The uniform version of Yau-Tian-Donaldson conjecture for singular Fano varieties
aa r X i v : . [ m a t h . DG ] S e p The uniform version of Yau-Tian-Donaldson conjecture forsingular Fano varieties
Chi Li, Gang Tian, Feng Wang
Abstract
We prove the following result: if a Q -Fano variety is uniformly K-stable, then it admits aK¨ahler-Einstein metric. We achieve this by modifying Berman-Boucksom-Jonsson’s strategywith appropriate perturbative arguments and non-Archimedean estimates. The idea of usingthe perturbation is motivated by our previous paper. Contents E NA . . . . . . . . . . . 224.3 Step 3: Perturbed L NA function . . . . . . . . . . . . . . . . . . . . . . . . . . 244.4 Step 4: Uniform Ding-stability of ( Y, B ǫ ) . . . . . . . . . . . . . . . . . . . . . 304.5 Step 5: Completion of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . 31 A Fano variety is defined to be a normal projective variety X such that its anticanon-ical divisor − K X is an ample Q -Cartier divisor. K-(poly)stability of Fano varieties wasintroduced by Tian in [41] and reformulated more algebraically by Donaldson [28]. TheYau-Tian-Donaldson (YTD) conjecture states that a smooth Fano manifold X admits aK¨ahler-Einstein metric if and only if X is K-polystable. Due to many people’s work, thisconjecture has been proved (see [41, 1, 20, 42]).In this paper, we are interested in the generalized Yau-Tian-Donaldson conjecture mean-ing that X is allowed to be singular. There are some previous works [40, 38] and [37] on xtending the YTD conjecture to special classes of singular Fano varieties. Berman’s workin [1] shows that the “only if” part of the conjecture is indeed true for any log Fano pair. Forthe “if” part, Note that, by [39] (see also [36]), a Fano variety X being K-semistable impliesthat X has at worst Klt singularities (see also [36]). We will call such Fano varieties to be Q -Fano varieties.When X has a discrete automorphism group, K-polystability is also called K-stability. Inthis case, the notion of uniform K-stability as defined in [12, 27] is the algebraic correspon-dent to the properness of Mabuchi energy and is a prior a strengthening of the K-stabilitycondition. The uniform K-stability is actually conjectured to be equivalent to K-stability.This is known in the smooth case through the solution of Yau-Tian-Donaldson conjecture.Uniform K-stability has recently been studied extensively. For example Fujita ([31]) provedthat there is a nice valuative criterion for uniform K-stability (see also [10, 15, 32]), andmoreover, uniformly K-stable Fano varieties with a fixed volume are parametrized by a goodmoduli stack (see [16]).In this paper, we will in fact deal with the more general case of log-Fano pairs ( X, D )(see Definition 3.10) and prove the following main result:
Theorem 1.1.
Assume that a log-Fano pair ( X, D ) is uniformly K-stable. Then the Mabuchienergy of ( X, D ) is proper. By the work of [4, 23, 26], if (
X, D ) has a discrete automorphism group, then (
X, D )has a K¨ahler-Einstein metric if and only if the Mabuchi energy is proper over the spaceof finite energy K¨ahler metrics (see Theorem 3.12). Moreover, the latter condition indeedimplies that (
X, D ) is uniformly K-stable (see [1, 13]). So we get the following version ofYau-Tian-Donaldson conjecture.
Corollary 1.2.
Assume that a log-Fano pair Q -Fano variety ( X, D ) has a discrete automor-phism group. Then ( X, D ) has a K¨ahler-Einstein metric if and only if ( X, D ) is uniformlyK-stable. We emphasize that the pair (
X, D ) in the above result is allowed to have any Klt singu-larities. As mentioned above, by the resolution of YTD conjecture in the smooth case, theabove results are known when X is smooth.The above theorem extends the work of Berman-Boucksom-Jonsson in [5, 6] to the classof singular Q -Fano varieties. Indeed our method of proof will be based on the strategyproposed by Berman-Boucksom-Jonsson in [5]. In particular, we depend on various tools frompluripotential theory, non-Archimedean K¨ahler geometry and birational algebraic geometry,but without using Cheeger-Colding-Tian’s theory and partial C -estimates in the originalsolution of the YTD conjecture. However, as explained in [6], there are technical difficultiesin applying their method directly when X is singular (see section 2.1). Here we use someideas of perturbative approach.It’s well known that to solve K¨ahler-Einstein metrics on singular varieties is equivalentto solve some degenerate Monge-Amp`ere equation on a resolution of the variety (see [29, 4]).It’s natural to study such degenerate Monge-Am`ere equation using an appropriate sequenceof non-degenerate Monge-Amp`ere equations to approximate the original equation, which isthe guiding principle in [37]. The perturbative approach used here is motivated by this ideabut is more on the non-Archimedean side.In the next section, we will first discuss Berman-Boucksom-Jonsson’s variational approachto YTD and our previous work in [37] which uses perturbation arguments to prove YTD fora class of singular Fano varieties. These will serve as comparisons to our new argument to getthe uniform version of YTD in the singular case, which we sketch in section 2.3 highlightingsome new ingredients about convergence of slope and non-Archimedean quantities. In section3, we recall the preliminary materials on space of K¨ahler metrics on singular varieties whichwere developed in [4, 23, 26]. We state the analytic criterion for the existence of KE metrics n singular Fano varieties as studied by Darvas and Di-Nezza-Guedj, and slightly refine itby using the approximation argument of Berman-Darvas-Lu. The reason for doing this isthe observation that the argument to get KE on singular Fano varieties would be easier ifwe know that the properness over Mabuchi energy over space of smooth K¨ahler potentialsimplies the existence of KE (see Remark 3.14 and 4.8). Since the latter is not known, weneed to work more in section 4.1.2 to get the convexity of Mabuchi energy along geodesicsegments connecting less regular positively curved Hermitian metrics. In section 3.3, werecall the definitions of K-stability and its equivalent Ding stability. In section 3.4, we willrecall the non-Archimedean formulation of stability conditions and the valuative criterion foruniform stability. We also observe that the valuative criterion still works when the boundarydivisor is non-effective at least when the ambient space is smooth. In section 4, we prove ourmain results by following the steps as sketched in section 2.3. Acknowledgement:
C. Li is partially supported by NSF (Grant No. DMS-1810867) andan Alfred P. Sloan research fellowship. G. Tian is partially supported by NSF (Grant No.DMS-1607091) and NSFC (Grant No. 11331001). F. Wang is partially supported by NSFC(Grant No.11501501). The first author would like to thank S. Boucksom and M. Jonssonfor helpful conversations, and Y. Liu, C. Xu for useful comments. We would like to thankR. Berman, T. Darvas for communications that help our proof of the convexity of Mabuchienergy, and Di Nezza and V. Guedj for clarifications on regularity of geodesics.
In this section, we first sketch and discuss the variational approach of Berman-Boucksom-Jonsson (BBJ) [5, 6] and the perturbative approach of Li-Tian-Wang ([37]). Then we sketchour proof which is a modification of BBJ’s approach by instilling some perturbative ideaand convergence results. In the following sketch we will only consider the case when theboundary is empty. We will also use the equivalence of uniform K-stability and uniformDing stability for any Q -Fano variety as proved in [5, 30]. See section 3 for the notationsused in the following sketch. We first sketch Berman-Boucksom-Jonsson’s proof of the smooth case of Theorem 1.1. As-sume a smooth Fano manifold X is uniformly K-stable. They proved the properness ofMabuchi energy using a proof by contradiction.1. Step 1: Assume on the contrary that the Mabuchi energy M is not proper, then onecan find a destabilizing geodesic ray Φ = { ϕ ( s ) } s ∈ [0 , ∞ ) in E := E ( X, − K X ) such that(a) M and the Ding energy D is decreasing along Φ. In particular, we have D ′∞ (Φ) := lim s → + ∞ D ( ϕ ( s )) s ≤ . (1)(b) With a smooth Hermitian metric ψ ∈ E , we have the following normalizationsup( ϕ ( s ) − ψ ) = 0 , E ψ ( ϕ ( s )) = − s. (2)2. Step 2: For m ≫
1, blow up the multiplier ideal sheaf J ( m Φ) to construct a sequenceof semi-ample test configurations X m whose associated psh-ray and non-Archimedeanmetric will be denoted by Φ m = { ϕ m ( s ) } and Φ NA m . oreover, Demailly’s regularization theorem ([24, Proposition 3.1]) implies that Φ m isless singular than Φ. This together with the monotonicity of E show that E ′∞ (Φ m ) = lim s → + ∞ E ( ϕ m ( s )) s ≥ lim s → + ∞ E ( ϕ ( s )) s =: E ′∞ (Φ) = − . (3)As noted in [6, Corollary 6.7], this may a priori be a strict inequality without knowingthat Φ is a maximal geodesic ray.3. Step 3: Prove the following expansion of L energy along Φ by generalizing [1] and usingthe valuative tools from [11]:lim s → + ∞ L ( ϕ ( s )) s = inf w ∈ W ( A X × C ( w ) − w (Φ)) − L NA (Φ NA ) , (4)where W is the set of C ∗ -invariant divisorial valuations w on X × C with w ( t ) = 1.Moreover, use Demailly’s regularization result and definition of multiplier ideals toprove that: lim m → + ∞ L NA (Φ NA m ) = L NA (Φ NA ) . (5)4. Step 4: Combine (10)-(5) to prove that Φ contradicts the uniform Ding-stability of X ,which is equivalent to the uniform K-stability.As pointed out in [6], a large part of the above arguments in [6] still applies to singular Q -Fano varieties. The difficulty in the singular case lies essentially in applying Demailly’sregularization directly on singular varieties. This regularization result is in general nottrue when the ambient space is singular and Berman-Boucksom-Jonsson suggested to find areplacement of this regularization result for singular varieties. The other difficulty may liein the study of non-Archimedean spaces over singular varieties. Theorem 1.1 has been proved in a special singular case in [37], which we will recall in thissubsection. Let X be any Q -Fano variety. Take a log resolution µ : Y → X such that thereduced exceptional divisor µ − ( X sing ) = P gk =1 E k is a simple normal crossing divisor. TheKlt condition allows one to write down the following identity: K Y = µ ∗ K X + g X k =1 a k E k = µ ∗ K X − g X i =1 b i E ′ i + g X j = g +1 a j E ′′ j , (6)where for i = 1 , . . . , g , E ′ i = E i , b i = − a i ∈ [0 , j = g + 1 , . . . , g , a j > E ′′ j = E j .It’s well known (e.g. [19, Lemma 2.2]) that we may and will assume that there exists alog resolution µ : Y → X such that for some θ k ∈ Q with 0 < θ ≪ k = 1 , . . . , gP := µ ∗ ( − K X ) − g X k =1 θ k E k is positive. (7)We can then rewrite the identity (6) in the following way: − K Y = 11 + ǫ (1 + ǫ ) µ ∗ ( − K X ) − ǫ X k θ k E k ! + X i ( b i + ǫ ǫ θ i ) E ′ i − X j ( a j − ǫ ǫ θ j ) E ′′ j = 11 + ǫ L ǫ + B ǫ , (8) here for simplicity we introduced the following notations for any ǫ ≥ L ǫ := (1 + ǫ ) µ ∗ ( − K X ) − ǫ g X k =1 θ k E k = µ ∗ ( − K X ) + ǫP ; B + ǫ := g X i =1 ( b i + ǫ ǫ θ i ) E ′ i , B − ǫ := g X j = g +1 ( a j − ǫ ǫ θ j ) E ′′ j B ǫ := B + ǫ − B − ǫ . (9)Note that B − = P gj = g +1 a j E j = 0 if and only if − < a k ≤ k = 1 , . . . , g . Oneintermediate result in [37] can be stated as follows: Theorem 2.1 ([37, Theorem 4.11]) . Let X be a Q -Fano variety. Assume that there is a logresolution µ : Y → X satisfying both (7) and B − = 0 . If X is uniformly K-stable, then thereexists a K¨ahler-Einstein metric on X . Let’s very briefly sketch the proof of the above result:1. Prove that (
Y, B ǫ ) is uniformly K-stable or equivalently uniformly Ding-stable. Notethat by assumption B ǫ = B + ǫ ≥ Y, B ǫ ) is a Klt pair. This is achieved by usingthe valuative criterion of uniform K-stability by Fujita.2. Adapt Berman-Boucksom-Jonsson’s result to the logarithmic setting to prove thatthe Mabuchi energy of ( Y, B ǫ ) is proper with slope constants that are uniform withrespect to ǫ . This in particular implies that there exists a K¨ahler-Einstein met-ric ω ǫ := √− ∂ ¯ ∂ϕ ǫ on the Klt pair ( Y, B ǫ ) where e − ϕ ǫ is an Hermitian metric on L ǫ = − (1 + ǫ )( K Y + B ǫ ).3. Prove the convergence of ϕ ǫ as ǫ → + by proving uniform estimates by comparingenergy functionals on X and ( Y, B ǫ ) with some rescaling argument and using uniformSobolev constants of K¨ahler-Einstein metrics with edge cone singularities.The difficulty in this perturbative approach for the general singular case seems more severe.Indeed if B − ǫ >
0, then B ǫ is not effective. Any K¨ahler-Einstein metric on the ineffective pair ( Y, B + ǫ − B − ǫ ), if it exists, would have edge cone singularities of cone angles bigger than 2 π along supp( B − ǫ ). It’s still not clear how to adapt Berman-Boucksom-Jonsson’s variationalapproach to construct such a singular K¨ahler-Einstein metric. For example, when ǫ > C -estimates for conical K¨ahler-Einstein metrics to get a full (K-polystable) versionof Yau-Tian-Donaldson conjecture in the special singular class. We now sketch the argument in our proof of Theorem 1.1. We will use the above notations(and notations from section 3) and prove by contradiction. So assume that the Q -Fanovariety X is uniformly K-stable.1. Step 1:Assume on the contrary that the Mabuchi energy M is not proper, then one canfind a destabilizing geodesic ray Φ = { ϕ ( s ) } s ∈ [0 , ∞ ) in E := E ( X, − K X ) such that(a) M and the Ding energy D is decreasing along Φ. In particular, we have D ′∞ (Φ) := lim s → + ∞ D ( ϕ ( s )) s ≤ . (10) b) With a smooth Hermitian metric ψ ∈ E , we have the following normalizationsup( ϕ ( s ) − ψ ) = 0 , E ψ ( ϕ ( s )) = − s. (11)2. Step 2: Fix a log resolution µ : Y → X satisfying (7). Consider the psh ray on L ǫ = µ ∗ ( − K X ) + ǫP given by Φ ǫ = µ ∗ Φ + ǫp ′∗ ψ P where ψ P is a smooth Hermitian metric on P = µ ∗ ( − K X ) − P gk =1 θ k D k whose cur-vature is a smooth K¨ahler form, and p ′ : Y × C → Y is the projection. Blow up themultiplier ideal sheaf J ( m Φ ǫ ) to construct test configurations ( Y ǫ,m , L ǫ,m ) of ( Y, L ǫ )whose associated psh-ray and non-Archimedan metric is denoted by Φ ǫ,m and Φ NA ǫ,m .Demailly’s regularization result on Y implies that (see (106)): E ′∞ L ǫ (Φ ǫ,m ) = lim s → + ∞ E ( ϕ ǫ,m ( s )) s ≥ lim s → + ∞ E ( ϕ ǫ ( s )) s =: E ′∞ L ǫ (Φ ǫ ) . (12)Moreover, we prove the following convergence (see (107)):lim ǫ → E ′∞ L ǫ (Φ ǫ ) = E ′∞ (Φ) = − . (13)3. Step 3: Prove an expansion of L ( Y,B ǫ ) along any psh ray on ( Y, L ǫ ) by adapting theproof in [5, 6] (see Proposition 4.11):lim s → + ∞ L ( Y,B ǫ ) ( ϕ ǫ ( s )) s = L NA(
Y,B ǫ ) (Φ NA ǫ ) . (14)Use Demailly’s regularization on Y to prove (see (121)):lim m → + ∞ L NA(
Y,B ǫ ) (Φ NA ǫ,m ) = L NA(
Y,B ǫ ) (Φ NA ǫ ) . (15)Moreover we prove the following convergence (see (122)):lim ǫ → L NA(
Y,B ǫ ) (Φ NA ǫ ) = L NA (Φ NA ) . (16)4. Step 4: Prove that the uniform K-stability of X implies the uniform Ding-stability of( Y, B ǫ ) for 0 < ǫ ≪ B ǫ is the not-necessarily effective Q -divisor in (9).5. Step 5: Combine (12)-(16) to prove that Φ ǫ contradicts the uniform Ding-stability of( Y, B ǫ ) for 0 < ǫ ≪ Y . Moreover, the perturbative partis indispensable. In particular, the convergences in the new arguments (13) and (16) arecrucial, and the Step 4 is directly analogous to the first step in [37]. The idea of usingperturbative approach here is suggested by our previous work in [37]. However, instead ofworking with the energy functional on the space of K¨ahler metrics as in [37], we will beworking more on the non-Archimedean side, which is more flexible in some sense due to thebirational-nature of the valuative criterions developed in [15, 30]. Equally important in ourarguments is the observation that some of the non-Archimedean arguments in [15, 6] workwell for the non-effective twisting at hand. Preliminaries
Let Z be an n -dimensional normal projective variety and Q a Weil divisor that is not neces-sarily effective. Assume that L is an ample Q -Cartier divisor. Choose a smooth Hermitianmetric e − ψ on L with a smooth semi-positive curvature form ω = √− ∂ ¯ ∂ψ ∈ πc ( L ).We will use the following spaces:PSH( ω ) := PSH( Z, ω ) = (cid:8) u.s.c. function u ∈ L ( Z ); ω u := ω + √− ∂ ¯ ∂u ≥ (cid:9) ; (17) H ( ω ) := H ( Z, ω ) = PSH( ω ) ∩ C ∞ ( Z ); (18)PSH bd ( ω ) := PSH bd ( Z, ω ) = PSH( ω ) ∩ { bounded functions on Z } ; (19)PSH( L ) := PSH([ ω ]) := { ϕ = ψ + u ; u ∈ PSH( ω ) } ; (20)PSH bd ( L ) := PSH bd ([ ω ]) := { ϕ = ψ + u ; u ∈ PSH bd ( ω ) } . (21)Note that PSH([ ω ]) is equal to the space of positively curved (possibly singular) Hermitianmetrics { e − ϕ = e − ψ − u } on the Q -line bundle L . Rigorously ψ + u is not a globally definedfunction, but rather a collection of local psh functions that satisfy the obvious compatiblecondition with respect to the transition functions of the Q -line bundle. However for thesimplicity of notations, we will abuse this notation.Note that we have weak topology on PSH( ω ) which coincides with the L -topology. If u j converges to u weakly, then sup( u j ) → sup( u ) by Hartogs’ lemma for plurisubharmonicfunctions. Proposition 3.1 ([18, Corollary C]) . For any u ∈ PSH(
Z, ω ) there exists a sequence ofsmooth functions u j ∈ PSH(
Z, ω ) which decrease pointwise on Z so that lim j → + ∞ u j = u on Z . For any u ∈ PSH(
Z, ω ), define: ω nu := lim j → + ∞ { u> − j } (cid:0) ω + √− ∂ ¯ ∂ max( u, − j ) (cid:1) n . (22)We will use the space E of finite energy ω -psh functions (see [33]): E ( ω ) := E ( Z, ω ) = (cid:26) u ∈ PSH(
Z, ω ); Z Z ω nu = Z Z ω n (cid:27) ; (23) E ( ω ) := E ( Z, ω ) = (cid:26) u ∈ E ( Z, ω ); Z Z | u | ω nu < ∞ (cid:27) ; (24) E ( L ) := E ( Z, L ) = (cid:8) ψ + u ; u ∈ E ( Z, ω ) (cid:9) . (25)We have the inclusion PSH bd ( ω ) ⊂ E ( ω ).For any ϕ ∈ PSH([ ω ]) such that ϕ − ψ ∈ E ( L ), we have the following important func-tional: E ( ϕ ) := E ψ ( ϕ ) = 1 n + 1 n X i =0 Z Z ( ϕ − ψ )( √− ∂ ¯ ∂ψ ) n − i ∧ ( √− ∂ ¯ ∂ϕ ) i . (26)Following [4], we endow E with the strong topology. Definition 3.2.
The strong topology on E is defined to as the coarsest refinement of theweak topology such that E is continuous. For any interval I ⊂ R . Denote the Riemann surface D I = I × S = { τ ∈ C ∗ ; s = log | τ | ∈ I } . efinition 3.3 (see [6, Definition 1.3]) . A ω -psh path, or just the psh path, on an openinterval I is a map U = { u ( s ) } : I → PSH( ω ) such that the U ( · , τ ) := U (log | τ | ) is a p ∗ ω -psh function on X × D I . A psh ray (emanating from u ) is a psh path on (0 , + ∞ ) (with lim t → u ( s ) = u ). Note in the literature, psh path (resp. psh ray) are also called subgeodesic(resp. subgeodesic ray).In the above situation, we also say that Φ( s ) = { ψ + u ( s ) } is a psh path (resp. a pshray). We will use geodesics connecting bounded potentials.
Proposition 3.4 ([26, Proposition 1.17]) . Let u , u ∈ PSH bd ( ω ) . Then U = sup (cid:8) u ; u ∈ PSH( Z × D [0 , , p ∗ ω ); U ≤ u , on ∂ ( Z × D [0 , ) (cid:9) . (27) is the unique bounded ω -psh function on Z × D [0 , that is the solution of the Dirichlet problem: ( ω + √− ∂ ¯ ∂U ) n +1 = 0 on Z × D [0 , , U | Z × ∂ D [0 , = u , . (28)We will call Φ = { ϕ ( s ) = ψ + U ( · , s ) } the geodesic segment joining ϕ = ψ + u and ϕ = ψ + u .For finite energy potentials, let u , u ∈ E ( ω ). Let u j , u j be bounded smooth ω -psh func-tions decreasing to u , u (see Proposition 3.1). Let u jt be the bounded geodesic connecting u j to u j . It follows from the maximum principle that j → u jt is non-increasing. Set: u t := lim j → + ∞ u jt . (29)Then U = { u t } is a finite-energy geodesic joining u to u as stated in the following result. Theorem 3.5 ([26, Proposition 4.6], [6, Theorem 1.7]) . For any u , u ∈ E ( ω ) , the pshgeodesic joining them exists, and defines a continuous map U : [0 , → E in the strongtopology. Generalizing Darvas’ result in the smooth case ([22]), the works in [23, 26] showed that E can be characterized as the metric completion of H ( ω ) under a Finsler metric d whichcan be defined as follows. Fix a log resolution µ : Y → Z and a K¨ahler form ω P > Y .Then ω ǫ := µ ∗ ω + ǫω P (30)is a K¨ahler form and one can define Darvas’ Finsler metric d ,ǫ on H ( Z, ω ǫ ). Note that u ∈ H ( Z, ω ) implies u ∈ H ( Y, ω ǫ ). One then defines (see [26, Definition 1.10]) d ( u , u ) = lim inf ǫ → d ,ǫ ( u , u ) . It is known that u j → u in E under the strong topology if and only if d ( u j , u ) = 0.Moreover in this case the Monge-Amp`ere measures ( √− ∂ ¯ ∂ ( ψ + u j )) n converges weakly to( √− ∂ ¯ ∂ ( ψ + u )) n . For any ϕ ∈ PSH([ ω ]) such that ϕ − ψ ∈ E ( L ), we also have the following well-studiedfunctionals: J ( ϕ ) := J ψ ( ϕ ) = Z Z ( ϕ − ψ )( √− ∂ ¯ ∂ψ ) n − E ψ ( ϕ ) , (31) I ( ϕ ) := I ψ ( ϕ ) = Z Z ( ϕ − ψ ) (cid:0) ( √− ∂ ¯ ∂ψ ) n − ( √− ∂ ¯ ∂ϕ ) n (cid:1) , (32)( I − J )( ϕ ) := ( I − J ) ψ ( ϕ ) = E ψ ( ϕ ) − Z Z ( ϕ − ψ )( √− ∂ ¯ ∂ϕ ) n . (33) wo properties we will use is the monotone and rescaling property of E functional: ϕ ≤ ϕ = ⇒ E ( ϕ ) ≤ E ( ϕ ); E λψ ( λϕ ) = λ n +1 · E ψ ( ϕ ) for any λ ∈ R > . (34)Moreover, we have the well-known inequality:1 n + 1 I ≤ J ≤ nn + 1 I . (35)Let µ : Y → Z be a log resolution of singularities such that µ − Z sing = P k E k is thereduced exceptional divisor, Q ′ := µ − ∗ Q is the strict transform of Q and Q ′ + P k E k hassimple normal crossings. We can write: K Y + Q ′ = µ ∗ ( K Z + Q ) + X k a k E k . (36) Definition 3.6. ( Z, Q ) is said to have sub-Klt singularities if there exists a log resolution ofsingularities as above such that a k > − for all k . If Q is moreover effective, then ( Z, Q ) issaid to have Klt singularities. Fix ℓ ∈ N ∗ such that ℓ ( K Z + Q ) is Cartier. If σ is a nowhere-vanishing holomorphicsection of the corresponding line bundle over a smooth open set U of Z , then there is apull-back meromorphic volume form on µ − ( U ): µ ∗ (cid:18) √− ℓ n σ ∧ ¯ σ (cid:19) /ℓ = Y i | z i | a i dV, (37)where z i are local holomorphic coordinates on Y . If ( Z, Q ) is sub-Klt, then the above volumeform is locally integrable.
Definition 3.7.
Assume L = λ − ( − K Z − Q ) is an ample Q -line bundle for λ > ∈ Q .Let ϕ ∈ PSH bd ( Z, L ) be a bounded Hermitian metric on the Q -line bundle L . The adaptedmeasure of e − ϕ is a globally defined measure: e − λϕ | s Q | := mes ϕ = (cid:18) √− ℓ n σ ∧ ¯ σ (cid:19) /ℓ | σ ∗ | /ℓ ℓ λϕ , (38) where σ ∗ is the dual nowhere-vanishing section of − ℓ ( K Z + Q ) . The Ding- and Mabuchi- functionals on E ( Z, L ) are defined as follows: L ( ϕ ) := L ( Z,Q ) ( ϕ ) = − Vλ · log (cid:18)Z Z e − λϕ | s Q | (cid:19) (39) D ( ϕ ) := D ( Z,Q ) ,ψ ( ϕ ) = D ψ ( ϕ ) = − E ψ ( ϕ ) + L ( Z,Q ) ( ϕ ) (40) H ( ϕ ) := H ( Z,Q ) ,ψ ( ϕ ) = Z Z log | s Q | ( √− ∂ ¯ ∂ϕ ) n e − λψ ( √− ∂ ¯ ∂ϕ ) n (41) M ( ϕ ) := M ( Z,Q ) ,ψ ( ϕ ) = M ψ ( ϕ ) = λ − H ( ϕ ) − ( I − J ) ψ ( ϕ ) . (42) Definition 3.8 ([4, Definition 1.3]) . A positive measure ν on Z is tame if ν puts no masson closed analytic sets and if there is a resolution of singularities µ : Y → Z such that thelift ν Y of ν to Y has L p density for some p > . The following compactness result is very important in the variational approach for solvingMonge-Amp`ere equations using pluripotential theory. heorem 3.9 ([4, Theorem 2.17]) . Let ν be a tame probability measure on Z . For any C > , the following set is compact in the strong topology: (cid:26) u ∈ E ( Z, ω ); sup Z u = 0 , Z Z log ω nu ν ω nu < C (cid:27) . In the rest of this subsection, we will assume that (
Z, Q ) = (
X, D ) is a log Fano pair inthe following sense:
Definition 3.10.
A pair ( X, D ) is called a log Fano pair if − ( K X + D ) is an ample Q -Cartierdivisor, D is effective and ( X, D ) has Klt singularities. We have the following well-known definition:
Definition 3.11.
We say that the energy F ∈ { D , M } is proper (sometimes called coercivein the literature) if there exist γ > and C ∈ R such that for any ϕ ∈ E ( X, L ) F ( ϕ ) ≥ γ · J ( ϕ ) − C. (43)We will use the following analytic criterion for the existence of K¨ahler-Einstein metricson log Fano varieties. Theorem 3.12 ([4], [23], [26]) . Let ( X, D ) be a log Fano pair with a discrete automorphismgroup. Assume L = λ − ( − K X − D ) with λ > ∈ Q . The following conditions are equivalent:(1) The Ding energy D is proper over E ( X, L ) .(2) The Ding energy D is proper over H ( X, L ) .(3) The Mabuchi energy M is proper over E ( X, L ) .(4) ( X, D ) admits a unique K¨ahler-Einstein metric with Ricci curvature λ . For later use, we need a refinement of the above properness for M . Fix a log resolution µ : Y → X and assume that the reduced exceptional divisor is given by µ − X sing = P k E k and write K Y = µ ∗ ( K X + D ) + µ − ∗ D + X k a k E k =: µ ∗ ( K X + D ) + B. (44)Choose a smooth reference metric ψ ∈ H ( X, ω ). By abuse of notations, we will identifyHermitian metrics on L := − ( K X + D ) with their pull-back metric on µ ∗ L . As a consequence,we identify E ( X, ω ) with E ( Y, µ ∗ ω ). So for any ϕ ∈ E ( X, ω ), we have the followingidentities: M ( ϕ ) = Z X log ( √− ∂ ¯ ∂ϕ ) n e − ψ | s D | ( √− ∂ ¯ ∂ϕ ) n + Z X ( ϕ − ψ )( √− ∂ ¯ ∂ϕ ) n − E ψ ( ϕ )= Z Y log ( √− ∂ ¯ ∂ϕ ) n e − ψ | s B | ( √− ∂ ¯ ∂ϕ ) n + Z Y ( ϕ − ψ )( √− ∂ ¯ ∂ϕ ) n − E ψ ( ϕ )= Z Y log ( √− ∂ ¯ ∂ϕ ) n Ω ( √− ∂ ¯ ∂ϕ ) n − Z Y log e − ψ | s B | Ω ( √− ∂ ¯ ∂ϕ ) n + Z Y ( ϕ − ψ )( √− ∂ ¯ ∂ϕ ) n − E ψ ( ϕ ) , (45)where for the last identity we used a fixed smooth volume form Ω on Y . Let s B be the definingsection of the Q -line bundle associated to the divisor B and choose a smooth Hermitian metricon this line bundle. Consider the space:ˆ H ( ω ) := ˆ H ( X, ω ) = { u ∈ PSH bd ( ω ); ( µ ∗ u ) | Y \ B ∈ C ∞ ( Y \ B ) , µ ∗ ( √− ∂ ¯ ∂ ( ψ + u )) n Ω ∈ C ∞ ( Y ) , and there exist α > , C > |√− ∂ ¯ ∂ ( µ ∗ u ) | ω ≤ C | s B | − α on Y \ B (cid:9) . ˆ H ( L ) := ˆ H ( X, L ) = { ψ + u ; u ∈ ˆ H ( X, ω ) } . (46) roposition 3.13. With the same notations as in the above theorem, the following condi-tions are equivalent:(1) The Mabuchi energy M is proper over E ( X, L ) .(2) The Mabuchi energy M is proper over ˆ H ( X, L ) .(3) ( X, D ) admits a unique K¨ahler-Einstein metrics with Ricci curvature λ .Proof. By the above theorem, we just need to show that (2) implies (1). For any ϕ ∈ E ( L ),we only need to show that there exists ϕ j ∈ ˆ H ( X, ω ) such that ϕ j d -converges to ϕ and M ( ϕ j ) → M ( ϕ ). To prove this, we carry verbatim the argument Berman-Darvas-Lu in[8, Proof of Lemma 3.1], which we just sketch here. We can assume that the entropy of( √− ∂ ¯ ∂ϕ ) n is finite. Set g = ( √− ∂ ¯ ∂ϕ ) n Ω and h k = min { k, g } . Then k h k − g k L (Ω) → Z Y h k (log h k )Ω −→ Z Y g (log g )Ω . (47)By using the density of C ∞ ( Y ) in L (Ω) and dominated convergence theorem, there is asequence of positive functions g k ∈ C ∞ ( Y ) such that k g k − h k k L ≤ k and (cid:12)(cid:12)(cid:12)(cid:12)Z Y h k (log h k )Ω − Z Y g k (log g k )Ω (cid:12)(cid:12)(cid:12)(cid:12) ≤ k . (48)As a consequence we get k g − g k k L → Z Y g k (log g k )Ω −→ Z Y g (log g )Ω = Z Y log ( √− ∂ ¯ ∂ϕ ) n Ω ( √− ∂ ¯ ∂ϕ ) n . (49)Using the Calabi-Yau theorem for degenerate complex Monge-Amp`ere equations as provedin [43, 34, 29, 25], we find potentials v k ∈ ˆ H ( ω ) with sup Y v k = 0 and ω nv k = C k · g k Ω. Byconstruction the entropy of ( √− ∂ ¯ ∂ ( ψ + v k )) n converges to the entropy of ( √− ∂ ¯ ∂ϕ ) n .On the other hand, Theorem 3.9 implies that v k converges strongly to some v ∈ E ( Y, ω ) = E ( X, ω ). In particular, the Monge-Amp`ere measures ω nv k converge weakly to ω nv . So weget the identity ω nv = g Ω. By the uniqueness of the solution to the complex Monge-Amp`ereequations, we get v = ϕ − ψ up to a constant. Then the convergence of the other part ofMabuchi energy follows as in the proof from [8, 4.2] (see also [23]). Remark 3.14.
Currently it is not known yet, over a singular Fano variety X , whether theproperness of Mabuchi energy over H ( X, ω ) imply the existence of KE metrics. The difficultyis that it is not clear whether this properness of Mabuchi energy can imply its properness over E ( X, ω ) . Indeed, there is an imprecision in the statement of [23, Theorem 2.2], and we wouldlike to thank T. Darvas for clarifications regarding this point. In this section we recall the definition of test configurations and stability of log Fano varieties.
Definition 3.15 ([41, 28], see also [36]) . Let ( Z, Q, L ) as before.(1) A test configuration of ( Z, L ) , denoted by ( Z , L , η ) or simply by ( Z , L ) , consists of thefollowing data • A variety Z admitting a C ∗ -action, which is generated by a holomorphic vectorfield η , and a C ∗ -equivariant morphism π : Z → C , where the action of C ∗ on C is given by the standard multiplication. A C ∗ -equivariant π -semiample Q -Cartier divisor L on Y such that there is an C ∗ -equivariant isomorphism i η : ( Z , L ) | π − ( C \{ } ) ∼ = ( Z, L ) × C ∗ .Let Q := Q Z denote the closure of Q × C ∗ in Z under the inclusion Q × C ∗ ⊂ Z × C ∗ i η ∼ = Z × C C ∗ ⊂ Z . We say that ( Z , Q , L ) is a test configuration of ( Z, Q, L ) .Denote by ¯ π : ( ¯ Z , ¯ Q , ¯ L ) → P the natural equivariant compactification of ( Z , Q , L ) → C obtained by using the isomorphism i η and then adding a trivial fiber over {∞} ∈ P .(2) A test configuration is called normal if Z is a normal variety. We will always considernormal test configurations in this paper.A test configuration ( Z , Q , L ) is called dominating if there exists a C ∗ -equivariant bi-rational morphism ρ : ( Z , Q ) → ( Z, Q ) × C .Two test configurations ( Z i , Q i , L i ) , i = 1 , are called equivalent, if there exists a family ( Z , Q ) that C ∗ -equivariantly dominates both test configurations via q i : ( Z , Q ) → ( Z i , Q i ) , i = 1 , and satisfies q ∗ L = q ∗ L . Note that any test configuration is equiv-alent to a dominating test configuration.(3) Assume L = λ − ( − K Z − Q ) . For any normal test configuration ( Z , Q , L ) of ( Z, Q, L ) ,define the divisor ∆ ( Z , Q , L ) to be the Q -divisor supported on Z that is given by: ∆ := ∆ ( Z , Q , L ) = − K Z / C − Q − λ · L . (50) (4) Assume that ( Z, Q ) is a log Fano pair and L = λ − ( − K Z − Q ) for some λ > ∈ Q .A test configuration of ( Z, Q, L ) is called a special test configuration, if the followingconditions are satisfied: • Z is normal, and Z is an irreducible normal variety; • L ∼ C λ − ( − K Z / C − Q ) , which is a π -ample Q -Cartier divisor; • ( Z , Z + Q ) has plt singularities. For any (dominating) normal test configuration ( Z , Q , L ) of (cid:0) Z, Q, L = λ − ( − K Z − Q ) (cid:1) ,we attach the following well-known invariants (where V = (2 π ) n L · n ): E NA ( Z , L ) = (cid:0) ¯ L · n +1 (cid:1) n + 1 , (51) J NA ( Z , L ) = (cid:0) ¯ L · ρ ∗ ( L × P ) · n (cid:1) − (cid:0) ¯ L · n +1 (cid:1) n + 1 , (52)CM( Z , Q , L ) = 1 λ K ( ¯ Z , ¯ Q ) / P · ¯ L · n + nn + 1 ¯ L · n +1 , (53) L NA ( Z , Q , L ) = Vλ · (lct ( Z , Q + ∆; Z ) − , (54) D NA ( Z , Q , L ) = − ¯ L · n +1 n + 1 + L NA ( Z , Q , L ) . (55)The following result is now well known: Proposition 3.16 (see [1, 13]) . Let ( Z , Q , L ) be a normal test configuration of ( Z, Q, L ) .Let Φ = { ϕ ( s ) } be a locally bounded and positively curved Hermitian metric on L . Then thefollowing limits hold true: lim s → + ∞ F ( ϕ ( s )) s = F NA ( Z , Q , L ) , (56) where the energy F is any one from { E , J , L , D } . efinition 3.17. (1) ( Z, Q ) is called uniformly K-stable if there exists γ > such that CM( Z , Q , L ) ≥ γ · J NA ( Z , L ) for any normal test configuration ( Z , Q , L ) of ( Z, Q, L ) .(2) ( Z, Q ) is called uniformly Ding-stable if there exists γ > such that D NA ( Z , Q , L ) ≥ γ · J NA ( Z , L ) for any normal test configuration ( Z , Q , L ) of ( Z, Q, L ) .For convenience, we will call γ to be a slope constant. Remark 3.18.
The rescaling parameter λ is included in our discussion to make the laterargument more flexible. By checking the rescaling properties of functionals, it is easy to seethat if the above statement holds for one λ then it holds for any other λ with the same slopeconstant. For any special test configuration ( Z s , Q s , L s ), its CM weight coincides with its D NA invariant, which coincides with the original Futaki invariant of the central fibre (as generalizedby Ding-Tian):CM( Z s , Q s , L s ) = D NA ( Z s , Q s , L s ) = − ( − K ( Z s , Q s ) / P ) · n +1 n + 1 = Fut ( Z s , Q s ) ( η ) . (57)By the work in [5, 30] (see also [36]), to test uniform K-stability, one only needs to teston special test configurations. As a consequence, Theorem 3.19 ([5, 30]) . For a log Fano pair ( X, D ) , ( X, D ) is uniformly K-stable if andonly if ( X, D ) is uniformly Ding-stable. Here we briefly recall the non-Archimedean formulation of K-stability/Ding-stability follow-ing [6, 12, 14]. Let (
Z, Q, L ) be the polarized projective pair as before. We denote by( Z NA , Q NA , L NA ) the Berkovich analytification of ( Z, Q, L ) with respect to the trivial abso-lute value on the ground field C . Z NA is a topological space, whose points can be consideredas semivaluations on Z , i.e. valuations v : C ( W ) ∗ → R on function field of subvarieties W of Z , trivial on C . The topology of Z NA is generated by functions of the form v v ( f ) with f a regular function on some Zariski open set U ⊂ Z . One can show that Z NA is compactand Hausdorff. Let Z div Q be the set of valuations over Z which are of the form λ · ord E whereord E ∈ Z div is a divisorial valuation over Z . Then Z div Q ⊂ Z NA is dense.For any v ∈ Z div Q , let G ( v ) denote the standard Gauss extension: for any f = P i ∈ Z f i t i ∈ C ( Z × C ) with f i ∈ C ( Z ), set G ( v ) X i f i t i ! = min i { v ( f i ) + i } . (58)In this paper, we will identify a non-Archimedean metrics with a function on Z div Q . Definition 3.20.
Let ( Z , L ) be a dominating test configuration of ( Z, L ) with ρ : Z → Z × C being a C ∗ -equivariant morphism. The non-Archimedean metric defined by ( Z , L ) isrepresented by the following function on Z div Q : φ ( Z , L ) ( v ) = G ( v ) ( L − ρ ∗ ( L × C )) . (59) The set of non-Archimedean metrics obtained in this way will be denoted as H NA ( Z, L ) . If ( Z , L ) is obtained as the normalized blowups of ( Z, L ) × C along some flag ideal sheaf I : Z = normalization of Bl I ( Z × C ) , L = π ∗ ( L × C ) − cE (60) for some c ∈ Q > , where π : Z → Z × C is the natural projection and E is the exceptionaldivisor of blowup, then we have: φ I ( v ) := − G ( v )( cE ) = − c · G ( v )( I ) , sup φ I = 0 . (61) efinition 3.21 ([6, section 4]) . A psh ray
Φ = { ϕ ( s ) } has linear growth if lim s → + ∞ s − sup X ( ϕ ( s ) − ψ ) < + ∞ . (62) In this case, we associate a function: Φ NA : Z div Q → R , by setting for any v ∈ Z div Q , Φ NA ( v ) = − G ( v )(Φ) , which is the negative of the generic Lelong number of Φ on suitable blow-up where the centerof G ( v ) is of codimension 1. Example 3.22.
For each normal test configuration ( Z , L ) of ( Z, L ) , one can find a pshray with linear growth Φ ( Z , L ) such that Φ NA( Z , L ) = φ ( Z , L ) . Indeed, it is well-known that for m ≫ sufficiently divisible, one has an equivariantly embedding ι m : Z → P N m × C with N m + 1 = dim C H ( Z, mL ) and ι ∗ m O P Nm (1) ∼ C m L where C ∗ acts on ( P N m , O P Nm (1)) via aone-parameter subgroup in GL ( N m +1 , C ) . Then the psh ray can be chosen to be ι ∗ m (cid:0) p ∗ m ψ FS (cid:1) where ψ FS is the canonical Fubini-Study metric on O P Nm (1) and p is the projection to thefirst factor. For any φ ∈ H NA ( Z, L ), the non-Archimedean functionals can be defined formally as(where V = (2 π ) n L · n ): E NA ( φ ) := E NA L ( φ ) = 1 n + 1 n X j =0 Z X NA φ ( ω NA φ ) j ∧ ( ω NA ) n − j , (63) J NA ( φ ) := J NA L ( φ ) = V · sup φ − E NA ( φ ) , (64) L NA ( φ ) := L NA(
Z,Q ) ( φ ) = Vλ · inf v ∈ Z div Q (cid:0) A ( Z,Q ) ( v ) + λφ ( v ) (cid:1) (65) D NA ( φ ) := D NA(
Z,Q ) ( φ ) = − E NA ( φ ) + L NA ( φ ) . (66)They recover the non-Archimedean functional for test configurations: for functional F ap-pearing in (51)-(55): F NA ( φ ( Z , L ) ) = F NA ( Z , Q , L ) . We will need the valuative criterion for the uniform Ding-stability studied in [15, 30]. Let Z be a projective variety with polarization L . For any divisorial valuation ord E over Z , let µ ′ : Y ′ → Z be a birational morphism such that E is an irreducible Weil divisor on Z . Setvol( L − xE ) := lim m → + ∞ h ( Y ′ , mµ ′∗ L − ⌈ mx ⌉ E ) m n /n ! for any x ∈ R ,S L ( E ) := 1 L n Z + ∞ vol( L − xE ) dx. Following [30, 10, 15], we define the stability threshold as: δ ( Z, Q ) := inf ord E ∈ Z div A ( Z,Q ) ( E ) S − K Z − Q ( E ) . (67)In light of the work [15], the following criterion could be derived by using the non-Archimedanversion of the equivalence between properness of Mabuchi energy and Ding energy. Notethat such type of criterion for K-(semi)stability by using valuations first appeared in thefirst author’s work [35] and also in [30]. Theorem 3.23. ([30, 31]) Let ( X, D ) be a log Fano pair. Then ( X, D ) is uniformlyDing-stable if and only if δ ( X, D ) > . . ([6, Theorem 7.3]) Assume ( Y, Q ) is a sub-Klt pair with Y smooth. Then ( Y, Q ) isuniformly Ding-stable if and only if δ ( Y, Q ) > . Moreover, if δ ( Y, Q ) > , then D NA(
Y,Q ) ≥ (cid:0) − δ ( Y, Q ) − /n (cid:1) J NA on H NA ( L ) . Fujita proved the first item using purely algebro-geometric techniques (e.g. MMP as usedin [36]) which also work well for any Q -Fano variety. The proof of the second item is basedon Boucksom-Jonsson’s non-Archimedean formulation ([14, 15]). We emphasize that the keyfeature of the second item is that Q is allowed to be non-effective, and this will be veryimportant for our argument. On the other hand, since in [6] the twisting is assumed to bea Klt current which is by definition quasi-positive, we give its proof following [6, Proof ofTheorem 7.3] to show that it indeed works for the non-effective Q at hand. Proof of 3.23.2.
Because of the remark 3.18, we can assume λ = 1 so that L = − K Y − Q .Assume δ := δ ( Y, Q ) >
1. By assumption A ( Y,Q ) ( v ) ≥ δS L ( v ) for any divisorial valuation v ∈ Y div Q . Pick any φ ∈ H NA ( Y, L ) with sup φ = 0. Since δ ≥ δ − φ ∈ H NA ( Y, L ). Thenby [14, Proposition 7.5], we have the identity: E NA L ( δ − φ ) ≤ V · inf v ∈ Y div Q (cid:0) S L ( v ) + δ − φ ( v ) (cid:1) . (68)Note that here we are working on a smooth Y as in [6]. So we get: L NA(
Y,Q ) ( φ ) = V · inf v ∈ Y div Q (cid:0) A ( Y,Q ) ( v ) + φ ( v ) (cid:1) ≥ V · inf v ∈ Y div Q ( δS L ( v ) + φ ( v )) ≥ δ E NA ( δ − φ ) . So we get: D NA(
Y,Q ) ( φ ) ≥ δ E NA ( δ − φ ) − E NA ( φ ) = δ J NA ( δ − φ ) − J NA ( φ ) ≥ (1 − δ − /n ) J NA ( φ ) . The last inequality is the non-Archimedean version of Ding’s inequality and is proved in [14,Lemma 6.17].Conversely, assume (
Y, Q ) is uniformly Ding-stable. By using the definition 3.17, thereexists γ > φ ∈ H NA ( Y, L ): L NA ( φ ) = D NA ( φ ) + E NA ( φ ) ≥ γ · J NA ( φ ) + E NA ( φ ) . As shown in [6, 15], the above inequality actually holds for any finite energy non-Archimdedeanmetrics. In particular, it holds for φ = φ v which satisfies φ v ( v ) = 0 and1 V E NA ( φ v ) = S L ( v ) , n S L ( v ) ≤ V J NA ( φ v ) ≤ nS L ( v ) . (69)On the other hand, A ( Y,Q ) ( v ) ≥ inf v ∈ Y div Q ( A ( Y,Q ) ( v ) + φ v ( v )) = V L NA ( φ v ). So we easily get: A ( Y,Q ) ( v ) ≥ (1 + γn − ) S L ( v ) . (70) The rest of this paper is to prove Theorem 1.1 following the argument sketched in section2.3. .1 Step 1: Constructing a destabilizing geodesic ray The argument in this section is the same as in [5, 6]. All energy functionals in this step areon X itself as defined in (26)-(42). Recall that by Proposition 3.13, we just need to provethat the Mabuchi energy M = M ψ (see (42)) is proper over ˆ H ( X, L ).Assume on the contrary that M = M ψ is not proper with slope constant γ . Then wecan pick a sequence { u j } ∞ j =1 ∈ ˆ H ( X, ω ) such that ϕ j = ψ + u j satisfies: M ( ϕ j ) ≤ γ J ( ϕ j ) − j. We will choose γ to be small in the last step of proof in section 4.5 to get a contradiction.We normalize ϕ j such that sup( ϕ j − ψ ) = 0. The inequality M ≥ C − n J implies J ( ϕ j ) → + ∞ , and hence E ( ϕ j ) ≤ − J ( ϕ j ) → −∞ .Denote V = (2 π ) n ( − K X − D ) · n . By Proposition 3.4 (see [23, 26]), we can connect ψ and ϕ j by a geodesic segment { ϕ j ( s ) } parametrized so that S j = − E ( ϕ j ) → + ∞ .For any s ∈ (0 , S j ], we have E ( ϕ j ( s )) = − s and sup( ϕ j ( s ) − ψ ) = 0. So J ( ϕ j ( s )) ≤ V · sup( ϕ j ( s ) − ψ ) − E ( ϕ j ( s )) = s ≤ S j and M ( ϕ j ) ≤ γ · S j − j ≤ γS j . By Proposition 4.1, M is convex along the geodesic segment { ϕ j ( s ) } . So M ( ϕ j ( s )) ≤ S j − sS j M ( ψ ) + sS j M ( ϕ j ) ≤ γs + C. (71)Using M ≥ H − n J , we get H ( ϕ j ( s )) ≤ ( γ + n ) s + C . So for any fixed S > s ≤ S , themetrics ϕ j ( s ) lie in the set: K S := { ϕ ∈ E ; sup( ϕ − ψ ) = 0 and H ( ϕ ) ≤ ( γ + n ) s + C } . This is a compact subset of the metric space ( E , d ) by Theorem 3.9 from [4]. So, by arguingas in [5], after passing to a subsequence, { ϕ j ( s ) } converges to a geodesic ray Φ := { ϕ ( s ) } s ≥ in E , uniformly for each compact time interval. { ϕ ( s ) } satisfiessup( ϕ ( s ) − ψ ) = 0 , E ( ϕ ( s )) = − s. (72)Moreover, D ( ϕ ( s )) ≤ M ( ϕ ( s )) ≤ γs + C for s ≥ . (73) In this section we will prove the convexity of Mabuchi energy along geodesic segments con-necting two metrics from ˆ H ( X, ω ) (see (46)), which is needed in the above construction ofdestabilizing geodesic ray. We fix a resolution of singularities µ : Y → X such that µ is anisomorphism over X reg , µ − ( X sing ) = P gk =1 E k is a simple normal crossing divisor and thatthere exist θ k ∈ Q > for k = 1 , . . . , g such that E θ := P gk =1 θ k E k satisfies P := P θ = µ ∗ L − E θ is an ample Q -divisor over Y . There is always such a resolution. We can then choose andfix a smooth Hermitian metric ψ P on P such that √− ∂ ¯ ∂ψ P >
0. Let D ′ = µ − ∗ D be thestrict transform of D under µ . Then we can write: − K Y = µ ∗ ( − ( K X + D )) + µ − ∗ D + X k b k E k = 11 + ǫ ( µ ∗ L + ǫ ( µ ∗ L − E θ ) + D ′ + X k (cid:18) b k + ǫ ǫ θ k (cid:19) E k = 11 + ǫ L ǫ + D ′ + B ′ ǫ = 11 + ǫ L ǫ + B ǫ , (74) here we have set: L ǫ = µ ∗ L + ǫP, B ′ := B ′ ǫ = X k (cid:18) b k + ǫ ǫ θ k (cid:19) E k , B ǫ = D ′ + B ′ ǫ . (75) Proposition 4.1.
Assume ϕ (0) , ϕ (1) ∈ ˆ H ( X, ω ) . Let Φ = { ϕ ( s ) , s ∈ [0 , } be the geodesicjoining ϕ (0) and ϕ (1) as constructed in Proposition 3.4. Then M ψ ( ϕ ( s )) is convex in s ∈ [0 , . The rest of this subsection is devoted to proving this proposition. Fix a smooth volumeform Ω on Y . Recall that we can write the Mabuchi energy on PSH bd ( L ) in the followingform (see (45)): M ψ ( ϕ ) = H ( ϕ ) − ( I − J ) ψ ( ϕ )= Z X log | s D | ( √− ∂ ¯ ∂ϕ ) n e − ψ ( √− ∂ ¯ ∂ϕ ) n − E ψ ( ϕ ) + Z X ( ϕ − ψ )( √− ∂ ¯ ∂ϕ ) n = Z Y log ( √− ∂ ¯ ∂ϕ ) n Ω ( √− ∂ ¯ ∂ϕ ) n − Z Y log e − ψ | s B | Ω ( √− ∂ ¯ ∂ϕ ) n + Z Y ( ϕ − ψ )( √− ∂ ¯ ∂ϕ ) n − E ψ ( ϕ )=: H Ω (( √− ∂ ¯ ∂ϕ ) n ) − T (( √− ∂ ¯ ∂ϕ ) n ) + Z Y ( ϕ − ψ )( √− ∂ ¯ ∂ϕ ) n − E ψ ( ϕ ) , where B = µ − ∗ D + B ′ = D ′ + P k b k E k . Now we want to compare this with the Mabuchienergy on PSH bd ( L ǫ ) for the effective pair ( Y, D ′ ): M ψ ǫ ( ϕ ) = Z Y log ( √− ∂ ¯ ∂ϕ ) n Ω ( √− ∂ ¯ ∂ϕ ) n − E Ric (Ω) ψ ǫ ( ϕ ) + E D ′ ψ ǫ | D ′ ( ϕ | D ′ ) + n ¯ S ǫ E ψ ǫ ( ϕ ) , (76)where ¯ S ǫ := − ( K Y + D ′ ) · L · n − ǫ L · nǫ = (cid:16) ǫ ( L + ǫP ) + B ′ ǫ (cid:17) · ( L + ǫP ) · n ( L + ǫP ) · n − (77)converges to 1 as ǫ → B ′ ǫ is exceptional). Moreover, for any η a smooth (1,1)-formand for any Q -Weil divisor Q on Y , we used the definition of the following functionals: E ηψ ǫ ( ϕ ) = 1 n Z Y ( ϕ − ψ ǫ ) n − X k =0 η ∧ ( √− ∂ ¯ ∂ψ ǫ ) k ∧ ( √− ∂ ¯ ∂ϕ ) n − − k , E Qψ ǫ | Q ( ϕ | Q ) = 1 n n − X k =0 Z Q ( ϕ − ψ ǫ )( √− ∂ ¯ ∂ψ ǫ ) n − − k ∧ ( √− ∂ ¯ ∂ϕ ) k . It is an easy exercise to show that there exists C ǫ ∈ R such that the following identity holdstrue: M ǫ ( ϕ ) + C ǫ = Z Y log | s B | ( √− ∂ ¯ ∂ϕ ) n e − ψ ( √− ∂ ¯ ∂ϕ ) n − ( n + 1 − n ¯ S ǫ ) E ψ ǫ ( ϕ ) + Z Y ( ϕ − ψ ǫ )( √− ∂ ¯ ∂ϕ ) n + nǫ E ω P ψ ǫ ( ϕ ) − n E B ′ ǫ ψ ǫ | B ′ ǫ ( ϕ | B ′ ǫ ) . (78)For simplicity of notations, we denote the right-hand-side of the above identity as:ˆ M ǫ ( ϕ ) = H ν (( √− ∂ ¯ ∂ϕ ) n ) + F ǫ ( ϕ ) , (79) here ν = e − ψ | s B | and the entropy part is given by H ν (( √− ∂ ¯ ∂ϕ ) n ) := Z Y log ( √− ∂ ¯ ∂ϕ ) n ν ( √− ∂ ¯ ∂ϕ ) n . (80)One sees immediately thatˆ M ( ϕ ) = M ( ϕ ) for all ϕ ∈ PSH bd ( L ) . (81)Fix any ϕ ∈ ˆ H ( ω ) with R Y ( ϕ − ψ ) ω n = 0. Set u = ϕ − ψ ∈ PSH bd ( ω ) and g := g ( ϕ ) = ( √− ∂ ¯ ∂ϕ ) n Ω ∈ C ∞ ( Y ) (see (46)). Solve the Monge-Amp`ere equation:( √− ∂ ¯ ∂ ( ψ + ǫψ P ) + √− ∂ ¯ ∂u ǫ ) n = d ǫ · g Ω , Z Y u ǫ ω n = 0 , (82)where d ǫ = (2 π ) n L · nǫ / ( R Y g Ω) convergs to d as ǫ →
0. Then, since ω ǫ = √− ∂ ¯ ∂ ( ψ + ǫψ P )is K¨ahler, there exists a solution u ǫ ∈ C ∞ ( Y ) by [43]. Moreover, we have the followingpartially uniform estimates: Proposition 4.2 (see [29, 25]) . There exist positive constants α > and C > independentof ǫ such that, for any ǫ > , we have the uniform estimates: | u ǫ | ≤ C , | ∂ ¯ ∂u ǫ | ω ≤ C | s E | α , (83) where ω is a K¨ahler metric on Y (see (30) ). Moreover u ǫ converges to u in L ( ω n ) over Y and locally uniformly over Y \ E . Lemma 4.3.
Set ϕ ǫ = ψ ǫ + u ǫ . Then we have the convergence: lim ǫ → E ψ ǫ ( ϕ ǫ ) = E ψ ( ϕ ) . (84) Proof.
First note that PSH( ω ) ⊂ PSH( ω ǫ ) and we can write: E ψ ǫ ( ϕ ǫ ) − E ψ ( ϕ ) = ( E ψ ǫ ( ψ ǫ + u ǫ ) − E ψ ǫ ( ψ ǫ + u )) + ( E ψ ǫ ( ψ ǫ + u ) − E ψ ( ψ + u ))=: P + P . (85)We will prove that both P and P converge to 0 as ǫ →
0. Note that E satisfies the co-cyclecondition. So we have: P = E ψ ǫ + u ( ψ ǫ + u ǫ )= E ψ ǫ + u ( ψ ǫ + u ǫ ) − Z Y ( u ǫ − u )( √− ∂ ¯ ∂ ( ψ ǫ + u )) n + Z Y ( u ǫ − u )(( √− ∂ ¯ ∂ ( ψ ǫ + u ) n )= − J ψ ǫ + u ( ψ ǫ + u ǫ ) + Z Y ( u ǫ − u ) (cid:0) √− ∂ ¯ ∂ ( ψ ǫ + u ) n ) − ( √− ∂ ¯ ∂ ( ψ + u )) n (cid:1) + Z Y ( u ǫ − u )( √− ∂ ¯ ∂ ( ψ + u )) n = − J ψ ǫ + u ( ψ ǫ + u ǫ ) + P ′ + P ′′ . Because u ǫ and u are uniformly bounded with respect to ǫ , and (recall that ψ ǫ = ψ + ǫψ P )( √− ∂ ¯ ∂ ( ψ ǫ + u )) n − ( √− ∂ ¯ ∂ ( ψ + u )) n = ǫ √− ∂ ¯ ∂ψ P ∧ n − X k =0 ( √− ∂ ¯ ∂ ( ψ ǫ + u )) k ∧ ( √− ∂ ¯ ∂ ( ψ + u )) n − − k (86) s a positive measure that converges to 0 weakly as ǫ →
0, we see that P ′ converges to 0 as ǫ →
0. Moreover, because u ǫ converges to u in L ( ω n ), P ′′ converges to 0 as ǫ →
0. So wejust need to consider the first term. By inequality (35), we have: J ψ ǫ + u ( ψ ǫ + u ǫ ) ≤ nn + 1 I ψ ǫ + u ( ψ ǫ + u ǫ )= nn + 1 Z Y (cid:0) ( u ǫ − u )(( √− ∂ ¯ ∂ ( ψ ǫ + u )) n − ( √− ∂ ¯ ∂ ( ψ ǫ + u ǫ )) n (cid:1) When ǫ →
0, the integral R Y ( u ǫ − u )( √− ∂ ¯ ∂ ( ψ ǫ + u )) n = P ′ + P ′′ converges to 0 asbefore. For the other integral, we use the method in [4, Proof of Theorem 2.17] as follows.By H¨older-Young inequality from [4, Proposition 2.15]: Z Y ( u ǫ − u )(( √− ∂ ¯ ∂ ( ψ ǫ + u ǫ )) n ≤ k u ǫ − u k L χ ∗ (Ω) · (cid:13)(cid:13)(cid:13)(cid:13) ( √− ∂ ¯ ∂ ( ψ ǫ + u ǫ )) n Ω (cid:13)(cid:13)(cid:13)(cid:13) L χ (Ω) = k u ǫ − u k L χ ∗ (Ω) · k d ǫ g k L χ (Ω) ≤ C k u ǫ − u k L χ ∗ (Ω) , (87)where χ ( s ) = ( s + 1) log( s + 1) − s , χ ∗ ( s ) = e s − s −
1, and the norm L χ ( ν ) (and similarly L χ ∗ ( ν )) for a measure ν and weight function χ is defined as: k f k L χ ( ν ) := inf (cid:26) λ > , Z Y χ (cid:0) λ − | f | (cid:1) ≤ (cid:27) . (88)To show that k u ǫ − u k L χ ∗ (Ω) converges to 0, by uing the inequality χ ∗ ( t ) ≤ te t , it is thenenough to show that, for any given λ > ǫ → Z Y | u ǫ − u | exp ( λ | u ǫ − u | ) Ω = 0 . (89)By [4, Proposition 1.4], R Y e − λu ǫ Ω and R Y e − λu Ω are uniformly bounded for some con-stant B independent of ǫ . Then (89) follows from the standard H¨older’s inequality and theconvergence of u ǫ → u in L ( ν ) for any tame measure ν (see [4, Proposition 1.4]).Finally, the P part in (85) converges to 0 by using the formula of E : P = Z Y u n X k =0 ( √− ∂ ¯ ∂ ( ψ + ǫψ P + u )) k ∧ ( √− ∂ ¯ ∂ ( ψ + ǫψ P )) n − k − n X k =0 ( √− ∂ ¯ ∂ ( ψ + u )) k ∧ ( √− ∂ ¯ ∂ψ ) n − k ! and the fact the positive measure in the bracket converges to 0 as ǫ → ϕ (0) , ϕ (1) ∈ ˆ H ( ω ) such that u (0) = ϕ (0) − ψ and u (1) = ϕ (1) − ψ satisfy R X u ( i ) ω n = 0 , i = 1 ,
2. Set g i = ( √− ∂ ¯ ∂ϕ ( i )) n Ω ∈ C ∞ ( Y ). We solve the same equationas in (82) to get approximations ϕ ǫ (0) = ψ ǫ + u ǫ (0) and ϕ ǫ (1) = ψ ǫ + u ǫ (1):( √− ∂ ¯ ∂ ( ψ ǫ + √− ∂ ¯ ∂u ǫ ( i )) n = d ǫ,i · g i Ω , Z Y u ǫ ( i ) ω n = 0 , (90)where d ǫ,i = (2 π ) n L · nǫ / ( R Y g i Ω).Let Φ ǫ = { ϕ ǫ ( s ) } s ∈ [0 , be the geodesic segment connecting Hermitian metrics ϕ ǫ (0) , ϕ ǫ (1) ∈H ( ω ǫ ). It is known that Φ ǫ ∈ C , ( Y × D [0 , ) (see [17, 21]). Moreover by [3] ˆ M ǫ ( ϕ ǫ ( s )) isconvex in s ∈ [0 , ǫ k = 2 − k which converges to 0 as k → + ∞ and consider the geodesicsegment ϕ ǫ k ( s ) joining ϕ ǫ k (0) = ϕ − k (0) and ϕ ǫ k (1) = ϕ − k (1). emma 4.4. ϕ ǫ k ( s ) − ψ ǫ k are uniformly bounded with respect to both k and s . As k → + ∞ , ϕ ǫ k ( s ) subsequentially converges pointwisely to the weak geodesic segment ϕ ( s ) connecting ϕ (0) and ϕ (1) . Moreover, E ψ ǫk ( ϕ ǫ k ( s )) = E ψ ( ϕ ( s )) = s. (91) Proof.
Set A = k u ǫ (1) − u ǫ (0) k L ∞ . Then by the definition of geodesics using envelopes (see(27)) and the maximal principle, we have the estimate (see [26, proof of Proposition 1.4]): u ǫ (0) − As ≤ u ǫ ( s ) ≤ u ǫ (0) + As for any s ∈ [0 ,
1] which gives the estimate: k u ǫ ( s ) − u ǫ (0) k ≤ k u ǫ (1) − u ǫ (0) k . (92)So the first statement follows from the uniform boundedness of u ǫ (0) and u ǫ (1) with respectto s .Recall that the entropy part of the Mabuchi energy is given by H ν (( √− ∂ ¯ ∂ϕ ǫ ( s )) n ) := Z Y log ( √− ∂ ¯ ∂ϕ ǫ ( s )) n ν ( √− ∂ ¯ ∂ϕ ǫ ( s )) n . (93)where ν = e − ψ | s B | . For i = 0 ,
1, it is easy to see that H ν (( √− ∂ ¯ ∂ϕ ǫ ( i )) n ) are uniformlybounded with respect to ǫ by using the equation (90).Recall the following formula from (94)-(79)ˆ M ǫ ( ϕ ǫ ( s )) = H ν ((( √− ∂ ¯ ∂ϕ ǫ ( s )) n ) + F ǫ ( ϕ ǫ ( s ))where F ǫ ( ϕ ( s )) = − ( n + 1 − n ¯ S ǫ ) E ψ ǫ ( ϕ ) + Z Y ( ϕ − ψ ǫ )( √− ∂ ¯ ∂ϕ ) n + nǫ E ω P ψ ǫ ( ϕ ) − n E B ′ ψ ǫ | B ′ ( ϕ | B ′ ) . By the above discussion, it is easy to see that ˆ M ǫ ( ϕ ǫ (0)) and ˆ M ǫ ( ϕ ǫ (1)) are uniformlybounded. By the convexity s ˆ M ǫ ( ϕ ǫ ( s )), we know that ˆ M ǫ ( ϕ ( s )) is uniformly boundedwith respect to s ∈ [0 , u ǫ ( s ) = ϕ ǫ ( s ) − ψ ǫ is uniformly bounded with respectto ǫ and s , we know that F ǫ ( ϕ ǫ ( s )) is uniformly bounded with respect to ǫ and s . As aconsequence we also get that the entropy H ν (( √− ∂ ¯ ∂ϕ ǫ ( s )) n are uniformly bounded.By the weak compactness of uniformly bounded quasi-psh functions, we know that ϕ ǫ k ( s ) = ϕ − k ( s ) → ˜ ϕ ( s ) in L ( ω n ), after passing to a subsequence. Then we can provethat E ψ ǫk ( ϕ ǫ k ( s )) → E ψ ( ˜ ϕ ( s )) by the same argument as in the proof of the convergence(84).Because ϕ ǫ ( s ) is a geodesic with uniformly bounded potentials, by using [26, Proposition1.4.(ii)] there exist C > ǫ and s such that: k ϕ ǫ ( s ) − ϕ ǫ ( s ) k L ∞ ≤ C | s − s | . (94)So we see that [0 , → ϕ ǫ ( s ) is equicontinuous in L ( ω n ). By Arzel`a-Ascoli, ϕ − k ( s ) sub-sequentially converges uniformly in L ( ω n ) topology to some ω -psh-path ˜ ϕ ( s ) joining ϕ (0)and ϕ (1) (see [6, Proposition 1.4]). In particular, the positive currents p ∗ ω + √− ∂ ¯ ∂ z,s ϕ ǫ converge to a positive current p ∗ ω + √− ∂ ¯ ∂ z,s ˜ ϕ . As a consequence, the positive currents p ∗ ω ǫ + √− ∂ ¯ ∂ z,s ϕ ǫ converge to a positive current p ∗ ω + √− ∂ ¯ ∂ z,s ˜ ϕ . Thus ˜ ϕ is also a ω -pshpath.Because E ψ ( ˜ ϕ ( s )), being the pointwise limit of the affine function E ψ − k ( ϕ − k ( s )), isalso affine in s , by [6, Corollary 1.8] which is also true in the current singular case, we knowthat ˜ ϕ ( s ) is nothing but the geodesic joining ϕ (0) and ϕ (1). emark 4.5. The above proof uses the boundedness of entropy to control the convergenceof geodesic segments. One should also be able to adapt [8, Proof of Proposition 4.3] to provethe above convergence results.
Lemma 4.6.
For any s ∈ [0 , , we have the convergence, after passing to a subsequence: lim ǫ → Z Y ( ϕ ǫ k ( s ) − ψ ǫ k )( √− ∂ ¯ ∂ϕ ǫ k ( s )) n = Z Y ( ϕ ( s ) − ψ )( √− ∂ ¯ ∂ϕ ( s )) n . (95) Proof.
With the above notations, u ǫ = u ǫ ( s ) = ϕ ǫ ( s ) − ψ ǫ . Then we can write: Z Y u ǫ ( √− ∂ ¯ ∂ϕ ǫ ) n − u ( √− ∂ ¯ ∂ϕ ) n = Z Y ( u ǫ − u )( √− ∂ ¯ ∂ϕ ǫ ) n ++ Z Y u (( √− ∂ ¯ ∂ϕ ǫ ) n − ( √− ∂ ¯ ∂ ( ψ ǫ + u )) n )+ Z Y u (( √− ∂ ¯ ∂ ( ψ ǫ + u )) n − ( √− ∂ ¯ ∂ ( ψ + u )) n ) . As in the proof of Lemma 4.3 (see (87)-(89)), letting ǫ = ǫ k = 2 − k we show that the firstintegral on the right-hand-side converges to 0 by using the uniform entropy bound and theweak convergence of u ǫ k to u (by Lemma 4.4). The last integral converges to 0 as ǫ k → α p = Z Y u (( √− ∂ ¯ ∂ϕ ǫ ) p ∧ ( √− ∂ ¯ ∂ ( ψ ǫ + u )) n − p . (96)Using integration by parts and Schwarz inequality, we get: | α p +1 − α p | ≤ C Z Y √− ∂ ( u ǫ − u ) ∧ ¯ ∂ ( u ǫ − u ) ∧ (( √− ∂ ¯ ∂ϕ ǫ ) p ∧ ( √− ∂ ¯ ∂ ( ψ ǫ + u ) n − p − )= C Z Y ( u ǫ − u )(( √− ∂ ¯ ∂ ( ψ ǫ + u )) n − ( √− ∂ ¯ ∂ ( ψ ǫ + u ǫ )) n ) = C · I ψ ǫ + u ( ψ ǫ + u ǫ ) . Then we easy to get that: | α n − α | ≤ n X p =1 | α p − α p − | ! / ≤ C ( I ψ ǫ + u ( ψ ǫ + u ǫ )) / . Finally with ǫ = ǫ k = 2 − k , the same argument as in the proof of 4.3 (see (87)-(89)) showsthat I ψ ǫk + u ( ψ ǫ k + u ǫ k ) converges to 0, after passing to a subsequence. Lemma 4.7.
For any s ∈ [0 , , we have the convergence: lim k → + ∞ E B ′ ψ ǫk | B ′ ( ϕ ǫ k | B ′ ) = 0 . (97) Proof.
Because u ǫ k = ϕ ǫ k − ψ ǫ k is uniformly bounded independent of ǫ , there exists C > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n − X i =0 Z B ′ ( ϕ ǫ k − ψ ǫ k )( √− ∂ ¯ ∂ψ ǫ k ) n − − i ∧ ( √− ∂ ¯ ∂ϕ ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C h L · n − ǫ k , B ′ i . The quantity on the right-hand-side converges to 0 as k → + ∞ because B ′ is exceptional. ompletion of the proof of Proposition 4.1. Under the convergence of geodesics ϕ ǫ ( s ) to ϕ ( s )(from Lemma 4.4), the entropy part of the Mabuchi energy converges for the end points (i.e. s = 0 ,
1) and is lower semicontinuous in the middle (i.e. s ∈ (0 , M ǫ k approximates M at end points: for i = 0 , k → + ∞ ˆ M ǫ k ( ϕ ǫ k ( i )) = ˆ M ( ϕ ( i )) = M ψ ( ϕ ( i )) . (98)For metrics in the middle of geodesics, we have:lim k → + ∞ ˆ M ǫ k ( ϕ ǫ k ( s )) ≥ ˆ M ( ϕ ( s )) = M ψ ( ϕ ( s )) . (99)By the convexity of ˆ M ǫ k ( ϕ ǫ ( s )), we get:ˆ M ǫ k ( ϕ ǫ k ( s )) ≤ S − sS ˆ M ǫ k ( ϕ ǫ (0)) + sS ˆ M ǫ k ( ϕ ǫ ( S )) . (100)The convexity of ˆ M ( ϕ ( s )) follows by letting ǫ k → Remark 4.8.
In Proposition 4.1, when ϕ (0) , ϕ (1) ∈ H ( X, ω ) (i.e. smooth K¨ahler metrics),R. Berman [2] showed me his earlier proof of the convexity along geodesics connecting smoothK¨ahler metrics. In this case, one can directly use the geodesics connecting ϕ (0) + ǫψ P and ϕ (1) + ǫψ P to approximate the geodesic connecting ϕ (0) , ϕ (1) . Then the convergence ofMabuchi energy at the end points ( (99) ) and the lower semicontinuity inequality (100) areeasier to prove because the smoothness of the ϕ ( i )+ ǫψ P and the easier convergence of geodesicsegments.The convexity of Mabuchi energy along geodesics connecting smooth K¨ahler metrics isenough for proving the properness of Mabuchi energy over H ( X, ω ) using the arguments inthis paper. However as discussed in Remark 3.14, there is still a difficulty to get KE metricwith this properness condition. NA As in section 4.1.2, we fix a resolution of singularities µ : Y → X such that µ is an isomor-phism over X reg , µ − ( X sing ) = P gk =1 E k is a simple normal crossing divisor and that thereexist θ k ∈ Q > for k = 1 , . . . , g such that E θ = P gk =1 θ k E k satisfies P := P θ = µ ∗ L − E θ isan ample Q -divisor over Y . Choose and fix a smooth Hermitian metric ψ P on P such that √− ∂ ¯ ∂ψ P >
0. For any ǫ ∈ Q > , define line bundles on Y by L ǫ := (1 + ǫ ) µ ∗ L − ǫE θ = µ ∗ L + ǫP. (101)Then L ǫ is a positive Q -line bundle on Y . Define a smooth reference metric on L ǫ by ψ ǫ = ψ + ǫψ P . In this section we will first construct a sequence of test configurations of( Y, L ǫ ) using the method from [5].Let Φ = { ϕ ( s ) } s ∈ [0 , ∞ ) be a geodesic ray in E ( X, L ) satisfying:sup X ( ϕ ( s ) − ψ ) = 0 , E ψ ( ϕ ( s )) = − s. (102)Denote by p ′ i , i = 1 , Y × C to the two factors. Define a singular and asmooth Hermitian metric on p ′∗ L ǫ byΦ ǫ := ¯ µ ∗ (Φ) + ǫ p ′∗ ( ϕ P ) , Ψ ǫ := p ′∗ ( µ ∗ ψ + ǫψ P ) . (103) here ¯ µ = µ × id : Y × C → X × C . Then √− ∂ ¯ ∂ Φ ǫ ≥ √− ∂ ¯ ∂ Ψ ǫ ≥
0. Consider themultiplier ideals J ( m Φ ǫ ) ⊂ O Y × C which is defined over any open set U ⊂ Y × C as: J ( m Φ ǫ )( U ) = J ( m Φ)( U ) := { f ∈ O Y × C ( U ); w ( f ) + A Y × C ( w ) − mw (Φ) ≥ w on Y × C } . Here we identity Φ with its pulled-back metric ¯ µ ∗ Φ on ¯ µ ∗ ( L × C ). The first identity holdstrue because ψ P is a smooth Hermitian metric on P . Denote Y C := Y × C and consider thefollowing coherent sheaf: F ǫ,m := O Y C ( p ′∗ ( mL ǫ ) ⊗ J ( m Φ ǫ )) . Fix a very ample line bundle H ′ over Y . Then for any 1 ≤ i ≤ n and j ≥
0, we can write: F ǫ,m ⊗ p ∗ H ′ j − i = O Y C ( p ′∗ ( K Y + mµ ∗ L + ( mǫP − K Y − ( n + 1) H ′ ) + ( j + n + 1 − i ) H ′ ) ⊗ J ( m ¯ µ ∗ Φ)) . Because P is positive, for m ≫ ǫ − and sufficiently divisible, mǫP − K Y − ( n + 1) H ′ is anample line bundle on Y . In this case, by Nadel vanishing theorem, for any j ≥ R j ( p ′ ) ∗ ( F ǫ,m ⊗ p ′∗ H ′− j ) = 0 . By the relative Castelnuovo-Mumford criterion, F ǫ,m is p ′ -globally generated.Let π ′ m : Y ǫ,m → Y C denote the normalized blow-up of Y × C along J ( m Φ ǫ ) = J ( m ¯ µ ∗ Φ),with exceptional divisor E ǫ,m and set L ǫ,m := π ′∗ m p ′∗ L ǫ − m E ǫ,m . (104)Then ( Y ǫ,m , L ǫ,m ) is a normal semi-ample test configuration for ( Y, L ǫ ) inducing a non-Archimedean metric φ ǫ,m ∈ H NA ( Y, L ǫ ) and φ ǫ,m ∈ H NA ( Y, L ǫ ) given by: φ ǫ,m ( v ) = − m G ( v )( J ( m ¯ µ ∗ Φ)) (105)for each divisorial valuation v on Y . See Definition 3.20.Let Φ ǫ,m be a locally bounded and positively curved Hermitian metric on L ǫ,m . ByDemailly’s regularization result ([24, Proposition 3.1]), Φ ǫ,m is less singular then Φ ǫ . By themonotonicity of E energy, we get: E ′∞ ψ ǫ (Φ ǫ,m ) := lim s → + ∞ E ψ ǫ ( ϕ ǫ,m ( s )) s ≥ lim s → + ∞ E ψ ǫ ( ϕ ǫ ( s )) s =: E ′∞ ψ ǫ (Φ ǫ ) . (106)The following key observation proves (13). Proposition 4.9.
With the above notations and assuming that Φ satisfies (102) , the follow-ing convergence holds: lim ǫ → E ′∞ ψ ǫ (Φ ǫ ) = lim s → + ∞ E ψ ( ϕ ( s )) s =: E ′∞ ψ (Φ) . (107) roof. Note that ϕ ǫ ( s ) − ψ ǫ = ϕ + ǫψ P − ( ψ + ǫψ P ) = ϕ ( s ) − ψ . So we get: E ψ ( ϕ ( s )) = 1 n + 1 X i Z X ( ϕ ( s ) − ψ )( √− ∂ ¯ ∂ϕ ( s )) i ∧ ( √− ∂ ¯ ∂ψ ) n − i =: f ( s ) E ψ ǫ ( ϕ ǫ ( s )) = 1 n + 1 n X i =0 Z X ( ϕ ǫ ( s ) − ψ ǫ )( √− ∂ ¯ ∂ϕ ǫ ( s )) i ∧ ( √− ∂ ¯ ∂ψ ǫ ) n − i = 1 n + 1 X i Z X ( ϕ ( s ) − ψ )( √− ∂ ¯ ∂ϕ + ǫ √− ∂ ¯ ∂ψ P ) i ∧ ( √− ∂ ¯ ∂ψ + ǫ √− ∂ ¯ ∂ψ P ) n − i = f ( s ) + Z X ( ϕ ( s ) − ψ ) T =: f ǫ ( s ) , where T = T ( ǫ ) is a positive ( n, n )-current which approaches 0 as ǫ →
0. We have thefollowing inequality: √− ∂ ¯ ∂ ( f ǫ ( s )) = Z X ( √− ∂ ¯ ∂ Φ ǫ ) n +1 ≥ , √− ∂ ¯ ∂ ( f ( s )) = Z X ( √− ∂ ¯ ∂ Φ) n +1 = 0 . So f ǫ ( s ) is a convex funtion (by using a standard regularization argument) and f ( s ) = − s is a linear function with respect to s ∈ [0 , + ∞ ).Because ϕ ( s ) − ψ ≤
0, we have f ǫ ( s ) ≤ f ( s ). So we get:lim s → + ∞ f ǫ ( s ) s ≤ lim s → + ∞ f ( s ) s = lim s → + ∞ E ψ ( ϕ ( s )) s = − . (108)On the other hand, it follows from the above expressions of f ǫ ( s ) that for any s ≥ ǫ → f ǫ ( s ) = f ( s ). Fix s ∗ >
0. By the convexity of f ǫ ( s ), we have:lim s → + ∞ f ǫ ( s ) s ≥ f ǫ ( s ∗ ) s ∗ . Letting ǫ →
0, we get lim s → + ∞ f ǫ ( s ) s ≥ f ( s ∗ ) s ∗ = − . (109)Now (107) follows from (108) and (109). NA function Recall that we have the identity (see (74)): K Y + D ′ = µ ∗ ( K X + D ) + g X k =1 a k E k = µ ∗ ( K X + D ) − g X i =1 b i E ′ i + g X j = g +1 a j E ′′ j , where D ′ = µ − ∗ D ; for i = 1 , . . . , g , E ′ i = E i , b i = − a i ∈ [0 , j = g + 1 , . . . , g , a j > E ′′ j = E j . Denote by ⌈ a j ⌉ the round up of a j and { a j } = ⌈ a j ⌉ − a j ∈ [0 , e-write the above identity as: − K Y + X j ⌈ a j ⌉ E j = µ ∗ ( − K X − D ) + D ′ + X i b i E ′ i + X j { a j } E ′′ j = 11 + ǫ (1 + ǫ ) µ ∗ ( − K X − D ) − ǫ X k θ k E k ! + D ′ + X i ( b i + ǫ ǫ θ i ) E ′ i + X j ( { a j } + ǫ ǫ θ j ) E ′′ j = 11 + ǫ ( µ ∗ ( − K X − D )) + ǫP ) + ∆ ǫ , where P = µ ∗ ( − K X − D ) − P k θ k E k and∆ ǫ = D ′ + X i b i E ′ i + X j { a j } E ′′ j + ǫ ǫ X k θ k E k =: ∆ + ǫ ǫ E θ . Note that ∆ ǫ is a simple normal crossing divisor with ⌊ ∆ ǫ ⌋ = 0. For simplicity of notations,we set G := P j ⌈ a j ⌉ E ′′ j and B ǫ = ∆ ǫ − G . Then we have: − K Y = 11 + ǫ ( µ ∗ ( − K X − D ) + ǫP ) + ∆ ǫ − G =: 11 + ǫ L ǫ + B ǫ . (110)Consider the Ding energy (40) associated to this decomposition. Denote V ǫ = (2 π ) n L · nǫ .For any ϕ ǫ ∈ E ( Y, L ǫ ), denote: D ǫ ( ϕ ǫ ) = − E ψ ǫ ( ϕ ǫ ) + L ( Y,B ǫ ) ( ϕ ǫ )where ψ ǫ = ψ + ǫψ P , B ǫ = ∆ ǫ − G and with λ = ǫ in (39)-(40) L ǫ ( ϕ ǫ ) := L ( Y,B ǫ ) ( ϕ ǫ ) = − V ǫ (1 + ǫ ) · log (cid:18)Z Y e − ϕǫ ǫ | s G | | s ∆ ǫ | (cid:19) . (111)The proof of the following lemma is similar to an argument from [4, Proof of Theorem 5.1]( ǫ = 0 case). Lemma 4.10.
With the above notations, let ǫ be sufficiently small such that ⌊ ∆ ǫ ⌋ = 0 .Assume that Φ ǫ = { ϕ ǫ ( s ) } is a psh ray in E ( Y, L ǫ ) . Then L ( Y,B ǫ ) ( ϕ ǫ ( s )) is convex in s = log | t | − .Proof. This essentially follows from Berndtsson’s convexity result from [9]. To see this, byusing (110) we set L ′ = ǫ L ǫ +∆ ǫ = − K Y + G to be a line bundle on Y . Let p ′ : Y × C → C be the natural projection. Then e − Φ ′ := e − Φ ǫ ǫ | s (∆ ǫ ) C | is a positively curved (singular)Hermitian metric on p ′∗ L ′ . Because K Y + L ′ = G = P j ⌈ a j ⌉ E ′′ j is exceptional, we see that H ( K Y + L ′ ) = C · s G C ∼ = C and L ( Y,B ǫ ) ( ϕ ǫ ( s )) is the Bergman kernel of K Y + L ′ with respectto the Hermitian metric e − Φ ′ . Moreover, we have H ( Y, K Y + L ′ ) = H ( Y, K Y + L ǫ + ∆ ǫ ) = 0by the Kawamata-Viehweg vanishing theorem. So all the conditions in [4, Theorem 11.1]are satisfied and, as proved there, Berndtsson’s convexity result implies that L ( Y,B ǫ ) ( ϕ ǫ ( s ))is indeed convex with respect to s = log | t | − . n the following discussion, let W denote the space of C ∗ -invariant divisorial valuationson Y C = Y × C such that w ( t ) = 1, and A Y C ( w ) is the log discrepancy of w over Y C . Thefollowing theorem can be proved by the similar method as [6, Theorem 3.1]. However, sincethere is a non-effective twisting in our case, we will give the details of proof. Proposition 4.11.
Fix ≤ ǫ ≪ . Let Φ ǫ = { ϕ ( s ) } s ∈ [0 , + ∞ ) be a psh ray in E ( Y, L ǫ ) normalized such that sup( ϕ ( s ) − ψ ǫ ) = 0 . With the above notations, we have the identity: V ǫ (1 + ǫ ) · lim s → + ∞ L ( Y,B ǫ ) ( ϕ ( s )) s = inf w ∈ W (cid:18) A Y C ( w ) −
11 + ǫ w (Φ ǫ ) − w ((∆ ǫ ) C ) + w ( G C ) (cid:19) − , (112) where (∆ ǫ ) C = ∆ ǫ × C and G C = G × C .Proof. For simplicity of notations, set ˜Φ := ǫ Φ ǫ . Then w ( ˜Φ) = ǫ w (Φ ǫ ). Since thefunction u ǫ ( t ) := L ( Y,B ǫ ) ( ϕ (log | t | − )) = − log (cid:18)Z Y e − ˜Φ | s G | | s ∆ ǫ | (cid:19) (113)is subharmonic on D by Lemma 4.10, its Lelong number ν at the origin coincides with thenegative of the left-hand-side of (112). We need to show that ν is equal to ζ := sup w ∈ W (cid:16) w ( ˜Φ) + w ((∆ ǫ ) C ) − w ( G C ) − A Y C ( w ) (cid:17) + 1 . (114)By [1, Proposition 3.8], ν is the infimum of all c ≥ Z U e − ( u ǫ ( t )+(1 − c ) log | t | ) √− dt ∧ d ¯ t = Z Y × U e − (˜Φ+(1 − c ) log | t | ) | s G C | | s (∆ ǫ ) C | √− dt ∧ d ¯ t < + ∞ . (115)Set p := ⌊ c ⌋ and r = c − p ∈ [0 , e − (˜Φ+(1 − c ) log | t | ) | s G C | | s (∆ ǫ ) C | √− dt ∧ d ¯ t = | t | p | s G C | e − (˜Φ+(1 − r ) log | t | +log | s (∆ ǫ ) C | ) dV. It follows from [11, Theorem 5.5] that Z Y × U e − (˜Φ+(1 − c ) log | t | ) | s G C | | s (∆ ǫ ) C | √− dt ∧ d ¯ t < + ∞ = ⇒ sup w ∈ W w ( ˜Φ) + w ((∆ ǫ ) C ) + (1 − r ) w ( t ) pw ( t ) + w ( G C ) + A Y C ( w ) ≤ , where w ranges over all divisorial valuations on Y C . By homogeneity and by the S -invarianceof Φ, it suffices to consider w that are C ∗ -invariant and normalized by w ( t ) = 1. We thenget: w (Φ) + 1 ≤ p + r + w ( G C ) − w ((∆ ǫ ) C ) + A Y C ( w ) . (116)So we get ζ ≤ ν .Conversely, [11, Theorem 5.5] shows that:sup w ∈ W w ( ˜Φ) + w ((∆ ǫ ) C ) + 1 − rp + w ( G C ) + A Y C ( w ) < ⇒ Z Y × U e − (Φ+(1 − c ) log | t | ) | s G C | | s (∆ ǫ ) C | √− dt ∧ d ¯ t < + ∞ . o prove ζ ≥ ν , it suffices to show that for any δ > a ≥ ζ + δ , if we let ⌊ a ⌋ = p and r = a − p ∈ [0 , w ∈ W w ( ˜Φ) + w ((∆ ǫ ) C + 1 − rp + w ( G C ) + A Y C ( w ) < . (118)For any w ∈ W , by (114) we have: a = p + r ≥ δ + ζ ≥ δ + w ( ˜Φ) + w ((∆ ǫ ) C ) − w ( G C ) − A Y C ( w ) + 1or equivalently: p + w ( G C ) + A Y C ( w ) ≥ δ + w ( ˜Φ) + w ((∆ ǫ ) C ) + 1 − r So we get: w ( ˜Φ) + w ((∆ ǫ ) C ) + 1 − rp + w ( G C ) + A Y C ( w ) ≤ − δp + w ( G C ) + A Y C ( w ) ≤ − δp + (1 + lct( G C )) A Y C ( w ) . On the other hand, because locally e − ˜Φ 1 | s (∆ ǫ ) C | ∈ L ( Y × D ∗ ) for ǫ ≪
1, by [6, Lemma5.5], there exist α ∈ (0 ,
1) and
C > w ( ˜Φ) + w ((∆ ǫ ) C ) ≤ (1 − α ) A ( w ) + C for any w ∈ W . So we have w ( ˜Φ) + w ((∆ ǫ ) C ) + 1 − rp + w ( G C ) + A Y C ( w ) ≤ w ( ˜Φ) + w ((∆ ǫ ) C ) + 1 A Y C ( w ) ≤ − α + 1 A Y C ( w ) . Now it’s easy to get the inequality (118).In the following discussion, let Φ be the destabilising geodesic ray constructed in section4.1. Then as in section 4.2, for any ǫ > ǫ = Φ + ǫψ P be a pshray in E ( Y, L ǫ ), and φ ǫ := Φ NA ǫ be the associated non-Archimedean metric (see Definition3.21). Let ( Y , L ǫ,m ) be the test configuration of ( Y, L ǫ ) constructed in (104)-(105)), and φ ǫ,m ∈ H NA ( Y, L ǫ ) be the associated non-Archimedean metric (see Definition 3.20).Set L NA(
Y,B ǫ ) ( φ ǫ ) := V ǫ (1 + ǫ ) · inf w ∈ W (cid:18) A Y C ( w ) −
11 + ǫ w (Φ ǫ ) − w ((∆ ǫ ) C + w ( G C ) (cid:19) − V ǫ (1 + ǫ ) · inf v ∈ Y div Q (cid:18) A Y C ( v ) + 11 + ǫ φ ǫ ( v ) − v ( B ǫ ) (cid:19) . With Proposition 4.11, the following result can be proved by the similar argument as in [5].Again since there is a non-effective twisting in our case, we give the details. We need thefollowing inequalities: for any psh ray Φ ǫ on L ǫ and w ∈ W , w ( J ( m Φ ǫ )) ≤ m w (Φ ǫ ) ≤ w ( J ( m Φ ǫ )) + A Y C ( w ) . (120)The first inequality holds because Φ m is less singular than Φ by Demailly’s regularizationresult. The second inequality follows from the definition of multiplier ideal J ( m Φ ǫ ). Proposition 4.12 (see [5]) . We have the identity: lim m → + ∞ L NA(
Y,B ǫ ) ( φ ǫ,m ) = L NA(
Y,B ǫ ) ( φ ǫ ) = lim s → + ∞ L ( Y,B ǫ ) ( ϕ ǫ ( s )) s . (121) roof. The second equality follows from Proposition 4.11. For simplicity of notations, denote T ′ = ǫ ) V ǫ L NA(
Y,B ǫ ) ( φ ǫ ) and T + := 1(1 + ǫ ) V ǫ lim sup m → + ∞ L NA(
Y,B ǫ ) ( φ ǫ,m ) ≥ T − := 1(1 + ǫ ) V ǫ lim inf m → + ∞ L NA(
Y,B ǫ ) ( φ ǫ,m ) . Using (120), we get, for any C ∗ -invariant valuation w on Y C with w ( t ) = 1, − m w ( J ( m Φ)) ≥ − w (Φ) . Adding A Y C ( w ) − w ((∆ ǫ ) C )+ w ( G C ) − ǫ ) V ǫ L NA(
Y,B ǫ ) ( φ ǫ,m ) ≥ T ′ for any m and hence T − ≥ T ′ .On the other hand, for any α >
0, there exists w ∈ W such that A Y C ( w ) − − w (Φ ǫ ) − w ((∆ ǫ ) C ) + v ( G C ) ≤ T ′ + α. So we get the inequality: − m w ( J ( m Φ ǫ )) ≤ (cid:18) − w (Φ ǫ ) + 1 m A Y C ( w ) (cid:19) ≤ T ′ + α − A Y C ( w ) + 1 + w ((∆ ǫ )) − v ( G C ) + 1 m A Y C ( w ) . Taking lim sup as m → + ∞ , we get T + ≤ T ′ + α . Since α > T + ≤ T ′ and hence T + = T − = T ′ as wanted.The following key proposition proves the convergence in (16). Proposition 4.13.
With the above notations, the following convergence holds true: lim ǫ → L NA(
Y,B ǫ ) ( φ ǫ ) = L NA ( φ ) . (122) Proof.
Since w (Φ + ǫψ P ) = w (Φ), by Proposition 4.11, we have: L NA(
Y,B ǫ ) ( φ ǫ ) = inf w ∈ W (cid:18) A Y C ( w ) −
11 + ǫ w (Φ) − w ((∆ ǫ ) C ) + w ( G C ) (cid:19) − . On the other hand, recall that (see (119)) L NA ( φ ) = inf w ∈ W (cid:0) A ( X C ,D C ) ( w ) − w (Φ) (cid:1) − . Note that since A ( X C ,D C ) ( w ) = A Y C ( w )+ w ( K Y C / ( X C ,D C ) ), we have the following identities: A Y C ( w ) −
11 + ǫ w (Φ) − w ((∆ ǫ ) C ) + w ( G C )= A Y C ( w ) − w ((∆ ) C ) + w ( G C ) −
11 + ǫ w (Φ) − ǫ ǫ w (( E θ ) C )= A ( X C ,D C ) ( w ) − w (Φ) + ǫ ǫ w (Φ) − ǫ ǫ w (( E θ ) C ) . This holds for any ǫ ≥
0. For the simplicity of notations, denote the above equivalentquantity by F ǫ ( w ) := A Y C ( w ) − w ((∆ ) C ) + w ( G C ) −
11 + ǫ w (Φ) − ǫ ǫ w (( E θ ) C ) . (123) hen, by definition, for any ǫ ≥ L NA ǫ ( ϕ ǫ ) = I ǫ − I ǫ := inf w ∈ W F ǫ ( w ) . So we need to prove lim ǫ → I ǫ = I .We first show that I ǫ is uniformly bounded for any ǫ ∈ [0 , A Y C ( w ) − w ((∆ ) C ) − w (Φ) − w (( E θ ) C ) ≤ F ǫ ( w ) = A Y C ( w ) − w ((∆ ) C ) + w ( G C ) −
11 + ǫ w (Φ) − ǫ ǫ w (( E θ ) C ) ≤ A Y C ( w ) − w ((∆ ) C ) + w ( G C ) − w (Φ) . We can assume θ i ≪ e − Φ 1 | s ∆0 · s Eθ | ∈ L ( Y × D ∗ ). By [6, Lemma 5.5], thereexist τ ∈ (0 ,
1) and a constant C > w ((∆ ) C ) + w (Φ) + w (( E θ ) C )) ≤ (1 − τ ) A ( w ) + C for all w ∈ W (124)So we easily get that there exists a constant C > ∈ R independent of ǫ ∈ [0 ,
1) such that | I ǫ | ≤ C. So we get I ǫ = inf F ǫ ( w ) ≤ C +1 F ǫ ( w ). Pick any w ∈ W such that F ǫ ( w ) ≤ A Y C ( w ) − w ((∆ ) C ) + w ( G C ) −
11 + ǫ w (Φ) − ǫ ǫ w (( E θ ) C ) < C + 1 . We can estimate, by using (124), that A Y C ( w ) ≤ C + 1 + w ((∆ ) C ) − w ( G C ) + 11 + ǫ w (Φ) + ǫ ǫ w (( E θ ) C ) ≤ C + 1 + w ((∆ ) C ) + w (Φ) + w (( E θ ) C ) ≤ C + 1 + C + (1 − τ ) A ( w ) . This implies A Y C ( w ) ≤ C ′ τ with C ′ = C + 1 + C . If we denote W ′ = { w ∈ W ; A Y C ( w ) ≤ C ′ τ } ,then we get: I ǫ = inf w ∈ W ′ F ǫ ( w ) . (125)For any w ∈ W ′ , we have, by using (124) again, that: F ǫ ( w ) = A Y C ( w ) − w ((∆ ) C ) + w ( G C ) − w (Φ) + ǫ ǫ w (Φ) − ǫ ǫ w (( E θ ) C ) ≤ F ( w ) + ǫ ǫ w (Φ) ≤ F ( w ) + ǫ ǫ ((1 − τ ) A ( w ) + C ) ≤ F ( w ) + ǫ ǫ (cid:18) (1 − τ ) C ′ τ + C (cid:19) . and also: F ǫ ( w ) = A Y C ( w ) − w ((∆ ) C ) + w ( G C ) − w (Φ) + ǫ ǫ w (Φ) − ǫ ǫ w (( E θ ) C ) ≥ A Y C ( w ) − w ((∆ ) C ) + w ( G C ) − w (Φ) − ǫ ǫ w (( E θ ) C ) ≥ F ( w ) − ǫ ǫ ((1 − τ ) A ( w ) + C ) ≥ F ( w ) − ǫ ǫ ((1 − τ ) C ′ τ + C ) . etting C ′′ = (1 − τ ) C ′ τ + C and taking infimum, we get:inf w ∈ W ′ F ǫ ( w ) − C ′′ ǫ ǫ ≤ inf w ∈ W ′ F ( w ) ≤ inf w ∈ W ′ F ǫ ( w ) + ǫ ǫ C ′′ . Now (122) follows by letting ǫ → ( Y, B ǫ ) Recall that we have the following identity from (110) − ( K Y + B ǫ ) = 11 + ǫ ( µ ∗ ( − K X − D ) + ǫP ) = 11 + ǫ L ǫ . (126)where B ǫ = D ′ + X k b k E k + ǫ ǫ X k θ k E k =: B + ǫ ǫ E θ . The following result is analogous to [37, Proposition 3.1]:
Proposition 4.14.
With the above notations, assume that ( X, − K X − D ) is uniformly Dingstable with δ ( X, D ) = δ > . Then there exists a constant C > and a constant ǫ ∗ > such that for any < ǫ ≪ ǫ ∗ , we have the following identity on H NA ( Y, L ǫ ) : D NA(
Y,B ǫ ) = − E NA L ǫ + L NA(
Y,B ǫ ) ≥ (cid:16) − ((1 − Cǫ ) δ ) − /n (cid:17) J NA L ǫ . Proof.
By Theorem 3.23, we just need to show that δ ( Y, B ǫ ) ≥ (1 − Cǫ ) δ . Consider thequantity: Θ( ǫ ) := A ( Y,B ǫ ) ( E )( − K Y − B ǫ ) n R ∞ vol Y ( − K Y − B ǫ − xE ) dx . (127)Then δ ( Y, B ǫ ) := inf E Θ( ǫ ). Moreover, by the definition of δ ( X, D ) in (67), we have:Θ(0) = A ( Y,B ) ( E )( − K Y − B ) n R ∞ vol Y ( − K Y − B − xE ) dx = A X ( E )( − K X − D ) n R ∞ vol( − K X − D − xE ) dx ≥ δ ( X, D ) = δ . So it is enough to prove that Θ( ǫ ) ≥ (1 − Cǫ )Θ(0). Consider the ratio: R ( ǫ ) := Θ( ǫ )Θ(0) = A ( Y,B ǫ ) ( E ) A ( Y,B ) ( E ) · R + ∞ vol Y ( − K Y − B − xE ) dx R ∞ vol Y ( − K Y − B ǫ − xE ) dx · ( − K Y − B ǫ ) n ( − K Y − B ) n = R · R · R . The second ratio R ≥ − B ǫ = − B − ǫ ǫ E θ ≤ − B and volume function isincreasing along effective divisors. The factor R , which does not depend on E , clearly goesto 1 as ǫ →
0. To estimate R , we use the decomposition B = ∆ − G with ⌊ ∆ ⌋ = 0 (see(110)) and estimate as follows: R = A ( Y,B ǫ ) ( E ) A ( Y,B ) ( E ) = A ( Y,B ) ( E ) − ǫ ǫ ord E ( E θ ) A ( Y,B ) ( E )= 1 − ǫ ǫ ord E ( E θ ) A Y ( E ) − ord E (∆ ) + ord E ( G ) ≥ − ǫ ǫ ord E ( E θ ) A Y ( E ) − ord E (∆ ) ≥ − ǫ ǫ (lct( Y, ∆ ; E θ )) − . So R ( ǫ ) ≥ − Cǫ for some C > E . This concludes the proof. .5 Step 5: Completion of the proof With the above notations and preparations, we can complete the proof of our main result.On the one hand, by (72)-(73), L NA ( φ ) = lim s → + ∞ L ( ϕ ( s )) s = lim s → + ∞ D ( ϕ ( s )) s + lim s → + ∞ E ( ϕ ( s )) s ≤ γ − . (128)On the other hand, by Proposition 4.14, we have, with ˜ δ ǫ = 1 − ((1 − Cǫ ) δ ) − /n , L NA(
Y,B ǫ ) ( φ ǫ,m ) = D NA(
Y,B ǫ ) ( φ ǫ,m ) + E NA L ǫ ( φ ǫ,m ) ≥ ˜ δ ǫ J NA L ǫ ( φ ǫ,m ) + E NA L ǫ ( φ ǫ,m )= (1 − ˜ δ ǫ ) E NA L ǫ ( φ ǫ,m ) = ((1 − Cǫ ) δ ) − /n E NA L ǫ ( φ ǫ,m ) ≥ ((1 − Cǫ ) δ ) − /n E ′∞ ψ ǫ (Φ ǫ ) . The second equality uses sup φ ǫ,m = 0 (see (61)). The last inequality uses (106). Taking m → + ∞ and using (121), we get the inequality: L NA(
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