ∧ ω n − j . For ϕ ∈ E ( X, ω ), define Ent( ϕ ) := V Z X log (cid:18) ω nϕ ω n (cid:19) ω nϕ , if ω ϕ n ≪ ω n , ∞ , otherwise . Let α be a smooth real (1 , X . We define the functional E α : E → R by E α ( ϕ ) := 1 nV n − X j =0 Z X ϕ α ∧ ω jϕ ∧ ω n − − j . LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 11
In particular, the Ricci energy is defined as E R := E − n Ric ω = − V n − X j =0 Z X ϕ Ric ω ∧ ω jϕ ∧ ω n − − j . Define the ˜ J functional as I − J , namely(3.1) ˜ J ( ϕ ) = E ( ϕ ) − V Z X ϕ ω nϕ . Note that(3.2) E ω = E + 1 n ˜ J .
Let M : E → ( −∞ , ∞ ] denote the Mabuchi functional: M ( ϕ ) = ¯ SE ( ϕ ) + Ent( ϕ ) + E R ( ϕ ) , where ¯ S is the average scalar curvature. In this paper, it is convenient to use a different normalization ofthe Mabuchi functional, so we define the twisted Mabuchi functional ˜ M : E ( X, ω ) → ( −∞ , ∞ ] as˜ M := M − ¯ SE = Ent + E R . Now assume that [ ω ] = c ( L ) for some ample line bundle L on X . Fix a smooth Hermitian metric h on L with c ( L, h ) = ω .For any k ∈ Z > , the Donaldson’s L k -functional ([Don05]) is defined as L k ( ϕ ) := − kV log det k · k Hilb k ( ϕ ) det k · k Hilb k (0) . Here Hilb k ( ϕ ) is the norm on H ( X, K X ⊗ L k ) defined by k s k k ( ϕ ) = Z X ( s, ¯ s ) h k e − kϕ . Definition 3.1.
Here our convention of the determinant follows that in [BE18], which differs from theconvention of [DX20] by a factor of 2.
Theorem 3.1.
For each k ≥ , the functional L k is convex along finite energy geodesics in E . This result is essentially Berndtsson’s convexity theorem ([Ber09a], [Ber09b])). See [DLR20, Proposi-tion 2.12] for details.3.2.
Radial functionals.
In this section, we assume that the Kähler class [ ω ] is in the integral Néron–Severigroup. Take an ample line bundle L on X so that [ ω ] = c ( L ). Fix a smooth positive metric h on L with c ( L, h ) = ω .Let R ( X, ω ) be the space of E ( X, ω ) geodesic rays emanating from 0. That is, a general element ℓ ∈ R is a map [0 , ∞ ) → E , such that ℓ = 0 and such that ℓ | [0 ,A ] is a (finite energy) geodesic in E for any A > R ∞ ( X, ω ) for the set of locally bounded geodesic rays emanating from 0.For F = Ent , E α , M, ˜ M , E, ˜ J , we define a corresponding radial functional F on R by F ( ℓ ) := lim t →∞ t F ( ℓ t ) . For each of them, the limit is well-defined by [BDL17, Proposition 4.5, Theorem 4.7].
Non-Archimedean functionals.
We write X an for the Berkovich analytification of X with respectto the trivial valuation on C . As a set, X an consists of all real semi-valuations (up to equivalence) extendingthe trivial valuation on C . There is a natural topology known as the Berkovich topology on X an . We alwaysendow X an with this topology (instead of the more natural G-topology). There is a continuous morphismof locally ringed spaces from X an to X with the Zariski topology. Let L an be the pull-back of L to X an . See[Ber12, Section 3.5]. We refer to [BJ18b] for the definition of E , NA = E ( X an , L an ). For ℓ ∈ R , we write ℓ NA for the corresponding potential in E , NA in the sense of [BBJ15], namely ℓ NA ( v ) := − G ( v )(Φ) , where G ( v ) is the Gauss extension of v and Φ is the potential on X × ∆ corresponding to ℓ . Recall thatthere is a natural embedding E , NA ֒ → R . We will often use this embedding implicitly. Geodesic rays inthe image of this embedding are known as maximal geodesic rays.For F = E, E α , ˜ J , we write F NA ( ℓ NA ) := F ( ℓ ) . When ℓ is a maximal geodesic ray. For explanation of this terminology, see [BHJ16] and [Li20, Proposi-tion 2.38].Let ψ ∈ E , NA , we write Ent NA ( ψ ) = Ent NA (MA( ψ )) := 1 V Z X an A X MA( ψ ) , where A X : X an → [0 , ∞ ] denotes the log discrepancy functional (see [JM12]) and MA( ψ ) denotes theChambert-Loir measure (see [CLD12], [CL06], [BJ18b]). We also write˜ M NA = E NA R + Ent NA . In the case of L k and ℓ is maximal, we write(3.3) L NA k ( ℓ NA ) = lim t →∞ t L k ( ℓ t ) . Recall the definition of δ -invariant:(3.4) δ = δ ([ ω ]) := inf v ∈ Val ∗ X A X ( v ) S L ( v ) , where Val ∗ X denotes the space of non-trivial real valuations of C ( X ) ([JM12]) and S L ( v ) := Z ∞ vol( L − tv ) d t and vol( L − tv ) = lim k →∞ n ! k n h ( X, L k ⊗ a tk )with a t being the ideal sheaf defined by the condition that v ≥ t . Recall that in the Fano setting, there isalways a quasi-monomial valuation that achieves the minimum in (3.4) (see [Xu20, Theorem 4.20], [BLZ19])).Recall that ([BJ18a, Section 2.9, Theorem 5.16])(3.5) δ ([ ω ]) = inf µ ∈M ( X an ) Ent NA ( µ ) E ∗ ( µ ) , where M ( X an ) denotes the set of Radon measures on X an with total mass V , E ∗ ( µ ) := sup ψ ∈E ( X an ,L an ) (cid:18) E ( ψ ) − Z X an ψ d µ (cid:19) . It is easy to see that for ϕ ∈ E , NA ,(3.6) E ∗ (MA( ϕ )) = ˜ J NA ( ϕ ) . LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 13
Flag ideals and test configurations.
Let X be a compact Kähler manifold of dimension n . Let L be a big and semi-ample line bundle on X . Let h be a smooth, non-negatively curved metric on L . Let ω = c ( L, h ). Definition 3.2. A flag ideal on X × C is a C ∗ -invariant coherent ideal sheaf of X × C that is cosupportedon the central fibre. Equivalently, a flag ideal is an ideal of the form(3.7) I = I + I t + · · · + I N − t N − + ( t N ) , where I ⊆ I ⊆ · · · ⊆ I N − ⊆ I N = O X are coherent ideal sheaves on X , t is the variable on C . Definition 3.3. A test configuration of ( X, L ) consists of a pair ( X , L ) consisting of a variety X and asemi-ample Q -line bundle L on X , a morphism Π : X → C , a C ∗ -action on X , L and an isomorphism( X , L| X ) ∼ = ( X, L ), so that(1) π is C ∗ -equivariant.(2) The fibration π is equivariantly isomorphic to the trivial fibration ( X × C ∗ , p ∗ L ) through an isomor-phism that extends the given on over 1. Here p denotes the projection to the first factor.A test configuration ( X , L ) can be compactified by gluing the trivial fibration over P \ { } . We write ( ¯ X , ¯ L )for the compactified test configuration. We will frequently omit the bars when we talk about compactifiedtest configurations. Definition 3.4 (Donaldson–Futaki invariant) . Let ( X , L ) be a test configuration of ( X, L ). Take r ∈ Z > so that L r is integral. For k ∈ Z > , define w ( rk ) as the weight of the C ∗ -action on H ( X , L rk | X ). Byequivariant Riemann–Roch theorem, we can write w ( rk ) = a ( rk ) n +1 + b ( rk ) n + O ( k n − ) . Define the
Donaldson–Futaki invariant of ( X , L ) asDF( X , L ) = n ! ¯ SV a − b n ! V .
Define the twisted Donaldson–Futaki invariant of ( X , L ) as f DF( X , L ) = − b n ! V .
Let ℓ be the Phong–Sturm geodesic ray associated to ( X , L ) and let φ = ℓ NA ∈ H NA be the non-Archimedean potential defined by ( X , L ). Proposition 3.2 ([BHJ16, Proposition 2.8],[Li20, Theorem 5.3]) . Assume that L is ample.(1) Let ℓ ∈ E , NA , then ˜ M NA ( ℓ NA ) ≤ ˜ M ( ℓ ) , Ent NA ( ℓ NA ) ≤ Ent ( ℓ ) . Equality holds if ℓ is the Phong–Sturm geodesic ray of some test configuration.(2) Let ( X , L ) be a (not necessarily normal) test configuration of ( X, L ) . Let p : ˜ X → X be the normal-ization. Let ˜ L = p ∗ L . Then (3.8) ˜ M ( ℓ ) = ˜ M NA ( φ ) = f DF( X , L ) − V (cid:0)(cid:0) ˜ X − ˜ X red0 (cid:1) · ˜ L n (cid:1) . The intersection-theoretic formulae of the Donaldson–Futaki invariant were obtained first in [Oda13] and[Wan12]. 4.
The theory of test curves
In this section, we review and extend the theory of test curves.
Ross–Witt Nyström correspondence.
Results in this section are contained in [RWN14], [DDNL18a]and [DX20]. The references work with ample line bundles and Kähler forms, but the readers can readilycheck that all arguments work for semi-ample line bundles and real semi-positive forms.Let X be a compact Kähler manifold of dimension n . Let ω be a real semi-positive form on X . Assumethat R X ω n >
0. Let PSH
Model ( X, ω ) denote the set of model potentials in PSH(
X, ω ). Definition 4.1. A test curve is a map ψ = ψ • : R → PSH
Model ( X, ω ) ∪ {−∞} , such that(1) ψ • is concave in • .(2) ψ is usc as a function R × X → [ −∞ , ∞ ).(3) lim τ →−∞ ψ τ = 0 in L .(4) ψ τ = −∞ for τ large enough.Let τ + := inf { τ ∈ R : ψ τ = −∞} . We say ψ is normalized if τ + = 0. The test curve is called bounded if ψ τ = 0 for τ small enough. Let τ − := sup { τ ∈ R : ψ τ = 0 } in this case.The set of bounded test curves is denoted by T C ∞ ( X, ω ), Remark . We remind the readers that our test curves correspond to maximal test curves in the literature.
Remark . In fact, it is more natural to define a test curve only on the interval ( −∞ , τ + ). But we adoptthe traditional definition here to facilitate the comparison with the literature. Definition 4.2.
The energy of a test curve ψ • is defined as(4.1) E ( ψ • ) := 1 V τ + + 1 V Z τ + −∞ (cid:18)Z X ω nψ τ − Z X ω n (cid:19) d τ . A test curve ψ is said to be of finite energy if E ( ψ ) > −∞ . We denote the set of finite energy test curves by T C ( X, ω ). Proposition 4.1.
Let ψ • be a test curve. Then(1) τ R X ω nψ τ is a continuous function for τ ∈ ( −∞ , τ + ) .(2) For any τ < τ + , R X ω nψ τ > .(3) The function τ log R X ω nψ τ is concave for τ ∈ ( −∞ , τ + ) .Proof. Part (1) and Part (2) follow from [DX20, Lemma 3.9]. Part (3) is a consequence of [DDNL19a,Theorem 6.1] and the monotonicity theorem [WN19]. (cid:3)
Definition 4.3.
Let ℓ ∈ R ( X, ω ). The
Legendre transform of ℓ is defined asˆ ℓ τ := inf t ≥ ( ℓ t − tτ ) , τ ∈ R . Let ψ ∈ T C ( X, ω ), the inverse Legendre transform of ψ is defined asˇ ψ t := sup τ ∈ R ( ψ τ + tτ ) , t ≥ . Theorem 4.2 ([DX20, Theorem 3.7]) . The Legendre transform and inverse Legendre transform establish abijection from R ( X, ω ) to T C ( X, ω ) . For ℓ ∈ R ( X, ω ) , We have sup X ℓ = τ + and E ( ℓ ) = E (ˆ ℓ ) .Moreover, under this correspondence, R ∞ corresponds to the set of bounded test curves. When ℓ ∈R ∞ , inf X ℓ = τ − . Now assume that ω = c ( L, h ) for some ample line bundle L and a strictly positively curved smoothHermitian metric h on L . Definition 4.4. An I -model test curve is a test curve ψ • such that for every τ < τ + , ψ τ is I -model. Theset of I -model test curves of finite energy is denoted by T C I ( X, ω ). Theorem 4.3 ([DX20, Theorem 3.7]) . The Legendre transform and inverse Legendre transform establish abijection between E , NA and T C I ( X, ω ) . LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 15
Test curves induced by filtrations.
Let X be a compact Kähler manifold of dimension n . Let L be a big and semi-ample line bundle on X . Let h be a smooth, non-negatively curved metric on L . Let ω = c ( L, h ). We use the notation R ( X, L ) := M k ∈ Z ≥ H ( X, L k ) . Definition 4.5. A filtration on R ( X, L ) is a decreasing, left continuous, multiplicative R -filtration F • onthe ring R ( X, L ) which is linearly bounded in the sense that there is
C >
0, so that F − kλ H ( X, L k ) = H ( X, L k ) , F kλ H ( X, L k ) = 0 , when λ > C .A filtration F is called a Z -filtration if F λ = F ⌊ λ ⌋ for any λ ∈ R .A Z -filtration F is called finitely generated if the bigraded algebra M λ ∈ Z ,k ∈ Z ≥ F λ H ( X, L k )is finitely generated over C .Recall that by [RWN14], a filtration induces a test curve in the following manner. Let F • be a filtration.For τ ∈ R , define(4.2) ψ τ := sup* k ∈ Z > k − sup* (cid:26) log | s | h k : s ∈ F kτ H ( X, L k ) , sup X | s | h k ≤ (cid:27) . By [DX20, Theorem 3.11], ψ τ is I -model or −∞ for each τ ∈ R . Lemma 4.4.
Let ψ • be the test curve induced by a filtration F • on R ( X, L ) . Let v be a real valuation of C ( X ) . Then v ( ψ τ ) = inf k ∈ Z > k − inf (cid:8) v ( s ) : s ∈ F kτ H ( X, L k ) (cid:9) . Proof.
For k ∈ Z > , let F k := sup* (cid:26) log | s | h k : s ∈ F kτ H ( X, L k ) , sup X | s | h k ≤ (cid:27) . For k, m ∈ Z > , F k + m ≥ F k + F m . So by Fekete lemma, ψ τ is the usc regularization of the increasing limit 2 − k F k . We conclude by themonotonicity and the upper semi-continuity of Lelong numbers. (cid:3) Lemma 4.5.
Let ψ • be the test curve induced by a Z -filtration F • on R ( X, L ) . Then (4.3) Z X ω nψ τ ≥ lim k →∞ n ! k n dim F kτ H ( X, L k ) . Equality holds if F • is finitely generated, τ < τ + . Note that the limit on the right-hand side exists by [LM09].
Proof.
By [DX20, Theorem 1.1], Z X ω nψ τ = lim k →∞ n ! k n h ( X, L k ⊗ I ( kψ τ )) . Each element in F kτ H ( X, L k ) is obviously square integrable with respect to kψ τ , (4.3) follows.Now assume that F • is finitely generated. Then it is the filtration induced by some test configuration( X , L ) of ( X, L ) by [BHJ17, Proposition 2.15]. Without loss of generality, we may assume that τ + = 0 forthe test curve ψ • . Then by [BHJ17, Section 5], the Duistermaat–Heckman measure of ( X , L ) is given by ν = − V dd τ vol( R ( τ ) ) , where vol( R ( τ ) ) := lim k →∞ n ! k n dim F kτ H ( X, L k ) . By [BHJ17, Lemma 7.3], the non-Archimedean Monge–Ampère energy of ( X , L ) is given by E NA ( X , L ) = Z ∞−∞ τ d ν ( τ ) = Z −∞ (cid:18) V vol R ( τ ) − (cid:19) d τ . On the other hand, by [DX20, Theorem 1.1], E NA ( X , L ) = Z −∞ (cid:18) V Z X ω nψ τ − (cid:19) d τ . Now by (4.3), Proposition 4.1 and [BHJ17, Theorem 5.3], we conclude that equality holds in (4.3) when τ < τ + . (cid:3) Let ( X , L ) be a test configuration of ( X, L ). It induces a filtration as follows: Take r ∈ Z > so that L r is integral. Then ( X , L ) induces a Z -filtration of R ( X, rL ) as follows: let s ∈ H ( X, rkL ), then s ∈ F λ H ( X, rkL ) iff t − λ s ∈ H ( X , L rk ). Here we have abused the notation by writing s for the equivariantextension of s as well. See [BHJ17]. The weight of the C ∗ -action on the central fibre of L rk is given by w ( rk ) = − Z ∞−∞ λ d dim F λ H ( X, L rk ) . Example 4.1.
Let I = I + I t + · · · + I N − t N − + ( t N ) be a flag ideal on X × P . Let X = Bl I X × P .Denote by Π :
X → X × P the natural morphism. Let E be the exceptional divisor. Let p : X × P → X be the natural projection. Assume that L := Π ∗ p ∗ L ⊗ O X ( − E ) is π -semiample.Write I k = Nk − X j =0 J k,j t j + ( t Nk ) . Then J k,j = X α ∈ N N , | α | = k, | α | ′ = j I α • . Here | α | ′ := P i iα i . Set J k,kN = O X .Let F • be the filtration on R ( X, L ) induced by ( X , L ) . Let ψ • be the corresponding test curve.We claim that ψ an τ ( v ) = − min α ∈ Q N ≥ , | α | =1 , | α | ′ = − τ X i α i v ( I i ) . In particular, ψ an0 ( v ) = − v ( I ) . Proof.
Let λ ∈ Z and s ∈ H ( X, L k ), then s ∈ F λ H ( X, L k ) iff t − λ s extends to a section of L k iff t − λ s ∈ H ( X × C , L k ⊗ I ⊗ k )iff s ∈ J k, − λ . Hence we have v ( ψ τ ) = inf k k inf { v ( s ) : s ∈ J k, −⌈ kτ ⌉ } . Observe that inf { v ( s ) : s ∈ J k, −⌈ kτ ⌉ } = min α ∈ N N , | α | = k, | α | ′ = −⌈ kτ ⌉ X i α i v ( I i ) . So v ( ψ τ ) ≥ min α ∈ Q N ≥ , | α | =1 , | α | ′ = − τ X i α i v ( I i ) . On the other hand, observe that the minimizer is indeed rational when ψ is rational, so the reverse inequalityalso holds. (cid:3) Observe that when τ < ψ τ has quasi-analytic singularities (Definition 2.8). LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 17
Example 4.2.
Let v = c ord F be a divisorial valuation of C ( X ) , where c ∈ Q > , F is a prime divisor over X . Then v induces a filtration F • v on R ( X, L ) : F λv H ( X, L k ) = ( H ( X, kL − λcF ) , λ ≥ ,H ( X, L k ) , λ < . Here we have omitted the pull-back of L to a model. Example 4.3.
Let ψ ∈ PSH(
X, ω ) be a potential with analytic singularities. Let π : Y → X be a logresolution of the singularities of ψ . Assume that π ∗ L − div Y ψ is semi-ample. Then ψ induces a testconfiguration of ( Y, π ∗ L ) by deformation to the normal cone with respect to div Y ψ . Then a section s ∈ H ( X, L k ) = H ( Y, π ∗ L k ) is in F λ iff t − λ s extends to the central fibre, that is, s ∈ O Y ( − ( k + λ ) div Y ψ ) .Hence F λψ H ( X, L k ) = ( H ( Y, kπ ∗ L − ( λ + k ) div Y ψ ) , λ ≤ ,H ( X, L k ) , λ > . The test curve ψ • defined in Section 2.7 is induced by this filtration. Example 4.4.
Let ψ ∈ PSH(
X, ω ) be a potential with analytic singularities. Let π : Y → X be a logresolution of the singularities of ψ . The deformation to the normal cone defined in Example 4.3 can beextended as follows: F λψ + H ( X, L k ) = ( H ( X, kπ ∗ L − λ div Y ψ ) , λ ≥ ,H ( X, L k ) , λ < . The test curve ψ + • defined in Section 2.6 is induced by this filtration. The Phong–Sturm geodesic ray.
Let X be a compact Kähler manifold of dimension n . Let ω bea real smooth semi-positive (1 , X . Let ( X , L ) be a semi-ample test configuration of ( X, L ). Fixa S -invariant smooth metric Φ on L with c ( L , Φ) = Ω, we may assume that Ω | X × S is the pull-back of ω . Let π : X → C be the natural map. Let X ◦ := π − (∆), where ∆ = { z ∈ C : | z | < } . Consider thehomogeneous Monge–Ampère equation(4.4) ( (Ω + dd c Ψ) n +1 = 0 on X ◦ , Ψ | X × S = 0 . By [CTW18], there is a unique bounded solution to (4.4) and the solution is C , outside the central fibre.Let ℓ be the geodesic ray in E ( X, ω ) corresponding to Ψ, then ℓ is known as the Phong–Sturm geodesicray induced by ( X , L ). This construction was first studied in [PS07] and [PS10]. Theorem 4.6 ([RWN14, Theorem 9.2]) . Let ( X , L ) be a semi-ample test configuration of ( X, L ) . Let ℓ bethe Phong–Sturm geodesic ray induced by ( X , L ) . Let F • be the filtration induced by ( X , L ) . Let ψ • be thetest curve induced by F • as in (4.2) . Then ˇ ψ = ℓ . Non-Archimedean analogue of Ross–Witt Nyström correspondence.
Assume that L is ample. Definition 4.6.
A function ψ : X an → [ −∞ , ∞ ) is called a good potential if there exists ϕ ∈ PSH(
X, ω )such that ψ = ϕ an .The set of good potential is denoted as PSH NAg ( X, ω ).See (2.1) for the definition of ϕ an . Proposition 4.7.
The map ψ ψ an is a bijection from PSH
Model I ( X, ω ) to PSH
NAg ( X, ω ) . This is obvious by definition.
Definition 4.7.
A test curve ψ ∈ T C ∞ ( X, ω ) is piecewise linear if ψ an is piecewise linear with finitely manybreaking points (i.e. non-differentiable points). Definition 4.8. A non-Archimedean test curve is a map ψ : ( −∞ , τ + ) → PSH
NAg ( X, ω ) for some τ + ∈ R ,such that(1) ψ is concave.(2) lim τ →−∞ ψ τ = 0 in L . We define τ − as in the Archimedean case.The non-Archimedean test curve ψ • is of finite energy if(4.5) E ( ψ • ) := 1 V τ + + 1 V Z τ + −∞ (cid:18)Z X ω nϕ τ − Z X ω n (cid:19) d τ > −∞ , where ϕ τ is the I -model potential in PSH( X, ω ) with ϕ an τ = ψ τ .The set of non-Archimedean test-curves of finite energy is denoted by T C , NA ( X, ω ). Proposition 4.8.
The map
T C I ( X, ω ) → T C , NA ( X, ω ) defined by ψ • ( ψ an τ ) τ<τ + is a bijection. This again is immediate by definition.
Remark . In Definition 4.8, we deliberately define ψ τ only for τ < τ + . This is because it is not alwaystrue that for a Archimedean test curve ψ • , ψ an τ + = lim τ → τ + − ψ an τ . Theorem 4.9 ([DX20, Proposition 3.13]) . The map ˇ: T C , NA ( X, ω ) → E , NA given by ψ an • sup τ<τ + ( ψ an τ + τ ) is a bijection. Moreover, when ψ • ∈ T C ( X, ω ) , ( ψ an • ) ˇ = (cid:16) ˇ ψ • (cid:17) NA , namely, the following diagram commutes: T C T C , NA R E , NAanˇ ˇNA
Remark . Ideally when we are considering only maximal geodesic rays, it should be possible to carryout the computations in Section 6 purely in terms of non-Archimedean test curves, without referring to themachinery of test curves, filtrations and test configurations. However, the difficulty is that we do not havea good understanding of the following non-Archimedean Monge–Ampère measureMA (cid:18) sup τ<τ + ( ψ an τ + τ ) (cid:19) . It is highly desirable to have a description of this measure in terms of certain real Monge–Ampère measureson some dual complexes. In the non-trivially valued case, a partial result is derived by Vilsmeier ([Vil20]).5.
Intersection theory of b-divisors
In this section, we apply the intersection theory of Shokurov’s b-divisors to the study of singularities ofpsh functions. Due to the technical assumptions in [DF20a] and [DF20b], we can not apply Dang–Favre’sintersection theory directly. Although it seems possible to remove the technical assumptions in Dang–Favre’stheory, we do not pursue this most general theory here.References to this section are [DF20a], [DF20b], [BFJ09], [BDPP13], [KK14].Let X be a projective manifold of dimension n .5.1. b-divisors. Recall that the Riemann–Zariski space of X is the locally ringed space defined by X := lim ←− Y Y , where Y runs over all birational models of X . Here the projective limit is taken in the category of locallyringed spaces. For valuative interpretation of X , see [Tem11]. We do not make use of the theory of Riemann–Zariski spaces in an essential way in this paper. Instead, we give an ad hoc treatment of divisors on X . LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 19
Definition 5.1.
By a
Weil divisor on X or a Weil b-divisor on X , we mean an element inbWeil( X ) := lim ←− Y Weil( Y ) , where Y runs over all (smooth) birational models of X and Weil( Y ) is the set of numerical classes of R -divisorson Y .By a Cartier divisor on X or a Cartier b-divisor on X , we mean an element inbCart( X ) := lim −→ Y Weil( Y ) , where Y runs over all (smooth) birational models of X .Both the limit and the colimit are taken in the category of topological vector spaces.There is a natural continuous injection bCart( X ) ֒ → bWeil( X ).5.2. Differentiability of the volume.
General references of results in this section are [BFJ09], [DP04].Let X be a compact Kähler manifold of dimension n . Let L be a big line bundle on X . Recall that thevolume of L is defined as vol( L ) := lim k →∞ n ! k n h ( X, L k ) . More generally, by requiring vol( L k ) = k n vol( L ) , we extend the definition of volume to all big Q -line bundles. By continuity, this definition further extendsto all pseudo-effective R -line bundles.When L is a nef R -line bundle, we have(5.1) vol( L ) = ( L n ) . Recall the following basic fact,
Theorem 5.1 ([BFJ09]) . The volume function vol is continuously differentiable in the big cone. Moreover,let L be a big and nef R -line bundle, let L ′ be a line bundle, then (5.2) dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 vol( L + ǫL ′ ) = n (cid:0) L n − · L ′ (cid:1) . Now assume that L is big and semi-ample. Fix a smooth semi-positive real (1 , ω ∈ c ( L ). Let ψ ∈ PSH(
X, ω ) be a potential with quasi-analytic singularities along a snc R -divisor div X ψ . Assume that ψ has positive mass. Recall that by Lemma 2.4, L − div X ψ is nef and big. Let L ′ be an R -line bundle on X . Now we define(5.3) D L ( ψ, L ′ ) = dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 vol( L − div X ψ + ǫL ′ ) = n (cid:0) ( L − div X ψ ) n − · L ′ (cid:1) . When ψ ∈ PSH(
X, ω ) has positive mass and there exists a birational model π : Y → X , ψ has quasi-analyticsingularities along a snc R -divisor div Y ψ , let L ′ be an R -line bundle on Y , we define(5.4) D L ( ψ, L ′ ) := D π ∗ L ( π ∗ ψ, L ′ ) . We formally set D L ( −∞ , L ′ ) = 0.5.3. Singularity divisors.
Let L be a semi-ample line bundle on X . Let h be a non-negatively curvedmetric on L . Let ω = c ( L, h ). Definition 5.2.
Let ψ ∈ PSH(
X, ω ). We define the singularity divisor of ψ as a Weil b-divisor div X ψ ∈ bWeil( X ): (div X ψ ) Y = div Y ψ . Here we have abused the notation by writing div Y ψ for the numerical class of the corresponding divisor,which is well-defined by [BFJ09, Proposition 1.3].We set vol( L − div X ψ ) := lim Y vol( π ∗ L − div Y ψ ) , where π : Y → X runs over all birational model of X . The net is decreasing, hence the limit is well-defined. Theorem 5.2.
Assume that ψ is I -model and of positive mass, then (5.5) Z X ω nψ = vol ( L − div X ψ ) . Proof.
Let ψ j be a quasi-equisingular approximation to ψ . By [DX20, Theorem 1.4], R X ω nψ j → R X ω nψ .Similarly, the right-hand side converges along ψ j as follows from [DF20a, Proof of Theorem 6(3)]. To bemore precise, it suffices to prove that for any ǫ >
0, any model π : Y → X , we can find j >
0, such that for j ≥ j , vol ( L − div X ψ ) ≤ vol (cid:0) L − div X ψ j (cid:1) ≤ vol ( π ∗ L − div Y ψ ) + ǫ . The first inequality is trivial. For the second inequality, observe that by Lemma 2.2, div Y ψ j → div Y ψ . Fixsome C >
0, depending on π , we may take j large enough, so that when j ≥ j , π ∗ L − div Y ψ j ≤ π ∗ L − div Y ψ + C − ǫπ ∗ ω . Then it follows that vol (cid:0) π ∗ L − div Y ψ j (cid:1) ≤ vol ( π ∗ L − div Y ψ ) + ǫ . Hence vol (cid:0) L − div X ψ j (cid:1) ≤ vol ( π ∗ L − div Y ψ ) + ǫ . (cid:3) In particular, this gives an additional characterization of I -model potentials. Corollary 5.3.
Let ψ ∈ PSH
Model ( X, ω ) be a model potential with positive mass. Then ψ is I -model iff Z X ω nψ = vol ( L − div X ψ ) . Radial functionals in terms of Legendre transforms
In this section, let X be a compact Kähler manifold of dimension n . Let L be a big and semi-ample linebundle on X . Let h be a smooth non-negatively curved metric on L . Let ω = c ( L, h ).From Section 6.2 on, we assume that L is an ample line bundle and h is strictly positively curved.6.1. Functionals on the space of test curves.
Let ψ • ∈ T C ( X, ω ). Recall that τ + := inf { τ ∈ R : ψ τ = −∞} .We have already defined the Monge–Ampère energy E ( ψ • ) in (4.1). For any real smooth (1 , α on X , define the α -energy of ψ • as(6.1) E α ( ψ • ) := τ + V Z X α ∧ ω n − + 1 V Z τ + −∞ (cid:18)Z X α ∧ ω n − ψ τ − Z X α ∧ ω n − (cid:19) d τ . The
Ricci energy of ψ • is defined as(6.2) E R ( ψ • ) := − nτ + V Z X Ric ω ′ ∧ ω n − − nV Z τ + −∞ (cid:18)Z X Ric ω ′ ∧ ω n − ψ τ − Z X Ric ω ′ ∧ ω n − (cid:19) d τ , where ω ′ denotes a Kähler form on X .The ˜ J -functional of ψ • is defined as(6.3) ˜ J ( ψ • ) = n E ω ( ψ • ) − n E ( ψ • ) = nV Z ∞−∞ (cid:18)Z X ω ∧ ω n − ψ τ − Z X ω nψ τ (cid:19) d τ . Remark . It is interesting to observe that E α ( ψ • ) depends only on the cohomology class of α .Assume that ψ • is I -model. The non-Archimedean L k -functional of ψ • is defined as(6.4) L NA k ( ψ • ) := 1 V Z ∞−∞ τ d h ( X, K X ⊗ L k ⊗ I ( kψ τ )) . Assume that ψ • is I -model, the entropy of ψ • is defined as(6.5) Ent( ψ • ) := Z ∞−∞ Ent([ ψ τ ]) d τ . Recall that Ent[ • ] is defined in Definition 2.11. LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 21
Definition 6.1.
Let ψ • ∈ T C ∞ ( X, ω ). We say ψ • is analytic if ψ τ has quasi-analytic singularities for any τ < τ + .We say ψ • is piecewise linear if ψ an • is piecewise linear with finitely many breaking points (non-differentiablepoints).We need the following observation. Lemma 6.1.
The test curves in Example 4.1 are analytic and piecewise linear.
Corollary 6.2.
The test curve induced by a test configuration is analytic and piecewise linear.Proof.
This follows from [Oda13, Proposition 3.10] † and Lemma 6.1. (cid:3) We observe the following obvious lemma.
Lemma 6.3.
Let ψ • be an analytic test curve. Then (6.6) Ent( ψ • ) = 1 V Z ∞−∞ D L ( ψ τ , K Y τ /X ) d τ + 1 V Z ∞−∞ D L ( ψ τ , red div Y ψ τ ) d τ , where π τ : Y τ → X is a log resolution of ψ τ . See Section 5.2 for the definition of D L .6.2. Monge–Ampère energy.
From this section on, we assume that L is ample and h is strictly positivelycurved, so that ω is a Kähler form. Theorem 6.4 ([DX20, Theorem 3.7]) . Let ℓ ∈ R . Then (6.7) E ( ℓ ) = E (ˆ ℓ ) . Recall that the right-hand side is defined in (4.1).6.3.
Non-archimedean L -functionals.Theorem 6.5 ([DX20, Theorem 1.1]) . Let ℓ ∈ E , NA . For each k ∈ Z > , (6.8) L NA k ( ℓ ) = L NA k (ˆ ℓ ) . The right-hand side is defined in (6.4) and the left-hand side is defined in (3.3).6.4. α -energy. Let α be a smooth real (1 , X . Lemma 6.6.
Let ϕ, ψ ∈ E ∞ , then dd s (cid:12)(cid:12)(cid:12)(cid:12) s =0 E α ( sψ + (1 − s ) ϕ ) = 1 V Z X ( ψ − ϕ ) α ∧ ω n − ϕ . Proof.
This result is well-known when ψ and ϕ are smooth. In general, it follows from a direct computationusing integration by parts ([Xia19], [Lu20]). (cid:3) Theorem 6.7.
Let ℓ ∈ E , NA or ℓ ∈ R ∞ . Then (6.9) E α ( ℓ ) = E α (ˆ ℓ ) . Proof.
Without loss of generality, we may assume that α is a Kähler form and sup X ℓ = 0.We first assume that ℓ ∈ R ∞ . We fix a few notations. Let ψ • be the Legendre transform of ℓ . Now foreach N ∈ N , M ∈ Z , t ≥
0, we introduceˇ ψ N,Mt := max k ∈ Z k ≤ M ( ψ k − N + tk − N ) . Let U N,Mt := n ˇ ψ N,M +1 t > ˇ ψ N,Mt o . † The statement of [Oda13, Proposition 3.10] needs to be corrected as follows: L r ( − E ) = f ∗ M + c B for some constant c ∈ Q . The mistake in the proof is on the fourth line, where we need to make sure that the isomorphism between h ∗ M s and L r extends to the generic point of the central fibre. Observe that on U N,Mt ,(6.10) ˇ ψ N,M +1 t = ψ ( M +1)2 − N + t ( M + 1)2 − N , ˇ ψ N,Mt = ψ M − N + tM − N . By Lemma 6.6, E α ( ˇ ψ N,M +1 t ) − E α ( ˇ ψ N,Mt ) = 1 V Z Z X (cid:16) ˇ ψ N,M +1 t − ˇ ψ N,Mt (cid:17) α ∧ ω n − s ˇ ψ N,M +1 t +(1 − s ) ˇ ψ N,Mt d s . By comparison principle ([DDNL18b, Proposition 3.5]), Z U N,Mt ( ˇ ψ N,M +1 t − ˇ ψ N,Mt ) α ∧ ω n − ψ N,M +1 t ≤ Z X ( ˇ ψ N,M +1 t − ˇ ψ N,Mt ) α ∧ ω n − s ˇ ψ N,M +1 t +(1 − s ) ˇ ψ N,Mt ≤ Z U N,Mt ( ˇ ψ N,M +1 t − ˇ ψ N,Mt ) α ∧ ω n − ψ N,Mt . We first deal with the upper bound, E α ( ˇ ψ N,M +1 t ) − E α ( ˇ ψ N,Mt ) ≤ − N V − t Z U N,Mt α ∧ ω n − ψ M − N . Set τ − := inf X ℓ . Take sum with respect to M from [ τ − ]2 N to −
1, we get − t [ τ − ] V − Z X ω n − ∧ α + E α ( ˇ ψ N, t ) ≤ − N V − t − X M =[ τ − ]2 N Z U N,Mt α ∧ ω n − ψ M − N ≤ − N V − t − X M =[ τ − ]2 N Z X α ∧ ω n − ψ M − N . Let N → ∞ and then t → ∞ , we get E α ( ℓ ) ≤ V Z τ − ] (cid:18)Z X α ∧ ω n − ψ τ − Z X α ∧ ω n − (cid:19) d τ . Now we deal with the lower bound part. We have E α ( ˇ ψ N,M +1 t ) − E α ( ˇ ψ N,Mt ) ≥ − N V − t Z U N,Mt α ∧ ω n − ψ ( M +1)2 − N + V − Z U N,Mt ( ψ ( M +1)2 − N − ψ M − N ) α ∧ ω n − ψ ( M +1)2 − N . Taking summation with respect to M from [ τ − ]2 N to −
1, we get1 t E α ( ˇ ψ N, t ) ≥ − N V − − X M =[ τ − ]2 N Z U N,Mt α ∧ ω n − ψ ( M +1)2 − N + V − t − − X M =[ τ − ]2 N Z U N,Mt ( ψ ( M +1)2 − N − ψ M − N ) α ∧ ω n − ψ ( M +1)2 − N + [ τ − ] V − Z X ω n − ∧ α . Note that as t → ∞ , U N,Mt → M < −
1. Hencelim t →∞ t E α ( ˇ ψ N, t ) ≥ − N V − − X M =[ τ − ]2 N Z X α ∧ ω n − ψ ( M +1)2 − N + lim t →∞ ( V t ) − − X M =[ τ − ]2 N Z U N,Mt ( ψ ( M +1)2 − N − ψ M − N ) α ∧ ω n − ψ ( M +1)2 − N + [ τ − ] V − Z X α ∧ ω n − . LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 23
Observe that ˇ ψ N, t ≤ ℓ t , so lim t →∞ t E α ( ˇ ψ N, t ) ≤ lim t →∞ t E α ( ℓ t ) = E α ( ℓ ) . Observe that U N,Mt \ S ⊆ (cid:8) N ( ψ ( M +1)2 − N − ψ M − N ) > − t (cid:9) \ S , where S is the pluripolar set { ψ M − N = −∞} .Let F N ( t ) := 2 − N − X M =[ τ − ]2 N Z { N ( ψ ( M +1)2 − N − ψ M − N ) > − t } α ∧ ω n − ψ ( M +1)2 − N . Then − X M =[ τ − ]2 N Z U N,Mt ( ψ ( M +1)2 − N − ψ M − N ) α ∧ ω n − ψ ( M +1)2 − N ≥ − N − X M =[ τ − ]2 N Z { N ( ψ ( M +1)2 − N − ψ M − N ) > − t } N ( ψ ( M +1)2 − N − ψ M − N ) α ∧ ω n − ψ ( M +1)2 − N = − − N − X M =[ τ − ]2 N Z t d a Z { − a ≥ N ( ψ ( M +1)2 − N − ψ M − N ) > − t } α ∧ ω n − ψ ( M +1)2 − N = − Z t (cid:0) F N ( t ) − F N ( a ) (cid:1) d a . Observe that F N is bounded and increasing, so we concludelim t →∞ t − − X M =[ τ − ]2 N Z U N,Mt ( ψ ( M +1)2 − N − ψ M − N ) α ∧ ω n − ψ ( M +1)2 − N ≥ . We conclude E α ( ℓ ) ≥ V Z τ − ] (cid:18)Z X α ∧ ω n − ψ τ − Z X α ∧ ω n − (cid:19) d τ . Now we deal with the case where ℓ ∈ E , NA . It suffices to write ℓ as a decreasing limit of a sequence ofPhong–Sturm geodesic rays ℓ j ∈ R ∞ as in [BBJ15] and apply the monotone convergence theorem and [Li20,(121)]. (cid:3) Corollary 6.8.
Let ℓ ∈ E , NA , let ψ = ˆ ℓ , then ˜ J ( ℓ ) = ˜ J ( ψ • ) . Proof.
This follows from Theorem 6.7, Theorem 6.4 and (3.2). (cid:3)
Corollary 6.9.
Let ℓ m ∈ E , NA ( m ∈ Z > ) be a decreasing sequence of maximal geodesic rays. Let ℓ ∈ R be its limit. Then E α ( ℓ m ) → E α ( ℓ ) as m → ∞ . This generalizes [Li20, (121)]. From our proof, it is easy to drop the condition that ℓ m be decreasing when ℓ ∈ R ∞ . Proof.
Without loss of generality, we may assume that α is a Kähler form. We may and do assume that ℓ m = 0, sup X ℓ m = 0.Observe that ℓ is maximal by the completeness of E , NA (see for example [DX20, Theorem 1.2], [Xia19,Example 3.3]). By Theorem 6.7, it suffices to prove that Z −∞ (cid:18)Z X α ∧ ω n − ℓ jτ − Z X α ∧ ω n − (cid:19) d τ → Z −∞ (cid:18)Z X α ∧ ω n − ℓ τ − Z X α ∧ ω n − (cid:19) d τ . By monotone convergence theorem, it suffices to prove that for almost all τ < Z X α ∧ ω n − ℓ jτ → Z X α ∧ ω n − ℓ τ . It suffices to show that ℓ τ is the d S -limit ([DDNL19b, Theorem 1.1]) of ℓ jτ for almost all τ <
0. In turn, itsuffices to show that R X ω nℓ jτ → R X ω nℓ τ for almost all τ <
0. This follows from the continuity of E NA alongdecreasing sequences and [DX20, Theorem 1.1]. (cid:3) Entropy, dreamy quasi-psh function.
Results in this section are special cases of the results ofSection 7, we will be sketchy here.Let ψ ∈ PSH(
X, ω ) be a dreamy (Definition 2.10) potential with analytic singularities. Let π : Y → X be a log resolution of ψ , which is a composition of blowing-ups with smooth centers. Assume that div Y ψ isintegral. Let ( X , L ) be the test configuration induced by ψ , namely X = P roj C M k ∈ Z ≥ M j ∈ Z ≥ t − j H ( Y, kL − j div Y ψ ) , where A ∈ Z > is a sufficiently divisible integer such that the algebra is generated in degree k = 1. Take L = O X (1). Recall that the test curve induced by ( X , L ) is ψ + • . By taking further blowing-ups, we mayassume that π also resolves the singularities of all ψ τ , for τ ∈ Q , τ < Psef( ψ ). The filtration induced by( X , L ) is F λ H ( X, L k ) = H ( Y, kπ ∗ L − λ div Y ψ )for λ ≥
0. We slightly reformulate Lemma 4.5 in our setting.
Lemma 6.10.
For any τ <
Psef( ψ ) , we have (6.11) Z X ω nψ + τ = vol( π ∗ L − τ div Y ψ ) . Corollary 6.11.
For any ≤ τ < Psef( ψ ) , the decomposition π ∗ L − τ div Y ψ = ( π ∗ L − div Y ψ + τ ) + N τ is the divisorial Zariski decomposition ([Bou04], [Nak04]). More precisely, ( π ∗ L − div Y ψ + τ ) is the movablepart of π ∗ L − τ div Y ψ .Proof. Recall that by our definition of ψ + τ , N τ is effective. So the result follows from Lemma 6.10 and[FKL16]. (cid:3) Remark . Using Lemma 4.5, one can easily generalize Corollary 6.11 to general test configurations. Weleave the details to the readers.In particular,(6.12) D L ( ψ + τ , red div Y ψ ) = D L ( ψ + τ , red div Y ψ + τ ) . See Section 5.2 for the definition of D L . Theorem 6.12.
Let ( X , L ) be as above, then f DF( X , L ) − E R ( ψ + ) = 1 V Z ∞−∞ D L ( ψ + τ , K Y/X ) d τ + 1 V Z ∞−∞ D L ( ψ + τ , div Y ψ ) d τ . When div Y ψ has a single irreducible component with coefficient , Ent NA ( ℓ NA ) = Ent ( ℓ ) = Ent( ψ + • ) . Proof.
For the first part, the computation is a generalization of those in [Fuj19]. As this method has beenelaborated in [DL20b, Section 3], we omit the proof.As for the second part, assume that div Y ψ = F for some prime divisor F over X . Then X is clearlyirreducible and reduced (see [DL20b, Lemma 3.2] for example). Recall that X is also regular in codimension1 by Lemma 2.8. Hence we conclude by Proposition 3.2. (cid:3) LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 25 Variational approach on Berkovich spaces
Let X be a compact Kähler manifold of dimension n . Let L be an ample line bundle on X . Fix a smoothstrictly positively-curved metric h on L . Let ω = c ( L, h ). Proposition 7.1.
Let φ ∈ E , NA . We have (7.1) lim ǫ → ǫ (cid:0) E NA ( P [ φ + ǫA X ]) − E NA ( φ ) (cid:1) ≥ V Z X an A X MA( φ ) . Here P [ • ] is usc-regularized supremum of all elements in E , NA lying below • . When φ ∈ H NA , the limitexists and equality holds.Remark . We expect that equality holds.
Proof.
Let Y run over the set of snc models of X × C C (( T )) (see [BJ18b, Section 1.3] for the precisedefinition). Let r Y : X an → ∆ Y be the natural retraction. Let f Y = A X ◦ r Y . Then ( f Y ) Y is an increasingnet of non-negative continuous functions converging to A X pointwisely. See [JM12] for details.Then for any ǫ > φ + ǫf Y ≤ φ + ǫA X , so E NA (cid:0) P [ φ + ǫf Y ] (cid:1) ≤ E NA ( P [ φ + ǫA X ]) . Hence lim ǫ → ǫ (cid:0) E NA ( P [ φ + ǫA X ]) − E NA ( φ ) (cid:1) ≥ V Z X an f Y MA( φ )by [BJ18b, Corollary 6.32]. By monotone convergence theorem ([Fol99, Proposition 7.12]), we conclude (7.1).Finally, let us deal with the case where φ is associated to some test configuration ( X , L ). We may assumethat X is snc. We claim that for any ǫ > P [ φ + ǫf X ] = P [ φ + ǫA X ] . We only have to prove(7.3) P [ φ + ǫf X ] ≥ P [ φ + ǫA X ] . By [BJ18b, Theorem 5.29], P [ φ + ǫA X ] ≤ P [ φ + ǫA X ] ◦ r X ≤ φ + ǫf X . Hence (7.3) follows and a fortiori equality holds in (7.1). (cid:3)
For ψ ∈ PSH(
X, ω ), each ǫ >
0, we define(7.4) ψ ǫ := sup* (cid:8) ϕ ∈ PSH(
X, ω ) : ϕ ≤ , ϕ an ≤ ψ an + ǫA X on X div Q (cid:9) . Note that ψ ǫ is increasing and concave in ǫ >
0. By [DDNL19a, Theorem 6.1], [WN19], log R X ω nψ ǫ is concavein ǫ . When ψ is I -model, the mass R X ω nψ ǫ is right-continuous at ǫ = 0. Lemma 7.2.
Let ψ ∈ PSH(
X, ω ) . Then for any birational model π : Y → X , − dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ div Y ψ ǫ ≤ X E A X ( E ) E ≤ red div Y ψ + K Y/X , where E runs over all irreducible divisors in div Y ψ .Proof. It follows from (7.4) that the only possible components of − dd ǫ (cid:12)(cid:12) ǫ =0+ div Y ψ ǫ are components ofdiv Y ψ . Also by (7.4), the multiplicity of each component E is bounded by A X ( E ), hence − dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ div Y ψ ǫ ≤ X E A X ( E ) E .
The second inequality is trivial. (cid:3)
Lemma 7.3.
Assume that ψ ∈ PSH(
X, ω ) is I -model and has positive mass. Then V dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ Z X ω nψ ǫ ≤ Ent([ ψ ]) . Proof.
By Theorem 5.2,(7.5) dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ log Z X ω nψ ǫ = dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ log vol ( L − div X ψ ǫ ) ≤ lim Y dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ log vol ( L − div Y ψ ǫ )= n R X ω nψ lim Y (cid:10) π ∗ L − div Y ψ ) n − (cid:11) · − dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ div Y ψ ǫ !! ≤ n R X ω nψ lim Y (cid:0)(cid:10) π ∗ L − div Y ψ ) n − (cid:11) · (cid:0) red div Y ψ + K Y/X (cid:1)(cid:1) = V R X ω nψ Ent([ ψ ]) . Here π : Y → X runs over all birational models of X , the second line follows from the log concavity of themasses of ψ ǫ in ǫ , the third line follows from [BFJ09, Theorem A], the fourth line follows from Lemma 7.2. (cid:3) Theorem 7.4.
Let ψ • ∈ T C ( X, ω ) be an I -model test curve. Let ℓ be the geodesic ray defined by ψ • , then Ent NA ( ℓ NA ) ≤ Ent( ψ • ) . Proof.
We may assume that Ent( ψ • ) < ∞ . We first assume that ψ τ + has positive mass.By Fatou’s lemma, we always have1 V Z ∞−∞ dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ Z X ω nψ ǫτ d τ ≤ lim ǫ → ǫ − Z ∞−∞ (cid:18) V Z X ω nψ ǫτ − V Z X ω nψ τ (cid:19) d τ . On the other hand, since log R X ω nψ ǫτ is concave in ǫ ≥
0, fix ǫ >
0, for ǫ ∈ (0 , ǫ ), we have V − Z ∞−∞ ǫ − (cid:18)Z X ω nψ ǫτ − Z X ω nψ τ (cid:19) d τ ≤ V − Z τ + −∞ ǫ − (cid:18)Z X ω nψ ǫ τ (cid:19) (cid:18) log Z X ω nψ ǫτ − log Z X ω nψ τ (cid:19) d τ . Thus by monotone convergence theorem,lim ǫ → ǫ − Z ∞−∞ (cid:18) V Z X ω nψ ǫτ − V Z X ω nψ τ (cid:19) d τ ≤ V Z τ + −∞ (cid:18)Z X ω nψ ǫ τ (cid:19) (cid:18)Z X Z X ω nψ τ (cid:19) − dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ Z X ω nψ ǫτ d τ . Let ǫ → ǫ → ǫ − Z ∞−∞ (cid:18) V Z X ω nψ ǫτ − V Z X ω nψ τ (cid:19) d τ ≤ V Z ∞−∞ dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ Z X ω nψ ǫτ d τ . Thus dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ E NA ( ψ ǫ • ) = dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ Z ∞−∞ (cid:18) V Z X ω nψ ǫτ − (cid:19) d τ = 1 V Z ∞−∞ dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ Z X ω nψ ǫτ d τ ≤ Z ∞−∞ Ent([ ψ τ ]) d τ , where the first equality follows from Theorem 6.4. By Theorem 4.9, the non-Archimedean potential associ-ated to ψ ǫ • is just P [ ℓ NA + ǫA X ], hence we can apply Proposition 7.1 to conclude.For a general ψ • , for each δ >
0, we define a new test curve ψ τ that agrees with ψ τ when τ ≤ τ + − δ andequals −∞ otherwise. We apply the previous step and the fact that Ent NA ( • ) is lsc. (cid:3) The same proof actually yields equality in the case of test configurations.
Corollary 7.5.
Let ( X , L ) be a test configuration of ( X, L ) . Let ℓ be the induced Phong–Sturm geodesic ray,let ψ = ˆ ℓ . Then Ent NA ( ℓ NA ) = Ent ( ℓ ) = Ent( ψ • ) . LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 27
Proof.
The first equality follows from Proposition 3.2. The inequality Ent NA ( ℓ NA ) ≤ Ent( ψ • ) follows fromTheorem 7.4.Now we prove the converse. Replacing L by (1 + δ ) L for a small δ ∈ Q > , we may assume that ψ τ + haspositive mass. We may assume that X = P b E E is snc and X dominates X × C by a map Π : X → X × C .Let D be a divisor supported on the central fibre and O ( D ) = L − p ∗ L , where p : X × C → X is the naturalmap.Observe that ψ ǫ • is the test curve defined by the (not necessarily finitely generated) Z -filtration F ǫ associ-ated to the model ( X , L + ǫK log X /X × C ). In fact, this follows from [BHJ17, Corollary 4.12], [BFJ16, Theorem 8.5]and (7.2) (see also discussions in [Li20] after Definition 2.7). By [BHJ17, Corollary 4.12], [BHJ17, Proof ofLemma 5.17], F ǫ is given by(7.6) F λǫ H ( X, L k ) = (cid:8) s ∈ H ( X, L k ) : r (ord E )( s ) + k ord E D + kǫA X ( r (ord E )) ≥ b E λ , ∀ E (cid:9) , where E runs over all components of X , r (ord E ) is the restriction of ord E to C ( X ). Recall that r (ord E )is a divisorial valuation ([BHJ17, Section 4.2]). Let π : Y → X be a birational model on which thedivisors corresponding to all r (ord E ) lie and which resolves the singularities of all ψ τ , which is possible byCorollary 6.2. Then π ∗ L − div Y ψ τ is nef by Lemma 2.4. Now by Lemma 4.5 and (7.6),dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ Z X ω nψ ǫτ ≥ dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0+ vol π ∗ L − div Y ψ τ + ǫ X F A X ( F ) F ! = n ( π ∗ L − div Y ψ τ ) n − · X F A X ( F ) F = n ( π ∗ L − div Y ψ τ ) n − · (cid:0) K Y/X + red div Y ψ τ (cid:1) , where F runs over all irreducible components of div Y ψ τ , the second step follows from Theorem 5.1. In thefirst and the last step, we applied the negativity lemma ([KM08, Lemma 3.39]). We conclude by the sameargument as above. (cid:3) Second order expansion of Donaldson’s L -functionals Let X be a compact Kähler manifold of dimension n . Let L be an ample line bundle on X . Let h be asmooth strictly positively-curved Hermitian metric on X . Let ω = c ( L, h ). Lemma 8.1.
Let ( ψ τ ) τ ∈ [0 , be a linear curve in PSH(
X, ω ) with ψ , ψ of quasi-analytic singularities andof positive masses. Assume that ψ is more singular than ψ . Assume that for each prime divisor E over X , either ψ an1 (ord E ) < ψ an0 (ord E ) or ψ an1 (ord E ) = 0 . Then Z (cid:0) h ( X, K X ⊗ L k ⊗ I ( kψ t )) − h ( X, K X ⊗ L k ) (cid:1) d t = Ak n + ( B + B + B ) k n − + o ( k n − ) , where A = 1 n ! Z (cid:18)Z X ω nϕ t − Z X ω n (cid:19) d t ,B = − n − · Z (cid:18)Z X Ric ω ∧ ω n − ϕ t − Z X Ric ω ∧ ω n − (cid:19) d t ,B = 1( n − · Z (cid:0) K Y/X · ( − div Y ϕ t ) n − (cid:1) d t ,B = 1( n − · (cid:16) G n − ( L − div Y ϕ , div Y ϕ − div Y ϕ ) · red(div Y ϕ − div Y ϕ ) (cid:17) . Here π : Y → X is a birational model such that the singularities of ψ and ψ are both determined on Y bya snc Q -divisor.Proof. Recall that ([Dem12, Proposition 5.8]) H ( X, K X ⊗ L k ⊗ I ( kψ τ )) ∼ = H ( Y, K Y ⊗ π ∗ L k ⊗ I ( kπ ∗ ψ τ )) . Hence by Lemma 2.3 and Nadel–Cao’s vanishing theorem ([Cao14, Theorem 1.4]), h ( X, K X ⊗ L k ⊗ I ( kϕ τ )) = χ ( Y, K Y ⊗ π ∗ L k ⊗ I ( kπ ∗ ψ τ )) = χ ( Y, K Y + kL − ⌊ k div Y ψ τ ⌋ ) . Write D τ := div Y ψ τ . Now we compute Z h ( X, K X ⊗ L k ⊗ I ( kψ τ )) d τ = 1 n ! Z ( kπ ∗ L − ⌊ kD τ ⌋ ) n d τ + 1( n − · Z (cid:0) K Y · ( kπ ∗ L − ⌊ kD τ ⌋ ) n − (cid:1) d τ + O ( k n − )= 1 n ! Z ( π ∗ L − D τ ) n d τ k n + 1( n − · Z (cid:0) K Y · ( π ∗ L − D τ ) n − (cid:1) d τ + n Z ( π ∗ L − D τ ) n − · red( D − D )) d τ ! k n − + o ( k n − ) . The last equality follows from Proposition A.1, Proposition A.2. (cid:3)
Now applying Lemma 8.1 to each piece of an analytic test curve, we obtain
Corollary 8.2.
Let ψ • be an analytic piecewise linear test curve, such that ψ τ + has positive mass. Then lim k →∞ k n − (cid:18) L NA k ( ψ • ) − n ! E ( ψ • ) k n (cid:19) = 1 n ! · E R ( ψ • ) + 1 n ! · ψ • ) . Here π : Y → X is a log resolution of all ψ τ where τ runs over the breaking points of ψ • . Conjecture 8.1.
For any ψ ∈ E ( X, ω ) , Ent( ψ ) < ∞ , then we have L k ( ψ ) = 1 n ! E ( ψ ) k n + 1 n ! · E R ( ψ ) + Ent( ψ )) k n − + o ( k n − ) . This conjecture is true when ψ is smooth and ω ψ is strictly positive by the well-known Bergman kernelexpansion ([MM07, Theorem 4.1.1]). We also know that the leading order asymptotic always hold, see[BB10]. See [Ber19, Conjecture 1.3] for a similar conjecture in 1 complex dimension. Corollary 8.3.
Let ℓ ∈ R ∞ ∩ E , NA . Let ψ = ˇ ℓ . Assume that ψ • is analytic and piecewise-linear. Assumethat Conjecture 8.1 holds, then Ent ( ℓ ) ≤ Ent( ψ • ) .Proof. Let ℓ be the geodesic ray induced by the test configuration ( X , L ). By Conjecture 8.1, we know thatfor any t ≥
0, lim k →∞ k n − (cid:18) L k ( ℓ t ) − n ! E ( ℓ t ) k n (cid:19) = 1 n ! · E R ( ℓ t ) + Ent( ℓ t )) . By the convexity of L k (Theorem 3.1), [DX20, Lemma 4.5] and Corollary 8.2, we get E R ( ℓ ) + Ent ( ℓ ) ≤ n ! · k →∞ k n − (cid:18) L NA k ( ψ • ) − n ! E NA ( ψ • ) k n (cid:19) = E R ( ψ • ) + Ent( ψ • ) . We conclude by Theorem 6.7. (cid:3) Stability thresholds
Let X be a compact Kähler manifold of dimension n . Let L be an ample line bundle on X . Fix a smoothstrictly positively-curved Hermitian metric h on X and let ω = c ( L, h ).We refer to Definition 2.12 and Definition 2.13 for the definitions of δ pp and δ ′ . Proposition 9.1.
We always have δ ′ ≥ δ .Proof. Let ψ ∈ PSH(
X, ω ) be an unbounded potential with analytic singularities. Let ℓ be the geodesicray induced by the generalized deformation to the normal cone with respect to ψ (see Section 2.7). ByTheorem 7.4 and Equation (A.2) in the appendix,(9.1) Ent NA (MA( ℓ NA )) ≤ Ent( ψ • ) = 1 V (cid:0) K Y/X · ( − div Y ψ ) n − (cid:1) + nV ( G n − ( L, div Y ψ ) · red div Y ψ ) . By Corollary 6.8, we have ˜ J ( ℓ ) = nV Z (cid:18)Z X ω ∧ ω n − τψ − Z X ω nτψ (cid:19) d τ . LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 29
By our definition (see also [Li20, Proposition 2.38]),˜ J NA ( ℓ NA ) = ˜ J ( ℓ ) . Hence by (3.5) and (3.6), δ ≤ Ent NA (MA( ℓ NA )) E ∗ (MA( ℓ NA )) ≤ ( K Y/X · ( − div Y ψ ) n − ) + n ( G n − ( L, div Y ψ ) · red div Y ψ ) n R (cid:16)R X ω ∧ ω n − τψ − R X ω nτψ (cid:17) d τ . (cid:3) Similarly, we have
Proposition 9.2.
We always have δ pp ≥ δ .Proof. Let ψ ∈ PSH(
X, ω ) be an I -model potential. Let ℓ be the geodesic ray induced by ψ + • . Then byTheorem 7.4, Ent NA (MA( ℓ NA )) ≤ Ent( ψ + • ) . While ˜ J NA ( ℓ ) = ˜ J ( ψ + • ) as in the previous proof. Hence δ ≤ Ent NA (MA( ℓ NA )) E ∗ (MA( ℓ NA )) ≤ Ent( ψ + • )˜ J ( ψ + • ) . We conclude. (cid:3)
Theorem 9.3.
Assume that X is Fano, L = − K X . If δ ≤ , then δ ≥ δ pp .Proof. Assume that δ <
1, by [BLZ19, Proof of Theorem 4.1], δ can be computed by a sequence of extractabledivisors, say E k in the sense that δ = lim k →∞ A X ( E k ) S L ( E k ) . Let π k : ( Y k , ∆ k ) → X be a dlt extraction of E k . Choose ǫ ′ ∈ Q > , so that π ∗ L − ǫ ′ E k is semi-ample. Take m ∈ Z > , so that m ( π ∗ L − ǫ ′ E k ) is base-point free. Take a basis s , . . . , s M of H ( Y k , m ( π ∗ L − ǫ ′ E k )),regarded as a subspace of H ( X, L m ). Let ψ k := 1 m log max i =1 ,...,M | s i | h m . The filtration induced by ψ k on R ( X, L m ) is the same as that defined by ord E k . So the geodesic ray ℓ k inducedby ψ k through the extended deformation to the normal cone construction is the same as the geodesic rayinduced by the filtration of ord E k . By Theorem 6.12 and Corollary 6.8, A X ( E k ) S L ( E k ) ≥ δ pp . Let k → ∞ , we conclude.Now assume that δ = 1 ‡ , by [BLZ19, Theorem 6.7], there is a prime divisor E over X computing δ ( X ).By [BLZ19, Proof of Theorem 4.5], E is extractable. So we can proceed as in the case δ < (cid:3) Remark . By slightly refining the argument, one finds that when δ ≤
1, there is always a qpsh functionwith analytic singularities that computes δ pp . Corollary 9.4.
Assume that X is Fano and L = − K X . Then δ pp ≥ (resp. δ pp > ) iff X is K-semistable(resp. uniformly K-stable). ‡ This part depends essentially on the assumption that X is smooth. Further problems
Minimizers.
Let X be a Fano manifold and L = − K X . We assume that δ <
1. Fix a smooth strictlypositively-curved Hermitian metric h on L . Let ω = c ( L, h ). In this case, it is well-known that δ is equalto the greatest Ricci lower bound R ( X ): R ( X ) := sup { t ∈ [0 ,
1] : ∃ ω ∈ c ( X ) s.t. Ric ω > tω } . This quantity was first explicitly introduced by Rubinstein in [Rub08], [Rub09]. See [Szé11] for furtherresults. This invariant also appears in an implicit form in [Tia92]. One could always solve Aubin’s continuitypath ([Aub84]) for t < R ( X ): ω nϕ t = e F − tϕ t ω n , where F is the Ricci potential of ω : Ric ω − ω = dd c F , R X (exp( F ) − ω n = 0.The following are known about ϕ t :(1) Blowing-up at the limit time: lim t → R ( X ) − sup X ϕ t = ∞ . See [Siu88], [Tia87].(2) There is a proper closed subvariety V ⊆ X , such that on each compact subset of X \ V , for anyincreasing sequence t i → R ( X ) and t i < R ( X ), up to passing to a subsequence, ω nϕ ti converges to 0uniformly ([Tos12]).(3) Tian’s partial C -estimate: let β m,t be the m -th Bergman kernel defined by ω ϕ t . Then there exists m ∈ Z > and C >
0, such that inf X ρ m,t ≥ C − for any t ∈ [0 , R ( X )). See [Szé16], [LS20],[Zha19], [CW20], [Bam18].It follows from the partial C -estimate that for any increasing sequence t i → R ( X ), t i < R ( X ), up tosubtracting a subsequence, there is G ∈ PSH(
X, ω ), such that ϕ t i − sup X ϕ t i → G in L . Moreover, G has the following type of singularities:1 m log N X j =1 λ j | S j | h m , where m ∈ Z > , λ j ∈ (0 , S j ∈ H ( X, K − mX ). Moreover, ω nG = 0. See [MT19] for details. The function G is known as a pluricomplex Green function of X .We make the following conjecture: Conjecture 10.1.
When δ < , the pluricomplex Green function G is a minimizer of δ pp . Moser–Trudinger type inequalities.
Let X be a compact Kähler manifold of dimension n . Let[ ω ] be a Kähler class on X with a representative Kähler form ω . Definition 10.1 ([Zha20, Definition 3.1, Proposition 3.5]) . We define the analytic δ -invariant δ A of [ ω ] as δ A ([ ω ]) := sup (cid:26) λ > Z X e − λ ( ϕ − E ( ϕ )) = O λ (1) for any ϕ ∈ H ( X, ω ) (cid:27) = sup (cid:8) λ > ϕ ) ≥ λ ˜ J ( ϕ ) − O λ (1) for any ϕ ∈ H ( X, ω ) (cid:9) . (10.1)Inequalities as in the first line of (10.1) are known as Moser–Trudinger type inequalities , they were firststudied in [BB11]. See [DGL20] for recent progress in Moser–Trudinger type inequalities. The equality of twolines in (10.1) follows essentially from [BBEGZ16, Proposition 4.11], as explained in [Zha20, Proposition 3.5].It is shown in [Zha20, Proposition 3.11], [RTZ20, Proposition 5.3] that δ A ([ ω ]) ≤ δ ( L ) when [ ω ] = c ( L )for some ample line bundle L . Hence in this case, δ A ≤ δ pp by Proposition 9.2. Moreover, both δ A and our δ pp make sense for a transcendental Kähler class. It is interesting to understand the exact relation between δ A and δ pp . LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 31
Non-Archimedean entropy in terms of test curves.
Let X be a compact Kähler manifold ofdimension n . Let ω be a Kähler form on X . Conjecture 10.2.
Let ℓ ∈ E , NA , let ψ = ˆ ℓ , then Ent NA ( ℓ NA ) = Ent ( ℓ ) = Ent( ψ • ) . Many special cases are known: when [ ω ] is integral, we know that Ent NA ( ℓ NA ) ≤ Ent ( ℓ ) (Proposition 3.2),Ent NA ( ℓ NA ) ≤ Ent( ψ • ) (Theorem 7.4). When ℓ is the Phong–Sturm geodesic ray of some test configuration,both equalities hold (Corollary 7.5).When [ ω ] is not integral, all three terms are still defined, but very few information is known.We summarize the information we know so far about various functionals in Table 1. Table 1.
Comparison of functionalsMaximal geodesic rays NA potentials Test curves Known facts E E NA E All equal E R E NA R E R All equal L NA k ? L NA k First=Third
Ent
Ent NA Ent First ≤ SecondFirst ≤ ThirdThis missing term in Table 1 is very likely to be given by a construction similar to the relative volumein [BE18, (0.1)]. One evidence of this is given by the analogy between [BGM20, Theorem 1.1] and [DX20,Theorem 1.2]. Note that every term on the third column is defined as an integral of some functional of pshsingularities along the test curve.hFinally, let us explain the relation between Conjecture 10.2 to the celebrated Yau–Tian–Donaldson (YTD)conjecture. Up to now, it is well-understood that in order to achieve the variational approach of the YTDconjecture, it suffices to show that for a maximal geodesic ray,
Ent ( ℓ ) is continuous along the approximationof Berman–Boucksom–Jonsson ([BBJ15], [Li20], [CC17], [CC18a], [CC18b]). If Conjecture 10.2 holds, upto some technical subtleties, the problem can be reduced to showing that Ent([ • ]) of a psh singularity iscontinuous along a suitable quasi-equisingular approximation.10.4. A possible partial solution to Conjecture 8.1.
Here we propose an approach to Conjecture 8.1when ϕ ∈ C , ,We ask the following two questions:(1) Is it true that for large enough k , L k is decreasing along the Calabi flow?(2) Does the Calabi flow have (classical) solution for arbitrary C , -initial values?Note that the second question has been conjectured by Chen. It is proved under slightly stronger conditionsby [HZ17].If both conjectures hold, one could run the Calabi flow ϕ t for a C , -initial value ϕ . Then we can applythe Bergman kernel expansion to ϕ ǫ and get at least one inequality in Conjecture 8.1. Appendix A. Integrals and summations
In order to minimize the repetition of computations, we collect here a few results concerning certaincomplicated integrals and summations.Let X be a compact Kähler manifold of dimension n . Let L be a Cartier divisor on X . Let D = X r a r E r , D = X r a r E r be effective Q -divisors on X . Here E r runs over the set of irreducible prime divisors in D and D . Set D t = tD + (1 − t ) D for t ∈ [0 , r , a r > a r . In particular, a r = 0 for any r .We define a polynomial(A.1) G n − ( A, B ) = n − X j =0 j + 1 (cid:18) n − j (cid:19) ( − j (cid:0) A n − − j · B j (cid:1) = 1 nB ( A n − ( A − B ) n ) . When A , B are divisors on X , G n − ( A, B ) is considered as an element in the Chow ring of X . We observethat when A, B , B ∈ R and if we set B t = tB + (1 − t ) B ( t ∈ [0 , Z ( A − B t ) n − d t = G n − ( A − B , B − B ) . Let D = P i a i D i be an effective divisor on X , we setred D = X a i > D i . Proposition A.1.
When the integer k → ∞ , Z ( kL − ⌊ kD t ⌋ ) n d t = Z ( L − D t ) n d t k n + 2 − n ( G n − ( L − D , D − D ) · red( D − D )) k n − + o ( k n − ) . Proof.
First of all, Z ( kL − ⌊ kD t ⌋ ) n d t = Z ( kL − kD t ) n d t + n Z (cid:0) ( kL − kD t ) n − · { kD t } (cid:1) d t + O ( k n − ) . It suffices to compute lim k →∞ Z (cid:0) ( L − D t ) n − · { kD t } (cid:1) d t . Fix r , write a r − a r = A/B , a r = D/B for some positive integers
A, B, D . Then Z (cid:0) ( L − D t ) n − · E r (cid:1) { ka r + kt ( a r − a r ) } d t = Z (cid:0) ( L − D t ) n − · E r (cid:1) (cid:26) kB ( D + tA ) (cid:27) d t . We decompose the integral into the following pieces: ( d − c ) / ( B/kA ) − X α =0 Z c +( α +1)( B/kA ) c + α ( B/kA ) + Z c + Z d , where c = BkA (cid:18)(cid:24) kDB (cid:25) − kDB (cid:19) , d = BAk (cid:22) kB ( D + A ) (cid:23) − DA .
Since we are only interested in the limit as k → ∞ , it suffices to concentrate on only the first part. Weexpand the first part further as the sum of ( j = 0 , . . . , n − I j := (cid:18) n − j (cid:19) ( − j (cid:0) ( L − D ) n − − j · ( D − D ) j · E r (cid:1) ( d − c ) / ( B/kA ) − X α =0 Z c +( α +1)( B/kA ) c + α ( B/kA ) t j (cid:26) kB ( D + tA ) (cid:27) d t . LURIPOTENTIAL-THEORETIC STABILITY THRESHOLDS 33
In order to simplify the notations, we introduce a = A/B .We compute ( d − c ) / ( B/kA ) − X α =0 Z c +( α +1)( B/kA ) c + α ( B/kA ) t j (cid:26) kB ( D + tA ) (cid:27) d t = ( d − c ) ka − X α =0 Z c +( α +1) /kac + α/ka t j ( ka ( t − c ) − α ) d t = 1 j + 2 ka ( d j +20 − c j +20 ) − j + 1 kac (cid:16) d j +10 − c j +10 (cid:17) − j + 1 ( d − c ) ka − X α =0 α (cid:0) ( c + ( α + 1) /ka ) j +1 − ( c + α/ka ) j +1 (cid:1) . Now ( d − c ) ka − X α =0 α (cid:0) ( c + ( α + 1) /ka ) j +1 − ( c + α/ka ) j +1 (cid:1) = j X β =0 (cid:18) j + 1 β (cid:19) ( d − c ) ka − X α =0 α ( c + α/ka ) β (1 /ka ) j +1 − β = j X β =0 (cid:18) j + 1 β (cid:19) β X γ =0 c γ (cid:18) βγ (cid:19) ( d − c ) ka − X α =0 α β − γ +1 (1 /ka ) j +1 − γ = j X β =0 (cid:18) j + 1 β (cid:19) β X γ =0 c γ (cid:18) βγ (cid:19) ( d − c ) β − γ +2 ( β − γ + 2) − ( ka ) β − j +1 − j X β =0 (cid:18) j + 1 β (cid:19) β X γ =0 c γ (cid:18) βγ (cid:19) ( d − c ) β − γ +1 ( ka ) β − j + O ( k − ) . The only terms that contribute to the limit are β = j and β = j −
1. The first is given by( j + 1) ka j X γ =0 c γ (cid:18) jγ (cid:19) ( d − c ) j − γ +2 ( j − γ + 2) − − ( j + 1)2 − j X γ =0 c γ (cid:18) jγ (cid:19) ( d − c ) j − γ +1 + O ( k − )=( j + 1) (cid:18) j + 2 ( d j +20 − c j +20 ) − c j + 1 ( d j +10 − c j +10 ) (cid:19) k − ( j + 1)2 − ( d − c ) d j + O ( k − ) . Similarly, the second term is given by (cid:18) j + 12 (cid:19) j − X γ =0 c γ (cid:18) j − γ (cid:19) ( d − c ) j − γ +1 ( j − γ + 1) − = (cid:18) j + 12 (cid:19) (cid:16) ( j + 1) − (cid:16) d j +10 − c j +10 (cid:17) − j − c ( d j − c j ) (cid:17) . Thus lim k →∞ I j = 12( j + 1) (cid:18) n − j (cid:19) ( − j (cid:0) ( L − D ) n − − j · ( D − D ) j · E r (cid:1) . We conclude. (cid:3)
Fix another divisor K on X . The following proposition is obvious. Proposition A.2. As k → ∞ , Z (cid:0) K · ( kL − ⌊ kD t ⌋ ) n − (cid:1) d t = Z (cid:0) K · ( L − D t ) n − (cid:1) d t k n − + O ( k n − ) . Let D be an effective Z -divisor on X . Proposition A.3. As k → ∞ , k − X j =0 ( kL − jD ) n = Z ( L − tD ) n d t · k n +1 − (cid:16) (( L − D ) n ) − ( L n ) (cid:17) k n + O ( k n − ) . Proof.
We compute k − X j =0 ( kL − jD ) n = n X b =0 k − X j =0 (cid:18) nb (cid:19) k b ( − j ) n − b (cid:0) L b · D n − b (cid:1) = n X b =0 (cid:18) nb (cid:19) k n +1 n − b + 1 ( − n − b (cid:0) L b · D n − b (cid:1) − n − X b =0 (cid:18) nb (cid:19) ( − n − b k n (cid:0) L b · D n − b (cid:1) + O ( k n − ) . (cid:3) Proposition A.4. As k → ∞ , k − X j =0 (cid:0) ( kL − jD ) n − · K (cid:1) = Z (cid:0) ( L − tD ) n − · K (cid:1) k n + O ( k n − ) . Proof.
We compute k − X j =0 (cid:0) ( kL − jD ) n − · K (cid:1) = n − X b =0 k b (cid:18) n − b (cid:19) k − X j =0 ( − j ) n − − b (cid:0) L b · D n − − b · K (cid:1) = k n n − X b =0 (cid:18) n − b (cid:19) n − b ( − n − − b (cid:0) L b · D n − − b · K (cid:1) . (cid:3) EFERENCES 35
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Department of Mathematics, Chalmers Tekniska Högskola, Göteborg
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