Continuity of Monge-Amp{è}re potentials in big cohomology classes
aa r X i v : . [ m a t h . DG ] F e b CONTINUITY OF MONGE-AMP`ERE POTENTIALS IN BIGCOHOMOLOGY CLASSES
QUANG-TUAN DANG
Abstract.
Extending DiNezza-Lu’s approach [DNL17] to the setting of bigcohomology classes, we prove that solutions of degenerate complex Monge-Amp`ere equations on compact K¨ahler manifolds are continuous on a Zariskiopen set. This allows us to show that singular K¨ahler-Einstein metrics on logcanonical varieties of general type have continuous potentials on the amplelocus outside of the non-klt part.
Contents
1. Introduction 12. Preliminaries 33. Regularity of solutions 8References 131.
Introduction
Finding canonical metrics on complex varieties is a fundamental problem ofcomplex geometry. As evidenced by recent developments in K¨ahler geometry inconnection with the Minimal Model Program, it is natural and necessary to allowthe varieties Y in question to be singular. Working on a desingularization π : X → Y one is led to consider degenerate complex Monge-Amp`ere equations of the form(1.1) h ( θ + dd c ϕ ) n i = e λϕ f dV, where θ is a smooth closed real (1 , α on X , λ ∈ { , ± } , and f is a density of the form f = e ψ + − ψ − , the functions ψ + , ψ − being quasi-plurisubharmonic on X . The integrability properties of f depend onthe singularities of Y .Finding a K¨ahler-Einstein metric on a stable variety Y of general type boils downto solving (1.1) on X for λ = 1 and f ∈ L − δ ( X, dV ) for some δ ∈ (0 , ϕ , inthe sense of [GZ07, BBGZ13]. When { θ } is additionally nef, they established thesmoothness of ϕ on a Zariski open set. As in the classical case of Yau [Yau78],the main difficulty lies in establishing an a priori C -estimate. Unfortunately the(normalized) solution ϕ to (1.1) is in general unbounded, so a natural idea is totry and bound such solution from below by a reference quasi-plurisubharmonicfunction. Date : February 5, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Complex Monge-Amp`ere equation, big cohomology class.The author is partially supported by the French ANR project PARAPLUI.
This motivates the following general question: for which density 0 ≤ f is thesolution ϕ locally bounded in some Zariski open subset of X ? The main result ofthis paper is the following: Theorem 1.1.
Let X be a compact K¨ahler manifold of dimension n and fix asmooth closed real (1 , -form θ which represents a big cohomology class. Let ϕ ∈E ( X, θ ) be the unique normalized solution to (1.2) ( θ + dd c ϕ ) n = f dV, sup X ϕ = 0 . Assume that f ≤ e − φ for some quasi-plurisubharmonic function φ on X . Then ϕ is continuous on Amp( θ ) \ E ( φ ) , where Amp( θ ) is the ample locus of θ and E ( φ ) = { x ∈ X : ν ( φ, x ) ≥ } , with ν ( φ, x ) being the Lelong number of φ at x . Let us recall that E ( φ ) which is called the Lelong super-level set of φ , is ananalytic subset of X by Siu’s result [Siu74]. We refer the reader to Section 2.3 forthe definition of the ample locus of a big cohomology class.Since µ = f dV is non-pluripolar, it is known [BEGZ10, Section 3] that there ex-ists a unique normalized solution ϕ ∈ E ( X, θ ), so the point is to study its regularity.The idea of the proof is that we first use Demailly’s equisingular approximation[Dem92, Dem15] (see Theorem 2.1) to replace φ by a quasi-psh function φ whichhas analytic singularities with polar locus Z contained in the set of points wherethe Lelong number of φ is greater than or equal to 1. We then adapt the approachof Di Nezza and Lu [DNL17] (see Theorem 3.2) to prove the continuity of ϕ in thecomplement of E ( φ ) in the ample locus of θ . We also prove a slightly more generalversion of Theorem 1.1 valid for less singular densities (see Theorem 3.1).When the density f is smooth in a Zariski open set (i.e. outside an analyticsubset), one expects the solution ϕ to be smooth in a Zariski open set (see [DGZ16,Question 21, 22]), but we are unable to prove this for the moment. When f issmooth on the whole of X , only the H¨older continuity of ϕ is known [DDG + { θ } is nef, the regularity propertiesfor the solutions for the degenerate Monge-Amp`ere equation (1.2) has been studiedby many authors (see [BEGZ10, BG14, DNL17] and the references therein). Thestrategy in these papers is that one first establishes a relative uniform estimatewhich allows to adapt classical ideas of Yau [Yau78] and Siu [Siu87] to obtain locallyuniform estimates for the Laplacian, and one finally uses Evans-Krylov’s generalregularity theory to conclude. In the above case functions in E ( X, θ ) have zeroLelong numbers (see [GZ07, Corollary 1.8], [DDNL18b, Theorem 1.1]). Using thisproperty Di Nezza and Lu [DNL17] have generalized Ko lodziej’s approach [Ko l98]to establish a relative uniform estimate. Let us mention that in the general case of abig class even the ”least singular” potential V θ may have positive Lelong numbers.To overcome this difficulty we exploit fine properties of quasi-plurisubharmonicenvelopes inspired by [DDNL20],[LN19].Our approach allows us to deal with non-nef data. As an application we provethat the unique singular K¨ahler-Einstein metric obtained in [BG14] is continuouson some Zariski open subset. More precisely, we have the following: Corollary 1.2.
Let ( Y, ∆) be a projective log canonical pair of general type, i.e.the canonical line bundle K Y + ∆ is big. Then there is unique a singular K¨ahler-Einstein metric ω on Y such that Ric( ω ) = − ω + [∆] in the weak sense of currents, and such that R Y ω n = ( K Y + ∆) n . Furthermore ω has continuous potentials on the ample locus of θ outside of the non-klt part. ONTINUITY OF MONGE-AMP`ERE POTENTIALS IN BIG COHOMOLOGY CLASSES 3
In [BG14, Theorem A, C], the authors proved more regularity properties of(singular) K¨ahler-Einstein metrics when Y is a projective complex variety withsemi-log canonical singularities such that K Y is ample (rather than big). Organization of the paper.
The paper is organized as follows. In Section 2we recall basic pluripotential theory that will be needed later on. The proof ofTheorem 1.1 is given in Section 3.1, while Corollary 1.2 is proved in Section 3.2.
Notations.
In the whole article we fix • X a n -dimensional compact K¨ahler manifold, • dV a smooth volume form on X , • α ∈ H , ( X, R ) a big cohomology class, and θ a smooth representative of α • a K¨ahler form ω so that ω ≥ θ . Acknowledgements.
I would like to express my gratitude to my advisors VincentGuedj and Chinh H. Lu for their help and various interesting discussions.2.
Preliminaries
The purpose of this section is to recall some essential material in pluripotentialtheory which will be used later.2.1.
Quasi-psh functions.
Recall that an upper semi-continuous function ϕ : X → R ∪ {−∞} is called quasi-plurisubharmonic ( quasi-psh for short) if it is locallythe sum of a smooth and a plurisubharmonic (psh for short) function. We say that ϕ is θ -plurisubharmonic ( θ -psh for short) if it is quasi-psh, and θ + dd c ϕ ≥ d c is normalized so that dd c = iπ ∂ ¯ ∂ .By the dd c -lemma any closed positive (1 , T cohomologous to θ can bewritten as T = θ + dd c ϕ for some θ -psh function ϕ which is furthermore unique upto an additive constant.We let PSH( X, θ ) denote the set of all θ -psh functions which are not identically −∞ . This set is endowed with the L ( X )-topology. By Hartog’s lemma ϕ sup X ϕ is continuous in this weak topology. Since the set of closed positive currentsin a fixed cohomology class is compact (in the weak topology), it follows that theset of ϕ ∈ PSH(
X, θ ) with sup X ϕ = 0 is compact.Quasi-psh functions are in general singular, and a convenient way to measuretheir singularities is the Lelong numbers. Recall that the Lelong number of a quasi-psh function ϕ at some point x ∈ X is defined to be ν ( ϕ, x ) := sup { γ ≥ ϕ ( z ) ≤ γ log | z | + O (1) , ∀ x ∈ V x } where ( V x , z ) are local holomorphic coordinates centered at x . In particular, if ϕ = log | f | in a neighborhood V x of x , for some holomorphic function f , then ν ( ϕ, x )is equal to the vanishing order ord x ( f ) := sup { k ∈ N : D γ f ( x ) = 0 , ∀ | γ | < k } . Wecan also define the Lelong super-level sets , for c > E c ( ϕ ) := { x ∈ X : ν ( ϕ, x ) ≥ c } . We also use the notation E c ( T ) for a closed positive (1 , T . A well knowresult of Siu [Siu74] asserts that the Lelong super-level sets E c ( ϕ ) are analyticsubsets of X . We refer the reader to [Dem92, Remark 3.2] (see also [Dem15,Corollary 3]) for a proof. QUANG-TUAN DANG
Demailly’s equisingular approximation.
We next recall the basic resulton the approximation of psh functions by psh functions with analytic singularities.For details about this, we refer the reader to [Dem92, Dem15].Following Demailly [Dem92], a closed positive (1 , T = θ + dd c ϕ andits global potential ϕ are said to have analytic singularities if there exists c > ϕ = c log N X j =1 | f j | + v, locally on X , where v is a smooth function and the f j ’s are holomorphic functions.Thanks to dd c -Lemma, the problem of approximating a positive closed (1 , Theorem 2.1 (Demailly’s equisingular approximation) . Let ϕ be a θ -psh functionon X . There exists a decreasing sequence of quasi-psh functions ( ϕ m ) such that (1) ( ϕ m ) converges pointwise and in L ( X ) to ϕ as m → + ∞ , (2) ϕ m have the same singularities as / m times a logarithm of a sum ofsquares of holomorphic functions, (3) θ + dd c ϕ m ≥ − ε m ω , where ε m > decreases to 0 as m → + ∞ , (4) R X e m ( ϕ m − ϕ ) dV < + ∞ ; (5) ϕ m is smooth outside the analytic subset E /m ( ϕ ) . We refer the reader to [Dem15, Theorem 8, Lemma 2] for a proof.2.3.
Big cohomology classes.
A cohomology class α ∈ H , ( X, θ ) is big if itcontains a
K¨ahler current , i.e. there is a positive closed current T ∈ α and ε > T ≥ εω . Theorem 2.1 enables us in particular to approximate a K¨ahlercurrent T inside its cohomology class by K¨ahler currents T m with analytic singu-larities, with a very good control of the singularities. A big class therefore containsplenty of K¨ahler currents with analytic singularities. Definition 2.2.
We let Amp ( α ) denote the ample locus of α , i.e. the Zariski opensubset of all points x ∈ X for which there exists a K¨ahler current T x ∈ α withanalytic singularities such that T x is smooth in a neighborhood of x . It follows from the work of Boucksom [Bou04, Theorem 3.17 (ii)] that one canfind a single K¨ahler current T ∈ α with analytic singularities such thatAmp( α ) = X \ Sing( T ) . In particular T is smooth in the ample locus Amp( α ).Given ϕ, ψ ∈ PSH(
X, θ ), we say that ϕ is less singular than ψ , and denote by ϕ (cid:22) ψ , if there exists a constant C such that ψ ≤ ϕ + C on X . We say that ϕ, ψ have the same singularity type , and denote by ϕ ≃ ψ if ϕ (cid:22) ψ and ψ (cid:22) ϕ . Definition 2.3. A θ -psh function is said to have minimal singularities if it is lesssingular than all the other ones. Such a function is not unique in general, only its class of singularities is. Follow-ing Demailly, one defines the extremal function V θ := { ϕ ∈ PSH(
X, θ ) : ϕ ≤ } . It is a θ -psh function with minimal singularities. By the analysis above V θ is locallybounded on the ample locus Amp( α ). Of course we have V θ ≡ θ is semi-positive. ONTINUITY OF MONGE-AMP`ERE POTENTIALS IN BIG COHOMOLOGY CLASSES 5
Non-pluripolar Monge-Amp`ere operator.
Let ϕ , · · · , ϕ n ∈ PSH(
X, θ )with minimal singularities. Then they are locally bounded on the ample locusAmp( α ). Following the construction of Bedford-Taylor [BT76, BT82] in local set-ting, the product ( θ + dd c ϕ ) ∧ · · · ∧ ( θ + dd c ϕ n )is thus well-defined as a positive Radon measure on Amp( α ) with finite total mass.One can then extend it trivially to the whole of X .It has been shown in [BEGZ10] that the current obtained by this trivial extensionis closed. In particular, if ϕ = · · · = ϕ n = ϕ then this procedure defines the(non-pluripolar) Monge-Amp`ere measure of ϕ . For a general ϕ ∈ PSH(
X, θ ), itscanonical approximants ϕ j := max( ϕ, V θ − j ), j > { ϕ>V θ − j } ( θ + dd c ϕ j )is increasing in j . Its limitMA θ ( ϕ ) = h ( θ + dd c ϕ ) n i := lim j → + ∞ ր { ϕ>V θ − j } ( θ + dd c ϕ j ) n is the non-pluripolar Monge-Amp`ere measure of ϕ . From now on, we let ( θ + dd c ϕ ) n or simply MA θ ( ϕ ) denote the non-pluripolar Monge-Amp`ere measure of ϕ . Inparticular the volume of α = { θ } is now given byVol( α ) = Z Amp( α ) MA θ ( V θ ) . We say that ϕ has full Monge-Amp`ere mass if R X MA θ ( ϕ ) = Vol( α ). We let E ( X, θ ) := (cid:26) ϕ ∈ PSH(
X, θ ) : Z X MA θ ( ϕ ) = Vol( α ) (cid:27) denote the set of θ -psh functions with full Monge-Amp`ere mass. Note that θ -pshfunctions with minimal singularities have full Monge-Amp`ere mass (see [BEGZ10,Theorem 1.16] for more details), but the converse is not true.We recall here the plurifine locality of the non-pluripolar product, which will beused several times in this paper. Lemma 2.4.
Assume that ϕ , ψ are θ -psh function such that ϕ = ψ on an open set U in the plurifine topology. Then U MA θ ( ϕ ) = U MA θ ( ψ ) . We stress in particular that sets of the form { u < v } , where u , v are quasi-pshfunctions, are open in the plurifine topology. Proof.
The proof for locally bounded functions can be found in [BT87, Corollary4.3] or [BEGZ10, Section 1.2]. For the general case we write ϕ (resp. ψ ) as thedecreasing limits of its canonical approximants ϕ t := max( ϕ, V θ − t ) (resp. ψ t :=max( ψ, V θ − t )). We observe that ϕ t (resp. ψ t ) is locally bounded on the amplelocus Amp( θ ). By the result for locally bounded functions we have U ∩ ( ϕ>V θ − t ) MA θ ( ϕ t ) = U ∩ ( ψ>V θ − t ) MA θ ( ψ t ) . Letting t → + ∞ , we conclude the proof. (cid:3) Capacities.
QUANG-TUAN DANG
The Monge-Amp`ere capacity.
For the convenience of the reader we recall herea few facts contained in [GZ17]. Let K be a Borel subset of X . The Monge-Amp`erecapacity isCap ω ( K ) := sup (cid:26)Z K ( ω + dd c u ) n : u ∈ PSH(
X, ω ) , − ≤ u ≤ (cid:27) . Lemma 2.5.
Let ν = gdV be a Radon positive measure with ≤ g ∈ L p ( dV ) forsome p > . Then there exists B > depending on n , p ω , dV and k g k L p ( dV ) suchthat, for all Borel subsets K of X , ν ( K ) ≤ B · Cap( K ) . We refer the reader to [EGZ09, Proposition 3.1] for a proof.2.5.2.
The generalized capacity.
We present here a generalization of this notionintroduced by Di Nezza and Lu [DNL17, DNL15] (see also [DDNL18a, Section 4.1]).
Definition 2.6.
Let ψ ∈ PSH(
X, θ ) . We define the ψ -relative capacity of a Borelsubset K ⊂ X by Cap θ,ψ ( K ) := sup (cid:26)Z K MA θ ( u ) : u ∈ PSH(
X, θ ) , ψ − ≤ u ≤ ψ (cid:27) . Note that when θ is K¨ahler, a related notion of capacity has been studied in[DNL15, DNL17]. The (generalized) Monge-Amp`ere capacity plays a vital rolein establishing uniform estimates for complex Monge-Amp`ere equation (see e.g.[EGZ09, BEGZ10, DNL15, DNL17] and the references therein). We shall use the ψ -capacity Cap θ,ψ in the proof of Theorem 3.2.The following results are important for the sequel. Lemma 2.7.
Fix ϕ ∈ E ( X, θ ) and ψ ∈ PSH(
X, θ ) . Then the function H ( t ) := Cap θ,ψ ( { ϕ < ψ − t } ) , t ∈ R , is right-continuous and H ( t ) → as t → + ∞ .Proof. The proof is almost the same as the one of [DNL17, Lemma 2.6] in theK¨ahler case, i.e. θ = ω is K¨ahler. For the reader’s convenience we give the proofhere. The right-continuity is straightforward . Fro the second statement, we firstassume that ψ ≤ V θ . Fix u ∈ PSH(
X, θ ) such that ψ − ≤ u ≤ ψ . The generalizedcomparison principle ([BEGZ10, Corollary 2.3]) yields Z { ϕ<ψ − t } MA θ ( u ) ≤ Z { ϕ
Let ψ be a quasi-psh function such that θ + dd c ψ ≥ δω for some δ ∈ (0 , . Then for any Borel set K ⊂ X , Cap ω ( K ) ≤ δ n Cap θ,ψ ( E ) . Proof.
Let u be a ω -psh function such that − ≤ u ≤
0. We then have that ϕ := ψ + δu is a candidate defining Cap θ,ψ . It follows that δ n Z K ( ω + dd c u ) n ≤ Z K ( θ + dd c ψ + dd c ( δu )) n ≤ Cap θ,ψ ( K ) , and taking the supremum over all u we get the desired estimate. (cid:3) ONTINUITY OF MONGE-AMP`ERE POTENTIALS IN BIG COHOMOLOGY CLASSES 7
Lemma 2.9.
Let ψ ∈ PSH(
X, θ ) and ϕ ∈ E ( X, θ ) . Then for all t > and < s ≤ we have s n Cap θ,ψ ( { ϕ < ψ − t − s } ) ≤ Z { ϕ<ψ − t } MA θ ( ϕ ) . Proof.
Let u be a θ -psh function such that ψ − ≤ u ≤ ψ . We then have { ϕ < ψ − t − s } ⊂ { ϕ < su + (1 − s ) ψ − t } ⊂ { ϕ < ψ − t } . Since s n MA θ ( u ) ≤ MA θ ( su + (1 − s ) ψ ) and ϕ has full Monge-Amp`ere mass, itfollows from the generalized comparison principle ([BEGZ10, Corollary 2.3]) that s n Z { ϕ<ψ − t − s } MA θ ( u ) ≤ Z { ϕ<ψ − t − s } MA θ ( su + (1 − s ) ψ ) ≤ Z { ϕ For a Borel function h , we let P θ ( h ) denote the largest θ -psh function lying below h : P θ ( h ) := (sup { ϕ ∈ PSH( X, θ ) : ϕ ≤ h on X } ) ∗ . Proposition 2.10. Fix ϕ ∈ E ( X, θ ) . Then for any b > , P ω ( bϕ − bV θ ) is a ω -pshfunction with full Monge-Amp`ere mass. The proof given below is inspired by [DDNL20, Lemma 4.3]. Proof. We first show that the function P ω ( bϕ − bV θ ) 6≡ −∞ for all b > j ∈ N we set ϕ j := max( ϕ, V θ − j ) and ψ j := P ω ( bϕ j − bV θ ). We observethat ( ψ j ) is a decreasing sequence of ω -psh functions, and ψ j ≥ − jb for each j .Therefore the proof would follow if we could show that lim j ψ j is not identically −∞ . We let for each j , D j := { ψ j = bϕ j − bV θ } denote the contact set. Fix t > { ψ j ≤ − t } ∩ D j = { ϕ j ≤ V θ − t/b } ⊂ { ϕ ≤ V θ − t/b } . Set ˜ ω := (cid:0) b + 1 (cid:1) ω . By Lemma 2.11 below and plurifine locality we have Z { ψ j ≤− t } ( ω + dd c ψ j ) n = Z { ψ j ≤− t } D j ( ω + dd c ψ j ) n ≤ b n Z { ψ j ≤− t } D j (˜ ω + dd c ϕ j ) n ≤ b n Z { ϕ ≤ V θ − t/b } (˜ ω + dd c ϕ j ) n = b n Z X (˜ ω + dd c ϕ j ) n − Z { ϕ>V θ − t/b } (˜ ω + dd c ϕ ) n ! , since ϕ j = ϕ on { ϕ > V θ − t/b } . Suppose by contradiction that sup X ψ j → −∞ as j → + ∞ . It then follows that { ψ j ≤ − t } = X for j large enough, t being fixed.Letting j → + ∞ , it follows from [DDNL18a, Theorem 2.3] that Z X ω n ≤ b n Z X (˜ ω + dd c ϕ ) n − Z { ϕ>V θ − t/b } (˜ ω + dd c ϕ ) n ! , (2.1) QUANG-TUAN DANG where we have used [DDNL18a, Theorem 2.3, Remark 2.5]: since ϕ j , ϕ ∈ E ( X, θ )hence (˜ ω − θ ) k ∧ ( θ + dd c ϕ j ) n − k → (˜ ω − θ ) k ∧ ( θ + dd c ϕ ) n − k in the weak senseof measures on X . Finally, letting t → + ∞ in (2.1) we obtain a contradiction.Consequently, since ψ j decreases to a ω -psh function, we have that P ω ( bϕ − bV θ ) isa ω -psh function for any b > A > b we have P ω ( bϕ − bV θ ) ≥ bA P ω ( Aϕ − AV θ ) . Using monotonicity of mass (see e.g. [WN19, Theorem 1.2]) we obtain Z X ( ω + dd c P ω ( bϕ − bV θ )) n ≥ (cid:18) − bA (cid:19) n Z X ω n + (cid:18) bA (cid:19) n Z X ( ω + dd c P ω ( Aϕ − AV θ )) n . Letting A → + ∞ we infer that P ω ( bϕ − bV θ ) has full Monge-Amp`ere mass. (cid:3) Lemma 2.11. Fix b > , ϕ and P ω ( bϕ − bV θ ) ∈ PSH( X, ω ) . Then the measure ( ω + dd c P ω ( bϕ − bV θ )) n is supported on the contact set D := { P ω ( bϕ − bV θ ) = bϕ − bV θ } , and D ( ω + dd c P ω ( bϕ − bV θ )) n ≤ b n D (cid:18)(cid:18) b (cid:19) ω + dd c ϕ (cid:19) n . Proof. We refer the reader to [DDNL20, Lemma 4.4] for a proof of the first state-ment.For the second one, set u = b P ω ( bϕ − bV θ ) + V θ . Then u is a ˜ ω := (cid:0) b + 1 (cid:1) ω -pshfunction, and u ≤ ϕ . It follows from [GZ17, Corollary 10.8] that { u = ϕ } (˜ ω + dd c u ) n ≤ { u = ϕ } (˜ ω + dd c ϕ ) n . (2.2)Furthermore the measure ( ω n + dd c P ω ( bϕ − bV θ )) n is supported on the contact set { bϕ − bV θ = P ω ( bϕ − bV θ ) } = { u = ϕ } , hence(2.3) 1 b n ( ω + dd c P ω ( bϕ − bV θ )) n = { u = ϕ } b n ( ω + dd c P ω ( bϕ − bV θ )) n ≤ { u = ϕ } (˜ ω + dd c u ) n . From (2.2) and (2.3) we obtain the desired estimate. (cid:3) Regularity of solutions Proof of the Main Theorem. In this section we prove Theorem 1.1. The keyingredient is an adaptation of Di Nezza-Lu’s approach [DNL17] (see also [DNL15]).Given a non-negative Radon measure µ whose total mass is Vol( α ), we considerthe Monge-Amp`ere equation(3.1) MA θ ( ϕ ) = µ. The systematic study of such equations in big cohomology classes has been initiatedin [BEGZ10]. It has been shown there that (3.1) admits a unique normalizedsolution ϕ ∈ E ( X, θ ) if and only if µ is a non pluripolar measure on X .Our goal is to prove the following: Theorem 3.1. Let ν = gdV be a Radon measure, with ≤ g ∈ L p ( dV ) for some p > . Assume that µ = f dν , with f ≤ e − φ for some quasi-psh function φ on X .Let ϕ ∈ E ( X, θ ) be the unique normalized solution to (3.1) . Then ϕ is continuousAmp ( α ) \ E /q ( φ ) , where q denotes the conjugate exponent of p . ONTINUITY OF MONGE-AMP`ERE POTENTIALS IN BIG COHOMOLOGY CLASSES 9 Note that Theorem 1.1 from the introduction is a particular case of Theorem 3.1.We first establish this result under an extra assumption. More precisely, we havethe following theorem, which is closely similar to [DNL17, Theorem 3.1] which dealswith the case when θ is K¨ahler. Theorem 3.2. Let ϕ ∈ E ( X, θ ) be normalized by sup X ϕ = 0 . Assume that MA θ ( ϕ ) ≤ e − φ gdV , for some quasi-psh function φ on X , and ≤ g ∈ L p ( dV ) ,with p > . Assume φ is locally bounded in an open set U ⊂ Amp( α ) . Then ϕ iscontinuous in U .Proof. We fix a θ -psh function χ on X such that θ + dd c ρ ≥ δ ω, for some small constant δ > 0. Subtracting a large constant, we can always assumethat ρ ≤ V θ . Moreover, we can choose ρ such that it is smooth in the ample locus,with analytic singularities thanks to [Bou04, Theorem 3.17 (ii)].We pick a > aφ belongs to PSH( X, δ ω ). Set ψ := ρ + aφ . Wethus have θ + dd c ψ ≥ δ ω , and ψ ≤ V θ + aφ . We claim that(3.2) ϕ ≥ ψ − A, for A > δ , p , dV , k g k L p ( dV ) , and R X e − P ω ( a − ϕ − a − V θ ) gdV . Remark 3.3. By Proposition 2.10 that for any b > P ω ( bϕ − bV θ ) ∈ PSH( X, ω )with full Monge-Amp`ere mass, hence it has zero Lelong numbers [GZ07, Corollary1.8]. Skoda’s integrability theorem [Sko72] ensures that e − P ω ( bϕ − bV θ ) belongs to L q ( dV ) for all q < + ∞ . In particular, R X e − P ω ( bϕ − bV θ ) gdV is finite for any b > s ∈ [0 , t > 0. Set dν = gdV , b = a − . We have bψ ≤ bV θ + φ . Using Lemma 2.9 and the assumption on MA θ ( ϕ ) we have(3.3) s n Cap θ,ψ ( { ϕ < ψ − t − s } ) ≤ Z { ϕ<ψ − t } MA θ ( ϕ ) ≤ Z { ϕ<ψ − t } e b ( ψ − ϕ ) e − φ gdV ≤ Z { ϕ<ψ − t } e − P ω ( bϕ − bV θ ) dν. Using H¨older inequality we have Z { ϕ<ψ − t } e − P ω ( bϕ − bV θ ) dν ≤ ( ν ( { ϕ < ψ − t } )) / (cid:18)Z X e − P ω ( bϕ − bV θ ) dν (cid:19) / . (3.4)By Lemma 2.5, one can find a constant B > n , p , ω , dV , and k g k L p ( dV ) such that ν ( · ) / ≤ B (Cap ω ( · )) . Since θ + dd c ψ ≥ δ ω it follows from Lemma 2.8 that Cap ω ≤ δ − n Cap θ,ψ , hence ν ( · ) / ≤ Bδ − n Cap θ,ψ ( · ) , (3.5)By (3.3), (3.4) and (3.5) we thus get(3.6) s n Cap θ,ψ ( { ϕ < ψ − s − t } ) ≤ C Cap θ,ψ ( { ϕ < ψ − t } ) , where C depends on ω , δ , n , p , k g k L p ( dV ) , and R X e − P ω ( bϕ − bV θ ) dν . Set H ( t ) := (cid:2) Cap θ,ψ ( { ϕ < ψ − t } ) (cid:3) /n , t > . By the estimate (3.6) we get sH ( t + s ) ≤ C /n H ( t ) . It follows from Lemma 2.7 that the function H is right-continuous and H (+ ∞ ) = 0.We can thus apply [EGZ09, Lemma 2.4] which yields H ( t + 2) = 0, where t > H ( t ) < C /n . Therefore for A = t + 2 we have ϕ ≥ ψ − A on X \ P for some Borel subset P suchthat Cap θ,ψ ( P ) = 0. By Lemma 2.8 we have Cap ω ( P ) = 0 so P is a pluripolar set.Hence ϕ ≥ ψ − A everywhere.Using H¨older’s inequality it follows from (3.3) (take s = 1) that H ( t ) n ≤ (cid:18)Z X e − P ω ( bϕ − bV θ )) dν (cid:19) / Z { ϕ<ψ − t +1 } dν ! / ≤ (cid:18)Z X e − P ω ( bϕ − bV θ ) gdV (cid:19) / (cid:18) t − Z X | ψ − ϕ | gdV (cid:19) / . The last integral is bounded by a uniform constant: using H¨older’s inequality againwe have R X | ψ − ϕ | gdV ≤ k g k L p ( dV ) (cid:0) k ψ k L q ( dV ) + k ϕ k L q ( dV ) (cid:1) with q = p/ ( p − ϕ belongs to the compact set of θ -psh functions normalized by sup X ϕ = 0,its L q norm is bounded by an absolute constant only depending on θ, dV and p .Consequently, we can choose t > dV , p , k g k L p ( dV ) , andan upper bound for R X e − P ω ( bϕ − bV θ ) dν .Let us complete the proof of Theorem 3.2. For convenience, we normalize ϕ sothat sup X ϕ = − 1. Let 0 ≥ h j be a sequence of smooth functions decreasing to ϕ . Then the sequence of θ -psh functions ϕ j := P θ ( h j ) decreases to ϕ as j → + ∞ .Indeed, since the operator P θ is monotone, the sequence ϕ j is decreasing to a θ -pshfunction u and since ϕ j ≥ ϕ for all j we have u ≥ ϕ . Moreover, u ( x ) ≤ ϕ j ( x ) ≤ h j ( x ) , ∀ x ∈ X , for all j , hence u ( x ) ≤ ϕ ( x ), as claimed. Furthermore, we havethat ϕ j is continuous in Amp( α ) for each j . In fact, [Ber19, Inequality 1.2] givesan upper bound on the Monge-Amp`ere measure of ϕ j :MA θ ( ϕ j ) ≤ D MA θ ( h j ) , D = { ϕ j = h j } . The equality also holds following the work of Di Nezza and Trapani [DNT19]. Inparticular MA θ ( h j ) has an L ∞ -density, hence [DDG + 14, Theorem D] shows that ϕ j is H¨older continuous in Amp( α ), as claimed.Fix λ ∈ (0 , j ∈ N set ψ j := λψ + (1 − λ ) ϕ j − ( A + 2) λ. If we pick ε j ≤ λ − λ ) δ for j > θ + dd c ψ j ≥ λ δ ω . We observeby definitions that ϕ j ≤ V θ , hence ψ j ≤ λψ + (1 − λ ) V θ ≤ V θ + λaφ . Set H j ( t ) := h Cap θ,ψ j ( { ϕ < ψ j − t } ) i /n , t > . For any s ∈ [0 , t > 0, we can argue as above to obtain sH j ( t + s ) ≤ C /n H ( t ) , for C > p , dV , k g k L p ( dV ) , δ , λ , and R X e − P ω ( cϕ − cV θ ) gdV ,with c = ( λa ) − . Let χ be an increasing convex weight such that χ (0) = 0, χ ( −∞ ) = −∞ , and ϕ has finite χ -energy (see [BEGZ10, Proposition 2.11]). Since ONTINUITY OF MONGE-AMP`ERE POTENTIALS IN BIG COHOMOLOGY CLASSES 11 ϕ j ≥ ϕ ≥ ψ − A we have ψ j ≤ ϕ j − λ . It follows from Lemma 2.9 (take s = λ , ψ = ψ j + λ ) that λ n Cap θ,ψ j { ϕ < ψ j } ≤ Z { ϕ<ψ j + λ } MA θ ( ϕ ) ≤ Z { ϕ<ϕ j − λ } MA θ ( ϕ ) ≤ − χ ( − λ ) Z X ( − χ ◦ ( ϕ − ϕ j )) MA θ ( ϕ )The latter converges to 0 as j → + ∞ since ϕ j decreases to ϕ , namely H j (0) goes to0 as j → + ∞ . We thus take j > H j (0) ≤ / (2 C /n ). It then followsfrom [EGZ09, Remark 2.5] that H j ( t ) = 0 if t ≥ t where t ≤ C /n H j (0). Wethen have ϕ ≥ λψ + (1 − λ ) ϕ j − ( A + 2) λ − C /n H j (0) . (3.7)Letting j → + ∞ , we thus obtainlim j → + ∞ inf K ( ϕ − ϕ j ) ≥ − λ (sup K | ψ | + A + 2) , (3.8)for any compact subset K of U . We observe by definition that ψ = ρ + aφ isbounded on K . Finally, letting λ → ϕ iscontinuous on K , hence on U since K was taken arbitrarily. (cid:3) Proof of Theorem 3.1. By rescaling, we may assume without loss of generality that φ is a ω -psh function. Thanks to Theorem 2.1, we can find a quasi-psh function φ m with analytic singularities such that ω + dd c φ m ≥ − ε m ω for ε m > m → + ∞ , and Z X e m ( φ m − φ ) dV < + ∞ . Note that φ m is smooth outside the analytic set E /m ( φ ) ⊂ X . We see thatMA θ ( ϕ ) ≤ e − φ m e ( φ m − φ ) gdV Set now g ′ = e ( φ m − φ ) g . We choose m = [ q ], where [ q ] denotes the integer part of q .We have 2 m > q , hence there is a constant p ′ > p ′ = p + m . It followsfrom generalized H¨older’s inequality that g ′ ∈ L p ′ ( dV ) for p ′ > 1. Observe that φ m is smooth in the complement X \ E /m ( φ ) of the analytic set E /m ( φ ) ⊂ X , inparticular it is locally bounded on Amp( α ) \ E /m ( φ ) ⊃ Amp( α ) \ E /q ( φ ). We canthus apply Theorem 3.2 to complete the proof.In particular if p = ∞ then we can choose m = 1 to conclude the proof ofTheorem 1.1. (cid:3) K¨ahler-Einstein metrics on log canonical pairs of general type. Log canonical singularities. A pair ( Y, ∆) is by definition a connected com-plex normal projective variety Y and an effective Weil Q -divisor ∆. We will saythat the pair ( Y, ∆) has log canonical singularities if K Y + ∆ is Q -Cartier, and iffor some (or equivalently any) log resolution π : X → Y , we have K X = π ∗ ( K Y + ∆) + X i a i E i , where E i are either exceptional divisors or components of the strict transform of∆, and the coefficients a i ∈ Q satisfy the inequality a i ≥ − j . The divisor P i E i has simple normal crossing support. We denote the singular set of Y by Y sing and let Y reg := Y \ Y sing .Let m be a positive integer such that m ( K Y +∆) is Cartier. If we choose σ a localgenerator of m ( K Y + ∆) defined on an open subset U of X , then ( i mn σ ∧ ¯ σ ) /m defines a smooth volume form on U ∩ ( Y reg \ ∆). If f i is a local equation of E i around a point π − ( U ), then we can see that π ∗ (cid:16) i mn σ ∧ ¯ σ (cid:17) /m = Y i | f i | a i dV locally on π − ( U ) for some local volume form dV .The previous construction leads to the following adapted measure which is intro-duced in [EGZ09, § Definition 3.4. Let ( Y, ∆) be a pair and let h be a smooth hermitian meric onthe Q -line bundle O Y ( K Y + ∆) . The corresponding adapted measure µ Y,h on Y reg is locally defined by choosing a nowhere vanishing section σ of m ( K Y + ∆) over asmall open set U and setting µ Y,h := ( i mn σ ∧ ¯ σ ) /m | σ | /mh m . The point of the definition is that the measure µ Y,h does not depend on thechoice of σ , so is globally defined.3.2.2. K¨ahler-Einstein metrics. Let ( Y, ∆) be a log canonical pair. A (singular)K¨ahler-Einstein metric ( KE for short) with negative curvature for ( Y, ∆) is a closedpositive current ω KE which is smooth in Y reg \ ∆ and such thatRic( ω KE ) = − ω KE + [∆] , and Z Y reg h ω nKE i = (( K Y + D ) n ) , on Y reg with n = dim C Y . Here [∆] is the integration current on ∆ | Y reg . Assume h isa smooth hermitian metric on the Q -line bundle O Y ( K Y +∆). Then η := − dd c log h is a smooth representative of c ( K Y +∆). Finding a singular K¨ahler-Einstein metricis equivalent to solving the following Monge-Amp`ere equation for an θ -psh ϕ withfull Monge-Amp`ere mass(3.9) ( η + dd c ϕ ) n = e ϕ + c µ Y,h . for some c ∈ R . Indeed, if we set ω := η + dd c ϕ , then on Y reg we have − Ric( ω ) = − dd c log ω n = − dd c ϕ − dd c log µ Y,h = − dd c ϕ − η + [∆]by definition of adapted measure µ Y,h and the Lelong-Poincar´e formula.We now prove Corollary 1.2 from the introduction. Assume the initial pair ( Y, ∆)is lc of general type, i.e. the canonical bundle K Y + ∆ is big. We consider the logresolution π : ( X, D ) → ( Y, ∆) of the pair. Here, D = P a i D i is a R -divisor withsimple normal crossing support (snc for short) on X , consisting of π -exceptionaldivisors with coefficients in ( −∞ , , ω KE for( X, D ), or equivalently the pull-back of the (singular) KE metric for ( Y, ∆) by π can be written as ω KE = θ + dd c ϕ where θ ∈ c ( π ∗ ( K Y + ∆)) is a smooth big formand ϕ is a θ -psh function solving the Monge–Amp`ere equation(3.10) MA θ ( ϕ ) = e ϕ dV Q i | s i | a i , where s i are non-zero sections of O X ( D i ), | · | i are smooth hermitian metrics on O X ( D i ), and dV is a smooth volume form on X . We let D nklt denote the non-kltpart of D , i.e. D nklt := ∪ a i =1 D i .As a consequence of Theorem 3.2 we have the following Corollary 3.5. The unique solution ϕ ∈ E ( X, θ ) to the equation (3.10) is contin-uous on the ample locus of θ outside of the non-klt part D nklt . ONTINUITY OF MONGE-AMP`ERE POTENTIALS IN BIG COHOMOLOGY CLASSES 13 Proof. The existence of a unique solution to the Monge-Amp`ere equation (3.10)follows from [BG14, Theorem 4.2]. It remains to prove the continuity of ϕ . Itis convenient to pull apart the ”klt part” from the ”non-klt part”, so we set g = Q a i < | s i | − a i ∈ L p for some p > 1, and φ = P a i =1 | s i | . We see that φ issmooth outside of the non-klt locus D nklt , in particular it is locally bounded onAmp( θ ) \ D nklt . Since X is projective we can take a K¨ahler metric ω on X as thepullback of the Fubini-Study metric on CP N . By rescaling we may assume that θ ≤ ω . Since ϕ is bounded from above, we can thus apply Theorem 3.2 to completethe proof. (cid:3) References [BBGZ13] Robert J. Berman, S´ebastien Boucksom, Vincent Guedj, and Ahmed Zeriahi, A vari-ational approach to complex Monge-Amp`ere equations , Publ. Math. Inst. Hautes´Etudes Sci. 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