aa r X i v : . [ m a t h . AG ] F e b K-stability and Fujita approximation
Chi Li
Abstract
This note is a continuation to the paper [26]. We derive a formula for non-ArchimedeanMonge-Amp`ere measures of big models. As applications, we derive a positive intersectionformula for non-Archimedean Mabuchi functional, and further reduces the (Aut(
X, L ) )-uniform Yau-Tian-Donaldson conjecture for polarized manifolds to a conjecture on the exis-tence of approximate Zariski decompositions that satisfy some asymptotic vanishing condi-tion. In an appendix, we also verify this conjecture for some of Nakayama’s examples thatdo not admit birational Zariski decompositions. Contents
Let (
X, L ) be a polarized projective manifold. The Yau-Tian-Donaldson (YTD) conjecturepredicts that the existence of constant scalar curvature K¨ahler (cscK) metrics in the K¨ahlerclass c ( L ) is equivalent to a K-stability condition for the pair ( X, L ). The K-stability condi-tion is usually expressed as a positivity condition on the Futaki invariants of test configura-tions. In a recent work [26], it was proved that the existence of cscK metrics is equivalent tothe uniform positivity of Mabuchi slopes along all maximal geodesic rays. Here the maximalgeodesic rays, as introduced by Berman-Boucksom-Jonsson [1], are essentially the geodesicrays in the space of (mildly singular) positive metrics in c ( L ) that can be algebraicallyapproximated by the data of test configurations. It is known that for test configurations,the Mabuchi slopes (of geodesic rays associated to test configurations) are the Futaki in-variants. So our result is of a Yau-Tian-Donaldson type. However the approximability ofMabuchi slopes of (maximal) geodesic rays, is not well-understood yet. In [26], we did apartial comparison between the Mabuchi slope with non-Archimedean Mabuchi functional nd reduced the ( G -)uniform version of YTD conjecture to a non-Archimedean version ofentropy regularization conjecture of Boucksom-Jonsson ([12]).Furthermore we carried out a partial regularization process (based on Boucksom-Favre-Jonsson’s work on Non-Achimedean Calabi-Yau theorems) and proved that K-stability formodel filtrations is a sufficient (and conjecturally also a necessary) condition for the existenceof cscK metrics. By a model filtration, we mean a filtration of the section ring R ( X, L ) = L + ∞ m =0 H ( X, mL ) induced by a model ( X , L ) of ( X, L ). See Definition 2.1 for the definitionof a model, for which the Q -line bundle L is not assumed to be semiample compared to a testconfiguration in the usual definition of K-stability (see [31, 19]). The main goal of this paper isto further reduce Boucksom-Jonsson’s non-Archimedean regularization conjecture and hencethe YTD conjecture to some purely algebro-geometric conjecture about big line bundles (seeConjecture 4.4, or more generally Conjecture 4.7, for the conjectural statements), whichcould be studied even without the background on K-stability or non-Archimedean geometry.More specifically, we will first derive a formula for the non-Archimedean Monge-Amp`eremeasure of big models, which implies a positive intersection formula for the non-ArchimedeanMabuchi functional of model filtrations. We refer to section 2 for definitions of terms in thefollowing statement of our main results. Theorem 1.1.
For any normal and big model ( X , L ) of ( X, L ) , if φ ( X , L ) denotes the asso-ciated non-Archimedean psh metric, then the following statements hold true.(i) If the central fibre is given by X = P Ii =1 b i E i , and x i = r ( b − i ord E i ) is the Shilov pointassociated to E i , then the non-Archimedean Monge-Amp`ere measure of φ ( X , L ) is givenby the formula: MA NA ( φ ( X , L ) ) = I X i =1 b i (cid:0) h ¯ L n i · E i (cid:1) δ x i , (1) where h ¯ L n i ∈ H n,n ( ¯ X ) is the positive intersection product of big line bundles (see section2.2).(ii) The non-Archimedean Mabuchi functional of any big model ( X , L ) is given by: M NA ( X , L ) = h ¯ L n i · (cid:18) K log¯ X / P + Sn + 1 ¯ L (cid:19) . (2)The non-Archimedean Monge-Amp`ere measure on Berkovich spaces were introducedby A. Chambert-Loir [13] and the formula (1) is a generalization of the formula of non-Archimedean Monge-Amp`ere measures for smooth semipositive non-Archimedean metrics.We refer to section 2.2 for the definition of positive intersection numbers that arise in thestudy of restricted volumes of big line bundles.The formula (2), which was announced in [26], generalizes the intersection formula fornon-Archimedean Mabuchi functional of a test configuration ([10, 32]) which coincides withthe CM weight when the central fibre of the test configuration is reduced (see [34, 29, 27]). Asmentioned above, it together with the work in [26] further reduce the proof of YTD conjectureto some algebraic conjecture (Conjecture 4.4). Here for the convenience of the reader werecall the main result from [26], which is the recent progress in the variational approach tothe YTD conjecture (as proposed in [1, 4]) and incorporates the analytic existence result ofChen-Cheng [14]. efinition 1.2. ( X, L ) is uniformly K-stable for models if there exists γ > such that forany model ( X , L ) , we have: M NA ( X , L ) ≥ γ · J NA ( X , L ) (3) where M NA and J NA are given in (23) - (24) . Theorem 1.3 ([26]) . If a polarized manifold ( X, L ) is uniformly K-stable for models, then ( X, L ) admits a cscK metric. We will see that the positive intersection formula (2) implies that it suffices to test theuniform K-stability for the models with reduced central fibres in which case K log¯ X / P = K X / P (see Proposition 3.4).The converse direction of Theorem 1.3 is expected to be true if Aut( X, L ) is discrete. In-deed, it is implied by Conjecture 4.4. Moreover there is a version in the case when Aut( X, L ) is not discrete (see [26] for details). As observed by Y. Odaka, such results can be appliedto get immediately the G-uniform version of Yau-Tian-Donaldson conjecture for polarizedspherical manifolds (see some beautiful refinement by T. Delcroix [16, 17] in this case andRemark 4.14).We end this introduction with the organization of this paper. In section 2.1, we recallthe construction of non-Archimedean psh metrics from models. In section 2.2 we recall theconcepts related to restricted volumes of big line bundles and positive intersection products,and important results from [6, 20] about the relation between them. In section 3, we proveTheorem 1.1. In the section 4, we propose a general conjecture which strengthens the usualFujita approximation theorem and (in the C ∗ -equivariant case) would imply the uniformYTD conjecture for cscK metrics. In the appendix, we verify this algebraic conjecture forsome of Nakayama’s examples that do not admits birational Zariski decompositions. Acknowledgement:
The author is partially supported by NSF (Grant No. DMS-1810867) and an Alfred P. Sloan research fellowship. Part of this paper was written whenthe author was working at Purdue University. I would like to thank members of algebraicgeometry group at Purdue, especially Sai-Kee Yeung, for their interests in this work, andLinquan Ma for helpful discussions. I am grateful to Sebastien Boucksom and Mattias Jons-son for their interest in this work and patient discussions about Conjecture 4.7, especially forsuggesting ways to approach it and pointing out some delicate difficulty (see Remark 4.8).
This paper is a following-up work of [26] and we will mostly follow the notations from thatwork.
Definition 2.1. • A model of ( X, L ) is a flat family of projective varieties π : X → C together with a Q -line bundle L satisfying:(i) There is a C ∗ -action on ( X , L ) such that π is C ∗ -equivariant;(ii) There is a C ∗ -equivariant isomorphism ( X , L ) × C C ∗ ∼ = ( X, L ) × C ∗ . • The trivial model of ( X, L ) is given by ( X × C , L × C ) =: ( X C , L C ) .Two models ( X i , L i ) , i = 1 , are called equivalent if there exists a model ( X , L ) andtwo C ∗ -equivariant birational morphisms µ i : X → X i such that µ ∗ L = µ ∗ L . If we forget about the data L and L , then we say that X is a model of X .If there is a C ∗ -equivariant birational morphism r X , X : X → X for two models X i , i = 1 , , then we say that X dominates X and write X ≥ X . If X ≥ X C , then wesay that X is dominating.If X is normal, we say that X is a normal model. We say a model X is a SNC (i.e.simple normal crossing) if ( X , X red0 ) is a simple normal crossing pair. • Let ( ¯ X , ¯ L ) be the canonical C ∗ -equivariant compactification of ( X , L ) over P by addingthe trivial ( X, L ) at ∞ ∈ P .We say that ( X , L ) is a big model if ¯ L is a big Q -line bundle over ¯ X and the stablebase ideal of m ¯ L is the same as the π -base ideal of m L for m ≫ . In particular, thestable base locus satisfies B ( L ) ⊆ X = π − ( { } ) . (This definition is motivated by [12,Lemma A.6].)In the following for simplicity of notations, if there is no confusion, we also just write ( X , L ) for ( ¯ X , ¯ L ) . • If L is semiample over C , then we call the model ( X , L ) to be a test configuration of ( X, L ) . Remark 2.2.
Rigorously speaking, the model of ( X, L ) should be called the model of ( X × C , L × C ) . In other words, with the language of [11], we used the base change from thetrivially valued case to the discrete valued case.In the original literature of K-stability, which we adopt in this paper, the line bundle L is assumed to be semi-ample. For us this is the only difference between the definition of testconfigurations and models. We refer to [7, 11] for the definition of Berkovich analytification ( X NA , L NA ) of ( X, L )with respect to the trivially valued field C and the definition of non-Archimedean psh metrics L NA which are represented by φ triv -psh functions on X NA (where φ triv is the metric associatedto the trivial test configuration).For each model ( X , L ) of ( X, L ), we can associate a non-Archimedean psh metric φ ( X , L ) in the following way. If b m denotes the π -relative base ideal of m L and µ m : X m → X isthe normalized blowup of b m with the exceptional divisor denoted by ˜ E m , then ( X m , L m = µ ∗ m L− m ˜ E m ) is a semiample test configuration. ( X m , L m ) defines a smooth non-Archimedeanmetric φ ( X m , L m ) ∈ H NA ( L ) and we set φ ( X , L ) = lim m → + ∞ φ ( X m , L m ) . (4)If the base variety X is clear, we just write φ ( X , L ) as φ L . It is easy to see that equivalentmodels define the same non-Archimedean psh metrics. Moreover, if L is semiample, then φ ( X , L ) = φ ( X m , L m ) for m sufficiently divisible.By resolution of singularities, we can assume that X is dominating via a C ∗ -equivariantbirational morphism ρ : X → X C . Write L = ρ ∗ L + D with D supported on X . Then L defines a model function f L on X div Q (the set of divisorial valuations on X ) given by: f L ( v ) = G ( v )( D ) , ∀ v ∈ X div Q (5)where G ( v ) : X div Q → ( X × C ) div Q is the Gauss extension, i.e. G ( v ) is a C ∗ -invariant valuationon X × C that extends v and satisfies G ( v )( t ) = 1. Set ˜ φ L = φ triv + f L . he φ triv -psh upper envelope of f L is defined as: P ( f L )( v ) = sup (cid:8) ( φ − φ triv )( v ); φ ∈ PSH NA ( L ) , φ − φ triv ≤ f L (cid:9) . (6)By [8, Theorem 8.5] we have the identity φ ( X , L ) = φ triv + P ( f L ) =: P ( ˜ φ L ). Moreover, by [8,Theorem 8.3], P ( f L ) is a continuous φ triv -psh function.Because ¯ L is ¯ π -big over the compactification ¯ X ¯ π → P , when c ≫
1, the Q -line bundle¯ L c := ¯ L + c X is big over ¯ X . Moreover, by [7, Lemma A.8], when c ≫
1, the π -relative baseideal of m ¯ L is the same as the absolute base ideal of m ¯ L for all m sufficiently divisible. Inother words we know that ( X , L c ) is a big model in the sense in Definition 2.1. Note thatwe have P ( ˜ φ L ) + c = P ( ˜ φ L c ) = φ triv + P ( f L c ). In this section, we (change the notation and) assume that X is a compact projective manifoldand L is a big line bundle over X of dimensional n + 1. Recall that the volume of L is definedas: vol X ( L ) = lim sup m → + ∞ h ( X , m L ) m n +1 / ( n + 1)! . (7)Denote by N ( X ) = Div( X ) / ≡ the N´eron-Severi group. Then the volume functionalextends to be a continuous function on N ( X ) R = N ( X ) ⊗ Q R . By Fujita’s approximationtheorem, this invariant can be calculated as the movable intersection number of L (see[18, 25]). In other words, if we let µ m : X m → X be the normalized blowup of b ( | mL | ) (orits resolution) with exceptional divisor ˜ E m and set L m = µ ∗ m L − m ˜ E m , thenvol X ( L ) = lim m → + ∞ L n +1 m . (8)As a consequence, the limsup in (7) is indeed a limit.Next we recall the notion of restricted volume ([33, 20, 6]) and the asymptotic intersectionnumber that calculates the restricted volume. Definition 2.3 ([20]) . For any irreducible ( d -dimensional) subvariety Z ⊂ X • The restricted volume of L along Z is defined as vol X | Z ( L ) = lim sup m → + ∞ dim C Im (cid:0) H ( X , m L ) → H ( Z, m L| Z ) (cid:1) m d /d ! . (9) • For any Z B ( L ) (the stable base locus of L ), the asymptotic intersection number of L and Z is defined as: kL d · Z k := lim sup m → + ∞ L dm · ˜ Z m , (10) where ˜ Z m is the strict transform of Z under the normalized blowup µ m : X m → X ofbase ideal of | m L| . Remark 2.4.
It is shown in [20] that the limsup in the formula (9) and (10) are actuallylimits.
Boucksom-Favre-Jonsson [6] proved that the restricted volume is equal to a positive in-tersection product. efinition 2.5 ([6, Definition 2.5]) . Let L be a big Q -line bundle. For any effective divisor D , define: hL n i · D = sup µ,E ( µ ∗ L − E ) n · µ ∗ D, (11) where supremum is taken over all birational morphism µ : ˜ X → X and an effective divisor E such that µ ∗ L − E is nef. If D = P i b i D i with b i ∈ R with D i effective, then we extendthe definition (11) linearly: hL n i · D = X i b i hL n i · D i . Remark 2.6.
In [6], Boucksom-Favre-Jonsson defined positive intersection product h ξ p i forany big class ξ ∈ N ( X ) R and ≤ p ≤ n + 1 , by developing an intersection theory on theRiemann-Zariski space. For example, when p = n + 1 , h ξ n +1 i = vol( ξ ) ; when p = 1 , h ξ i is thecollection of positive parts of divisorial Zariski decomposition of π ∗ ξ for all smooth blowups π : X π → X . We refer to [6] for details on these more general definitions.Moreover an analytic definition of the positive intersection product was defined even ear-lier in [5, Theorem 3.5] (called movable intersection product there). For each semipositiveclass α ∈ H , ( X , R ) , define: hL n i · α = sup T,µ { β n · µ ∗ α } (12) where T ranges over all K¨ahler currents in c ( L ) that have logarithmic poles and µ : ˜ X → X ranges over the set of those log resolutions satisfying µ ∗ T = { E } + β (with { E } an effectivedivisor and β smooth and semipositive). By Poincar´e duality the class hL n i is uniquelydefined as a semipositive class in H n,n ( X , R ) . In the above definitions, we see that the left-hand-side of (11) depends only on thenumerical class of L and D .Recall that the augmented base locus of L is defined as (see [20]): B + ( L ) = \ L = A + E Supp( E ) , (13)where the intersection is over all decompositions of L = A + E into Q -divisors with A ampleand E effective. It is know that the augmented base locus depends only on the numericalclass of L (see [20] and reference therein). We will use the following important results: Theorem 2.7. If L → X is a big line bundle, and Z ⊂ X is a prime divisor, then thefollowing statements are true:1. ([20, Theorem 2.13, Theorem C]) If Z B + ( L ) then vol X | Z ( L ) = kL n · Z k . If Z ⊆ B + ( L ) , then vol X | Z ( L ) = 0 .2. ([6, Theorem B]) There is an identity vol X | Z ( L ) = hL n i· Z . As a consequence, vol X | Z ( L ) depends only on the numerical class of L and Z . As a consequence of these results, we know that in the definition of positive intersectionnumber in (11), it suffices to take the supremum along the sequence µ m : X m → X which isthe normalized blowup of b ( | mL | ) (or its resolution) with exceptional divisor ˜ E m . In otherwords, if we set L m = µ ∗ m L − m ˜ E m , then for any divisor D , we have the identity: hL n i · D = lim m → + ∞ L nm · µ ∗ D. (14) Positive intersection formula
Let π : ( X , L ) → C be a big model of ( X, L ). By resolution of singularities, we can assumethat ( X , X red0 ) is a dominating and SNC model of ( X, L ). From now on, for simplicity ofnotation, we still denote by ( X , L ) its natural compactification over P .Because L is big over X (= ¯ X ) (by the definition of big model), B + ( L ) = X , there existsa fiber X t = π − ( { t } ) for some t ∈ P \ { } such that X t B + ( L ). In particular, X t B ( L ).We then apply Theorem 2.7 to get hL n i · X t = vol X | X ( L ) = kL n · X k = V. (15)Because hL n i · X t depends only on numerical classes of L and X t (see definition 2.5), wecan use X t ≡ X = P Ii =1 b i E i to get: V = hL n i · X t = I X i =1 b i ( hL n i · E i ) . (16)Now we can prove the formula (1) for the non-Archimedean Monge-Amp`ere measure of non-Archimedean metrics associated to model filtrations. This result refines and generalizes [7,Lemma 8.5]. Proof of Theorem 1.1.(i).
We will use the notations in section 2.1. Via the resolution ofsingularity, we can first replace X by any SNC model X ′ that dominates X via π ′ : X ′ → X and replace L by L ′ = π ′∗ L . For simplicity of notations, we will still use the notation ( X , L )instead of ( X ′ , L ′ ).Because the sequence of continuous metrics φ m := φ ( X m , L m ) ∈ H NA increases to thecontinuous metric φ L = φ triv + P ( f L ), by Dini’s theorem we know that φ m converges to φ L uniformly. In particular, φ m converges to φ L in the strong topology and MA NA ( φ m )converges strongly, and hence also weakly, to MA NA ( φ L ).Set ν X ,m = ( r X ) ∗ (MA NA ( φ m )) and ν X = ( r X ) ∗ MA NA ( φ L ) where r X : X → ∆ X is thenatural retraction to the dual complex of X (see [11]). Then they are supported on ∆ X and it is easy to see that ν X ,m converges to ν X weakly. By Portmanteau’s theorem for weakconvergence of measures (see [2, Theorem 2.1]), we have:lim sup m → + ∞ ν X ,m ( { x i } ) ≤ ν X ( { x i } ) . (17)On the other hand, we clearly have ν X ,m ( { x i } ) = ( r X ) ∗ MA NA ( φ m )( { x i } )= MA NA ( φ m )(( r X ) − { x i } ) ≥ MA NA ( φ m )( { x i } ) . So we combine the above two inequalities to get:lim sup m → + ∞ MA NA ( φ m )( { x i } ) ≤ ν X ( { x i } ) =: V i . (18)We consider two cases:1. If E i B + ( L ), then E i is not contained in B ( L ). By the formula of non-ArchimedeanMonge-Amp`ere measures of test configurations (see [11, section 3.4]) we get that: b i L nm · ˜ E i = MA NA ( φ m )( { x i } ) , (19) here ˜ E i is the strict transform of E i under µ m . So by (19) and (10) we getlim sup m → + ∞ MA NA ( φ m )( { x i } ) = lim sup m → + ∞ b i L nm · ˜ E i = b i kL n · E i k . (20)So by Theorem 2.7 and the inequality (18) we have b i hL n i · E i = b i · vol X | E i ( L ) = b i kL n · E i k ≤ V i . (21)2. If E i ⊆ B + ( L ), then b i hL n i · E i = 0 ≤ V i .Combining these with (16), we have: V = X i b i hL n i · E i ≤ X i V i = X i ν X ( { x i } ) ≤ V. So the inequalities in the above chain are actually equalities. So b i hL n i · E i = V i = ν X ( { x i } )for i = 1 , . . . , I and ν X = ( r X ) ∗ MA NA ( φ L ) is supported on the finite set { x i ; i = 1 , . . . , I } .In other words, we have ( r X ) ∗ MA NA ( φ L ) = N X i =1 b i h ( L n i · E i ) δ x i . (22)But we have said that ( X , L ) can be replaced by any SNC model that dominates X . Moreover,the pairs { ( x i , V i ); V i = 0 } do not depend on the choice of such SNC models. By usingthe homemorphism X NA = lim ←− ∆ X , it is then easy to conclude that the Radon measureMA NA ( φ L ) is indeed only supported on the finite set { x i ; i = 1 , . . . , I } and the identity (1)holds true. Remark 3.1.
Although our work on K-stability is the through the study of non-Archimedeangeometry in the trivially valued case (which is base-changed to the discretely valued case,following [11, 12]), the proof of formula for non-Archimedean Monge-Amp`ere measure alsoholds true for more general non-trivially valued case.
We recall that the formula for non-Archimedean functionals following the works in [9, 11,12] (see also [26]). For any continuous continuous psh metric φ on L NA , the non-ArchimedeanMabuchi functional is given by: M NA ( φ ) = H NA ( φ ) + ( E K X ) NA ( φ ) + S E NA ( φ ) (23)where the terms on the right-hand-side are given by the following non-Archimedean integrals: H NA ( φ ) = Z X NA A X ( x )MA NA ( φ ) , ( E K X ) NA ( φ ) = n − X i =0 Z X NA ( φ − φ triv )dd c ψ ∧ MA NA ( φ [ i ]triv , φ [ n − − i ] ) E NA ( φ ) = 1 n + 1 n +1 X i =0 Z X NA ( φ − φ triv )MA NA ( φ [ i ]triv , φ [ n − i ] ) , where in the second identity ψ is a Hermitian metric on K NA X . We also recall the J NA -functional: J NA ( φ ) = L n · sup( φ − φ triv ) − E NA ( φ ) . (24) roposition 3.2. With the above notation, we have: H NA ( φ L ) = hL n i · K log X /X P . (25) Proof.
Note that we have the identity: K log X /X P = K X + X red0 − ( K X P + X ) = X i ( A X P ( E i ) − b i ) E i = X i b i ( A X P ( b − i ord E i ) − E i = X i b i A X ( x i ) E i . So we can use (1) to get the identity: H NA ( φ L ) = Z X NA A X ( x )MA NA ( φ L ) = X i A X ( x i ) b i hL n i · E i = hL n i · K log X /X P . Proposition 3.3.
With the above notation, we have the following identities: R NA ( φ L ) = hL n i · ρ ∗ K X , (26) E NA ( φ L ) = 1 n + 1 hL n +1 i = 1 n + 1 hL n i · L . (27) Proof.
Because φ m converges to φ L strongly, by [11] we have: R NA ( φ L ) = lim m → + ∞ R NA ( φ m ) = lim m → + ∞ L nm · µ ∗ m ρ ∗ K X . (28)Write ρ ∗ K X = A − A with A , A very ample. Moreover we can choose A i , i = 1 , A i , i = 1 , b m for all m . Then the strict transforms of A i , i = 1 , µ m : X m → X are the same as thetotal transform of A i , i = 1 ,
2. By using Theorem 2.7 we see that the right-hand-side of (28)is equal to kL n · A k − kL n · A k = hL n i · ( A − A ) = hL n i · ρ ∗ K X . (29)For the first equality in (27), we can again use φ m = φ ( X m , L m ) (for which (27) is known tobe true) to approximate and directly apply the Fujita approximation result in [25, Theorem11.4.11]. The last equality in (27) follows from the orthogonality property proved in [5,Corollary 4.5] or [6, Corollary 3.6].We can complete the proof the formula for the non-Archimedean Mabuchi functional. Proof of Theorem 1.1.(ii).
The formula (2) follows immediately from the decomposition M NA = H NA + R NA + S E NA and the formula for each part in (25), (26) and (27).As an application of the positive intersection formula, we get: Proposition 3.4.
To check the ( G -) unform K-stability for models (see Definition 1.2 and[26]), it suffices to consider models with reduced central fibres. roof. Let ( X , L ) be any big model. We can take a base change ( X ( d ) , L ( d ) ) = ( X , L ) × C ,t t d C such that its normalization ˜ X has reduced central fibers. Let f : ˜ X → X be the naturalfinite morphism and set ˜ L = f ∗ L . Then we have the identity K log( ˜ X , ˜ X ) := K ˜ X + ˜ X = f ∗ ( K X + X red0 ) = f ∗ K log( X , X ) . It is known that volumes of big line bundles are multiplicative under generic finite mor-phisms (see [21, Lemma 4.3]). So we get the identity h ( ˜ L + ǫK log( ˜ X , ˜ X ) ) n +1 i = d · h ( L + ǫK log( X , X ) ) n +1 i . (30)Taking derivative with respect to ǫ at ǫ = 0, we also get: h ˜ L n i · K log( ˜ X , ˜ X ) = d · hL n i · K log( X , X ) . (31)Moreover (30) for ǫ = 0 gives E NA ( φ ˜ L ) = d · E NA ( φ L ). On the other hand, it is known wehave the formula (see [11])( φ ˜ L − φ triv )( x ) = d · ( φ L − φ triv )( d − x ) , for all x ∈ X NA . (32)So we get the identity J NA ( φ ˜ L ) = d · J NA ( φ L ) by (24). Combining these identities with thepositive intersection formula (2), the statement now follows easily. In view of the above intersection formula, it seems natural to consider the following invariantfor big line bundles.
Definition 4.1.
Let L be a big line bundle over a projective manifold X of dimension n + 1 .The first Riemann-Roch coefficient (1st-RR coefficient) of L is defined to be: r ( X , L ) = hL n i · K X . (33) If the base manifold X is clear, we just write r ( X , L ) as r ( L ) . The zero-th Riemann-Roch coefficient is of course the volume of L : r ( X , L ) := vol X ( L ) = hL n +1 i . (34)One would hope that r ( X , L ) is the second order coefficients in the expansion of h ( X , m L ).This is true if L is big and nef by Fujita’s vanishing theorem. But due to the example in[15], this does not seem to be true for general big line bundles. Lemma 4.2. If µ : Y → X is a birational morphism between smooth projective manifold,which is a composition of blowups along smooth subvarieties. Then we have: r ( L ) = r ( µ ∗ L ) . (35) roof. Write K X as the difference of very ample divisors A − A and arguing as in the proofof Proposition 3.3, we see that: hL n i · K X = h µ ∗ L n i · µ ∗ K X . (36)Let E i be the exceptional divisor of µ . We just need to show that h µ ∗ L n i· E i = vol Y| E i ( µ ∗ L n ) =0. This can be seen by the inclusion:Im (cid:0) H ( Y , mµ ∗ L ) → H ( E i , mµ ∗ L| E i ) (cid:1) ⊆ H ( E i , mµ ∗ L| E i ) = H ( µ ∗ ( E i ) , m L| µ ∗ ( E i ) ) (37)and using the fact that the right-hand-side is equal to o ( m n ) because dim( µ ∗ ( E i )) < n .If we consider L as a Cartier b -divisor in the sense of Shokurov, then because of identity(35), r ( L ) is an invariant of the Cartier b -divisor L .The following lemma follows immediately from the results in [28, section 3.1]. Lemma 4.3.
Let L = P + N be the divisorial Zariski decomposition of L . Then we have: r ( L ) = r ( P ) . (38) Moreover if L admits a Zariski decomposition, i.e. if P is nef, then we have: r ( L ) = P n · K X . (39)We propose the following main conjecture. Conjecture 4.4.
Let ( X , L ) be a big model of ( X, L ) . Then there exists a sequence of blowups µ m : X m → X along C ∗ -equivariant ideal sheaves cosupported on X and decompositions into Q -divisors µ ∗ m L = L m + E m with L m semiample and E m effective supported on the exceptionaldivisor of µ m such that: lim m → + ∞ vol X m ( ¯ L m ) = vol X ( ¯ L ) and lim m → + ∞ r ( ¯ L m ) = r ( ¯ L ) . (40)Because of the positive intersection formula in (2) and the reduction in [26], this indeedimplies Boucksom-Jonsson’s regularization conjecture. Moreover by the following lemma andthe work in [26], it would complete the solution of Yau-Tian-Donaldson conjecture for cscKmetrics. Lemma 4.5.
For any big model ( X , L ) , Conjecture 4.4 implies that there exists φ m ∈ H NA such that φ m converges to φ ( X , L ) in the strong topology and M NA ( φ m ) → M NA ( φ ( X , L ) ) .Proof. By the same base change construction as in the proof of Proposition 3.4, we canassume that X has a reduced central fibre.For simplicity of notations, we denote by φ = φ ( X , L ) (resp. φ m ) the non-Archimedeanmetrics associated to L (resp. L m ). Then because E m is effective, we have φ ≥ φ m . We claimthat φ m → φ strongly. Indeed, by [12, Proposition 6.26], it suffices to show the followingnon-negative quantity converges to 0 as m → + ∞ : J NA φ ( φ m ) = Z X NA ( φ m − φ )MA NA ( φ m ) − E NA ( φ m ) + E NA ( φ ) . (41)This follows immediately from φ m ≤ φ and (27):0 ≤ J NA φ ( φ m ) ≤ − E NA ( φ m ) + E NA ( φ ) = vol X ( L ) n + 1 − vol X m ( L m ) n + 1 . By the positive intersection formula (2) the second identity in (40) implies M NA ( φ m ) → M NA ( φ ). e hope the conjecture 4.4 can be studied by using the geometric tools introduced in thestudy of Fujita’s approximation theorem. We recall the following definition Definition 4.6 (see [25, Definition 11.4.3]) . Let L be a big line bundle. A Fujita approx-imation of L consists of a projective birational morphism µ : X ′ → X with X ′ irreducibletogether with a decomposition µ ∗ L = A + E in N ( X ) Q such that A is big and semiampleand E is effective. In the more general context of big line bundles, we conjecture the following result:
Conjecture 4.7.
Let L be a big line bundle over a smooth projective manifold X . Thenthere exists a sequence of Fujita approximations ([25, Definition 11.4.3]), i.e. birationalmorphisms µ m : X m → X and decompositions into Q -divisors µ ∗ m L = L m + E m with L m ample and E m effective, such that: lim m → + ∞ vol X m ( L m ) = vol X ( L ) and lim m → + ∞ r ( L m ) = r ( L ) . (42) Remark 4.8.
Sebastien Boucksom pointed out to me that this conjecture could be formulatedusing the language of b-divisors. Such a formulation has some consequences and (hopefully)might be useful for studying this problem.
Let’s recall an orthogonality estimate by Boucksom-Demailly-Pˇaun-Peternell (see also[25, Theorem 11.4.21]):
Theorem 4.9 ([5, Theorem 4.1]) . Fix any ample line bundle H on X . There exists aconstant C = C ( X , H ) > such that any Fujita decomposition ( µ : X ′ → X , µ ∗ L = A + E ) satisfies the estimate: ( A n · E ) ≤ C · (vol X ( L ) − vol X ′ ( A )) . (43)We observe an immediate consequence of this estimate. Lemma 4.10.
Let µ m : X m → X be a sequence of birational morphisms such µ ∗ m L = L m + E m where L m is ample and E m is effective. Assume that the following conditions are satisfied:1. lim m → + ∞ vol X m ( L m ) = vol X ( L ) .2. lim m → + ∞ L nm · µ ∗ m K X = hL n i · K X .Then lim m → + ∞ r ( L m ) = r ( L ) if and only if lim m → + ∞ L nm · K X m / X = 0 . (44) In particular, if there exists a constant
C > independent of m such that for any irreduciblecomponent F of E m we have ord F ( K X m / X ) ≤ C · ord F ( E m ) , then we have the convergence: lim m → + ∞ r ( L m ) = r ( L ) . Remark 4.11.
The above lemma suggests that the techniques from birational algebraic geom-etry might be useful for achieving (44) . Indeed, our hope is that the MMP techniques (basedon the work of Birkar-Casini-Hacon-McKernan) could be used to extract suitable exceptionaldivisors satisfying the conditions in the above lemma. Note that such type of techniques hasprove to be very powerful in the study of K-stability for Fano varieties (see for example [3]). y the works in [18, 25] and [20, 6], the sequence {L m } that satisfy the first two conditionscan be obtained by blowing up base ideals. Moreover one can also get L m by blowing upappropriate asymptotic multiplier ideals, which satisfy the important Nadel-vanishing andglobal generation properties. We review the construction in [25, 11.4.B, Proof of Theorem11.4] for the reader’s convenience. Fix a very ample bundle H on X such that G := K X +( n + 2) H is very ample. For m ≥
0, set M m = m L − G . Given ǫ > m ≫ M m ) ≥ m n +1 (vol( L ) − ǫ ). Set J = J ( X , k M m k ), let µ m : X m → X be acommon resolution of J such that µ ∗ m J = O ( − ˜ E m ). Then L m := L − m ˜ E m is semiampleand L n +1 m ≥ vol( L ) − ǫ (see [25, 11.4.B] for more details). By letting ǫ →
0, we see that thefirst condition in Lemma 4.10 is thus satisfied.Now we claim that in this construction, the second condition in Lemma 4.10 can also besatisfied. This fact will be used in the calculations of appendix A. To see this, we use somesimilar argument as in [20, Proof of Theorem 2.13]. Choose m ≫ m L − G = N ′ is effective. Fix a very ample divisor H such that N := N ′ + H is ample. Then we have theinclusion b ( | ( m − m ) L| ) O X ( − N ) ⊆ b ( | ( m − m ) L ) O ( − N ′ ) ⊆ b ( | m L − G | ) = b ( | M m | ) ⊆ J ( k M m k ) . We can also assume that µ m is both resolutions of b ( | M m | ) and b ( | ( m − m ) L| ) satisfyingthe identities µ ∗ m b ( | m L − G | ) = O X m ( − ˜ F m ) and µ ∗ m b ( | ( m − m ) L| ) = O X m ( − ˜ Q m ). Set L ′ m = µ ∗ m ( L − Gm ) − m ˜ F m and L ′′ m = µ ∗ m L − m − m ˜ Q m .Fix any effective divisor D on X . Let ˜ D m be the strict transform of D under µ m . Thenthe above inclusion implies:vol(( L m + Nm ) | ˜ D m ) ≥ vol(( L ′ m + Nm ) | ˜ D m ) ≥ vol( L ′′ m | ˜ D m ) . Because L m and L ′′ m are both semiample, this implies( L m + Nm ) n · ˜ D m ≥ L ′′ nm · ˜ D m . (45)Now we can fix a very ample line bundle H such that µ ∗ m ( mH ) − L m = mµ ∗ ( H − L ) + ˜ E m is effective. Then for any 1 ≤ i ≤ n , we have:lim sup m → + ∞ m n ( m L n − im ) · N i · ˜ D m ≤ lim sup m → + ∞ m n ( mH ) n − i · N i · ˜ D m = 0 . By expanding the left-hand-side of (45), this implies hL n i · D ≥ lim sup m → + ∞ L nm · ˜ D m ≥ lim sup m → + ∞ L ′′ nm · ˜ D m = kL n · D k which, by using Theorem 2.7, implies the equalitylim m → + ∞ L nm · µ ∗ m D = kL n · D k = hL n i · D. Writing − K X = D − D with D , D effective, we then see that the second condition ofLemma 4.10 is satisfied too.Finally we point out that Conjecture 4.7 holds true any for any big line bundle thatadmits a birational Zariski decomposition (in the sense of Cutkosky-Kawamata-Moriwaki). nfortunately not all big line bundles admit such birational Zariski decomposition by thecounterexamples of Nakayama [30]. On the other hand, we verify in the appendix thatconjecture 4.7 indeed holds for some of Nakayama’s examples. Indeed, we will show thatin these examples the bound of discrepancies in the above lemma is indeed satisfied. So itseems to be very interesting to know whether (44) can be achieved in general. Definition 4.12.
We say a big line bundle L admits a birational Zariski decomposition ifthere is a modification µ : ˜ X → X , a nef R -divisor P and an R -effective divisor N on ˜ X with the following properties: • µ ∗ L = P + N . • For any positive integer m > , the map H ( ˜ X , O ˜ X ( ⌊ mP ⌋ )) → H ( ˜ X , O ˜ X ( m L )) (46) induced by the section e m is an isomorphism, where e m is the canonical section of ⌈ m N ⌉ . Lemma 4.13.
If a big line bundle L admits a birational Zariski decomposition, then theconjecture 4.7 for L is true.Proof. By Lemma 4.2 and (39), we have r ( L ) = r ( µ ∗ L ) = r ( P ) = P n · K ˜ X . Choose anyample divisor A on ˜ X . Because P is big and nef, we know that for k ≫ k P − A = ∆ k iseffective. So we get: ( m + k ) P = m P + A + E k , (47)which implies the decomposition over ˜ X : µ ∗ L = P + N = 1 m + k ( m P + A ) + 1 m + k ∆ k + N . (48)By perturbing the coefficients of A , we can assume that m P + A is a Q -divisor. Set L m = m + k ( m P + A ). Then it is easy to see that (42) holds true. Remark 4.14. If ( X, L ) is a polarized spherical manifold, it is known that its models in thesense of Definition 2.1 is a Mori dream space (see the appendix A by Y. Odaka to [16]). SinceZariski decomposition of big lines bundles always exist on Mori dream spaces, the above lemmain the C ∗ -equivariant setting gives an explanation why the Yau-Tian-Donaldson conjectureholds for polarized spherical manifolds. See [16, Appendix A] for a slightly different proof ofthis fact (again based on Theorem 1.3). A Conjecture 4.7 for Nakayama’s examples with-out birational Zariski decomposition
In this appendix, we will use Lemma 4.10 to show that conjecture 4.7 is indeed true forsome examples of big line bundles that do not have a birational Zariski decomposition.Such examples were first discovered by Nakayama [30]. Here we will do a case study basedon the construction of Fujita approximation in [25, Theorem 11.4.4] and the calculation ofasymptotic multiplier ideals for Nakayama’s examples in the work of Koike [23]. e first write down some notations. Set S = E × E for an elliptic curve E withoutcomplex multiplication. Then the pseudoeffective cone PE( S ) coincides with the nef coneNef( S ). Fix a point p ∈ E and consider in N ( S ) R three classes: f = [ { P } × E ] =: [ F ] , f = [ E × { P } ] =: [ F ] , δ = [∆]where ∆ ⊂ E × E is the diagonal. Then N ( S ) R is spanned by { f , f , δ } and the descriptionof the nef cone is known (see [24, Lemm 1.5.4]): α = x · f + y · f + z · δ ∈ N ( S ) R is nef ifand only if xy + xz + yz ≥ , x + y + z ≥ . (49)By standard linear algebra, we can use the following linear transformation to diagonalizethe above relation: l = 16 ( f + f − δ ) , l = 16 ( −√ f + √ f ) ,
16 ( f + f + δ ) a = x + y − z, b = −√ x + √ y, c = 2( x + y + z ) . such that α = al + bl + cl ∈ N ( S ) R is nef if and only if c ≥ a + b , c ≥ . (50)Let L i , i = 0 , , S . Set X = P ( O S ⊕ ( L − L ) ⊕ ( L − L )) ∼ = P ( L ⊕ L ⊕ L ) . (51)Denote by H = O X (1) the tautological line bundle.We use the description of X as a toric bundle over S as in [30]. Let Σ denote the standardfan of P , i.e. the fan generated by three cones: σ = Cone { e , e } , σ = Cone { e , − ( e + e ) } , σ = Cone {− ( e + e ) , e } . Let h : R → R be the piecewise linear function on Σ satisfying h ( e ) = h ( e ) = 0 and h ( − ( e + e )) = −
1. Then X is the toric bundle associated to Σ and h determines the linebundle H . Set L = π ∗ L + D h . By a result of Cutkosky (see [23, Lemma 6.1]), L is a big linebundle if and only if there exists ( k , k , k ) ∈ N such that L k ⊗ L k ⊗ L k is an ample linebundle over S . Moreover it is well-known that the canonical line bundle of the projectivebundle X is given by: K X = π ∗ ( K S + ( L − L ) + ( L − L )) − H = L + L − L − H. (52)The last identity uses the triviality of K S .We will consider the example in [23, Example 6.5]. Set L = 4 F + 4 F + ∆, L = O V , L = O V ( − F + 9 F + ∆). Then c ( L ) = 6( l + 3 l ) , c ( L ) = 0 , c ( L ) = 6 l + 10 √ l + 18 l . (53)Because c ( L ) is in the interior of the nef cone, L is ample. Note that H is relativelyample. So it is easy to see that there exist a, b ∈ Z > such that aL + bH − K X n +1 is a very ampleline bundle. Set G = aL + bH and M p = p L − G = p ( L + H ) − ( aL + bH ) = ( p − a ) L + ( p − b ) H = ( p − b ) (cid:18) p − ap − b L + H (cid:19) =: ( p − b )( Q p + H ) . et J p := J ( X, k M p k ). Let µ p : Y p → X be the normalized blowup of J p with E p := µ ∗ p J p = O Y p ( − P i c p,i E p,i ). Set A p = µ ∗ p ( L ) − p E p .By the discussion after Lemma 4.10, we known that ( Y p , A p ) satisfies the first two con-ditions of Lemma 4.10. So, by Lemma 4.10, it suffices to show that there exists C > A X ( E i,p ) ≤ Cp − c p,i for any i, p .For any Q -line bundle L on S and with h as above, define a compact convex set following[30, § (cid:3) ( L, h ) = { ( x, y ) ∈ R ≥ ; x + y ≤ L + x ( L − L ) + y ( L − L ) ∈ PE( S ) } . (54)Then it is straight-forward to use (50) and (53) to get: (cid:3) ( Q p , h ) = (cid:26) ( x, y ) ∈ R ≥ ; x + 52 √ p − bp − a y ≤ (cid:27) . Let ϕ p, min be the metric of minimal singularity on M p . Then it is known that J ( ϕ p, min ) = J ( k M p k ) = J p . For each fan σ i , i = 0 , ,
2, there exists an open set U i ∼ = S × C which is anaffine toric bundle over S . Applying the result in [23, 5.2], J p is trivial on U , U , and over U we can choose the canonical affine coordinate ( z , z ) on C such that the multiplier idealis generated by monomials: J p ( U ) = (cid:10) z m z m ; ( m + 1 , m + 1) ∈ Int(( p − b ) S p ) ∩ Z (cid:11) (55)where S p = ( x, y ) ∈ R ≥ ; x + y − √ p − a )5( p − b ) ≥ , (56)is generated by the exponents (0 , , (1 , , (0 , − √ p − a )5( p − b ) ), which are the images of (0 , , , √ p − a )5( p − b ) ) under the linear map ( a, b ) ( h ( a, b ) − m σ , v i i ) i =1 , = ( a, − a − b ) with m σ = (0 , , v = (1 , , v = ( − , −
1) (see [23, Definition 4.1]). In particular, the multiplierideal sheaf is co-supported on P ( L ) ⊂ P ( L ⊕ L ⊕ L ). Moreover, as pointed out in [23], byusing [30, Theorem 2.10], we know that the line bundle L does not admit birational Zariskidecomposition.Equivalently, we have: J p ( U ) = (cid:28) z m z m ; ( m , m ) ∈ Z ≥ , m α p + m β p > (cid:29) . (57)where α p = p − bd p , β p = (1 − √ ) p + √ a − bd p , d p = 1 − p − b − − √ ) p + √ a − b . We only needto know that there exists C = . − √ > α p ≥ Cp, β p ≥ Cp for p ≫ d p = 1 + O ( p − ).Because the multiplier ideal is monomial, we can use the result about Rees valuationsof monomial ideals to see that the blow-up of J p corresponds to the sides of the Newton-polygons of J p (see [22, 15.4]). Indeed, such blowup also corresponds to a subdivision of thecone σ .Now denote the sides of the Newton polygon be given by P i − P i , ≤ i ≤ r with P i =( x i , y i ). Then it is easy to see that P = (0 , ⌊ β ⌋ + 1) and P r = ( ⌊ α ⌋ + 1 , a i , b i ) = ( y i − − y i , x i − x i − ) ∈ R > is a normal vector of P i − P i . The monomial valuation rd E i that corresponds to the side P i P i +1 can be chosen to be given by the weighted blowupwith weights ( a i , b i ). Set τ i = b i /a i >
0. It is easy to see that: w ( E i ) := A ( E i )ord E i ( J p ) = a i + b i a i x i − + b i y i − = 1 + τ i x i − + y i − τ i = a i + b i a i x i + b i y i = 1 + τ i x i + y i τ i . As a consequence, we have: w ( E i ) − w ( E i +1 ) = 1 + τ i x i + y i τ i − τ i +1 x i + y i τ i +1 = ( y i − x i )( τ i +1 − τ i )( x i + y i τ i )( x i + y i τ i +1 ) . (58)From this identity, we easily see that max { w ( E i ); 1 ≤ i ≤ r } = max { w ( E ) , w ( E r ) } . Nownote that τ − is at most the absolute value of the slope of the line P P ′ where P ′ is the point(1 , − βα + β ) one the line connecting ( α,
0) and (0 , β ), which gives the inequality: w ( E ) = 1 + τ ( ⌊ β ⌋ + 1) τ = 1 ⌊ β ⌋ + 1 + 1( ⌊ β ⌋ + 1) τ ≤ β + 1 ⌊ β ⌋ + 1 ( ⌊ β ⌋ + 1 − β + βα )= 1 β + ⌊ β ⌋ + 1 − β ⌊ β ⌋ + 1 + β ⌊ β ⌋ + 1 1 α = O ( p − ) . By the same argument (or just by symmetry), we also get w ( E r ) = O ( p − ). According tothe previous discussion, the verification of 3rd condition in Lemma 4.10 is complete. Remark A.1.
It is easy to see that the above arguments, which reduce the problem to the es-timates for Rees valuations of monomial ideals, works for many more examples of Nakayamatype.
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Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019.
E-mail address: [email protected]@rutgers.edu