INTRINSIC CAPACITIES ON COMPACT KA¨HLER MANIFOLDS 5 0 0 2 VINCENTGUEDJ & AHMED ZERIAHI n a Abstract. We study fine properties of quasiplurisubharmonic func- J tionsoncompactK¨ahlermanifolds. Wedefineandstudyseveralintrin- 0 sic capacities which characterize pluripolar sets and show that locally 1 pluripolar sets are globally ”quasi-pluripolar”. ] V 2000 Mathematics SubjectClassification: 32W20, 32F05, 32U15, 32Q15. C . h Introduction t a Since the fundamental work of Bedford and Taylor [4],[5], several authors m have developed a ”Pluripotential theory” in domains of Cn (or of Stein [ manifolds). This theory is devoted to the fine study of plurisubharmonic 2 (psh) functions and can be seen as a non-linear generalization of the classi- v cal potential theory (in onecomplex variable), wheresubharmonicfunctions 2 0 and the Laplace operator ∆ are replaced by psh functions and the complex 3 Monge-Amp`ere operator (ddc)n. Here d,dc denote the real differential op- 1 erators d := ∂ +∂, dc := i [∂ ∂] so that ddc = i∂∂ ; the normalization 40 being chosen so that the p2πosit−ive measure ddc1 loπg[1+ z 2] n has total 0 mass 1 in Cn. We refer the reader to [3],[9],[282], [29] for||a||survey of this / (cid:0) (cid:1) h local theory. at Our aim here is to develop a global Pluripotential theory in the context m of compact Ka¨hler manifolds. It follows from the maximum principle that : there are no psh functions (except constants) on a compact complex man- v ifold X. However there are usually plenty of positive closed currents of i X bidegree (1,1) (we refer the reader to [17], chapter 3, for basic facts on posi- r tivecurrents). Given ω arealclosedsmoothformofbidegree(1,1) onX, we a may consider every positive closed current ω′ of bidegree (1,1) on X which is cohomologous to ω. When X is Ka¨hler, it follows from the ”ddc-lemma” that ω′ can be written as ω′ = ω + ddcϕ, where ϕ is a function which is integrable with respect to any smooth volume form on X. Such a function ϕ will be called ω-plurisubharmonic (ω-psh for short). It is globally defined on X and locally given as the sum of a psh and a smooth function. We let PSH(X,ω) denote the set of ω-psh functions. Such functions were intro- duced by Demailly, who call them quasiplurisubharmonic (qpsh). These are the main objects of study in this article. Thereare several motivations to study qpsh functions on compact Ka¨hler manifolds. First of all they arise naturally in complex analytic geometry as positive singular metrics of holomorphic line bundles (see section 4) whose study is central to several questions of complex algebraic geometry. Solving Monge-Amp`ere equations associated to ω-psh functions has been used to produce metrics with prescribed singularities (see [15]). It is also related to 1 2 VINCENTGUEDJ&AHMEDZERIAHI the existence of canonical metrics in Ka¨hler geometry (see [39]). Important contributions have been made in this direction by Blocki (see [6], [7]) and also by Kolodziej usingtechniques fromlocal Pluripotential theory (see [30], [31]). Quasiplurisubharmonicfunctions have also been used in [23] to define a notion of ω-polynomial convexity and study the fine approximation of positive currents by rational divisors. Last but not least, such functions are of constant use in complex dynamics in several variables (see [19], [20], [24], [25], [35]). It seems to us appropriate to develop a theory of qpsh functions of its own rather than view these functions as particular cases of the local theory. Although thetwo theories look quitesimilar, thereareimportantdifferences which make the ”compact theory” both simpler and more difficult than the local one. Here are some examples: Thereisnopluriharmonicfunctions(exceptconstants)onacompact • manifold, hence each ω-psh function ϕ is canonically associated (up to normalization) to its curvature current ω := ω+ddcϕ 0. This ϕ ≥ yields compactness properties of subsets of PSH(X,ω) (see section 1)which arequiteuseful(e.g. incomplex dynamics, seesection 6.2). Integration byparts(of constant useinsuchtheories)is quitesimple • in the compact setting since there is no boundary. As an illustra- tion, we obtain transparent proofs of Chern-Levine-Nirenberg type inequalities (see example 1.8 and section 2). A successful applica- tion of this simple observation has been made in complex dynamics in [25]. On the other hand one looses homogeneity of Monge-Amp`ere op- • erators in the compact setting. They do have uniformly bounded mass (by Stokes theorem), but there is no performing ”comparison principle”, which is a key tool in the local theory. This is a source of difficulty when, for example, one wishes to solve Monge-Amp`ere equations on compact manifolds (see [30], [26]). Weshalldevelopourstudyinaseriesofarticles. Inthepresentonewedefine and study several intrinsic capacities which we shall use in our forthcoming articles. Let us now describe more precisely the contents of the article. In section 1 we define ω-psh functions and gather useful facts about them (especially compacity results such as proposition 1.7). For locally bounded ω-psh functions ϕ we define the complex Monge-Amp`ere operator ωn in ϕ section 2. We establish Chern-Levine-Nirenberg inequalities (proposition 2.1) and study the ”Monge-Amp`ere capacity” Cap (definition 2.4). As ω in the local theory, ω-psh functions are quasicontinuous with respect to Cap (corollary 2.8). The capacity Cap is comparable to the local Monge- ω ω Amp`ere capacity of Bedford and Taylor (proposition 2.10) and moreover enjoys invariance properties (proposition 2.5). In section 3 we define a rel- ative extremal function h∗ and establish a useful formula (theorem 3.2) E,ω Cap∗(E) = ( h∗ ) ω+ddch∗ n. ω − E,ω E,ω ZX (cid:0) (cid:1) INTRINSIC CAPACITIES ON COMPACT KA¨HLER MANIFOLDS 3 ThisistheglobalversionofthefundamentallocalformulaofBedford-Taylor [5], Cap(E,ω) = (ddcu∗)n. Ω E In section 4 we study yet another capacity (the Alexander capacity T , ω R definition 4.7) which is defined by means of a (global) extremal function (definition 4.1). When ω is a Hodge form, it can be defined as well in terms of Tchebychev constants: these are the contents of section 5 (theorem 5.2) where we further give a geometrical interpretation of T when X = ω CPn is the complex projective space and ω is the Fubini-Study Ka¨hler form (theorem 5.4), following Alexander’s work [1]. In section 6 we show that locally pluripolar sets can be defined by ω-psh functions when ω is Ka¨hler: this is our version of a result of Josefson (theorem 6.2). We then give an application in complex dynamics which illustrates how invariance properties of these capacities can be used. Finally in an Appendix we show how to globally regularize ω-psh functions, following ideas of Demailly. This paper lies at the border of Complex Analysis and Complex Geome- try. We have tried to make it accessible to mathematicians from both sides. This has of course some consequences for the style of presentation. We have includedproofsof someresultswhich may beseenas consequences of results from the local pluripotential theory. We have spent some efforts defining, regularizingandapproximatingpositivesingularmetricsofholomorphicline bundles,although someofthesefacts may beconsideredas classical bycom- plex geometers. Altogether we hope the paper is essentially self-contained. Our efforts will not be vain if for instance we have convinced specialists of the (local) pluripotential theory that the right point of view in studying the Lelong class (Cn) of psh functions with logarithmic growth in Cn is to consider qpshLfunctions on the complex projective space CPn. We also think this paper should be useful to people working in complex dynamics in several variables where pluripotential theory has become an important tool. Warning. Inthewholepaperpositivity (likee.g. inpositive metricandpos- itive current) has to be understood in the weak (french, i.e. non-negativity) sense of currents, except when we talk of a positive line bundle L, in which case it means that L admits a smooth metric whose curvature is a Ka¨hler form. 1. Quasiplurisubharmonic functions In the sequel, unless otherwise specified, Lp-norms will always be com- puted with respect to a fixed volume form on X, which is a compact con- nected Ka¨hler manifold. Let ω be a closed real current of bidegree (1,1) on X. Weassumethroughoutthearticlethatω hascontinuouslocalpotentials. Definition 1.1. Set PSH(X,ω) := ϕ L1(X,R )/ddcϕ ω and ϕ is u.s.c. . { ∈ ∪{−∞} ≥ − } The set PSH(X,ω) is the set of ”ω-plurisubharmonic” functions. ObservethatPSH(X,ω)isnonemptyifandonlyifthereexistsapositive closed currentof bidegree (1,1) onX whichis cohomologous to ω. Onethen saysthatthecohomology class[ω]ispseudoeffective. Inthesequelwealways 4 VINCENTGUEDJ&AHMEDZERIAHI assume this property holds. We also always assume that ω has continuous local potentials. This guarantees that ω-plurisubharmonic functions (ω- psh for short) are upper semi-continuous (u.s.c.), so they are locally hence globally bounded from above. We endow PSH(X,ω) with the L1-topology. Observe that PSH(X,ω) is a closed subspace of L1(X). Example 1.2. The most fundamental example which may serve as a guide- line to everything that follows is the case where X = CPn is the complex projective space and ω = ω is the Fubini-Study Ka¨hler form. There is FS then a 1-to-1 correspondence between PSH(CPn,ω ) and the Lelong class FS 1 (Cn) := ψ PSH(Cn)/ψ(z) log[1+ z 2]+C ψ L ∈ ≤ 2 | | (cid:26) (cid:27) which is given by the natural mapping ψ(x) 1log[1+ x2] if x Cn ψ (Cn) ϕ(x) = − 2 | | ∈ ∈ L 7→ (cid:26) limy∈Cn→x(ψ(y)− 12 log[1+|y|2]) if x∈ H∞, where H denotes the hyperplane at infinity. One can easily show that this ∞ mapping is bicontinuous for the L1 topology. loc The Lelong class (Cn) of plurisubharmonic functions with logarithmic growthinCn hasbeeLnintensivelystudiedinthelastthirtyyears. Itseemsto usthatthepropertiesof (Cn)aremoreeasily seenwhen (Cn)isviewedas PSH(CPn,ω ). FurtheLr we shall see hereafter that theLclass PSH(X,ω) FS of ω-psh functions enjoys several properties of (Cn) when ω is Ka¨hler. We L start by observing (proposition 1.3.1 & 1.3.2 below) that PSH(X,ω) and PSH(X,ω′) are comparable if ω,ω′ are both Ka¨hler. Proposition 1.3. 1) If ω ω then PSH(X,ω ) PSH(X,ω ). 1 2 1 2 2) A ≤R∗, PSH(X,Aω) = A ⊂PSH(X,ω). ∀ ∈ + · 3) If ω′ is cohomologous to ω, ω′ = ω+ddcχ, then PSH(X,ω′) = PSH(X,ω)+χ. 4) If ϕ,ψ PSH(X,ω) then ∈ ϕ+ψ max(ϕ,ψ) , , log[eϕ +eψ] PSH(X,ω) 2 ∈ Proof. Assertions 1),2),3) follow straightforwardly from the definition. Ob- serve that 1.3.4 says that PSH(X,ω) is a convex set which is stable un- der taking maximum and also under the operation (ϕ,ψ) log[eϕ +eψ]. 7→ Theseareall consequences of thecorrespondinglocal propertiesof pshfunc- tions. We nevertheless give a proof, in the spirit of this article. That (ϕ+ψ)/2 PSH(X,ω) follows by linearity. The latter assertion is a con- ∈ sequence of the following computation eϕddcϕ+eψddcψ eϕ+ψd(ϕ ψ) dc(ϕ ψ) ddclog[eϕ+eψ] = + − ∧ − , eϕ +eψ [eϕ +eψ]2 usingthatdf dcf 0. Thiscomputationmakessenseifforinstanceϕ,ψare ∧ ≥ smooth. The general case follows then by regularizing ϕ,ψ (see Appendix). Finally observe that max(ϕ,ψ) = limj−1log[ejϕ+ejψ] PSH(X,ω). (cid:3) ∈ INTRINSIC CAPACITIES ON COMPACT KA¨HLER MANIFOLDS 5 Itfollows from1.3.3 thatPSH(X,ω) essentially dependsonthecohomol- ogy class [ω]. In the same vein we have the following: Proposition 1.4. Let (X) denote the set of positive closed currents ω′ [ω] T of bidegree (1,1) on X which are cohomologous to ω. Then PSH(X,ω) (X) R. [ω] ≃ T ⊕ Proof. The mapping Φ :ϕ PSH(X,ω) ω := ω+ddcϕ (X) ϕ [ω] ∈ 7→ ∈ T is a continuous affine mapping whose kernel consists of constants mappings: indeed ω = ω implies that ϕ ψ is pluriharmonic hence constant by the ϕ ψ − maximum principle. Moreover Φ is surjective: if ω′ 0 is cohomologous to ω then ω′ = ω + ddcϕ for some ϕ L1(X,R) -t≥his is the celebrated ∈ ddc-lemma on Ka¨hler manifolds (see e.g. lemma 8.6, chapter VI in [17]). Thus ϕ coincides almost everywhere with a function of PSH(X,ω) and ω′ = Φ(ϕ). (cid:3) Remark 1.5. The size of PSH(X,ω) is therefore related to that of (X) [ω] T hence only depends on the positivity of the cohomology class [ω]. The more positive [ω], the bigger PSH(X,ω). When [ω] is Ka¨hler then PSH(X,ω) is large: if e.g. χ is any 2-function C on X then εχ PSH(X,ω) for ε > 0 small enough. We will see (theorem ∈ 6.2) that PSH(X,ω) characterizes locally pluripolar sets when [ω] is Ka¨hler. It follows from proposition 1.3 that PSH(X,ω) and PSH(X,ω′) have the same ”size” if ω and ω′ are both Ka¨hler. Note on the other hand that PSH(X,ω) R when ω is cohomologous to ≃ [E], the current of integration along the exceptional divisor of a smooth blow up. Indeed let π :X X˜ be a blow up with smooth center Y, codimCY 2 → ≥ (see e.g. chapter 2 of [17] for the definition of blow-ups). Let E = π−1(Y) denote the exceptional divisorand ω = [E] be the current of integration along E. If ϕ PSH(X,[E]) then ddc(ϕ π−1) 0 in X˜ Y. Since codimCY 2, ∈ ◦ ≥ \ ≥ ϕ π−1 extends trivially through Y has a global psh function on X˜. By the ◦ maximum principle ϕ π−1 is constant hence so is ϕ. Alternatively there is ◦ no positive closed current of bidegree (1,1) on X which is cohomologous to [E] except [E] itself. It follows from previous proposition that any set of ”normalized” ω-psh functions is in 1-to-1 correspondence with (X) which is compact for [ω] T the weak topology of currents. This is the key to several results to follow: normalized ω-psh functions form a compact family in L1(X). N Proposition 1.6. Let (ϕ ) PSH(X,ω) . j ∈ 1) If (ϕ ) is uniformly bounded from above on X, then either ϕ converges j j uniformly to on X or the sequence (ϕ ) is relatively compact in L1(X). j −∞ 2) If ϕ ϕ in L1(X), then ϕ coincides almost everywhere with a unique j → function ϕ∗ PSH(X,ω). Moreover ∈ supϕ∗ = lim supϕ . j X j→+∞ X 3) In particular if ϕ is decreasing, then either ϕ or ϕ= limϕ j j j → −∞ ∈ PSH(X,ω). Similarly, if ϕ is increasing and uniformly bounded from j 6 VINCENTGUEDJ&AHMEDZERIAHI above then ϕ := (limϕ )∗ PSH(X,ω), where ∗ denotes the upper-semi- j ∈ · continuous regularization. Proof. This is a straighforward consequence of the analogous local result for sequences of psh functions. We refer the reader to [17], chapter 1, for a proof. Note that 1.6.2 is a special case of a celebrated lemma attributed to Hartogs. (cid:3) The next result is quite useful (see [42], [43] for a systematic use). Proposition 1.7. The family := ϕ PSH(X,ω)/ supϕ = 0 0 F { ∈ } X is a compact subset of PSH(X,ω). If µ is a probability measure such that PSH(X,ω) L1(µ) then ⊂ := ϕ PSH(X,ω)/ ϕdµ = 0 µ F { ∈ } ZX is a relatively compact subset of PSH(X,ω). In particular there exists C µ such that ϕ PSH(X,ω), ∀ ∈ C +supϕ ϕdµ supϕ. µ − ≤ ≤ X ZX X Proof. It follows straightforwardly from proposition 1.6.1 that is a rel- 0 F atively compact subset of PSH(X,ω). Moreover is closed by Hartogs 0 F lemma (1.6.2). N Let (ϕ ) . Then ψ := ϕ sup ϕ which is relatively j ∈ Fµ j j − X j ∈ F0 compact. Assume first µ is smooth. Then ( ψ dµ) is bounded: this X j is because if ψ ψ in L1(X) then ψ µ ψµ in the weak sense of jk → jk →R (negative) measures hence ψ dµ ψdµ > . Now ψ dµ = X jk → X −∞ X j ϕ dµ sup ϕ dµ = sup ϕ thus(sup ϕ )isboundedandwecan X j − X X j −R X j R X j R apply the previous proposition to conclude that (ϕ ) is relatively compact ( j R R it cannot converge uniformly to since ϕ dµ = 0). −∞ X j Whenµisnotsmooth,itonlyremainstoprovethat( ψ dµ)isbounded. R X j Assume on the contrary that ψ dµ . Extracting a subsequence if X j → −∞ R necessary we can assume ψ dµ 2j. Set ψ = 2−jψ . This is a X Rj ≤ − j≥1 j decreasing sequence of ω-psh functions, hence ψ PSH(X,ω) or ψ . R ∈ P ≡ −∞ Now it follows from the previous discussion that ψ dV C if dV X j ≥ − denotes some smooth probability measure on X. Thus ψdV > R X −∞ hence ψ PSH(X,ω). We obtain a contradiction since by the Monotone convergen∈ce theorem, ψdµ = 2−j ψ dµ = . R (cid:3) X j≥1 X j −∞ Example 1.8. It wasRpart of oPur definitiRon 1.1 that ω-psh functions are integrable with respect to a fixed volume form. Therefore PSH(X,ω) ⊂ L1(µ) for every smooth probability measure µ on X. More generally if µ is a probability measure on X such that (1) µ = Θ+ddc(S), where Θ is smooth and S is a positive current of bidimension (1,1) on X, then PSH(X,ω) L1(µ) for any smooth ω. Indeed let ϕ in PSH(X,ω), ⊂ INTRINSIC CAPACITIES ON COMPACT KA¨HLER MANIFOLDS 7 ϕ 0. If ϕ is smooth, it follows from Stokes theorem that ≤ 0 ( ϕ)dµ = ( ϕ)Θ+ ( ϕ)ddcS ≤ − − − ZX ZX ZX C ϕ + S ( ddcϕ) Θ L1 ≤ || || ∧ − ZX C ϕ + S ω < + , Θ L1 ≤ || || ∧ ∞ ZX where the last inequality follows from S 0 and ddcϕ ω. The general ≥ − ≤ case follows by regularizing ϕ (see Appendix). Probability measures satisfying (1) naturally arise in complex dynamics (see [25]). Observe also that Monge-Amp`ere measures arising from the local theory ofBedford andTaylor [5]dosatisfy(1): ifuispshandlocally bounded near e.g. the unit ball B of Cn, we can extend it to Cn as a global psh function with logarithmic growth considering u(z) if z B ∈ U(z) := max(u(z),Alog+ z sup u 1) if z (1+ε)B B  Alog+| z|− supB| u|− 1 if z ∈ Cn (1+\ε)B  | |− B| |− ∈ \ where log+ z := max(log z ,0) and with A large enough. We assume A= 1  | | | | for simplicity. Now ϕ := U 1 log[1+ z 2]+C extends as a bounded function inPSH(CPn,ω), where ω i−st2he Fubin|i-|StudyKa¨hler formon CPn, so ϕ 0 ≥ if C > 0 is large enough. To conclude note that, setting ω := ω+ddcϕ 0, ϕ ≥ we get ωn = (ddcu)n in B and ϕ n−1 ωn =ωn+ddcS, where S = ϕ ωj ωn−1−j 0. ϕ ϕ∧ ≥ j=0 X The Monge-Amp`ere operator ωn will be defined in the next section. ϕ Example 1.9. If µ is a probability measure on X = CPn and ω denotes as before the Fubini-Study Ka¨hler form, then x y ϕ (x):= log || ∧ || dµ(y) µ ZCPn (cid:18)||x||·||y||(cid:19) defines a ω-psh function on CPn. Such functions have been considered by Molzon, Shiffman and Sibony [34], [33] in order to define capacities on CPn. However they do not characterize pluripolar sets. 2. Monge-Amp`ere capacity We assume in this section that ω is a Ka¨hler form on X. Let T be a positive closed current of bidegree (p,p) on X, 0 p n= dimCX. It can ≤ ≤ be thought of as a closed differential form of bidegree (p,p) with measure coefficients whose total variation is controlled by T := T ωn−p. || || ∧ ZX We refer the reader to chapter 3 of [17] for basic properties of positive currents. Given ϕ PSH(X,ω) we write ϕ L1(T) if ϕ is integrable with ∈ ∈ respect to each (measure) coefficient of T. This is equivalent to ϕ being 8 VINCENTGUEDJ&AHMEDZERIAHI integrable with respect to the trace measure T ωn−p. In this case the ∧ current ϕT is well defined, hence so is ω T := ω T +ddc(ϕT). ϕ ∧ ∧ Thisisagain apositive closed currenton X, of bidegree(p+1,p+1). Indeed positivity is a local property which is stable under taking limits. One can locally regularize ϕ and approximate ω T by the currents ω T which ϕ∧ ϕε ∧ are positive since ω are smooth positive forms. ϕε When ϕ PSH(X,ω) L∞(X) then ϕ L1(T) for any positive closed ∈ ∩ ∈ j current T of bidegree (p,p). One can thus inductively define ω T, 1 ϕ ∧ ≤ j n p, for ϕ PSH(X,ω) L∞(X). For T = 0 and j = n one obtains ≤ − ∈ ∩ the complex Monge-Amp`ere operator, ωn. It follows from the local theory ϕ that the operator ϕ ωn is continuous undermonotone sequences (see [5]). 7→ ϕ The proof of these continuity properties is simpler in our compact setting. We refer the reader to [26] where this is proved in a more general global context. Proposition 2.1 (Chern-Levine-Nirenberginequalities). Let T be a positive closed current of bidegree (p,p) on X and ϕ PSH(X,ω) L∞(X). ∈ ∩ Then ω T = T . Moreover if ψ PSH(X,ω) L1(T), then ϕ || ∧ || || || ∈ ∩ ψ L1(T ω ) and ϕ ∈ ∧ ψ ψ +[2supψ+supϕ infϕ] T . || ||L1(T∧ωϕ) ≤ || ||L1(T) X X − X || || Proof. By Stokes theorem, ddcϕ T ωn−p−1 = 0, hence X ∧ ∧ ω T := ωR T ωn−p−1 = T ωn−p := T . ϕ ϕ || ∧ || ∧ ∧ ∧ || || ZX ZX Consider now ψ L1(T). Since T has measure coefficients, this simply ∈ means that ψ is integrable with respect to the total variation of these mea- sures. Assume first ψ 0, ϕ 0 and ϕ,ψ are smooth. Then ≤ ≥ ψ := ( ψ)T ω ωn−p−1 = ψ + ( ψ)T ddcϕ ωn−p−1. || ||L1(T∧ωϕ) − ∧ ϕ∧ || ||L1(T) − ∧ ∧ ZX ZX Now it follows from Stokes theorem that ( ψ)T ddcϕ ωn−p−1 = ϕT ( ddcψ) ωn−p−1 − ∧ ∧ ∧ − ∧ ZX ZX ϕT ωn−p supϕ T ωn−p, ≤ ∧ ≤ ∧ ZX X ZX where the forelast inequality follows from ϕT ωn−p 0 and ddcψ ω. ∧ ≥ − ≤ This yields ψ ψ +supϕ T . || ||L1(T∧ωϕ) ≤ || ||L1(T) || || X Thegeneralcasefollows byregularizingϕ,ψ, observingthatωϕ =ωϕ′ where ϕ′ = ϕ inf ϕ 0, and decomposing ψ = ψ′ + sup ψ with ψ′ = ψ sup ψ −0. X ≥ X (cid:3)− X ≤ Remark 2.2. The fact that the L1-norm of ψ with respect to the probability measure T ω ωn−p−1 is controlled byits L1-norm with respect to T ωn−p ϕ ∧ ∧ ∧ INTRINSIC CAPACITIES ON COMPACT KA¨HLER MANIFOLDS 9 is similar to the phenomenon already encountered in example 1.8: one can write T ω ωn−p−1 = T ωn−p+ddcS, S = (ϕ infϕ)T ωn−p−1 0. ϕ ∧ ∧ ∧ − X ∧ ≥ This type of estimates is usually referred to as ”Chern-Levine-Nirenberg in- equalities”, inreferenceto[10]where simpler-butfondamental- L∞-estimates were established (with ψ = constant). Estimates involving the L1-norm of ψ were first proved in the local context by Cegrell [8] and Demailly [14]. A straightforward induction yields the following Corollary 2.3. Let ψ,ϕ PSH(X,ω) with 0 ϕ 1. Then ∈ ≤ ≤ 0 ψ ωn ψ ωn +n[1+2supψ] ωn. ≤ | | ϕ ≤ | | ZX ZX X ZX FollowingBedford-Taylor[5]andKolodziej[31]weintroducethefollowing Monge-Amp`ere capacity. Definition 2.4. Let K be a Borel subset of X. We set Cap (K) := sup ωn/ϕ PSH(X,ω),0 ϕ 1 . ω ϕ ∈ ≤ ≤ (cid:26)ZK (cid:27) Note that this definition only makes sense when the cohomology class [ω] is big, i.e. when [ω]n > 0, and when it admits locally bounded potentials ϕ. This implies, by a regularization result of Demailly [14], that [ω] is big and nef. To simplify we actually assume that ω (hence [ω]) is Ka¨hler. Proposition 2.5. 1) If K K′ X are Borel subsets then ⊂ ⊂ Vol (K):= ωn Cap (K) Cap (K′) Cap (X) = Vol (X). ω ω ω ω ω ≤ ≤ ≤ ZK 2) If K are Borel subsets of X then Cap ( K ) Cap (K ). More- j ω j ω j ∪ ≤ over Cap ( K ) = limCap (K ) if K K . ω j ω j j j+1 ∪ ⊂ P 3) If ω ω then Cap () Cap (). For all A 1, Cap () 1 ≤ 2 ω1 · ≤ ω2 · ≥ ω · ≤ Cap () AnCap (). In particular if ω,ω′ are two Ka¨hler forms then Aω ω · ≤ · there exists C 1 such that ≥ 1 Capω() Capω′() C Capω(). C · ≤ · ≤ · · 4) If f :X X is holomorphic then for all Borel subset K of X, → Capω(f(K)) Capf∗ω(K). ≤ In particular Cap (f(K)) = Cap (K) for every ω-isometry f. ω ω Proof. That Vol () Cap () is a straightforward consequence of the def- ω ω · ≤ · inition (since ω = ω). It then follows from Stokes theorem that ωn = 0 X ϕ ωn for every ϕ PSH(X,ω) L∞(X), thus Vol (X) = Cap (X). X ∈ ∩ ω ω R Property 2) is a straightforward consequence of the definitions. R If ω ω then PSH(X,ω ) PSH(X,ω ) hence Cap () Cap (). 1 ≤ 2 1 ⊂ 2 ω1 · ≤ ω2 · Fix A 1. If ψ PSH(X,Aω) is such that 0 ψ 1 then ψ/A ≥ ∈ ≤ ≤ ∈ PSH(X,ω) with 0 ψ/A 1/A 1. Moreover (Aω +ddcψ)n = An(ω + ≤ ≤ ≤ ddc(ψ/A))n. This shows Cap () AnCap (). Aω ω · ≤ · 10 VINCENTGUEDJ&AHMEDZERIAHI In particular if ω,ω′ are both Ka¨hler then A−1ω ω′ Aω for some ≤ ≤ constant A 1, hence C−1Capω() Capω′() C Capω() with C = An. ≥ · ≤ · ≤ · · It remains to prove 4). It follows from the change of variables formula that if ϕ PSH(X,ω) with 0 ϕ 1 then ∈ ≤ ≤ ωϕn ≤ f∗ωϕn = (f∗ω+ddc(ϕ◦f))n ≤ Capf∗ω(K) Zf(K) ZK ZK since ϕ f PSH(X,f∗ω) with 0 ϕ f 1. We infer Cap (f(K)) ω ◦ ∈ ≤ ◦ ≤ ≤ Capf∗ω(K). When f is a ω-isometry, i.e. f Aut(X) with f∗ω = ω, then ∈ the mapping ϕ ϕ f is an isomorphism of u PSH(X,ω)/0 u 1 , whence Capω(f7→(K))◦= Capf∗ω(K) = Capω(K{).∈ ≤ ≤ }(cid:3) Let PSH−(X,ω) denote the set of negative ω-pshfunctions. A set is said tobePSH(X,ω)-polar ifitisincludedinthe locusofsomefunctionψ −∞ ∈ PSH(X,ω), ψ . As we shall soon see, the sets of zero Monge-Amp`ere 6≡ −∞ capacity are precisely the PSH(X,ω)-polar sets. We start by establishing the following: Proposition 2.6. If P is a PSH(X,ω)-polar set, then Cap (P) = 0. More ω precisely, if ψ PSH−(X,ω) then ∈ 1 Cap (ψ < t) ( ψ)ωn +nVol (X) , t> 0. ω ω − ≤ t − ∀ (cid:20)ZX (cid:21) Proof. Fix ϕ PSH(X,ω) such that 0 ϕ 1. Fix t > 0 and set ∈ ≤ ≤ K = x X/ψ(x) < t . By Chebyshev’s inequality, t { ∈ − } 1 ωn ( ψ/t)ωn ( ψ)ωn +nVol (X) , ϕ ≤ − ϕ ≤ t − ω ZKt ZX (cid:20)ZX (cid:21) where the last inequality follows from corollary 2.3. Taking supremum over all ϕ′s yields the claim. (cid:3) Observe that the previous proposition says that Cap∗(P) = 0, where ω Cap∗(E) := inf Cap (G)/G open with E G , ω { ω ⊂ } is the outer capacity associate to Cap . ω Our aim is now to show that ω-psh functions are quasicontinuous with respect to Cap (corollary 2.8). We first need to show that decreasing ω sequences of ω-psh functions converge ”in capacity”. Proposition 2.7. Letψ,ψ PSH(X,ω) L∞(X) such that (ψ ) decreases j j ∈ ∩ to ψ. Then for each δ > 0, Cap ( ψ > ψ+δ ) 0. ω j { } → Proof. We can assume w.l.o.g. that Vol (X) = 1 and 0 ψ ψ 1. Fix ω j ≤ − ≤ δ > 0 and ϕ PSH(X,ω), 0 ϕ 1. By Chebyshev inequality, it suffices ∈ ≤ ≤ to control (ψ ψ)ωn uniformlyin ϕ. Itfollows from Stokes theorem that X j− ϕ (ψ R ψ)ωn = (ψ ψ)ω ωn−1 d(ψ ψ) dcϕ ωn−1. j − ϕ j − ∧ ϕ − j − ∧ ∧ ϕ ZX ZX ZX Now by Cauchy-Schwartz inequality, 1/2 1/2 df dcϕ ωn−1 df dcf ωn−1 dϕ dcϕ ωn−1 , j ∧ ∧ ϕ ≤ j ∧ j ∧ ϕ · ∧ ∧ ϕ (cid:12)ZX (cid:12) (cid:18)ZX (cid:19) (cid:18)ZX (cid:19) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)