aa r X i v : . [ m a t h . AG ] J un VALUATIVE MULTIPLIER IDEALS
ZHENGYU HU
Abstract.
The main goal of this paper is to construct an algebraic analogueof quasi-plurisubharmonic function (qpsh for short) from complex analysisand geometry. We define a notion of qpsh function on a valuation spaceassociated to a quite general scheme. We then define the multiplier ideals ofthese functions and prove some basic results about them, such as subaddi-tivity property, the approximation theorem. We also treat some applicationsin complex algebraic geometry. Introduction
Given a line bundle L on a smooth projective complex variety, a classicaltheorem of Kodaira asserts that if L carries on a smooth metric with positivecurvature, then L is ample, or equivalently the global sections of a multipleof L give an embedding to a projective space and hence induce such a metricon L . More generally, global sections of a multiple of L induce a semi-positivesingular metric. Conversely, given a semi-positive singular metric h , the localweight function ϕ , which is plurisubharmonic (psh for short), should be relatedto sections of multiples of L , or perhaps of a small perturbation of L . See [27]for more details.On the other hand, if we work locally near the origin of C n , then [5, Section 5]shows that we can transform a psh germ ϕ to a formal psh function b ϕ on quasi-monomial valuations centered at the origin. This valuative transform usuallyloses much information on the original psh function, however, it preserves theinformation on the singularity of ϕ . In particular, they give the same multi-plier ideals which essentially means that they characterize the same singularitybecause of the Demailly’s approximation. The idea of studying psh functionsusing valuations was systematically developed in [5] and its predecessors [19],[20], [21]. The main purpose of this paper is to define a similar notion of qpshfunctions on a separated, regular, connected and excellent schemes over Q , andwe then study these functions.Although we don’t discuss Berkovich spaces in this paper, our work shouldbe related to the qpsh functions (or metrics on line bundles) on the Berkovichspace associated to a smooth projective variety over a trivially valued field. See[6] and [7].Let us briefly introduce some terminologies. Roughly speaking, we considera function ϕ on divisorial valuations on a scheme X such that ϕ ( t ord E ) = Date : June 15, 2018.2010 MSC: 14F18 (Primary) 12J20 (Secondary). tϕ (ord E ) and sup E | ϕ (ord E ) | A (ord E ) < + ∞ where E runs over all prime divisors over X . We prove that such functions form a Banach space BH( X ) if we equip itwith the norm k ϕ k = sup E | ϕ (ord E ) | A (ord E ) (see Proposition 3.2). By convention weset log | a | (ord E ) = − ord E ( a ) for a nonzero coherent ideal sheaf a , and one caneasily check that log | a | is a valuative function in BH( X ). We define the set ofqpsh function QPSH( X ) to be the closed convex cone generated by functionsof the form log | a | . We then define the multiplier ideal J ( ϕ ) of a qpsh function ϕ to be the largest ideal a such that sup E − ord E ( a ) − ϕ (ord E ) A (ord E ) <
1. This definitionis reasonable because of Proposition 4.3 and Corollary 4.14.Our first main result is that a qpsh function is a decreasing limit of a sequenceof qpsh functions of the form c k log | b k | . In complex analysis and geometry,such a regularization is crucial. See [12], [13]. Moreover, we prove that wecan actually choose b k = J ( kϕ ) which satisfies the subadditivity property. SeeProposition 4.22(1). Readers can compare this result with [14]. Theorem 1.1 (cf. Theorem 4.24) . Let ϕ be a bounded homogeneous function.Then ϕ is qpsh if and only if ϕ is the limit function, in the norm, of a decreasingsequence of qpsh functions of the form c k log | b k | . Furthermore, we can choose c k = k and b k = J ( kϕ ) which form a subadditive sequence of ideals. Given an ideal a on a scheme X , the log canonical threshold lct( a ) is afundamental invariant both in singularity theory and birational geometry (see[25], [24], etc.). The log canonical threshold admits the following description interms of valuations: lct( a ) = inf E A (ord E )ord E ( a )where E runs over all prime divisors over X and A (ord E ) = ord E ( K Y/X ) + 1.In fact in the above formulae one can take the infimum over all real valuationscentered on X . It is well-known that if Y is a log resolution of a , then thereexists some prime divisor E on Y such that ord E computes the log canonicalthreshold, that is, lct( a ) = A (ord E )ord E ( a ) . Given a qpsh function ϕ , we can define thelog canonical threshold lct( ϕ ) as the limit of c k lct( a k ) where c k log | a k | convergesto ϕ strongly in the norm. We show thatlct( ϕ ) = inf E A (ord E ) − ϕ (ord E ) . Unfortunately, there might be no divisorial valuation which computes the logcanonical threshold in general. However, we can prove that there exists a realvaluation which computes the log canonical threshold. This has been heavilystudied in [22], [23] and other references. It is conjectured by [22, Conjecture B]that a valuation which computes the norm is quasi-monomial (see Conjecture5.9). Equivalently we consider the reciprocal of the log canonical thresholdwhich is exactly the norm of ϕ by definition. More generally, for a nonzeroideal q we consider k ϕ k q := sup E − ϕ (ord E ) A (ord E )+ord E ( q ) , and we prove that there existsa real valuation which compute this norm. The proof in this paper mainlyfollows the strategy of [22]. A similar result appears in [23]. ALUATIVE MULTIPLIER IDEALS 3
Theorem 1.2 (=Theorem 5.2) . Let ϕ ∈ QPSH( X ) be a qpsh function and let q be a nonzero ideal on X . Then there exists a nontrivial tempered valuation v which computes k ϕ k q . If X is a complex projective variety, then we can provide QPSH( X ) withmore structures. Namely, given a Q -line bundle L on X , we say that thefunction λ log | a | is L -psh if λ is a nonnegative rational number and L ⊗ a λ issemi-ample. We can then define PSH( L ) ⊆ QPSH( X ) as the closure of theset of such functions. We also define the set of pseudo L -psh functions asPSH σ ( L ) := T ǫ> PSH( L + ǫA ), where A is an ample line bundle. See Section6.1 for more details.Under the above setting, we show that there exists the maximal L -psh func-tion ϕ which can be written explicitly as ϕ ( v ) = − v ( k L k ), and there ex-ists the maximal pseudo L -psh function φ which can be written explicitly as φ ( v ) = − σ v ( k L k ) (see Proposition 6.10 and 6.11). As an immediate corollarywe generalize [27, Theorem 6.14] as follows. See [27] for the definition of theperturbed ideal and the diminished ideal. Theorem 1.3 (=Theorem 6.16) . Let D be a pseudo-effective divisor. Assumethat φ max is the maximal pseudo D -psh function. Then, the perturbed ideal J σ, − ( D ) = J − ( φ max ) , and the diminished ideal J σ ( D ) = J ( φ max ) . In particular,we can write J σ ( D ) explicitly as Γ( U, J σ ( L )) = { f ∈ Γ( U, O X ) | v ( f ) + A ( v ) − σ v ( k L k ) > for all v ∈ V ∗ U } . Further, a nonzero ideal q ⊆ J σ ( k L k ) if and onlyif v ( q ) + A ( v ) − σ v ( k L k ) > for all v ∈ V ∗ X . In the last subsection of this paper, we prove the finite generation of a diviso-rial module as another application. The proposition below can also be obtainedusing minimal model theory (see Remark 6.21). Note that our proof here avoidsusing ’the length of extremal rays’ (see [11]).
Proposition 1.4 (=Proposition 6.18) . Let ( X, B ) be a log canonical pair. As-sume that K X + B is Q -Cartier and abundant, and that R ( K X + B ) is finitelygenerated. Then, for any reflexive sheaf F , M p F ( K X + B ) is a finitely generated R ( K X + B ) -module. From an easy observation, the above proposition can be slightly generalized(cf. Proposition 6.24).
Acknowledgements.
This paper is based on part of the author’s PhD the-sis. I would like to express my deep gratitude to my supervisor Professor KefengLiu for numerous conversations and encouragement from him. I am indebted toProfessor Hongwei Xu and many other professors in Zhejiang Unversity becauseI have greatly benefited from the courses and discussions with them. I wouldalso like to thank Professor S´ebastien Boucksom, Mattias Jonsson and MirceaMustat¸˘a for patiently answering my questions and providing many valuablecomments. The author is supported by the EPSRC grant EP/I004130/1.
ZHENGYU HU Valuation spaces
Throughout this paper, all schemes are assumed to be separated, regular,connected and excellent schemes over Q . All rings are assumed to be integral,regular and excellent rings containing Q . An ideal on a scheme means a co-herent ideal sheaf on a scheme. A birational model of a scheme is a schemebirational to and proper over this scheme, and a divisor over a scheme is adivisor on a birational model of the scheme. For definitions and properties ofvaluations, multiplier ideals, singularities in birational geometry, etc., we referto [25],[22] and [24]. Real valuations.
Let X be a scheme, and let K ( X ) be its function field. A real valuation v is a function v : K ( X ) ∗ −→ R such that v ( f g ) = v ( f ) + v ( g )and v ( f + g ) ≥ min { v ( f ) , v ( g ) } . By convention we set v (0) := + ∞ . Let O v := { f | v ( f ) ≥ } be its valuation ring. If there exists a point ξ ∈ X such that the morphism O X,ξ ֒ → O v is a local homomorphism, then ξ is calledthe centre of v on X and denoted by c X ( v ). Note that ξ is unique since X is separated, and also note that the centre always exists provided that X iscomplete. A real valuation with centered on X is called a real valuation on X or simply a valuation on X , and we denote by Val X the set of valuations on X .The set of valuations Val X is independent of the choice of a birational modelof X . More precisely, if Y → X is a proper birational morphism of schemes,then Val X = Val Y . A valuation v on X is said to be the trivial valuation if itscentre c X ( v ) is the generic point of X . We denote by Val ∗ X ⊆ Val X the set ofnontrivial valuations on X .The set Val X can be equipped with an induced topology defined by the maps v −→ v ( f ) for all rational functions f ∈ K ( X ) ∗ . For every nonzero ideal a ,we have that v ( a ) is well defined and v ( a ) = v ( a ) where a denotes the integralclosure of a . Note that the topology on Val X defined by pointwise convergenceon ideals on X is equivalent to that on functions in K ( X ). Readers can consult[22, Section 1] for more details.Under the above topology, the map c X : Val X −→ X is anti-continuous. Thatis, the inverse image of an open subset is closed. More precisely, if U ⊆ X is anopen subset and m is the defining ideal of X \ U , then Val U = { v ∈ Val X | v ( m ) =0 } and Val U is closed in Val X .For two valuations v , w on X , we say that v ≤ w if v ( a ) ≤ w ( a ) for everynonzero ideal a . This is equivalent to that the centre η := c X ( w ) ∈ c X ( v ) andthat v ( f ) ≤ w ( f ) for every nonzero local function f ∈ O X,η . Quasi-monomial valuations.
Let X be a scheme, let ξ ∈ X be a point, andlet x = ( x , . . . , x r ) be a regular system of parameters at ξ . If f ∈ O X,ξ is a localregular function, then f can be expressed as f = P β c β x β in [ O X,ξ with eachcoefficient c β either zero or a unit. For each α = ( α , . . . , α r ) ∈ R r ≥ , we define areal valuation by val ξ,α ( f ) = min { < α, β > | c β = 0 } where < α, β > := P i α i β i ,which is called a monomial valuation on X . ALUATIVE MULTIPLIER IDEALS 5
A pair (
Y, D ) is called log smooth if Y is a scheme and D is a reduced divisorwhose components are regular subschemes intersecting each other transversally.A pair ( Y, D ) is called a log resolution of X if there is a birational projectivemorphism π : Y → X and ( Y, D + K Y/X ) is log smooth. Let ( Y ′ , D ′ ) beanother log resolution of X , we say ( Y ′ , D ′ ) (cid:23) ( Y, D ) if Y ′ is projective over Y and the support of D ′ contains the support of the pull-back of D . Note thatlog resolutions of X form an inverse system.Let ( Y, D ) be a log resolution of X , and let η be the generic point of anirreducible component of the intersection of some prime components of D . Wedenote by QM η ( Y, D ) the set of real valuations which can be defined as a mono-mial valuation at η . Note that η ∈ c X ( v ) and QM η ( Y, D ) ∼ = R r ≥ as topologicalspaces. We also define QM( Y, D ) = S η QM η ( Y, D ) where η runs over everygeneric point of some component of the intersection of some prime componentsof D . A real valuation v is said to be quasi-monomial if there exists a logresolution ( Y, D ) such that v ∈ QM(
Y, D ). Remark 2.1.
Let Γ v = v ( K ( X ) ∗ ) ⊆ R be the value group of v . Let ratrk( v ) =dim Q (Γ v ⊗ Z Q ) be the rational rank of v , and let k v , k ( ξ ) be the residuefields O v , O X,ξ respectively, where ξ = c X ( v ). If we denote by trdeg X v =trdeg( k v /k ( ξ )) the transcendental degree of v over X , we have the Abyankarinequality ratrk( v ) + trdeg X v ≤ dim( O X,ξ ). A result asserts that the quasi-monomial valuations are exactly the ones that give equality in the Abhyankarinequality (cf. [22, Proposition 3.7]).Let v ∈ Val X be a quasi-monomial valuation. A log smooth pair ( Y, D ) is saidto be adapted to v if v ∈ QM(
Y, D ). We say (
Y, D ) is a good pair adapted to v if { v ( D i ) | v ( D i ) > } are rationally independent. The following useful lemmais established as [22, Lemma 3.6]. Lemma 2.2.
Let v ∈ Val X be a quasi-monomial valuation. There exists a goodpair ( Y, D ) adapted to v . If ( Y ′ , D ′ ) (cid:23) ( Y, D ) and ( Y, D ) is a good pair adaptedto v , then ( Y ′ , D ′ ) is also a good pair adapted to v . An important class of valuations are divisorial valuations. A valuation iscalled divisorial if it is positively proportional to ord E for some prime divisor E over X , where ord E is the vanishing order along E . One easily verifies thatthe trivial valuation is quasi-monomial of rational rank zero, and a divisorialvaluation is quasi-monomial of rational rank one. Let ( Y, D ) be a log smoothpair adapted to v . It can be verified that v is divisorial if and only if R ≥ [ v ] ⊆ QM η ( Y, D ) ∼ = R r ≥ is a rational ray, that is, R ≥ [ v ] contains some rational pointin R r ≥ .For every log resolution ( Y, D ) we can define the retraction map r Y,D : Val X −→ QM(
Y, D )by taking v to a quasi-monomial valuation in QM( Y, D ) with r Y,D ( v )( D i ) = v ( D i ). Note that r Y,D is continuous and v ≥ r Y,D ( v ) with equality if and onlyif v ∈ QM(
Y, D ). Furthermore, if ( Y ′ , D ′ ) (cid:23) ( Y, D ) is another resolution, then
ZHENGYU HU the retraction map r Y,D : QM( Y ′ , D ′ ) −→ QM(
Y, D ) (by abuse of notaion ifwithout confusion) is a surjective mapping which is integral linear on everyQM η ′ ( Y ′ , D ′ ) and we have that r Y,D ◦ r Y ′ ,D ′ = r Y,D . Therefore we can naturallyregard QM(
Y, D ) as a subset of QM( Y ′ , D ′ ), and hence of the set of quasi-monomial valuations on X . Also note that v ( a ) ≥ r Y,D ( v )( a ) for an ideal a on X , with equality if ( Y, D ) is a log resolution of a . (cf. [25], [22, Corollary 4.8]) Tempered valuations.
We first introduce the log discrepancy on an ar-bitrary scheme. Let π : Y −→ X be a birational proper morphism. The 0 th fitting ideal Fitt (Ω Y/X ) is a locally principle ideal with its corresponding ef-fective divisor denoted by K Y/X (cf. [22, Section 1.3]). For a quasi-monomialvaluation v ∈ QM(
Y, D ), we define the log discrepancy A X ( v ) = X v ( D i ) · A X (ord D i ) = X v ( D i ) · (1 + ord D i ( K Y/X )) . We simply denote this by A when the scheme X is obvious. Note that A isstrictly positive linear on every QM η ( Y, D ), and in particular continuous onevery QM η ( Y, D ) (or is weakly continuous according to Definition 3.4). Alsonote that if ( Y ′ , D ′ ) (cid:23) ( Y, D ) and v ∈ QM( Y ′ , D ′ ), then A ( v ) ≥ A ( r Y,D ( v )) andequality holds only when v ∈ QM(
Y, D ). For an arbitrary valuation v ∈ Val X ,we define A ( v ) = sup ( Y,D ) A ( r Y,D ( v )) ∈ [0 , + ∞ ] . Note that A is lower-semicontinuous (lsc) as a valuative function. Definition 2.3.
A valuation v is said to be tempered if A ( v ) < ∞ . Thevaluation space V X of X is defined to be the space of tempered valuations as asubspace of Val X .We similarly denote by V ∗ X the subset of nontrivial tempered valuations. If f : X ′ → X is a proper birational morphism, then A X ( v ) = A X ′ ( v ) + v ( K X ′ /X )(cf. [22, Proposition 5.1(3)]) and hence V X ′ = V X . Since V X is a topologicalsubspace of Val X , it is naturally a subspace of of the Berkovich space X an . See[22, Section 6.3] for a comparison.With the aid of the log discrepancy, we can normalize V ∗ X by letting A ( v ) = 1,that is, we define Λ X := { v ∈ V ∗ X | A ( v ) = 1 } . In particular, we normalize everycone complex QM( Y, D ) by setting ∆(
Y, D ) := { v ∈ QM(
Y, D ) | A ( v ) = 1 } . It isclear that ∆( Y, D ) naturally possess the structure of a simplicial complex, andby convention we say that ∆(
Y, D ) is a dual complex . Readers can compare theconstructions here with [5], [6] and [7].The following lemma allows us to compare v and ord ξ where ξ = c X ( v ) whichis quite useful (see [25], [22, Section 5.3] for the definition of ord ξ ). See [22,Proposition 5.10] for a proof. Recently S. Boucksom, C. Favre and M. Jonssongave a refinement of the following lemma in [8]. Lemma 2.4 (Izumi’s type inequality) . Let ξ = c X ( v ) and m ξ be the definingideal of { ξ } . Then we have v ( m ξ )ord ξ ≤ v ≤ A ( v )ord ξ . Passing to the completion.
A morphism f : X ′ → X is regular if it isflat and its fibres are geometrically regular (cf. [22,Section 1.1]). The following ALUATIVE MULTIPLIER IDEALS 7 lemma on log discrepancy is essential for finding a valuation which computesthe log canonical threshold or norms in Section 5.
Lemma 2.5 (22, Proposition 5.13) . Let f : X ′ −→ X be a regular morphism,and let f ∗ : Val X ′ −→ Val X be the induced map. If v ′ ∈ Val X ′ is a valuation on X ′ , then A ( v ′ ) ≥ A ( f ∗ ( v ′ )) . If we assume further that X ′ = Spec d O X,ξ and v ′ iscentered at the closed point of X ′ , then A ( v ′ ) = A ( f ∗ ( v ′ )) . Definition 2.6. If ξ ∈ X is a point, then we define V X,ξ := c − X ( ξ ) as a subspaceof V X . We can normalize V X,ξ by letting v ( m ) = 1, where m is the definingideal of { ξ } . More precisely we define V X,ξ := { v ∈ V X,ξ | v ( m ) = 1 } . Let M > V X,ξ,M := { v ∈ V X,ξ | A ( v ) ≤ M } .According to [22, Proposition 5.9] the space V X,ξ,M is compact. If X = Spec A and m is the defining ideal of { ξ } , we often use the notation V A, m instead of V X,ξ .Let ( R, m ) be a local ring. Given a tempered valuation v ∈ V R, m , we define v ′ ( f ) = lim k →∞ v ( a k ) for every f ∈ b R where a k · b R = ( f ) + b m k . This is well-definedsince v ( a k ) ≤ A ( v )ord ξ ( a k ) ≤ A ( v )ord ξ ′ ( f ) < ∞ by Lemma 2.4. The abovedefinition leads to a correspondence between the valuation spaces of Spec R andSpec b R as follows. Throughout this paper we will use the notations v and v ′ toindicate that v = f ∗ v ′ for simplicity if without confusion. Proposition 2.7.
Let ( R, m ) be a local ring, and let ( b R, b m ) be its m -adic com-pletion. If we denote by f : Spec b R −→ Spec R the canonical morphism, thenthe induced map f ∗ : V b R, b m −→ V R, m is bijective. If K ′ is a compact subspaceof V b R, b m , then f ∗ is a homeomorphism from K ′ to its image. In particular, V b R, b m ,M ∼ = V R, m ,M for any positive number M > .Proof. The bijectivity of f ∗ follows from [22, Corollary 5.11], and we will provethe latter statement. Let K = f ∗ ( K ′ ). It suffices to show that K ′ is homeomor-phic to K . Let h ∈ b R be a nonzero function. We have that max v ′ ∈ K ′ v ′ ( h ) = α < ∞ since K ′ is compact. If g ∈ R is a nonzero function such that g − h ∈ b m n in b R for some n > α . Then v ′ ( g − h ) ≥ nv ′ ( b m ) > v ′ ( h ) for all v ′ ∈ K ′ . It fol-lows that v ( g ) = v ′ ( g ) = v ′ ( h ) for all v ′ ∈ K ′ and hence they induce the sametopology. (cid:3) Functions on valuation spaces
In this section we will discuss various classes of functions on valuation spacewith an emphasis on the quasi-plurisubharmonic (qpsh for short) functions.3.1.
Bounded homogeneous functions.
Let X be a scheme and V X be itsvaluation space. A valuative function ϕ is homogeneous if ϕ ( tv ) = tϕ ( v ) for all v ∈ V X and t ∈ R + . A valuative function ϕ is bounded if sup v ∈ V ∗ X | ϕ ( v ) | A ( v ) < ∞ .The set of bounded homogeneous functions forms an R -linear space, which can ZHENGYU HU be equipped with the norm k ϕ k = sup v ∈ V ∗ X | ϕ ( v ) | A ( v ) , and is denoted by BH( X ). If q is a nonzero ideal on X , then we define the q -norm as k ϕ k q = sup v ∈ V ∗ X | ϕ ( v ) | A ( v )+ v ( q ) .We also define k ϕ k + q := sup v ∈ V ∗ X ϕ ( v ) A ( v ) + v ( q )and k ϕ k − q := sup v ∈ V ∗ X − ϕ ( v ) A ( v ) + v ( q ) . Clearly, k ϕ k + q = k − ϕ k − q and k · k q = max {k · k + q , k · k − q } . Lemma 3.1.
Given two nonzero ideals p , q on X , the p -norm and the q -normare equivalent.Proof. We first assume that p = O X . Then we have the inequality k · k q ≤k · k ≤ (1 + sup v ∈ V ∗ X v ( q ) A ( v ) ) k · k q . Note that sup v ∈ V ∗ X v ( q ) A ( v ) = max D i ord Di ( q ) A (ord Di ) < ∞ where D i runs over all irreducible components of D on a birational model Y such that ( Y, D ) is a log resolution of q . This implies that 1 + sup v ∈ V ∗ X v ( q ) A ( v ) < ∞ and leads to the conclusion. (cid:3) Proposition 3.2.
Given a scheme X , BH( X ) is a Banach space. If f : X ′ −→ X is a regular morphism and f ∗ : V X ′ −→ V X is the induced map, then theinduced map f ∗ : BH( X ) −→ BH( X ′ ) by taking ϕ to ϕ ◦ f ∗ is a boundedlinear operator of Banach spaces. More precisely, k f ∗ ( ϕ ) k q ·O X ′ ≤ k ϕ k q for anynonzero ideal q on X .Proof. Note that a bounded homogeneous function ϕ is also a function on Λ X := { v ∈ V ∗ X | A ( v ) = 1 } with the norm sup v ∈ Λ X | ϕ ( v ) | < ∞ . If { ϕ m } is a Cauchysequence in BH( X ), then ϕ m converges pointwisely to a homogeneous function ϕ . Since sup v ∈ Λ X | ϕ ( v ) | ≤ sup v ∈ Λ X | ϕ ( v ) − ϕ m ( v ) | + sup v ∈ Λ X | ϕ m ( v ) | < ∞ , ϕ isa bounded homogeneous function. This proves that BH( X ) is a Banach space.For the second statement, simply note that k f ∗ ( ϕ ) k q ·O X ′ = sup v ′ ∈ V ∗ X ′ | ϕ ( v ) | A ( v ′ ) + v ′ ( q · O X ′ ) ≤ sup v ∈ V ∗ X | ϕ ( v ) | A ( v ) + v ( q ) = k ϕ k q by Lemma 2.5. (cid:3) Remark 3.3.
Let ϕ be a bounded homogeneous function such that ϕ ( v ) = − v ( a ) for some nonzero ideal a on X . It is easy to see that the norm k ϕ k q isexactly the Arnold multiplicity Arn q a , and its reciprocal is the log canonicalthreshold lct q a . We will discuss this type of functions in detail later. Definition 3.4.
A bounded homogeneous function ϕ is said to be weakly con-tinuous if ϕ is continuous on every dual complex ∆( Y, D ). Example 3.5. (1). As we already mentioned, the log discrepancy A is a weaklybounded homogeneous function.(2). If { ϕ k } is a sequence of continuous bounded homogeneous functions whichconverges to a function ϕ strongly in the norm, then ϕ is weakly continuous. ALUATIVE MULTIPLIER IDEALS 9
Ideal functions and qpsh functions.
Given a nonzero ideal a , we define | a | ( v ) = − e v ( a ) by convention. It is obvious that log | a | is a continuous boundedhomogeneous function. Definition 3.6.
A bounded homogeneous function ϕ is said to be an idealfunction if there exists a finite number of nonzero ideals a j and positive realnumbers c j such that ϕ = P lj =1 c j log | a j | . Lemma 3.7.
Let ϕ = P lj =1 c j log | a j | be an ideal function on X and q be anonzero ideal. Then, k ϕ k q = max E { P lj =1 c j ord E a j A (ord E ) + ord E q } for some prime divisor E over X .Proof. Let (
Y, D ) be a log resolution of q · ( Q lj =1 a j ), and let D i ’s be the irre-ducible components of D . By an easy computation, we see that k ϕ k q = max D i { P lj =1 c j ord D i a j A (ord D i ) + ord D i q } where D i runs over all irreducible components of D . (cid:3) Lemma 3.8.
Let ϕ be a bounded homogeneous function is determined on somedual complex ∆( Y, D ) in the sense of ϕ = ϕ ◦ r Y,D . Assume that ϕ is affineon each face of the dual complex ∆( Y, D ) and that ( Y ′ , D ′ ) (cid:23) ( Y, D ) . Then ϕ = ϕ ◦ r Y ′ ,D ′ which is also affine on each face of the dual complex ∆( Y ′ , D ′ ) .Proof. The assumption that ϕ is affine on each face of the dual complex ∆( Y, D )is equivalent to that ϕ is linear on each simplicial cone of QM( Y, D ). Theconclusion follows from the fact that r Y,D is linear on each simplicial cone ofQM( Y ′ , D ′ ). (cid:3) Definition 3.9.
A bounded homogeneous function ϕ is said to be a quasi-plurisubharmonic ( qpsh for short) function if there exists a sequence of idealfunctions which converges to ϕ strongly in the norm. The set of qpsh functions,which is a closed convex cone in BH( X ), is denoted by QPSH( X ). Definition 3.10.
The support of a qpsh function is defined to be the set { x ∈ X | x = c X ( v ) for some nontrivial tempered valuation v such that ϕ ( v ) < } .If ϕ = P lj =1 c i log | a i | is an ideal function, then the support of ϕ is the unionof the vanishing loci V ( a j ) and hence proper closed. We will see that thesupport of is a qpsh function is a countable union of proper closed subsets. SeeCorollary 4.26. Proposition 3.11.
Let ϕ ∈ QPSH( X ) be a qpsh function. Then, ϕ is convex oneach face of every dual complex ∆( Y, D ) . Moreover, ϕ ◦ r Y,D form a decreasingnet of continuous functions which converges to ϕ strongly in the norm. Inparticular, ϕ is weakly continuous and upper-semicontinuous (usc for short). Proof.
To show that ϕ is convex on each face of every dual complex ∆( Y, D ),it suffices to prove this when ϕ is an ideal function. We can assume that ϕ = c log | a | . Let η be a generic point of the intersection of D , . . . , D l . We willprove that ϕ is convex on QM η ( Y, D ) which essentially implies the convexity on∆(
Y, D ). To this end, assume that v = P kj =1 λ j v j such that v , v j ∈ QM η ( Y, D ), λ j > j and P kj =1 λ j = 1. Assume further that a · O Y is principlenear η generated by f . If we consider the local coordinates y = { y , . . . , y l } with the origin η , then v and v j can be represented by α = ( α , . . . , α l ) and α j = ( α j , . . . , α lj ) with the values v ( f ) = min { < α, β > | f = P c β y β } and v j ( f ) = min { < α j , β > | f = P c β y β } . Obviously, v ( f ) ≥ P kj =1 λ j v j ( f ) and weobtain the required convexity. If a · O Y is not principle, then ϕ is the maximumof a finite number of convex functions and hence convex.Given an arbitrary qpsh function ϕ , the functions ϕ ◦ r Y,D form a decreasingnet because v ≤ r Y,D ( v ), and ϕ is continuous on ∆( Y, D ) because it is theuniform limit function of continuous functions. It suffices to show that ϕ ◦ r Y,D converges to ϕ strongly in the norm. To this end, consider a sequence of idealfunctions ϕ j = c j log | a j | which converges to ϕ strongly in the norm. We thenobtain that k ϕ − ϕ ◦ r Y,D k ≤ k ϕ − ϕ j k + k ϕ j − ϕ ◦ r Y,D k ≤ k ϕ − ϕ j k if ( Y, D ) is a log resolution of a j which completes the proof. (cid:3) Remark 3.12.
The proposition above implies that a qpsh function is uniquelydetermined by its values on divisorial valuations. In fact, if ϕ and φ have thesame values on divisorial valuations, then ϕ = φ on every dual complex ∆( Y, D )by the continuity and hence ϕ = φ on V X .The following example shows that the pointwise limit of a decreasing sequenceof ideal functions is not qpsh in general. Example 3.13.
Let X = Spec k [ x ] be an affine line, and let φ k = P kj =1 log | f j | where f j = x − j . We see that φ k is a decreasing sequence of ideal functions andthe pointwise limit function ϕ exists. But ϕ is not qpsh because k ϕ − φ k ≥ φ .If f : X ′ → X is a regular morphism and ϕ is a qpsh function on X , then f ∗ ϕ is a qpsh function on V X ′ by Proposition 3.2. In particular if f : U → X isan open inclusion (resp. f : Spec O X,ξ → X ), we say that f ∗ ϕ is the restriction(resp. germ) of ϕ , and denote by ϕ | U (resp. ϕ ξ ). Also, restrictions to theneighborhoods of a point ξ induce a map QPSH( X ) → lim −−→ U ∋ ξ QPSH( U ), and theimage of ϕ is also said to be the germ of ϕ , and denoted by ϕ | ξ .If ξ is not contained in the support of a qpsh function ϕ , then ϕ ξ = 0 byProposition 3.11. However, the following example shows that it could happenthat the germ of ϕ is nonzero in the set lim −−→ U ∋ ξ QPSH( U ). ALUATIVE MULTIPLIER IDEALS 11
Example 3.14.
Let X = Spec k [ x ] be an affine line, and let φ k = P km =1 12 m log | f m | where f m = x − m . It is easy to see that φ k converges to a function φ stronglyin the norm. Note that the origin is not contained in the support of φ , but thegerm of φ in lim −−→ U ∋ QPSH( U ) is nonzero.From the above example we see that if we define k ϕ | ξ k := inf U ∋ ξ k ϕ | U k , then k · k is only a semi-norm. Proposition 3.15.
There is a surjective map of convex cones r : lim −−→ U ∋ ξ QPSH( U ) −→ QPSH(Spec O X,ξ ) which preserves the semi-norm, and also preserves k · k + and k · k − .Proof. If ϕ = c log | a | and ϕ ′ = c ′ log | a | , then we claim that k ϕ | ξ − ϕ ′ | ξ k = k ϕ ξ − ϕ ′ ξ k . To this end, let µ : ( Y, D ) → X be a log resolution of a · a ′ , and let a · O Y = O Y ( − F ) and a ′ · O Y = O Y ( − F ′ ). One can easily check that k ϕ | ξ − ϕ ′ | ξ k = max D i ∈S | ord D i F − ord D i F ′ | A (ord D i )where S consists of irreducible components D i of D such that µ ( D i ) contains ξ in its support. This implies the claim.Given a qpsh function ϕ ξ ∈ QPSH(Spec O X,ξ ), there exists a sequence of idealfunctions ϕ ξ,k = c i log | a ξ,i | which converges to ϕ ξ strongly in the norm. Let a i be ideals on X such that a i · O X,ξ = a ξ,i . We have that ϕ k = c i log | a i | convergesto a qpsh function in lim −−→ U ∋ ξ QPSH( U ) strongly in the norm due to the previousclaim. Therefore we obtain the surjectivity of r .Finally, for two qpsh function ϕ and ϕ ′ on an open neighborhood of ξ , theequality k ϕ | ξ − ϕ ′ | ξ k = k ϕ ξ − ϕ ′ ξ k follows from the claim in the first paragraph.Apply a similar argument to k · k + and k · k − , we obtain the conclusion. (cid:3) From the discussion above, we see that ϕ | ξ provides more information whileit is not a valuative function. We sometimes identify ϕ | ξ and ϕ ξ as the germ of ϕ at ξ . 4. Multiplier ideals
In this section we will discuss the multiplier ideals of qpsh functions. Recallthat a graded sequence of ideals a • is a sequence of ideals which satisfies a m · a n ⊆ a m + n . By convention we put a = O X , and we say a • is nontrivial if a m = 0 forsome positive integer m . Note that in this case there are infinitely many m suchthat a m = 0. A subadditive sequence of ideals b • is a one-parameter family b t satisfying b s · b t ⊇ b s + t for every s , t ∈ R + . Similarly, we put b = O X and wesay that b • is nontrivial if b t = 0 for all t ∈ R + . Throughout this paper, everysequence of ideals is assumed to be nontrivial. We define v ( a • ) = inf m ≥ v ( a m ) m and v ( b • ) = sup t> v ( b t ) t as in [16]. We similarly define | a • | ( v ) = e − v ( a • ) and | b • | ( v ) = e − v ( b • ) for a graded sequence and a subadditive sequence of idealsrespectively.4.1. Multiplier ideals.Definition 4.1.
For a bounded homogeneous function ϕ ∈ BH( X ), the multi-plier ideal J ( ϕ ) of ϕ is the largest ideal in the set of nonzero ideals { a |k log | a | − ϕ k + < } . If the above set is empty, then we define J ( ϕ ) = (0). Remark 4.2.
We will see that the above set is always nonempty when ϕ isqpsh and J ( ϕ ) is therefore nonzero (cf. Remark 4.21). Moreover, we havethe inequality ϕ ≤ log |J ( ϕ ) | (cf. Remark 4.21), and hence the inequality k log |J ( ϕ ) | − ϕ k < Proposition 4.3. If ϕ is an ideal function and we write ϕ = P li =1 c i log | a i | ,then J ( ϕ ) = J ( Q li =1 a i c i ) .Proof. Let π : ( Y, D ) −→ X be a log resolution of Q li =1 a i , and a i · O Y = O Y ( − F i ) with F i being supported in D . Since J ( Q li =1 a i c i ) = π ∗ O Y ( K Y/X − x P li =1 c i F i y ), it is easy to check that k log |J ( Q li =1 a i c i ) | − ϕ k + < J ( ϕ ) ⊇ J ( Q li =1 a i c i ).Conversely, if f ∈ Γ( U, J ( ϕ )) is a regular function on an affine open subset U ,then k log | f |− ϕ | U k + <
1. It follows that v ( f )+ A ( v )+ ϕ ( f ) > v on U . In particular, ord E f + ord E K Y/X + 1 > − ϕ (ord E )for any prime divisor E on π − U . Thus f ∈ Γ( U, J ( Q li =1 a i c i )) and it followsthat J ( ϕ ) ⊆ J ( Q li =1 a i c i ). (cid:3) The lemmas below will be frequently used in this paper.
Lemma 4.4.
Given a nonzero ideal q and a qpsh function ϕ ∈ QPSH( X ) , q ⊆ J ( λϕ ) if and only if λ − > k ϕ k q . Thus k ϕ k − q = min { t | q * J ( tϕ ) } Proof. If q ⊆ J ( λϕ ), then k log | q |− λϕ k + < v ∈ V ∗ X − v ( q ) − λϕ ( v ) A ( v ) <
1. This implies that − v ( q ) − λϕ ( v ) ≤ (1 − ε ) A ( v ) for every v ∈ V ∗ X . Thus − λϕ ( v ) A ( v )+ v ( q ) ≤ (1 − ε ) A ( v )+ v ( q ) A ( v )+ v ( q ) ≤ (1 − ε ) + ε k log | q |k q < λ − > k ϕ k q by definition.Conversely we assume that k ϕ k q = sup v ∈ V ∗ X − λϕ ( v ) A ( v )+ v ( q ) <
1. Then − λϕ ( v ) ≤ (1 − ε )( A ( v ) + v ( q )). Therefore − v ( q ) − λϕ ( v ) A ( v ) ≤ − ε − ε v ( q ) A ( v ) ≤ − ε for asufficiently small ε which leads to the conclusion q ⊆ J ( λϕ ). (cid:3) Lemma 4.5.
Let ξ be a point of a scheme X , and ϕ be a qpsh function. Assumethat the multiplier ideal J ( ϕ ) is nonzero. (In fact, this assumption automati-cally holds by Lemma 4.20 and Remark 4.21). Then,(1). J ( ϕ | U ) = J ( ϕ ) · O U . ALUATIVE MULTIPLIER IDEALS 13 (2). J ( ϕ ξ ) = J ( ϕ ) · O X,ξ .(3). Set λ − := k ϕ k q . If ξ ∈ V( J ( λϕ ) : q ) , then k ϕ k q = k ϕ ξ k q ·O X,ξ .Proof. (1). Since k log |J ( ϕ ) · O U | − ϕ | U k + ≤ k log |J ( ϕ ) | − ϕ k + <
1, we have J ( ϕ ) · O U ⊆ J ( ϕ | U ). On the other hand, if we denote by m the defining ideal of X \ U , then there exists a sufficiently large integer k such that v ( J ( ϕ )) ≤ v ( m k )for all valuations v centered in X \ U . Now we extend J ( ϕ | U ) to X and stilldenote it by J ( ϕ | U ). Therefore k log |J ( ϕ | U ) · m k | − ϕ k + < J ( ϕ | U ) ⊆ J ( ϕ ) · O U .(2). First note that k log |J ( ϕ ) · O X,ξ | − ϕ ξ k + ≤ k log |J ( ϕ ) | − ϕ k + < J ( ϕ ) · O X,ξ ⊆ J ( ϕ ξ ). For the inverse inclusion, we seethat if f ∈ J ( ϕ ξ ), then there exists an open neighborhood U of ξ such that k log | f |− ϕ | U k + < f ∈ J ( ϕ | U ) ·O X,ξ = J ( ϕ ) ·O X,ξ .(3). It is obvious that k ϕ k q ≥ k ϕ ξ k q ·O X,ξ by Proposition 3.2. If ξ ∈ V( J ( λϕ ) : q ), then ( J ( λϕ ξ ) : q · O X,ξ ) = ( J ( λϕ ) : q ) · O X,ξ = O X,ξ . Therefore q · O X,ξ * J ( λϕ | ξ ) and λ − ≤ k ϕ ξ k q ·O X,ξ by Lemma 4.4. (cid:3)
Algebraic qpsh functions.Definition 4.6.
A qpsh function ϕ ∈ QPSH( X ) is algebraic if it is the limitfunction of an increasing sequence of ideal functions ϕ = lim m →∞ ϕ m (in the norm).Note that ϕ being algebraic implies that tϕ is algebraic for any t ∈ R > , andthat ϕ + ψ is algebraic provided ψ is another algebraic qpsh function. Thus theset of algebraic qpsh functions is a convex subcone of QPSH( X ), and denotedby QPSH a ( X ).An algebraic function is lower-semicontinuous (lsc) on V X by its definition,and it is usc by Proposition 3.11, so it is continuous. We will see that in theabove definition the phrase ’in the norm’ is not necessary, that is, the pointwiselimit of an increasing sequence of ideal functions is algebraic qpsh (cf. Lemma4.15). One can compare this fact with Remark 4.25. The following exampleshows that a qpsh function is not necessarily algebraic. Example 4.7.
Let X = Spec k [ x , x ] be the affine plane. If we set φ k = k P l =1 12 l log | f l | where f l = x + x l , then φ k converges to a qpsh function φ stronglyin the norm. However, the qpsh function φ is not algebraic since there is noideal function ϕ ≤ φ .The following lemma shows that a graded system of ideals naturally inducesan algebraic qpsh functions. Lemma 4.8.
Let a • be a graded sequence of ideals. If we define log | a • | ( v ) = − v ( a • ) , then log | a • | is an algebraic qpsh function.Proof. It suffices to show that there exists a subsequence of { a m } such that { ϕ k := m k log | a m k |} is an increasing sequence of ideal functions which convergesto a qpsh function strongly in the norm. Let b • be a sequence of ideals suchthat b t = J ( a t • ) (cf. [25]). Note that b • is subadditive of controlled growth (cf. [22, Section 2, Section 6, Appendix]). Now we fix an integer m suchthat a m = 0. Since b m ⊇ J ( a m ) ⊇ a m , we have v ( b m ) ≤ v ( a m ). Since v ( b m ) + A ( v ) − k v ( a mk ) > v where k isa sufficiently divisible integer, we have v ( a mk ) mk < v ( b m ) m + A ( v ) m . From the inequality v ( b m ) m ≤ v ( a mk ) mk < v ( b m ) m + A ( v ) m , we have that k mk log | a mk | − mkl log | a mkl |k < m for every positive integer l . As we multiply m , we obtain the desired sequenceof ideal functions. (cid:3) Definition 4.9.
Let ϕ ∈ BH( X ) be a bounded homogeneous function. Itsenvelope ideal a ( ϕ ) is the largest ideal in the set { a | log | a | ≤ ϕ } if this set isnonempty. If it is empty, we set a ( ϕ ) = 0. Proposition 4.10. If ϕ is qpsh and a ( ϕ ) is nonzero, then the envelope idealcan be written explicitly as Γ( U, a ( ϕ )) := { f ∈ O X ( U ) | v ( f ) + ϕ ( v ) ≥ for every v ∈ V ∗ U } on every open subset U .Proof. Since the question is local, we can assume that X = Spec A is affine. Itsuffices to prove that the ideal a , defined by a ( U ) := { f ∈ O X ( U ) | v ( f ) + ϕ ( v ) ≥ v ∈ V ∗ U } on every open subset U , is coherent. To this end, we write I := a ( X ), and we will prove that a ( U g ) = I g for any nonzero regular function g ∈ A , where U g denotes the affine open subset defined by g . Since a ( U g ) ⊇ I g by definition, we only need to prove the converse inclusion. Note that thereexists a large integer k such that kv ( g ) ≥ v ( a ( ϕ )) for every nontrivial temperedvaluation v centered in the locus V ( g ), and hence k log | g | ( v ) ≤ ϕ ( v ). If f isa regular function on U g such that v ( f ) + ϕ ( v ) ≥ v ∈ V ∗ U g , then v ( f g k ) + ϕ ( v ) ≥ v ∈ V ∗ X which implies that f ∈ I g . (cid:3) If we set a ( ϕ ) m = a ( mϕ ), then { a ( ϕ ) • } is a (possibly trivial) graded sequenceof ideals. The following lemma shows that every algebraic qpsh function is ofthe form log | a • | . Lemma 4.11. If ϕ ∈ QPSH a ( X ) is an algebraic qpsh function, then ϕ =log | a ( ϕ ) • | .Proof. Given an arbitrary small positive number ε , there exist an ideal a on X and an integer m such that m log | a | ≤ ϕ and k m log | a | − ϕ k < ε . We have m log | a ( ϕ ) m | ≥ m log | a | by definition and the conclusion follows. (cid:3) By combining Lemma 4.3, Lemma 4.8 and Lemma 4.11, we see that a boundedhomogeneous function is algebraic qpsh if and only if it is derived from a gradedsequence of ideals. Readers can compare the following theorem with Theorem4.24.
Theorem 4.12.
Let ϕ be a bounded homogeneous function. Then the followingsare equivalent.(1). ϕ ∈ QPSH a ( X ) is algebraic qpsh.(2). There exists a graded sequence of ideals a • such that ϕ = log | a • | .(3). The graded system of ideals a ( ϕ ) • is nontrivial and ϕ = log | a ( ϕ ) • | . ALUATIVE MULTIPLIER IDEALS 15
Proof.
If we assume (1), then (3) holds by Lemma 4.11. Note that (3) implies(2) if we simply put a • = a ( ϕ ) • . Finally, (1) follows from (2) by Lemma 4.8. (cid:3) We will use the following easy lemma. For the convenience of readers wepresent a proof here.
Lemma 4.13.
Let ϕ ∈ QPSH a ( X ) be an algebraic qpsh function.(1). Assume that { ϕ m } is an increasing sequence of qpsh functions which con-verges to ϕ strongly in the norm. Then J ( ϕ ) = J ( ϕ m ) for m sufficiently large.(2). Assume that f : X ′ −→ X is a regular morphism of schemes. Then f ∗ ϕ is algebraic qpsh.Proof. (1). We see that k log |J ( ϕ ) | − ϕ k + = 1 − ε for some positive number ε . If k ϕ − ϕ m k < ε , then k log |J ( ϕ ) | − ϕ m k + < J ( ϕ ) ⊆ J ( ϕ m ). Theinverse inclusion J ( ϕ ) ⊇ J ( ϕ m ) is obvious because ϕ ≥ ϕ m .(2). Assume ϕ m is an increasing sequence of ideal functions which convergesto ϕ strongly in the norm. Then f ∗ ϕ m is also an increasing sequence of idealfunctions which converges to f ∗ ϕ strongly in the norm by Proposition 3.2. Thisimplies that f ∗ ϕ is algebraic qpsh. (cid:3) By combining Lemma 4.8 and Proposition 4.13(1), we see that the definitionof valuative multiplier ideals of algebraic functions coincides with the ’classicaldefinition’ of asymptotic multiplier ideals.
Corollary 4.14.
Let a • be a graded sequence of ideals. If we write ϕ = log | a • | ,then J ( ϕ ) = J ( a • ) . General qpsh functions.Lemma 4.15. If { ϕ λ } is a family of (algebraic) qpsh functions, then sup λ ϕ λ is an (algebraic) qpsh function. Therefore, the convex cone QPSH( X ) (resp. QPSH a ( X ) ) is closed under taking the supremum.Proof. We firstly assume that { ϕ λ } is a family of algebraic qpsh functions, andwe write ψ = sup λ ϕ λ . Since ψ ≥ ϕ λ for every λ , a ( ψ ) m ⊇ a ( ϕ λ ) m . It followsthat log | a ( ψ ) • | ≥ log | a ( ϕ λ ) • | = ϕ λ . Therefore ψ = log | a ( ψ ) • | is algebraicqpsh.Now we treat the case when { ϕ λ } is a family of general qpsh functions. Foreach λ , we assume that { ϕ λ,m } is a sequence of ideal functions which convergesto ϕ λ strongly in the norm such that k ϕ λ − ϕ λ,m k < m . If we set ψ m = sup λ ϕ λ,m which is algebraic qpsh by the previous argument, then k ψ − ψ m k ≤ m and itfollows that { ψ m } is a sequence which converges to ψ strongly in the norm. (cid:3) Since the convex cones QPSH( X ) and QPSH a ( X ) are closed under takingthe supremum by Lemma 4.15, we can introduce the following definition. Definition 4.16.
Let ϕ be a bounded homogeneous function. Assume that theset { ψ ∈ QPSH( X ) | ψ ≤ ϕ } is nonempty. Then we say the maximal functionin this set the qpsh envelope function. We similarly define the algebraic qpshenvelope function of ϕ if it exists. In general, we cannot ensure the sets defined as above are nonempty. Forinstance, the function in Example 3.13 is bounded homogeneous but its qpshenvelope function does not exist. Also note that the function φ in Example 4.7is qpsh itself but its algebraic qpsh envelope function does not exist. Lemma 4.17.
Let ϕ be a bounded homogeneous function that is determinedon some dual complex ∆( Y, D ) in the sense of ϕ = ϕ ◦ r Y,D . Then, its qpshenvelope function ψ exists. Further, ψ is algebraic qpsh.Proof. Let Z ⊆ X be the image of the reduced divisor D on X , and m bethe defining ideal of Z . Since log | m | is strictly negative on ∆( Y, D ) and ϕ is bounded on ∆( Y, D ), there exists an integer k such that k log | m | ≤ ϕ on∆( Y, D ). Because ϕ is determined on the dual complex ∆( Y, D ) in the senseof ϕ = ϕ ◦ r Y,D , we have that k log | m | ≤ ϕ on V X . It follows that its algebraicqpsh envelope function φ exists. In particular, its qpsh envelope function ψ exists.Now we will show that ψ = φ . Set µ = max v ∈ ∆( Y,D ) | v ( m ) | and µ =min v ∈ ∆( Y,D ) | v ( m ) | . For any small number ε >
0, we choose δ ≪ µ µ ) δ < ε and an ideal function φ ′ such that k φ ′ − ψ k < δ . Note that forevery valuation v ∈ ∆( Y, D ) we have ψ ( v ) > φ ′ ( v ) − δµ v ( m ) ≥ φ ′ ( v ) − δµ µ > ψ ( v ) − (1 + µ µ ) δ. After replacing φ ′ by φ ′ + δµ log | m | , we can assume that φ ′ ≤ ψ and | ψ ( v ) − φ ′ ( v ) | < ε on ∆( Y, D ). Because ϕ = ϕ ◦ r Y,D , we obtain that φ ′ ≤ ϕ . It followsthat φ ′ ≤ φ ≤ ψ by the definition of the qpsh envelope function. Since ε canbe chosen arbitrary small, this forces φ = ψ on ∆( Y, D ). If we pick any higherlog resolution ( Y ′ , D ′ ), we can show that φ = ψ on ∆( Y ′ , D ′ ) by the sameargument. The conclusion therefore follows from Proposition 3.11. (cid:3) The above lemma leads to the following definition.
Definition 4.18.
Let ϕ ∈ QPSH( X ) be a qpsh function. We denote by ϕ Y,D the qpsh envelop function of ϕ ◦ r Y,D . Lemma 4.19.
Let ϕ be a qpsh function. Then there exists a decreasing sequenceof algebraic functions which converges to ϕ strongly in the norm.Proof. Let { ϕ m } be a sequence of ideal functions which converges to ϕ stronglyin the norm. We can assume that ϕ m = c m log | a m | and k ϕ − ϕ m k < m . Let( Y, D ) be a log resolution a . It is easy to see that k ϕ ◦ r Y,D − ϕ k <
1, andtherefore k ϕ Y,D − ϕ k <
1. We deduce that k ϕ Y,D − ϕ k <
2. Now we replace ϕ by ϕ Y,D and continue this process. Note that if ( Y ′ , D ′ ) (cid:23) ( Y, D ), then ϕ Y ′ ,D ′ ≤ ϕ Y,D by Proposition 3.11. We easily obtain the required decreasingsequence of algebraic functions. (cid:3)
Lemma 4.20.
Let { ϕ m } be a sequence of qpsh functions which converges toa qpsh function ϕ strongly in the norm. Then J ( ϕ ) = J ((1 + ε ) ϕ m ) for asufficiently small positive real number ε > and a sufficiently large integer m . ALUATIVE MULTIPLIER IDEALS 17
Proof.
First we prove that J ( ϕ ) ⊆ J ((1 + ε ) ϕ m ) for a sufficiently small number ε > m . To this end, we pick a sufficiently smallnumber ε > J ( ϕ ) = J ((1+ ε ) ϕ ). Since J ((1+ ε ) ϕ ) ⊆ J ((1+ ε ) ϕ m )provided that m is sufficiently large, we have J ( ϕ ) ⊆ J ((1+ ε ) ϕ m ). Conversely,we pick a sufficiently large integer m such that k ϕ − ϕ m k < − ε . ApplyingLemma 4.4 again, we see that if f ∈ J ((1 + ε ) ϕ m ), then k ϕ m k f < ε andhence k ϕ k f ≤ k ϕ m k f + k ϕ − ϕ m k f < f ∈ J ( ϕ ). (cid:3) Remark 4.21.
Note that for an algebraic qpsh function φ , we always have φ ≤ log |J ( φ ) | by [22, Proposition 6.2 and 6.5]. It follows by Lemma 4.19 andLemma 4.20 that J ( ϕ ) is nonzero and (1 + ε ) ϕ ≤ (1 + ε ) ϕ m ≤ log |J ( ϕ ) | where { ϕ m } is a decreasing sequence of algebraic functions which converges to a qpshfunction ϕ strongly in the norm. Since ε can be chosen arbitrary small, weimmediately obtain that ϕ ≤ log |J ( ϕ ) | .Now we discuss the multiplier ideals of general qpsh functions. Proposition 4.22.
Let ϕ ∈ QPSH( X ) be a qpsh function on X .(1). Assume that ψ is another qpsh function on X . Then, J ( ϕ + ψ ) ⊆ J ( ϕ ) · J ( ψ ) . (2). Assume that f : X ′ −→ X is a regular morphism of schemes. Then, J ( ϕ ) · O X ′ = J ( f ∗ ϕ ) . Proof. (1). By Lemma 4.19 we can assume that there are decreasing sequencesof algebraic functions { ϕ m } and { ψ m } convergent to ϕ and ψ strongly in thenorm respectively. Then for some sufficiently large integer m , by Lemma 4.20we have J ( ϕ + ψ ) = J ((1 + ε )( ϕ m + ψ m )) ⊆ J ((1 + ε ) ϕ m ) · J ((1 + ε ) ψ m ) = J ( ϕ ) · J ( ψ ) since ϕ m + ψ m converges to ϕ + ψ strongly in the norm. Theinclusion appeared in the above inequality follows from [22, Theorem A.2].(2). Since f is regular, for any ideal function φ = P i c i log a i , we have J ( φ ) · O X ′ = J ( Y i a c i i ) · O X ′ = J ( Y i ( a i · O X ′ ) c i ) = J ( f ∗ φ )by the argument of [22, Proposition 1.9]. If { ϕ m } is a sequence of ideal functionswhich converges to ϕ strongly in the norm, then f ∗ ϕ m is a decreasing sequenceof ideal functions which converges to f ∗ ϕ strongly in the norm by Proposition3.2. Therefore we have J ( ϕ ) · O X ′ = J ((1 + ε ) ϕ m ) · O X ′ = J ((1 + ε ) f ∗ ϕ m ) = J ( f ∗ ϕ ). (cid:3) Recall from [22] that if b • is subadditive, then the limit v ( b • ) := lim m →∞ m v ( b m ) ∈ [0 , + ∞ ]is well-defined. For the purpose of constructing a ”good” valuative function, weintroduce the notion of a subadditive sequence of ideals of controlled growth asfollows. Definition 4.23 (22, Definition 2.9) . A subadditive sequence of ideals b • is ofcontrolled growth if v ( b t ) t > v ( b • ) − A ( v ) t for every nontrivial tempered valuation v and every t > v ( b • ) := lim m →∞ m v ( b m ) < + ∞ for every nontrivial temperedvaluation v . Furthermore, if we define log | b • | ( v ) = − v ( b • ), then log | b • | isapproximated by m log | b m | strongly in the norm and hence qpsh. Given a qpshfunction, if we define J ( ϕ ) t := J ( tϕ ), then J ( ϕ ) • is a subadditive sequenceof controlled growth by Proposition 4.22, Definition 4.1 and Remark 4.2. Thisallows us to give a characterization of qpsh functions as follows. Readers couldcompare the following theorem with Theorem 4.12. Theorem 4.24.
Let ϕ be a bounded homogeneous function. Then the followingsare equivalent.(1). ϕ is qpsh.(2). There is a subadditive sequence of ideals b • of controlled growth such that ϕ = log | b • | .(3). The ideal J ( tϕ ) is nonzero for every t > and ϕ = log |J ( ϕ ) • | .Proof. If we assume (1), then (3) follows from the previous argument togetherwith Definition 4.1 and Remark 4.2. Note that (3) implies (2) if we simply put b • = J ( ϕ ) • . Finally, (1) follows from (2) by the previous argument. (cid:3) Remark 4.25.
From the above theorem we see that every qpsh function ϕ canbe approximated by a decreasing sequence of ideal functions ϕ k in the norm.Indeed, we can take ϕ k = k log |J (2 k ϕ ) | . However, if ϕ is only the pointwiselimit of a decreasing sequence of ideal functions on V X , then ϕ is not necessarilyqpsh (cf. Example 3.13).An immediate application of the above discussion is the following result onthe support of a qpsh function. Corollary 4.26.
Let ϕ be a qpsh function. Then its support Supp ϕ is a count-able union of proper Zariski closed subsets of X . Remark 4.27.
Readers can compare the constructions here with [5]. If we workon X = Spec b R where R is the localization of C [ x , . . . , x n ] at the origin, thenour definition of qpsh functions coincides the notion of formal psh functions .A brief argument is as follows. Given a formal psh function g , we have asubadditive sequence of ideals {L ( tg ) } t> in b R by [5, Theorem 3.10] whichsatisfies that v ( L ( tg )) + A ( v ) + (1 + ǫ ) g ( v ) ≥ v centered at the origin and an arbitrary small ǫ by [5, Theorem 3.9].It follows that {L ( tg ) } t> form a subadditive sequence of ideals of controlledgrowth which induces to a qpsh function ϕ on X . Therefore ϕ ( v ) = g ( v ) forevery divisorial valuation v centered at the origin. Conversely, a qpsh functioncan be naturally viewed as a formal psh function by definition. Therefore weconstruct an one-to-one correspondence. The details are left to the readers. ALUATIVE MULTIPLIER IDEALS 19
Remark 4.28.
Recall from complex geometry that a function ϕ : X → [ −∞ , + ∞ )from a complex manifold is qpsh if it is locally equal to the sum of a smoothfunction and a psh function. If X is a smooth complex variety, then we shouldbe able to define the valuative transform of ϕ which is expected to be a qpshfunction on the valuation space V X as defined in this paper. This was donelocally in [5] and its predecessors [19], [20], [21]. However, the global situationis not fully understood by us at this point.5. Computing norms
Generalities.Definition 5.1.
Let ϕ be a bounded homogeneous function and q be a nonzeroideal on X . We say a nontrivial tempered valuation v ∈ V ∗ X computes k ϕ k q ifthe equality k ϕ k q = | ϕ ( v ) | A ( v )+ v ( q ) holds.The main result of this section is the following theorem. Theorem 5.2.
Let ϕ ∈ QPSH( X ) be a qpsh function and let q be a nonzeroideal on X . Then there exists a nontrivial tempered valuation v which computes k ϕ k q . Before we prove this theorem, we need some preparations.
Proposition 5.3.
Let ϕ be a bounded homogeneous function that is determinedon some dual complex ∆( Y, D ) in the sense of ϕ = ϕ ◦ r Y,D . Assume that ϕ is weakly continuous (cf. Definition 3.4). Then there exists a quasi-monomialvaluation v which computes k ϕ k q . If we assume further that ϕ is affine on eachface of ∆( Y, D ) , then there exists a divisorial valuation v which computes k ϕ k q .Proof. For every nontrivial tempered valuation v ∈ V ∗ X , we have | ϕ ( v ) | A ( v ) + v ( q ) ≥ | ϕ ◦ r Y,D ( v ) | A ( r Y,D ( v )) + r Y,D ( v )( q )with equality if and only if v ∈ QM(
Y, D ). Thus k ϕ k q = sup v ∈ QM(
Y,D ) | ϕ ( v ) | A ( v ) + v ( q ) = sup v ∈ ∆( Y,D ) | ϕ ( v ) | v ( q ) . Since ϕ is weakly continuous, the function v → | ϕ ( v ) | A ( v )+ v ( q ) is continuous onQM( Y, D ). Therefore the function v → | ϕ ( v ) | v ( q ) is continuous on the dual complex∆( Y, D ) and hence achieves its maximum in ∆(
Y, D ).Assume that ϕ is affine on ∆( Y, D ), and we denote by D i ’s the irreduciblecomponents of D . After replacing ( Y, D ) by some higher log resolution, wecan assume that (
Y, D ) is a log resolution of q by Lemma 3.8. Then we have k ϕ k q = max D i | ϕ (ord Di ) | A (ord Di )+ord Di ( q ) where D i runs over all irreducible components of D since the functions ϕ , A and log | q | are all affine on ∆( Y, D ). (cid:3) Computing norms of qpsh functions.
This subsection is devoted tothe proof of Theorem 5.2. The proof here follows from the strategy of [22]. Wefirst consider the local case.
Lemma 5.4.
Let ( R, m ) be a local ring, let ϕ ∈ QPSH(Spec R ) be a qpsh func-tion, and let q be a nonzero ideal on Spec R . We set λ − = k ϕ k q and assume that p ( J ( λϕ ) : q ) = m . If we define another qpsh function ψ = max { ϕ, p log | m |} for a sufficiently large integer p , then k ϕ k q = k ψ k q . Moreover, if a nontrivialtempered valuation v computes k ψ k q , then v also computes k ϕ k q .Proof. Since p (( J ( λϕ ) : q ) = m , we have m n · q ⊆ J ( λϕ ) for some integer n . Set λ ′− = k ϕ k m n · q , it follows that λ ′ > λ by Lemma 4.4. Pick an integer p > n/ ( λ ′ − λ ), and fix a sufficiently small number ε < ≪ p >n/ ((1 − ε ) λ ′ − λ ). Observe that k ψ k q = sup v ∈ V ∗ R min {− ϕ ( v ) , pv ( m ) } A ( v ) + v ( q ) ≥ sup v ∈ V ∗ ε min {− ϕ ( v ) , pv ( m ) } A ( v ) + v ( q )where V ∗ ε is the set of v ∈ V ∗ R satisfying − ϕ ( v ) A ( v )+ v ( q ) ≥ (1 − ε ) /λ .By the definition of λ ′ we have nv ( m ) − ϕ ( v ) ≥ λ ′ − A ( v )+ v ( q ) − ϕ ( v ) for every nontrivialtempered valuation v . This implies that k ψ k q ≥ sup v ∈ V ε − ϕ ( v ) A ( v ) + v ( q ) min { , pn ( λ ′ − A ( v ) + v ( q ) − ϕ ( v ) ) }≥ sup v ∈ V ε − ϕ ( v ) A ( v ) + v ( q ) min { , pn ( λ ′ − λ − ε ) } = sup v ∈ V ε − ϕ ( v ) A ( v ) + v ( q ) = k ϕ k q . Moreover, if a nontrivial tempered valuation v computes k ψ k q , then from theabove inequality we see that v also computes k ϕ k q . (cid:3) Lemma 5.5.
Let ( R, m ) be a local ring, let ϕ be an ideal function on Spec R suchthat ϕ ≥ p log | m | for some integer p , and let q be a nonzero ideal on Spec R .Then there exists a nontrivial tempered valuation v ∈ V R, m ,M (cf. Definition2.6) which computes k ψ k q provided that M > p · k ϕ k − q .Proof. If we write c = p/M , then 0 < c < k ψ k q . For every v ∈ V R, m suchthat − ϕ ( v ) A ( v )+ v ( q ) > c , we have A ( v ) ≤ A ( v ) + v ( q ) < p/c = M . Thus k ϕ k q =sup v ∈ V R, m ,M − ϕ ( v ) A ( v )+ v ( q ) . Note that V R, m ,M is compact. Since the function v → − ϕ ( v ) A ( v )+ v ( q ) is usc as the valuative function A is lsc, the maximum can be achieved in V R, m ,M . (cid:3) Lemma 5.6.
Let ϕ ∈ QPSH( X ) be a qpsh function on X and { ϕ m } be adecreasing sequence of algebraic functions which converges to ϕ strongly in thenorm. Set λ − = k ϕ k q and λ − m = k ϕ m k q . Then, J ( λϕ ) ⊆ J ( λ m ϕ m ) for everysufficiently large integer m .Proof. If f ∈ J ( λϕ ), then k ϕ k f < (1 − ε ) /λ for a sufficiently small number ε >
0. Since λ m < λ/ (1 − ε ) for sufficiently large m , we have k ϕ m k f ≤ k ϕ k f < (1 − ε ) /λ < λ − m . It follows that f ∈ J ( λ m ϕ m ) by Lemma 4.4. (cid:3) ALUATIVE MULTIPLIER IDEALS 21
Lemma 5.7.
Let ( R, m ) be a local ring, let ϕ be a qpsh function on Spec R suchthat ϕ ≥ p log | m | , and let q be a nonzero ideal on Spec R . Then there existsa nontrivial tempered valuation v ∈ V R, m ,M which computes k ϕ k q provided that M > p · k ϕ k − q .Proof. Assume that { ϕ m } is a decreasing sequence of ideal functions whichconverges to ψ strongly in the norm. Then m n · q ⊆ J ( λϕ ) ⊆ J ( λ m ϕ m ) forevery sufficiently large integer m by Lemma 5.6. We set λ ′− = k ϕ k m n · q and λ ′− m = k ϕ m k m n · q . Note that M > p · λ m for every sufficiently large integer m .Therefore for every sufficiently large integer m , there exists v m ∈ V R, m ,M whichcomputes k ϕ m k q by Lemma 5.5. By passing { ϕ m } to a subsequence, we canassume { v m } is a sequence of points which converges to some point v ∈ V R, m ,M .Note that − λϕ ( v ) A ( v ) + v ( q ) ≥ − λϕ m ( v ) A ( v ) + v ( q ) ≥ − λϕ m ( v n ) A ( v n ) + v n ( q ) − δ ≥ − k λϕ m − λ n ϕ n k q − δ ≥ − λ k ϕ m − ϕ n k q − δ − ( λ n − λ ) k ϕ k q where the second inequality holds because the function v → − λϕ m ( v ) A ( v )+ v ( q ) is usc.Since k ψ m − ψ n k q , δ and λ n − λ can be chosen arbitrary small, we have that − λψ ( v ) A ( v )+ v ( q ) ≥ (cid:3) Now we turn to treat the global case.
Proof of Theorem 5.2.
Pick a generic point ξ of V ( J ( λϕ ) : q ). Note that k ϕ k q = k ϕ ξ k q ·O X,ξ by Lemma 4.5(3). After replacing X and ϕ by Spec O X,ξ and ϕ ξ , respectively, we reduce the global case to the local case. After replac-ing ϕ by max { ϕ, p log | m |} for a sufficiently large integer p by Lemma 5.4, wecan assume that ϕ ≥ p log | m | . Finally by Lemma 5.7, there exists a valuation v ∈ V X,ξ,M which computes k ϕ k q . (cid:3) An immediate consequence of Theorem 5.2 is the following corollary.
Corollary 5.8.
Let ϕ be a qpsh function on X . Then, on every open subset U ,we can explicitly write Γ( U, J ( ϕ )) = { f ∈ Γ( U, O X ) | v ( f ) + A ( v ) + ϕ ( v ) > forevery v ∈ V ∗ U } . Let q be a nonzero ideal on X . Then, q ⊆ J ( ϕ ) if and only if v ( q ) + A ( v ) + ϕ ( v ) > for every v ∈ V ∗ X . The following conjecture was raised as [22, Conjecture B] (cf. [22, Theorem7.8]). It is already known for several special cases (cf. [22, Section 8 and 9]).
Conjecture 5.9.
Let ϕ be a qpsh function on X and q be a nonzero ideal on X . Then there exists a nontrivial quasi-monomial valuation v which computes k ϕ k q . Conversely, if a nontrivial tempered valuation v computes the norm ofsome qpsh function, then v is quasi-monomial. Applications If X is a smooth complex projective variety, then we are interested in asso-ciating a qpsh function to a line bundle which plays the role of a semi-positivesingular metric. The starting point is the following easy observation. Given apseudo-effective line bundle L , an ideal a together with a nonnegative rationalnumber λ such that L ⊗ a λ is semi-ample corresponds to a semi-positive singularmetric h in the sense that they give the same multiplier ideals J ( a λm ) = J ( h ⊗ m )for every integer m >
0. However in general, this correspondence become quitemysterious since many analogue notions cannot be constructed. This has beenstudied in many relevant references such as [3], [16], [17], [18], [27], [28]. Wewill discuss the qpsh function associated to a line bundle in detail within thissection. Besides, it might be possible to generalize the results in this section tovarieties with mild singularities such as klt singularities (cf. [9], [10]).Throughout this section X will be a projective smooth variety over C forsimplicity. The term ’divisor’ will always refer to a Q -Cartier Q -divisor. Givena section s ∈ H ( X, L ) of a line bundle, the notation log | s | denotes the qpshfunction defined locally by a regular function corresponding to s .6.1. D -psh functions.Definition 6.1. Let D be a divisor. We define the set L D := { k log | a | | kmD ⊗ a m is globally generated for every sufficiently divisible m } . We then define set of D -psh functions to be the closure PSH( D ) = L D in thenorm. Lemma 6.2. (1).
PSH( D ) is compact and convex in QPSH( X ) ;(2). PSH( tD ) = t PSH( D ) for any t ∈ Q > ;(3). PSH( D ) + PSH( D ′ ) ⊆ PSH( D + D ′ ) .(4). If A is a semiample divisor, then PSH( D ) ⊆ PSH( D + A ) .Proof. We firstly prove (1). To prove that PSH( D ) is convex, it suffices to showthat L D is convex. Given qpsh functions ϕ , ϕ ′ ∈ L D and a rational number0 < λ <
1, we will show that λϕ + (1 − λ ) ϕ ′ ∈ L D . If we write ϕ = k log | a | , ϕ ′ = k ′ log | a ′ | and λ = q/p , then λϕ + (1 − λ ) ϕ ′ = 1 kp log | a q | + 1 k ′ p log | a ′ p − q | = 1 kk ′ p log | a qk ′ · a ′ k ( p − q ) | . It is easy to check that kk ′ pmL ⊗ a mqk ′ · a ′ mk ( p − q ) is globally generated for everysufficiently divisible integer m and the conclusion follows. Note that (2), (3)and (4) can be proved in a similar way. (cid:3) Question 6.3.
Let ϕ be a general qpsh function. Does there exist a divisor D such that ϕ ∈ PSH( D )? ALUATIVE MULTIPLIER IDEALS 23
Definition 6.4.
The set of pseudo D -psh functions is defined to be PSH σ ( D ) := T ε> PSH( D + εA ) where A is an ample divisor.Note that the above definition is independent of the choice of the ampledivisor A , and that the set PSH σ ( D ) also satisfies the properties listed in Lemma6.2. Theorem 6.5 (Nadel Vanishing) . Let L be a line bundle on a smooth projectivevariety X and L ≡ A + D where A is a nef and big Q -divisor. Assume that ϕ ∈ PSH σ ( D ) . Then H i ( X, ( K X + L ) ⊗ J ( ϕ )) = 0 for all i > .Proof. First by Kodaira Lemma A − δE is ample for some effective divisor E and every sufficiently small number δ >
0. If we write ϕ E = log |O X ( − E ) | ,then by semicontinuity of multiplier ideals we have J ( ϕ ) = J ( ϕ + δϕ E ) forevery sufficiently small number δ >
0. After replacing A and ϕ with A − δE and ϕ + δϕ E , respectively, we can assume that A is ample.By definition we can assume that there exists a sequence of ideal functions { ϕ k } which converges to ϕ strongly in the norm, such that ϕ k ∈ L D + ǫ k A and ǫ k → ε ≪ A − εD is ample. We see that J ( ϕ ) = J ((1 + ε ) ϕ k ) for every sufficiently large integer k by Lemma 4.20. Note that(1 + ε ) ϕ k ∈ L (1+ ε )( D + ǫ k A ) . For a sufficient large integer k , A − εD − (1 + ε ) ǫ k A is ample. After replacing A and ϕ by A − εD − (1 + ε ) ǫ k A and (1 + ε ) ϕ k ,respectively, we reduce to the classical Nadel vanishing (cf. [25]). (cid:3) As an application of the above theorem, one can easily deduce the followingtheorem by letting G = K X + ( n + 1) H where H is a hypersurface of X and n = dim X , with the aid of the Castelnuovo-Mumford regularity. Theorem 6.6 (Global generation) . Let D be a divisor on X and ϕ be a qpshfunction. Then, ϕ ∈ PSH σ ( D ) if and only if there exists a line bundle G suchthat ( mD + G ) ⊗ J ( mϕ ) is globally generated for all m ∈ Z + with mD integral. Given a qpsh function ϕ , a positive real number λ is said to be the (higher)jumping number of ϕ if J (( λ − ǫ ) ϕ ) ) J ( λϕ ) for every positive real number ǫ . Definition 6.7.
Let ϕ be a qpsh function. We define the ideal J − ( ϕ ) to bethe largest ideal in the set { a |k log | a | − ϕ k ≤ } . One can see that J − ( ϕ ) canbe written explicitly asΓ( U, J − ( ϕ )) = { f ∈ O X ( U ) | v ( f ) + A ( v ) + ϕ ( v ) ≥ for every v ∈ V ∗ U } for every open subset U . Lemma 6.8. If ϕ is D -psh for some divisor D , then the descending chain ofideals J ((1 − ǫ ) ϕ ) stabilizes as ǫ → . Further, J ((1 − ǫ ) ϕ ) = J − ( ϕ ) for ǫ ≪ . It follows that the set of its (higher) jumping numbers is discrete. Proof.
By adding an ample divisor to D , we can assume that D is Cartier. ByTheorem 6.5 and the Castelnuovo-Mumford regularity there exists an ample linebundle G such that O X ( D + G ) ⊗ J ((1 − ǫ ) ϕ ) is globally generated for ǫ ≪ H ( X, O X ( D + G ) ⊗ J ((1 − ǫ ) ϕ ))will stabilize as ǫ → J ((1 − ǫ ) ϕ ) will stabilize.The reader can find more details in [27, Theorem 4.2].Fix a sufficiently small number ǫ ′ . Since k log |J ((1 − ǫ ′ ) ϕ ) | − (1 − ǫ ) ϕ k < ǫ , we see that k log |J ((1 − ǫ ′ ) ϕ ) | − ϕ k ≤ J ((1 − ǫ ) ϕ ) ⊆ J − ( ϕ ). To prove the converse inclusion, simplynotice thatΓ( U, J − ( ϕ )) = { f ∈ O X ( U ) | v ( f ) + A ( v ) + ϕ ( v ) ≥ for every v ∈ V ∗ U } and hence J ((1 − ǫ ) ϕ ) ⊇ J − ( ϕ ) for ǫ ≪ (cid:3) To investigate the structure of the sets PSH( D ) and PSH σ ( D ), we needthe following construction. Given an integer k , a divisor D and a qpsh func-tion ϕ , we define the linear system V m ( D, ϕ, t ) := { L ∈ | x mD y || m log | s L | ≤ t log |J − ( tϕ ) |} where s L is the section associated to the divisor L and ǫ ≪
1. Ifwe set a ( D, ϕ, t ) m := b ( V m ( D, ϕ, t )) where b ( V m ( D, ϕ, t )) denotes the base idealof the linear system V m ( D, ϕ, t ), then a ( D, ϕ, t ) • is a graded sequence of ideals.Moreover, we define ϕ Dt := log | a ( D, ϕ, t ) • | for every positive rational number t . Lemma 6.9.
Let D be a divisor on X and ϕ be a qpsh function. Then, ϕ ∈ PSH( D ) if and only if ϕ = lim t →∞ ϕ Dt pointwisely.Proof. First assume that ϕ ∈ PSH( D ). Let { ϕ m } be a sequence of ideal func-tions which converges to ϕ such that each ϕ m ∈ L D . If t is not a (higher)jumping number of ϕ , then, by Lemma 4.20 we have J − ( tϕ ) = J (( t − ǫ ) ϕ ) = J (( t − ǫ + ǫ ′ ) ϕ m ) ⊇ J − ( tϕ m )and J − ( tϕ ) = J ( tϕ ) = J (( t + ǫ ) ϕ m ) ⊆ J − ( tϕ m )for every sufficiently large integer m . It follows that J − ( tϕ ) = J − ( tϕ m ) and ϕ Dt = ϕ Dm,t . Note that ϕ Dm.t ≥ ϕ m , and hence t log |J − ( tϕ ) | ≥ ϕ Dt ≥ ϕ . If t is a (higher) jumping number, then ϕ Dt ≥ ϕ Dt − ǫ ≥ ϕ . Therefore, we have k ϕ Dt − ϕ k ≤ t and hence ϕ = lim t →∞ ϕ Dt .Conversely, we assume that ϕ = lim t →∞ ϕ Dt . Since ϕ Dt is algebraic from a ( D, ϕ, t ) • for each t , ϕ Dt is D -psh for every t >
0. Since t log |J − ( tϕ ) | ≥ ϕ Dt and ϕ Dt has adecreasing subsequence, ϕ Dt converges to ϕ strongly in the norm which impliesthe conclusion immediately. (cid:3) For every nontrivial tempered valuation v , we define v ( k D k ) = v ( a • ) where a m = b ( | x mD y | ). Proposition 6.10.
The set
PSH( D ) is closed under taking the supremum. Themaximal D -psh function ϕ max can be written explicitly as ϕ max ( v ) = − v ( k D k ) for all v ∈ V ∗ X . ALUATIVE MULTIPLIER IDEALS 25
Proof.
Let ϕ λ be a family of D -psh functions. By Lemma 6.9 ϕ λ = lim t →∞ ϕ Dλ,t .Note that ϕ Dλ,t = log | a ( D, ϕ λ , t ) • | , where a ( D, ϕ λ , t ) m = b ( V m ( D, ϕ λ , t )).If we write ϕ = sup λ ϕ λ , then J − ( tϕ λ ) ⊆ J − ( tϕ ) for every λ and every t . Itfollows that b ( V m ( D, ϕ λ , t )) ⊆ b ( V m ( D, ϕ, t )) for every m , λ and t . We deducethat sup λ ϕ Dλ,t ≤ ϕ Dt and hence ϕ ( v ) = sup λ lim t →∞ ϕ Dλ,t ( v ) ≤ lim t →∞ sup λ ϕ Dλ,t ( v ) ≤ lim t →∞ ϕ Dt ( v )for every v ∈ V ∗ X . Note that the pointwise limits appeared in the above in-equality exist because we can take decreasing subsequences which are boundedfrom below. Since t log |J − ( tϕ ) | ≥ ϕ Dt , we obtain the equality ϕ = lim t →∞ ϕ Dt and ϕ is D -psh by Lemma 6.9.Now we prove that ϕ max ( v ) = − v ( k D k ) for all v ∈ V ∗ X . Let ϕ be a qpshfunction such that ϕ ( v ) = − v ( k D k ). Because ϕ is algebraic from a • where a m = b ( | x mD y | ), ϕ is D -psh. It suffices to show that ϕ max ≤ ϕ . For each t , ϕ D max ,t = log | a ( D, ϕ max , t ) • | where a ( D, ϕ max , t ) m = b ( V m ( D, ϕ max , t )). Itfollows that a ( D, ϕ max , t ) m ⊆ a m , and hence ϕ D max ,t ≤ ϕ . Therefore, ϕ max =lim t →∞ ϕ D max ,t ≤ ϕ which forces ϕ max = ϕ . (cid:3) For every nontrivial tempered valuation v , we define σ v ( k D k ) := lim ε → v ( k D + εA k ) for some ample divisor A . Note that [28] verifies that this definition isindependent of the choice of the ample divisor A . Proposition 6.11.
The set
PSH σ ( D ) is closed under taking the supremum.The maximal pseudo D -psh function φ max can be expressed explicitly as φ max ( v ) = − σ v ( k D k ) for all v ∈ V ∗ X .Proof. Let ϕ λ be a family of pseudo D -psh functions, and let ϕ = sup λ ϕ λ . ByTheorem 6.6 there exists an ample divisor G such that ϕ λ,k ∈ PSH( D + k G )where ϕ λ,k = k log |J ( kϕ λ ) | . We have sup λ ϕ λ,k ∈ PSH( D + k G ) for every k byProposition 6.10. Since Σ λ J ( kϕ λ ) ⊆ J ( kϕ ), we have ϕ k ≥ sup λ ϕ λ,k ≥ ϕ . Itfollows that ϕ = lim k →∞ (sup λ ϕ λ,k ) ∈ PSH σ ( D ) . Now we prove that φ max ( v ) = − σ v ( k D k ) for all v ∈ V ∗ X . Let φ ( v ) = − σ v ( k D k ),and let ϕ ǫ max be the maximal ( D + ǫA )-psh function for every ǫ ≪
1. We see that φ = lim ǫ → ϕ ǫ max pointwisely. Because ϕ ǫ max is decreasing as ǫ → J ( mϕ ǫ max )form a descending chain of ideals as ǫ →
0+ for every integer m >
0. If wefix an integer m and a sequence ǫ > ǫ > . . . such that lim k →∞ ǫ k = 0, then thedescending chain stabilizes when k ≫ G such that mD + G is Cartier and O X ( mD + G ) ⊗J ( mϕ ǫ k max ) is globally generatedfor every k ≫
0. It follows that k ϕ ǫ k max − ϕ ǫ k ′ max k < m for all sufficiently large k and k ′ . Equivalently, ϕ ǫ k max form a Cauchy sequence with respect to the norm.Therefore ϕ ǫ k max converges to φ strongly in the norm, and hence φ ∈ PSH σ ( D ).Note that φ max ≤ ϕ ǫ max , and hence φ max ≤ φ which implies the conclusion. (cid:3) The question below was raised by B. Lehmann (cf. [27, Question 6.15]).
Question 6.12.
Is the maximal pseudo D -psh function algebraic?Abundant divisors, introduced by [28] and [4], form a class of pseudo-effectivedivisors with nice asymptotic behavior. We denote by κ σ ( D ) the numericalKodaira dimension. A pseudo-effective divisor D is said to be abundant if κ ( D ) = κ σ ( D ). We present the following easy corollary for the reader’s conve-nience. Corollary 6.13. (1). The set
PSH( D ) is nonempty if and only if D is Q -effective.(2). ∈ PSH( D ) if and only if D is nef and abundant;(3). The set PSH σ ( D ) is nonempty if and only if D is pseudo-effective.(4). ∈ PSH σ ( D ) if and only if D is nef.(5). Let ϕ max be the maximal D -psh function, and φ max be the maximal pseudo D -psh fucntion. Then, D is abundant if and only if ϕ max = φ max .Proof. The first statement is trivial. The second is a consequence of the mainresult of [29], and (4) follows from (2) immediately. If D is not pseudo-effective,then PSH σ ( D ) is empty from (1). We prove (3) as follows. If D pseudo-effective,then PSH σ ( D ) is nonempty by Proposition 6.11. To prove (5), simply noticethat D is abundant if and only if v ( k D k ) = σ v ( k D k ) for every divisorial valu-ation v by [27, Proposition 6.18] and the last statement follows by Proposition6.10 and Proposition 6.11. (cid:3) Question 6.14.
Assume that the divisor D is abundant. Is the set PSH( D )equal to the set PSH σ ( D )?We introduce the following definition of the perturbed ideal and the dimin-ished ideal as [27, Definition 4.3 and Definition 6.2]. We use the notation J σ, − ( D ) instead of J − ( D ) to avoid that readers may confuse it with the nota-tion J − ( ϕ ). Definition 6.15.
Let D be a pseudo-effective divisor. We define the perturbedideal J σ, − ( D ) to be the smallest ideal in the finite descending chain {J ( k L + m A k ) } ∞ m =1 , and we define the diminished ideal J σ ( D ) to be the largest ideal inthe set {J σ, − ((1 + ǫ ) D ) } ǫ> .Finally, we obtain a generalization of [27, Theorem 6.14]. Theorem 6.16.
Let D be a pseudo-effective divisor. Assume that φ max isthe maximal pseudo D -psh function. Then, the perturbed ideal J σ, − ( D ) = J − ( φ max ) , and the diminished ideal J σ ( D ) = J ( φ max ) . In particular, we canwrite J σ ( D ) explicitly as Γ( U, J σ ( L )) = { f ∈ Γ( U, O X ) | v ( f )+ A ( v ) − σ v ( k L k ) > for all v ∈ V ∗ U } . Further, a nonzero ideal q ⊆ J σ ( k L k ) if and only if v ( q ) + A ( v ) − σ v ( k L k ) > for all v ∈ V ∗ X . ALUATIVE MULTIPLIER IDEALS 27
Proof.
The equality J σ, − ( D ) = J − ( φ max ) follows from [27, Proposition 4.7]. Toprove the second equality, note that by definition J σ ( D ) = J ((1 + ǫ ) ϕ δ max ),where ϕ δ max denotes the maximal ( D + δA )-psh function for an ample divisor A , for sufficiently small ǫ and sufficiently small δ = δ ( ǫ ). From the proof ofProposition 6.11, ϕ δ max converges to φ max strongly in the norm. Therefore,Lemma 4.20 asserts that J ( φ max ) = J ((1 + ǫ ) ϕ δ max ) = J σ ( D ) as δ → (cid:3) Remark 6.17.
It should not be too difficult to generalize most results in thissubsection from Q -divisors to R -divisors, that is, one can define D -psh functionsfor an R -divisor D and obtain similar results.6.2. Finite generation.
The goal of this subsection is to prove the finite gen-eration proposition below as an application of qpsh functions. For definitionsand properties of different types of Zariski decompositions, divisorial algebrasand modules, we refer to [28].
Proposition 6.18.
Let ( X, B ) be a log canonical pair. Assume that K X + B is Q -Cartier and abundant, and that R ( K X + B ) is finitely generated. Then, forany reflexive sheaf F , M p F ( K X + B ) is a finitely generated R ( K X + B ) -module. Before we prove the above proposition, we first prove the following lemma.
Lemma 6.19 (Global division) . Let X be a smooth projective variety of di-mension n . Consider line bundles L and D , a linear system V ⊆ | L | which isspanned by the sections { s , . . . , s l } , and a D -psh function ϕ . If we denote by φ V the L -psh function max ≤ j ≤ l log | s j | . Then, for every integer m ≥ n + 2 , anysection σ in H ( X, O X ( K X + mL + D ) ⊗ J ( mφ V + ϕ )) can be written as a linear combination P j s j g j of sections g j in H ( X, O X ( K X +( m − L + D )) .Proof. Let { ϕ k ∈ L D } be a sequence of ideal functions which converges to ϕ strongly in the norm. Since J ( mφ V + ϕ k ) ⊇ J ( mφ V + ϕ ), the section σ vanishes along the ideal J ( mφ V + ϕ k ). If we denote by a the base ideal b ( V ),then φ V = log | a | . Apply [18, Theorem 4.1] and we deduce the conclusion. (cid:3) Remark 6.20.
In the statement of [18, Theorem 4.1], one can verify that theassumption that D ⊗ b λ is nef and abundant implies that λ log | b | is D -psh.Note that Lemma 6.19 is not a generalization of [18, Theorem 4.1] because wedid not obtain that every g j is in H ( X, O X ( K X + ( m − L + D ) ⊗ J (( m − φ V + ϕ )). Nonetheless, it should be possible to generalize in the sense that g j ∈ H ( X, O X ( K X + ( m − L + D ) ⊗ J (( m − φ V + ϕ )), if one can develop atheory on the restriction of qpsh functions to subvarieties (see the proof of [18,Theorem 3.2]). Proof of Proposition 6.18.
We can assume that (
X, B ) is log smooth of dimen-sion n , K X + B is a Q -Cartier Q -divisor, and F = O X ( A ) is a very ample linebundle by [1, Theorem 1.1]. Since R = R ( K X + B ) is finitely generated, after a possible truncation we can assume that R is generated by R = H ( m ( K X + B ))for some integer m such that m ( K X + B ) is Cartier (see [1, Remark 2.2 and2.3]). If we set a = b ( | m ( K X + B ) | ) and L := m ( K X + B ), then φ := log | a | isthe maximal L -psh function. The rest of the proof is an analogue of [15, Section3]. Let m be a sufficiently large integer (to be specified later), and let σ be aglobal section of m ( K X + B ) + A . We have that m ( K X + B ) + A = K X + ( n + 2) L + D where D := B + ( m − ( n + 2) m − K X + B + m A ) + m ( n +2)+1 m A . If we set ϕ = ψ m + ( m − ( n + 2) m − ϕ m where ψ m is ( B + m ( n +2)+1 m A )-psh such that k ψ m k <
1, and ϕ m is the maximal( K X + B + m A )-psh function. Notice that k log | σ | − ( n + 2) φ − ϕ k + ≤ k ( m ( n + 2) + 1) ϕ m − ( n + 2) φ − ψ m k + . We will show that ( m ( n + 2) + 1) ϕ m ≤ ( n + 2) φ for m sufficiently large whichimplies that k log | σ | − ( n + 2) φ − ϕ k + < σ vanishes along J (( n + 2) φ + ϕ ). Since φ is determined on some dual complex ∆( Y, D ), itsuffices to prove that ( m ( n + 2) + 1) ϕ m ≤ ( n + 2) φ on ∆( Y, D ). Further, wecan assume that φ is affine on ∆( Y, D ). It suffices to check the above inequalityat vertices because ϕ m is convex on the dual complex. From the argumentof Proposition 6.11, we see that m ϕ m converges to φ strongly in the normsince K X + B is abundant. Therefore for m sufficiently large the inequality m ( n +2)+1 n +2 ϕ m ≤ φ holds at vertices of ∆( Y, D ), and hence for every nontrivialtempered valuation. Finally, σ can be written as a linear combination P j s j g j of sections g j in H ( X, O X (( m − m )( K X + B ) + A ) by Lemma 6.19, whichcompletes the proof. (cid:3) Remark 6.21.
The above finite generation proposition can be proved in an-other way as follows. Since the conclusion that M p F ( K X + B ) is a finitelygenerated R ( K X + B )-module is equivalent to that ( X, B ) has a good minimalmodel by [1, Theorem 1.3], it suffices to prove that (
X, B ) has a good minimalmodel. By [11, Theorem 5.3] we conclude that (
X, B ) has a log minimal model( X ′ , B ′ ). Since the positive part of the CKM-Zariski decomposition is semi-ample, the log minimal model ( X, B ) is good. We here give a different proofwithout using the minimal model theory, in particular the length of extremalrays.Proposition 6.18 can be slightly generalized as follows.
Definition 6.22 (2, Definition 3.6.4 and 3.6.6) . Let D be a divisor on X .A normal projective variety Z is said to be the ample model of D if there isa rational map g : X Z and an ample R -divisor H on Z such that if p : W → X and q : W → Z resolve g then q is a contraction and we can write p ∗ D = q ∗ H + N , where N ≥ R -divisor and for every B ∼ Q p ∗ D then B ≥ N . Let ( X, B ) be a pair. A normal variety Z is said to be the log canonicalmodel of ( X, B ) if it is the ample model of K X + B . ALUATIVE MULTIPLIER IDEALS 29
Lemma 6.23.
Let D be an abundant divisor on a normal projective variety X .Assume that D has the ample model. Then, R ( D ) is finitely generated.Proof. After replacing X by a log resolution, we can assume that g : X Z isa morphism and D = P + N = g ∗ H + N where H is an ample R -divisor on theample model Z and N ≥ R -divisor such that for every B ∼ Q D we have B ≥ N . Note that D = P + N is a CKM-Zariski decomposition. Since D isabundant, we have that Fix k D k = N σ ( D ) ≤ N ≤ Fix k D k by [27, Proposition6.18] and hence P = P σ ( D ). Furthermore, we can assume that there exist asmooth projective variety T and a big Q -divisor G on T such that µ : X → T is a contraction and P σ ( D ) = P σ ( µ ∗ G ) by [26, Theorem 6.1, Theorem 5.7]. Itfollows that Z is also the ample model of G . Notice that the rational map h : T Z is birational. Therefore H = p ∗ G is an R -Cartier Q -divisor andhence Q -Cartier which completes the proof. (cid:3) Finally, we obtain the proposition below by combining Proposition 6.18 andLemma 6.23.
Proposition 6.24.
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