Domains of Definition of Monge-Ampère Operators on Compact Kähler Manifolds

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1 Syracuse University SURFACE Mathematics Faculty Scholarship Mathematics Domains of Definition of Monge-Ampère Operators on Compact Kähler Manifolds Dan Coman Syracuse University Vincent Guedj Universite Paul Sabatier Ahmed Zeriahi Universite Paul Sabatier Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Coman, Dan; Guedj, Vincent; and Zeriahi, Ahmed, "Domains of Definition of Monge-Ampère Operators on Compact Kähler Manifolds" (2007). Mathematics Faculty Scholarship. Paper This Article is brought to you for free and open access by the Mathematics at SURFACE. It has been accepted for inclusion in Mathematics Faculty Scholarship by an authorized administrator of SURFACE. For more information, please contact

2 DOMAINS OF DEFINITION OF MONGE-AMPÈRE OPERATORS ON COMPACT KÄHLER MANIFOLDS ariv: v1 [math.cv] 31 May 2007 DAN COMAN, VINCENT GUEDJ, AND AHMED ZERIAHI Abstract. Let (, ω) be a compact Kähler manifold. We introduce and study the largest set DMA(, ω) of ω-plurisubharmonic (psh) functions on which the complex Monge-Ampère operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set PSH(, ω) of all ω-psh functions. We prove that certain twisted Monge-Ampère operators are well defined for all ω-psh functions. As a consequence, any ω-psh function with slightly attenuated singularities has finite weighted Monge-Ampère energy. Introduction It is well known that the complex Monge-Ampère operator is not well defined for arbitrary plurisubharmonic (psh) functions. Bedford and Taylor [BT3] found a way to define it for locally bounded psh functions. Later the definition was extended to classes of unbounded functions (see [Sib], [D2], [FS]). Whenever defined, the Monge-Ampère operator was shown to be continuous along decreasing sequences, but it is discontinuous along sequences in L p. The natural domain of definition of the Monge-Ampère operator on open sets in C n was recently characterized in [C2], [Bl1], [Bl2]. We consider here the problem of defining Monge-Ampère operators on a compact Kähler manifold of complex dimension n. Let PSH(,ω) denote the set of ω-plurisubharmonic (ω-psh) functions on. Here, and throughout the paper, ω is a fixed Kähler form on. Recall that an upper semicontinuous function ϕ L 1 () is called ω-psh if the current ω ϕ := ω + dd c ϕ is positive. Motivated by the results of [BT3], [C2], [Bl1], [Bl2], it is natural to define the domain of the Monge-Ampère operator as follows: Definition. Let DMA(,ω) be the set of functions ϕ PSH(,ω) for which there is a positive Radon measure M A(ϕ) with the following property: If {ϕ j } is any sequence of bounded ω-psh functions decreasing to ϕ then (ω + dd c ϕ j ) n MA(ϕ), in the weak sense of measures. We set ω n ϕ = (ω + dd c ϕ) n := MA(ϕ). According to this definition, DMA(,ω) is the largest set of ω-psh functions on which the Monge-Ampère operator (ω + dd c ) n can be defined so that it is continuous with respect to decreasing sequences of bounded ω-psh functions. It includes all the classes in which the operator was previously 2000 Mathematics Subject Classification. Primary: 32W20. Secondary: 32U15, 32Q15. Dan Coman was partially supported by the NSF Grant DMS

3 2 DAN COMAN, VINCENT GUEDJ, AND AHMED ZERIAHI defined, either as a consequence of the local theory, or genuinely in the compact setting (the class E(,ω) from [GZ2]). The set DMA(,ω) is a proper subset of PSH(,ω) (see Examples 1.3 and 1.4). We show in Proposition 1.1 that the operator is continuous under decreasing sequences in its domain. Moreover, if ϕ DMA(,ω) then we prove in Proposition 1.2 that the set of points where ϕ has positive Lelong number is at most countable. There are several properties of DMA(,ω) that we expect to hold. We discuss these in section 1.2, and we introduce a few subclasses of interest, especially the class DMA(,ω) (Definition 1.7): a function ϕ PSH(,ω) belongs to this class if it is in DMA(,ω) and moreover for any sequence of bounded ω-psh function ϕ j decreasing towards ϕ, and for any test function u PSH(,ω) L (), u(ω + dd c ϕ j ) n u(ω + dd c ϕ) n. This convergence property interpolates inbetween the two natures of the Monge-Ampère measure (ω + dd c ϕ) n : on one hand it is stronger than the weak convergence in the sense of positive Radon measures (any smooth test function is Cω-psh for some constant C > 0), on the other hand it is weaker than the convergence in the sense of Borel measures. We show (Theorem 1.9) that a generalized comparison principle holds in DM A(, ω), and that all concrete classes under consideration are subsets of this class (see Corollary 2.3 and Theorem 3.2). In [GZ2] a class E(,ω) PSH(,ω) was introduced, on which the Monge-Ampère operator is well defined and continuous along decreasing sequences, hence E(,ω) DMA(,ω). Defining this class requires that one works globally on a compact manifold, hence many of its properties have no analogue in the local context (see [GZ2]). We study in Section 2 more general classes E(T,ω) of ω-psh functions with finite energy with respect to a closed positive current T. These help us in studying twisted Monge-Ampère operators, which happen to be well defined in all of PSH(,ω) (Theorems 2.4 and 2.5). As a consequence, we show that any ω-psh function with slightly attenuated singularities has finite energy (Corollary 2.6): Theorem. Let χ : R R be a smooth convex increasing function, with χ ( 1) 1, χ ( ) = 0. Fix ϕ PSH(,ω) with sup ϕ 1. Then χ ϕ E(,ω) DMA(,ω), hence its Monge-Ampère measure does not charge pluripolar sets. Note that it is necessary to slightly attenuate the singularities of ϕ (condition χ ( ) = 0), since functions in E(,ω) have zero Lelong numbers at all points (see Lemma 3.5). In Sections 3 and 4 we consider the set DMA loc (,ω) on which the Monge-Ampère operator is defined as a consequence of the local theory ([Bl1], [Bl2]). We prove in Theorem 3.2 that this local domain can be characterized in terms of energy classes. Moreover, DMA loc (,ω) is a proper subset of DMA(,ω) and consists of functions whose gradient is square integrable (Proposition 4.2). In Proposition 4.6 we show that ω-psh functions, bounded in a neighborhood of an ample divisor, belong to DMA loc (,ω).

4 DEFINITION OF THE COMPLE MONGE-AMPÈRE OPERATOR 3 In Proposition 4.1 we prove that the measure χ (ϕ)dϕ d c ϕ ω n 1 has density in L 1 () for every ϕ PSH(,ω), where χ is any convex increasing function. It is an interesting problem to study the stability of subclasses in DM A(, ω) under standard geometric constructions. We show that DMA loc (,ω) is not preserved by blowing up, but behaves well under blowing down. We conclude this paper by analyzing further the connection between DMA(,ω) and various energy classes in some concrete cases of Kähler surfaces. The motivation comes from the fact that energy classes have important properties, such as convexity and stability under taking maximum. Hence an equivalent description of DMA(,ω) in terms of such energy classes would be very useful. Let E(ω,ω) be the class of ω-psh functions on such that the trace measure ω ϕ ω does not charge the set {ϕ = }. Then both DMA loc (,ω) and E(,ω) are contained in E(ω,ω). We give some evidence that E(ω,ω) might be equal to DMA(,ω) when dim C = 2. Acknowledgement. D. Coman is grateful to the Laboratoire Emile Picard for the support and hospitality during his visits in Toulouse. 1. General properties and examples 1.1. Lelong numbers and other constraints. It is important to require in the definition that the convergence holds on any sequence of bounded ω- psh functions (see Example 1.3). This allows us to show that the operator is continuous under decreasing sequences in DMA(,ω). We have in fact the following more general property. Proposition 1.1. Let ǫ j 0, ǫ j 0, and let ϕ j DMA(,(1 + ǫ j )ω) be a decreasing sequence towards ϕ DM A(, ω). Then in the weak sense of measures. ((1 + ǫ j )ω + dd c ϕ j ) n (ω + dd c ϕ) n Proof. We first assume that all ǫ j = 0. Fix χ a test function and set, for an integer k > 0, ϕ k j := max(ϕ j, k) PSH(,ω) L (). Since ϕ j DMA(,ω), we can find an increasing sequence k j so that (ω + dd c ϕ k j j )n,χ (ω + dd c ϕ j ) n,χ 2 j. Thus ϕ j := max(ϕ j, k j ) is a sequence of bounded ω-psh functions decreasing towards ϕ, hence ωϕ ñ j,χ ωϕ n,χ. The desired convergence follows. In the general case, subtracting a constant we may assume that ϕ 1 < 0. The sequence of measures ((1 + ǫ j )ω + dd c ϕ j ) n has bounded mass, so by passing to a subsequence we may assume that it converges weakly to a measure µ. By taking another subsequence we may assume that ǫ j decreases to 0. Then ϕ j = ϕ j/(1 + ǫ j ) DMA(,ω) is a decreasing sequence to ϕ, so ωϕ n ω ϕ n. We conclude that µ = ωn ϕ. j If ϕ PSH(,ω), let ν(ϕ,x) be the Lelong number of ϕ at x, and E ε (ϕ) = {x /ν(ϕ,x) ε}, E + (ϕ) = ε>0e ε (ϕ).

5 4 DAN COMAN, VINCENT GUEDJ, AND AHMED ZERIAHI Proposition 1.2. If ϕ DMA(,ω) then ωϕ({p}) n ν n (ϕ,p), for all p. Moreover, the set E + (ϕ) is at most countable and ν n (ϕ,x) ω n. x E + (ϕ) Proof. Observe that the measure ω n ϕ has at most countably many atoms. We are going to show that if ν(ϕ,x) > 0 then ω n ϕ has an atom at x. Let g = χ log dist(,x), where χ 0 is a smooth cut off function such that χ = 1 in a neighborhood of x. Since ω is Kähler, dd c g 0 near x, and g is smooth on \ {x}, we can find ε 0 > 0 such that ε 0 g PSH(,ω). Consider ϕ j := max(ϕ,ε 0 g j) PSH(,ω) L loc ( \ {x}). Proposition 4.6 from Section 4 implies ϕ j DMA(,ω). Since ϕ j is in the local domain of definition of the Monge-Ampère operator, we have by [D2] ω n ϕ j ({x}) ν n (ϕ j,x) = (min(ε 0,ν(ϕ,x))) n. Using Proposition 1.1, we infer that ωϕ n has a Dirac mass at x. By [D1], there exist sequences ε j,δ j ց 0, and j PSH(,(1 + δ j )ω), j ց ϕ, such that j is smooth in \ E εj (ϕ) and sup x ν( j,x) ν(ϕ, x) 0. Using Siu s theorem [Siu] and the previous discussion, we conclude that E εj (ϕ) is a finite set and (1 + δ j ) n ω n ν n ( j,x), ((1 + δ j )ω + dd c j ) n ({p}) ν n ( j,p), x E + (ϕ) where p. We conclude by Proposition 1.1, letting j +. Proposition 1.2 provides examples of functions not in DMA(,ω). Example 1.3. Let = P 2, ω be the Fubini-Study Kähler form, and ϕ PSH(P 2,ω) be so that ω ϕ = d 1 [C], where [C] is the current of integration along an algebraic curve C of degree d 1. By Proposition 1.2, ϕ / DMA(P 2,ω) since E + (ϕ) is not countable. Alternatively, we can construct two sequences of functions in DMA(P 2,ω) decreasing to ϕ with constant Monge-Ampère measures that are different. Indeed, if L is a generic line, ϕ L j := max(ϕ,u L j) PSH(P 2,ω), where ω+dd c u L = [L], then ϕ L j is continuous outside the finite set L C, and ϕ L j ց ϕ. Hence ϕl j DMA(P2,ω) and (ω + dd c ϕ L j )2 = ω ϕ [L] = 1 δ p d p L C is independent of j. Here δ p is the Dirac mass at p, and the first equality follows easily since the currents involved have local potentials which are pluriharmonic away from their ( )-locus. Using sequences ϕ L j,ϕl j, for lines L L, we conclude by Proposition 1.1 that ϕ / DMA(P 2,ω). The previous construction can be generalized to exhibit examples of functions ϕ / DMA(,ω) with zero Lelong numbers at all but one point. Example 1.4. Assume ϕ PSH(P 2,ω) is such that {ϕ = } is a closed proper subset of P 2, and the positive current ω ϕ is supported on {ϕ = }. We claim that ϕ / DMA(P 2,ω). Indeed, let p / {ϕ = } and q 1,q 2 be

6 DEFINITION OF THE COMPLE MONGE-AMPÈRE OPERATOR 5 distinct points in {ϕ = }. The lines L k = (pq k ) intersect {ϕ = } in compact subsets of C P 1 L k, hence both sequences ϕ L k j := max(ϕ,u Lk j) DMA(P 2,ω), where ω + dd c u Lk = [L k ], decrease towards ϕ. Now (ω + dd c ϕ L k j ) 2 = ω ϕ [L k ] are distinct measures (independent of j), hence ϕ / DMA(P 2,ω). A concrete function ϕ as in Example 1.4 is obtained as follows. Fix a P 2 and a line L P 1, a / L. Let π denote the projection from a onto L: this is a meromorphic map which is holomorphic in P 2 \ {a}. Fix ν a probability measure on L with no atom, whose logarithmic potential is exactly on the support of ν (one can consider for instance the Evans potential of a compact polar Cantor set). Then π ν = ω ϕ, where ϕ PSH(P 2,ω) satisfies our assumptions. Note that ν(ϕ,x) = 0, x a, and ν(ϕ,a) = 1. Further examples can be obtained by considering functions ϕ such that ω ϕ is a laminar current, whose transverse measure is an Evans measure Special subclasses. There are several properties of DM A(, ω) that we expect to hold. We analyze here some of these and introduce interesting subclasses of DMA(,ω) Intermediate Monge-Ampère operators. It is natural to expect that if a function ϕ PSH(,ω) has a well defined Monge-Ampère measure (ω+dd c ϕ) n, then the currents (ω+dd c ϕ) l are also well defined for 1 l n. Unfortunately we are unable to prove this (except of course in dimension 2), hence the following: Definition 1.5. We let DMA l (,ω), where 1 l n, be the set of functions ϕ PSH(,ω) for which there is a positive closed current MA l (ϕ) of bidegree (l,l) with the following property: If {ϕ j } is any sequence of bounded ω-psh functions decreasing to ϕ then ω l ϕ j MA l (ϕ), in the weak sense of currents. We set ω l ϕ = (ω + dd c ϕ) l := MA l (ϕ). We also set DMA k (,ω) = k DMA l (,ω). We let the reader check that Proposition 1.1 holds for ϕ DMA l (,ω) and 1 l n. Clearly l=1 DMA n (,ω) DMA n (,ω) = DMA(,ω), with equality when n = 2 since ϕ ω ϕ is well defined for all ω-psh functions. We expect the equality to hold also when n 3. It is clear from the definition that ϕ DMA(,ω) if and only if λϕ DMA(,λω), where λ > 0. One would expect moreover that, if ϕ DMA(,ω), then λϕ DMA(,ω) for 0 λ 1, as the function λϕ is slightly less singular than ϕ. It is also natural to expect that the class DMA(,ω) is stable under taking maximum: ϕ DMA(,ω) and PSH(,ω)? = max(ϕ,) DMA(,ω).

7 6 DAN COMAN, VINCENT GUEDJ, AND AHMED ZERIAHI If this property holds, then applying it to = λϕ 0, 0 λ 1, shows that λϕ = max(ϕ,λϕ) DMA(,ω) as soon as ϕ DMA(,ω). An alternative and desirable property is DMA(,ω)? = DMA(,ω ) PSH(,ω), when ω ω. All these properties are related to convexity properties of DMA(,ω) The non pluripolar part. It was observed in [GZ2] that if ϕ is ω-psh and ϕ j := max(ϕ, j), then j µ j (ϕ) := 1 {ϕ> j} (ω + dd c ϕ j ) n is an increasing sequence of positive Radon measures, of total mass uniformly bounded above by ωn. Thus {µ j (ϕ)} converges to a positive Radon measure µ(ϕ) on. It is generally expected (see [BT4] for similar considerations in the local context) that µ(ϕ) should correspond to the non pluripolar part of (ω + dd c ϕ) n, whenever the latter makes sense. We can justify this expectation in two special cases: Proposition 1.6. Fix ϕ DM A(, ω). Then µ(ϕ) 1 {ϕ> } (ω + dd c ϕ) n, with equality if exp ϕ is continuous, or if ωϕ n is concentrated on {ϕ = }. Proof. We can assume wlog that ϕ 0. Set u s := max(ϕ/s+1,0). Note that u s are bounded ω-psh functions which increase towards 1 {ϕ> }. Moreover {u s > 0} = {ϕ > s} and u s = 0 elsewhere. We infer, when j > s, that u s (ω + dd c ϕ j ) n = u s 1 {ϕ> j} (ω + dd c ϕ j ) n, where ϕ j = max(ϕ, j) are the canonical approximants. When e ϕ is continuous, then so is u s, hence passing to the limit yields u s (ω+dd c ϕ) n = u s µ(ϕ). Letting s +, we infer µ(ϕ) = 1 {ϕ> } µ(ϕ) = 1 {ϕ> } (ω + dd c ϕ) n. In the general case, we only get u s µ(ϕ) u s (ω+dd c ϕ) n, since u s is uppersemi-continuous. This yields µ(ϕ) = 1 {ϕ> } µ(ϕ) 1 {ϕ> } (ω + dd c ϕ) n, whence equality if (ω + dd c ϕ) n is concentrated on {ϕ = }. To overcome this difficulty in the general case, we introduce interesting subclasses of DMA(,ω): Definition 1.7. Fix 1 l n. We let DMA l (,ω) (resp. DMA l (,ω)) denote the set of functions ϕ DMA l (,ω) (resp. DMA l (,ω)) such that for any sequence ϕ j PSH(,ω) L () decreasing to ϕ, u(ω + dd c ϕ j ) l ω n l u(ω + dd c ϕ) l ω n l, for all u PSH(,ω) L (). Note that this convergence property is stronger than the usual convergence in the weak sense of Radon measures: any smooth test function is Cω-psh for some constant C > 0. The following corollary can be proved exactly like Proposition 1.6.

8 DEFINITION OF THE COMPLE MONGE-AMPÈRE OPERATOR 7 Corollary 1.8. If ϕ DMA(,ω) then µ(ϕ) = 1 {ϕ> } (ω + dd c ϕ) n. Moreover, if ϕ j = max(ϕ, j) and B {ϕ > } is any Borel set then ωϕ n = lim ωϕ n j + j. B B {ϕ> j} As a consequence, we obtain the following generalized comparison principle: Theorem 1.9. Let ϕ, PSH(,ω) and set ϕ := max(ϕ,). Then Moreover if ϕ, DM A(, ω) then ω n 1 {ϕ>} µ(ϕ) = 1 {ϕ>} µ(ϕ ). {ϕ<} {ϕ<} {ϕ= } Proof. Set ϕ j = max(ϕ, j) and j = max(, j). Recall from [BT4] that the desired equality is known for bounded psh functions, ω n ϕ. 1 {ϕj > j+1 }(ω + dd c ϕ j ) n = 1 {ϕj > j+1 }(ω + dd c max(ϕ j, j+1 )) n. Observe that {ϕ > } {ϕ j > j+1 }, hence 1 {ϕ>} 1 {ϕ> j} (ω + dd c ϕ j ) n = 1 {ϕ>} 1 {ϕ> j} (ω + dd c max(ϕ,, j)) n = 1 {ϕ>} 1 {ϕ > j} (ω + dd c max(ϕ, j)) n. Note that the sequence of measures 1 {ϕ> j} (ω + dd c ϕ j ) n converges in the strong sense of Borel measures towards µ(ϕ) and the sequence 1 {ϕ > j} (ω+ dd c (ϕ, j)) n converges in the strong sense of Borel measures towards µ(ϕ ). Hence, since {ϕ > } {ϕ > } {ϕ > }, it follows from Corollary 1.8 that 1 {ϕ>} µ(ϕ) = 1 {ϕ>} µ(ϕ ). Now let ϕ, DMA(,ω) and assume first that is bounded. Then it follows from Corollary 1.8 that (ω + dd c ) n = (ω + dd c max(,ϕ)) n. {ϕ<} {ϕ<} Since (ω + ddc max(,ϕ)) n = (ω + ddc ϕ) n it follows that (ω+dd c ) n (ω+dd c ϕ) n (ω+dd c ϕ) n = {ϕ<} {ϕ>} {ϕ } (ω+dd c ϕ) n. If DMA(,ω) is not bounded, we apply the previous inequality to ϕ and j := max(, j) for j N. Then we get for any j N (ω + dd c j ) n (ω + dd c ϕ) n. {ϕ< j } {ϕ j } Since {ϕ < } { > j} {ϕ < j } for any j and {ϕ j } is a decreasing sequence of Borel sets converging to the Borel set {ϕ }, by

9 8 DAN COMAN, VINCENT GUEDJ, AND AHMED ZERIAHI taking the limit in the previous inequalities and applying Corollary 1.8, we obtain (ω + dd c ) n (ω + dd c ϕ) n. {ϕ<} {ϕ } Now take 0 < ε < 1 and apply the previous result with ϕ + ε and. Then let ε 0 to obtain the required inequality. Remark Let ϕ PSH(,ω) and ϕ j := max(ϕ, j). It follows as in [GZ2] that µ l j (ϕ) = 1 {ϕ> j}(ω + dd c ϕ j ) l is an increasing sequence of positive currents of mass at most ωn, so it converges in the strong sense to a positive current µ l (ϕ) on. We leave it to the reader to check that Proposition 1.6, Corollary 1.8 and Theorem 1.9 have analogues in the classes DMA l (,ω). For instance, if ϕ DMA l (,ω) and e ϕ is continuous, or if ϕ DMA l (,ω), then µ l (ϕ) = 1 {ϕ> } ωϕ. l Moreover, if ϕ, DMA l (,ω) then ω l ωn l ωϕ l ω n l. {ϕ<} {ϕ<} {ϕ= } Remark Observe that in dimension n = 1, PSH(,ω) = DMA(,ω) = DMA(,ω). The latter equality follows easily by integrating by parts and using the monotone convergence theorem. We expect that the equality DMA(,ω) = DMA(,ω) continues to hold in higher dimension The class E(,ω). We consider here { E(,ω) = ϕ PSH(,ω)/µ(ϕ)() = ω n }. Equivalently, ϕ E(,ω) if and only if (ω + dd c ϕ j ) n ({ϕ j}) 0. This class of functions was studied in [GZ2], where it was shown that (i) E(,ω) DMA(,ω); (ii) E(,ω) is convex and stable under maximum; (iii) E(,ω) is the largest subclass of DMA(,ω) on which the comparison principle holds. The rough idea is that a ω-psh function ϕ should belong to E(,ω) iff it belongs to DM A(, ω) and its Monge-Ampère measure does not charge pluripolar sets. This is partly justified by the following result: Proposition A function ϕ belongs to E(,ω) if and only if it belongs to DMA(,ω) and (ω + dd c ϕ) n does not charge pluripolar sets. Proof. The inclusion E(,ω) DMA(,ω) will follow from Theorem 2.1 below. Assume conversely that ϕ DMA(,ω) is such that (ω + dd c ϕ) n does not charge pluripolar sets. Then µ(ϕ) = 1 {ϕ> } (ω + dd c ϕ) n = (ω + dd c ϕ) n has full mass, hence ϕ E(,ω).

10 DEFINITION OF THE COMPLE MONGE-AMPÈRE OPERATOR 9 Remark The previous characterization of E(, ω) is related to the question of uniqueness of solutions to the equation (ω + dd c ) n = µ. Indeed assume ϕ DMA(,ω) is such that µ := (ω + dd c ϕ) n does not charge pluripolar sets. It was shown in [GZ2] that there exists E(,ω) so that µ = (ω+dd c ) n. It is expected that the solution is unique up to an additive constant. If such is the case, then ϕ + constant belongs to E(,ω). 2. Finite energy classes In this section we establish further properties of the class E(, ω) and we consider energy classes with respect to a fixed current T Weighted energies. Let T be a positive closed current of bidimension (m,m) on. In the sequel T will be of the form T = (ω + dd c u m+1 ) (ω + dd c u n ), where u j PSH(,ω) L (). Let χ : R R be a convex increasing function such that χ( ) =. Following [GZ2] we consider { } E χ (T,ω) := ϕ PSH(,ω)/ sup ( χ) ϕ j ωϕ m j T < +, j where ϕ j := max(ϕ, j) denote the canonical approximants of ϕ. We let the reader check that [GZ2] can be adapted line by line, showing that the Monge-Ampère measure (ω + dd c ϕ) m T is well defined for ϕ E χ (T,ω) and that χ ϕ L 1 ((ω + dd c ϕ) m T). When m = n, i.e. T = [] is the current of integration along, then the classes E χ (T,ω) = E χ (,ω) yield the following alternative description of E(,ω) (see [GZ2, Proposition 2.2]): E(,ω) = { E χ (,ω), E χ (,ω) := ϕ E(,ω) : χ ϕ L 1 (ωϕ) n }, χ W where W = {χ : R R /χ convex, increasing, χ( ) = }. We set E(T,ω) := χ W E χ (T,ω). Theorem 2.1. Fix χ W. Let ϕ j be a sequence of ω-psh functions decreasing towards ϕ E χ (T,ω). Then ϕ j E χ (,ω) and ( χ) ϕ j (ω + dd c ϕ j ) m T ( χ) ϕ(ω + dd c ϕ) m T. Moreover for any u PSH(,ω) L (), u(ω + dd c ϕ j ) m T u(ω + dd c ϕ) m T. Proof. When ϕ j is the canonical sequence of approximants, the theorem follows by a similar argument as in [GZ2, Theorem 2.6]. Moreover, it also follows from [GZ2, Theorem 2.6] that the first convergence in the statement holds for an arbitrary decreasing sequence, if there exists a weight function χ such that χ = o( χ) and ϕ E χ (T,ω). It turns out that such a weight always exists. Indeed, since χ(ϕ) L 1 (ω m ϕ T), it follows from standard measure theory arguments that there exists a convex increasing function h : R + R + such that lim t + h(t)/t = + and h( χ(ϕ)) L 1 (ω m ϕ T).

11 10 DAN COMAN, VINCENT GUEDJ, AND AHMED ZERIAHI Note that we can assume that h has polynomial growth, which implies that ϕ E χ (T,ω), where χ := h( χ) W W + M (see [GZ2] for the definition of this latter class). We now prove the second assertion. Set ϕ k j := max(ϕ j, k) and ϕ k := max(ϕ, k). Then uωϕ m j T uωϕ m T = uωϕ m j T uω m T ϕ k j + uω m T uω m ϕ k ϕ T j k + uω m ϕ T uω m k ϕ T. It suffices to prove that uωm ϕ j T uωm ϕ k j T converges to 0 as k + uniformly in j. This is a consequence of the following estimate, uωϕ m j T uω m T ϕ k j 2 u L () ( χ(ϕ j ))ωϕ m χ( k) j T. The latter integral is uniformly bounded since ϕ E χ (T,ω). Let E 1 (ω n p,ω) denote the class E χ (T,ω) for T = ω n p and χ(t) = t. Corollary 2.2. Let 1 p n 1 and let {ϕ j } be a decreasing sequence of ω-psh functions converging to ϕ E 1 (ω n p,ω). Then for any u PSH(,ω) L () we have lim j + uω p+1 ϕ j ω n p 1 = In particular, E 1 (ω n p,ω) DMA p+1 (,ω). uω p+1 ϕ ω n p 1. Proof. For simplicity, we consider the case p = n 1. The general case follows along the same lines. We want to prove that if ϕ j ց ϕ E 1 (ω,ω) then for any u PSH(,ω) L () we have lim uωϕ n j + j = uωϕ. n Observe that since ωϕ n j ωϕ n in the weak sense of Radon measures on, the above equality holds when u is continuous on. We claim that for any E 1 (ω,ω), we have (1) ( u)ω n = ( u)ω ω n 1 + ( )ω u ω n 1 + ω ω n 1. Observe that this identity is just integration by parts which clearly holds when u is a smooth test function on. The identity also holds when u, PSH(,ω) L () (see [GZ2]). Fix u a bounded ω-psh function and set j := max{, j} for j N. Applying (1) to u and j, it follows immediately that E 1 (ω u,ω) = E 1 (ω,ω). Hence the corollary follows at once from Theorem 2.1, as soon as (1) is established for E 1 (ω,ω).

12 DEFINITION OF THE COMPLE MONGE-AMPÈRE OPERATOR 11 To prove (1) we can assume that u 0. Note that ω n j ω n weakly. Using the upper semicontinuity of u and applying (1) to u and j, it follows from Theorem 2.1 that (2) ( u)ω n lim inf ( u)ω n j + j = ( u)ω ω n 1 + ( )ω u ω n 1 + ω ω n 1. Next let u j ց u be a sequence of smooth ω-psh functions (see [BK]). Note that u j ω n 1 uω n 1 in the weak sense of currents, hence ω uj ω n 1 ω u ω n 1 weakly in the sense of measure. Applying (1) to u j and, it follows by monotone convergence and the upper semicontinuity of that (3) ( u)ω n = ( u)ω ω n 1 + lim ( )ω uj ω n 1 j + + ω ω n 1 ( u)ω ω n 1 + ( )ω u ω n 1 + ω ω n 1. The identity (1) now follows from the inequalities (2) and (3). Corollary 2.3. E(,ω) DMA n (,ω). Proof. Fix ϕ E(,ω). Then there exists χ W such that ϕ E χ (,ω). Recall now that for any 1 p n 1, E χ (,ω) E χ (ω p,ω), as follows from simple integration by parts (see [GZ2]). We can thus apply Theorem 2.1 several times to conclude Twisted Monge-Ampère operators. We show here that certain Monge-Ampère operators with weights are always well defined. Theorem 2.4. Let η : R R + be a continuous function with η( ) = 0 and 1 l n. Let ϕ PSH(,ω) and {ϕ j } be any sequence of bounded ω-psh functions decreasing to ϕ. Then the twisted Monge-Ampère currents M l η (ϕ j) := η ϕ j (ω + dd c ϕ j ) l converge weakly towards a positive current Mη(ϕ), l which is independent of the sequence {ϕ j }. The twisted Monge-Ampère operator Mη l is well defined on PSH(,ω) and is continuous along any decreasing sequences of ω-psh functions. If ϕ DMA l (,ω) then M l η(ϕ) = η ϕ(ω + dd c ϕ) l. Proof. It is a standard fact that the operator M l η(ϕ) = η ϕ(ω + dd c ϕ) l is well defined and continuous under decreasing sequences in the subclass of bounded ω-psh functions [BT3]. As it was observed in Remark 1.10, the sequence k 1 {ϕ> k} (ω + dd c max(ϕ, k)) l

13 12 DAN COMAN, VINCENT GUEDJ, AND AHMED ZERIAHI is an increasing sequence of positive currents of total mass bounded by ωn. Hence this sequence converges in a strong sense to a positive current µ l (ϕ) which puts no mass on pluripolar sets and satisfies (see [GZ2, Theorem 1.3]) (4) 1 {ϕ> k} (ω + dd c max(ϕ, k)) l = 1 {ϕ> k} µ l (ϕ). Since η ϕ is a bounded positive Borel function on, we can define a positive current on M l η (ϕ) := η(ϕ)µl (ϕ). Note that when ϕ is bounded then µ l (ϕ) = ωϕ l hence Ml η (ϕ) = η(ϕ) ωl ϕ. To prove the continuity of this operator, let ϕ j be any sequence of ω-psh functions decreasing to ϕ, and set ϕ k j := max(ϕ j, k), ϕ k := max(ϕ, k) for j,k N. Since the forms of type (n l,n l) on have a basis consisting of forms hψ, where h are smooth functions and Ψ are positive closed forms, it suffices to prove that η(ϕ j )µ l (ϕ j ) Ψ η(ϕ)µ l (ϕ) Ψ weakly as measures on. For simplicity, we may assume that Ψ = ω n l. Then hη(ϕ j )µ l (ϕ j ) ω n l hη(ϕ)µ l (ϕ) ω n l = hη(ϕ j )µ l (ϕ j ) ω n l hη(ϕ k j )µ l (ϕ k j ) ω n l + hη(ϕ k j )µl (ϕ k j ) ωn l hη(ϕ k )µ l (ϕ k ) ω n l + hη(ϕ k )µ l (ϕ k ) ω n l hη(ϕ)µ l (ϕ) ω n l. We claim that the first term in the sum tends to 0 uniformly in j as k +. Indeed, we have by (4) hη(ϕ j )µ l (ϕ j ) ω n l hη(ϕ k j )µl (ϕ k j ) ωn l h η(ϕ j )µ l (ϕ j ) ω n l + h η(ϕ k j )µ l (ϕ k j ) ω n l {ϕ j k} 2 η( k) h ω n, {ϕ j k} where η( k) := sup{η(s) : s k} 0 as k +. In the same way we see that the last term tends to 0 as k +. Now for fixed k, it follows from the uniformly bounded case that the second term tends to 0 as j +. The desired continuity result follows. Theorem 2.5. Let η : R R + be an increasing function of class C 1 and 0 l n 1. Let ϕ PSH(,ω) and {ϕ j } be any sequence of bounded ω-psh functions decreasing to ϕ. Then the currents S l η (ϕ j) := η ϕ j dϕ d c ϕ j (ω + dd c ϕ j ) n l 1 converge weakly towards a positive current Sη l (ϕ) which is independent of the sequence {ϕ j }. The operator Sη l is well defined on PSH(,ω) and is continuous along any decreasing sequences of ω-psh functions.

14 DEFINITION OF THE COMPLE MONGE-AMPÈRE OPERATOR 13 Proof. By subtracting a constant, we may assume that η( ) = 0. Fix l and ϕ PSH(,ω). We can assume that ϕ < 0 and η(0) = 1. Observe that if θ : R R is any C 1 function with 0 θ 1 then for any u PSH(,ω) L () and any positive closed current R on of bidimension (1,1), we have d(θ(u)d c u R) = θ (u)du d c u R + θ(u)ω u R θ(u)ω R. Using Stokes formula and the fact that 0 θ(u) 1, we get the following uniform bound (5) θ (u)du d c u R ω R. Assume that ϕ 0 and set ϕ k := max(ϕ, k) for k 0. We want to prove that the sequence of positive currents 1 {ϕ> k} η (ϕ k )dϕ k d c ϕ k ω n l 1 ϕ k converges to a positive current of bidimension (l,l) which will be denoted by S l η(ϕ). It follows from the quasi-continuity of bounded psh functions [BT4] that for j k 0, 1 {ϕ> k} η (ϕ k )dϕ k d c ϕ k ω n l 1 ϕ k This implies that = 1 {ϕ> k} η (ϕ j )dϕ j d c ϕ j ω n l 1 ϕ j. 1 {ϕ> k} η (ϕ k )dϕ k d c ϕ k ω n l 1 ϕ k is an increasing sequence of positive currents of bidimension (l, l), with uniformly bounded mass ωn by (5). Therefore it converges in a strong sense to a positive current S l η(ϕ) on which satisfies the following equation (6) 1 {ϕ> k} S l η (ϕ) = 1 {ϕ> k}η (ϕ k )dϕ k d c ϕ k ω n l 1 ϕ k. Observe that if ϕ is bounded then Sη l(ϕ) = η (ϕ)dϕ d c ϕ ωϕ n l 1. Now we want to prove that for any decreasing sequence {ϕ j } which converges to ϕ, the currents Sη l(ϕ j) converge to the current Sη l (ϕ) in the sense of currents on. As before it is enough to prove that the sequence of positive measures Sη(ϕ l j ) ω l converges weakly to the positive measure Sη(ϕ) l ω l on. Let h be a continuous function on. Proceeding as in the proof of the previous theorem, it suffices to show that hsη l (ϕ j) ω l hη (ϕ k j )dϕk j dc ϕ k j ωn l 1 ω l 0 ϕ k j as k, uniformly in j, where ϕ k j = max(ϕ j, k). Indeed, if θ = η then η = 2 η θ, so it follows at once from the definition of S l η that S l η(ϕ j ) = 2 η(ϕ j )S l θ (ϕ j). Since η (ϕ k j ) 2 η( k)θ (ϕ k j ) on the set {ϕ j k}, we

15 14 DAN COMAN, VINCENT GUEDJ, AND AHMED ZERIAHI have by (6) that hsη l (ϕ j) ω l hη (ϕ k j )dϕk j dc ϕ k j ωn l 1 ω l ϕ k j 2 η( k) h Sθ l (ϕ j) ω l {ϕ j k} +2 η( k) h θ (ϕ k j)dϕ k j d c ϕ k j ω n l 1 ω l ϕ {ϕ j k} k j 4 η( k) h ω n, which tends to 0 as k + uniformly in j Attenuation of singularities. The goal of this section is to show that a very small attenuation of singularities transforms a function ϕ PSH(,ω) into a function χ ϕ E(,ω). In particular most functions in the class E(,ω) do not belong to the local domain of definition DMA loc (,ω), which we consider in Section 3. Let χ : R R be a convex increasing function of class C 2 and such that χ( ) =, χ ( ) = 0 and χ ( 1) 1. Define for x 1 η l (x) = (χ (x)) l (1 χ (x)) n l, ε l (x) = χ (x) 0 t n l 1 (1 t) l dt. Corollary 2.6. If ϕ PSH(,ω), ϕ 1, then χ ϕ E(,ω) and n ( ) n 1 n (7) ωχ ϕ n = ( ) n 1 Mη l l l (ϕ) ω n l + n Sε l l l (ϕ) ω l, l=0 where MA l η l and S l ε l are the operators defined in Theorems 2.4 and 2.5. Proof. Note first that if ϕ is bounded then l=0 (ω + dd c χ ϕ) n = [ (1 χ (ϕ))ω + χ (ϕ)ω ϕ ] n + nχ (ϕ)dϕ d c ϕ [ (1 χ (ϕ))ω + χ (ϕ)ω ϕ ] n 1, which equals the measure from (7). In the general case, let ϕ s = max(ϕ,s), s < 0. As s we have, by the proofs of Theorems 2.4 and 2.5, that ( ε l (ϕs )dϕ s d c ϕ s ωϕ n l 1 s ωl) ({ϕ s}) 0, ( η l (ϕ s )ωϕ l s ωn l) ({ϕ s}) 0. Since max(χ(ϕ), j) = χ(ϕ s j ), where s j = χ 1 ( j) as j +, we conclude by formula (7) applied to ϕ s j that (ω + dd c max(χ(ϕ), j)) n ({χ(ϕ) j}) 0, so χ ϕ E(,ω). Moreover, using the continuity of the operators in Theorems 2.4 and 2.5, it follows that (7) holds for ϕ. Examples ) For χ(t) = ( t) p, Corollary 2.6 shows that ( ϕ) p E(,ω) DMA(,ω) for all p < 1, although it usually does not have gradient in

16 DEFINITION OF THE COMPLE MONGE-AMPÈRE OPERATOR 15 L 2 () when p 1/2. Functions with less attenuated singularities can be obtained by considering for instance χ(t) = t/log(m t). 2) Recall from Example 1.4 that there are functions ϕ / DMA(,ω) such that the current ω ϕ has support in {ϕ = }. The smoothing effect of the composition with χ is quite striking: indeed, by Corollary 2.6 and Proposition 4.1, (ω + dd c χ ϕ) n = [1 χ (ϕ)] n ω n + n[1 χ (ϕ)] n 1 χ (ϕ)dϕ d c ϕ ω n 1 is a measure with density in L 1 (,ω n ). 3. The local vs. global domains of definition Cegrell found in [C2] the largest class of psh functions on a bounded hyperconvex domain on which the Monge-Ampère operator is well defined, stable under maximum and continuous under decreasing limits. Later, Blocki gave in [Bl2] a complete characterization of the domain of definition of the Monge-Ampère operator on any open set in C n, n 2. For an open subset U C n, the domain of definition D(U) PSH(U) of the Monge-Ampère operator on U is given by (n 1) local boundedness conditions on weighted gradients [Bl2]. In particular, for n = 2, D(U) = PSH(U) W 1,2 loc (U), where W 1,2 loc (U) is the Sobolev space of functions in L2 loc (U) with locally square integrable gradient [Bl1]. We describe here the class of ω-psh functions on which locally belong to the domain of definition of the Monge-Ampère operator. As we shall see, it is smaller than the global domain DMA(,ω). Definition 3.1. Let DMA loc (,ω) be the set of functions ϕ PSH(,ω) such that locally, on any small open coordinate chart U, the psh function ϕ U + ρ U D(U), where ρ U is a psh potential of ω on U. The goal of this section is to establish the following: Theorem 3.2. We have Moreover DMA loc (,ω) = 1 p n 1 E p (ω p,ω). DMA loc (,ω) E 1 (ω,ω) DMA n (,ω). Here E p (ω p,ω) denotes the class E χ (T,ω) for T = ω p and χ(t) = ( t) p. Proof. We first show that DMA loc (,ω) = 1 p n 1 Ep (ω p,ω). By [Bl2], a ω-psh function ϕ DMA loc (,ω) if and only if (8) sup ( ϕ j ) p 1 dϕ j d c ϕ j ωϕ n p 1 j ω p < +, 1 p n 1, j where {ϕ j } is any sequence of bounded ω-psh functions decreasing to ϕ on. For our first claim we need to prove that (8) is equivalent to the following (9) sup ( ϕ j ) p ωϕ n p j ω p < +, 1 p n 1, j

17 16 DAN COMAN, VINCENT GUEDJ, AND AHMED ZERIAHI This follows by integration by parts. Indeed for PSH(,ω) L (), 1 and 1 p n 1, we have ( ) p ω n p ω p = ( ) p ω n p 1 ω p+1 + p By iterating the above formula we get ( ) p ω n p ω p = ( ) p ω n + p n p 1 k=0 ( ) p 1 d d c ω n p 1 ω p. ( ) p 1 d d c ω n p k 1 ω p+k. This yields (8) (9), so DMA loc (,ω) = 1 p n 1 Ep (ω p,ω). By Corollary 2.2 we have E 1 (ω p,ω) DMA n p+1 (,ω), for any 1 p n 1. Therefore the class { } E 1 (ω,ω) := ϕ PSH(,ω)/ sup j ϕ j (ω + dd c ϕ j ) n 1 ω < + is contained in DMA n (,ω) (here ϕ j := max(ϕ, j) denote as usually the canonical approximants), since E 1 (ω,ω) = E 1 (ω p,ω). 1 p n 1 This equality also shows that DMA loc (,ω) E 1 (ω,ω). Remark 3.3. A class similar to E 1 (ω,ω) has been very recently considered by Y. ing in []. Note that in dimension n = 2, the class E 1 (ω,ω) = DMA loc (,ω) is simply the set of ω-psh function whose gradient is in L 2 (). However when n 3, the class E 1 (ω,ω) is strictly larger than DMA loc (,ω), as the following example shows: Example 3.4. Observe that E 1 (,ω) E 1 (ω,ω). We are going to exhibit an example of a function ϕ such that ϕ E 1 (,ω), but ϕ / L 2 (ω ϕ ω n 1 ). This will show that ϕ / DMA loc (,ω) when n 3. Assume = P n 1 P 1 and ω(x,y) := α(x) + β(y), where α is the Fubini-Study form on P n 1 and β is the Fubini-Study form on P 1. Fix u PSH(P n 1,α) C (P n 1 ) and v E(P 1,β). The function ϕ defined by ϕ(x,y) := u(x) + v(y) for (x,y) belongs to E(,ω). Moreover ω ϕ = α u + β v and for any 1 l n, we have Therefore ω l ϕ = αl u + lαl 1 u β v. ϕ L p (ω n ϕ) v L p (β v ) ϕ L p (ω l ϕ ω n l ). Thus choosing v L 1 (ω v )\L 2 (ω v ), we obtain an example of a ω-psh function ϕ such that ϕ E 1 (,ω) E 1 (ω,ω) but ϕ / DMA loc (,ω).

18 DEFINITION OF THE COMPLE MONGE-AMPÈRE OPERATOR 17 We finally observe that there are functions in DMA loc (,ω) which do not belong to the class E(, ω), since the latter cannot have positive Lelong numbers. Lemma 3.5. Let ϕ PSH(,ω) be such that ϕ clog dist(,p), for some c > 0 and p. Then ϕ DMA loc (,ω), and ϕ E(,ω) if and only if ν(ϕ,p) = 0. Proof. If p PSH(,ω) is a function comparable to clog dist(,p), then by Proposition 4.6, ϕ, p DMA loc (,ω). We can assume without loss of generality that p ϕ 0. Note that the positive Radon measure ω ϕ ω n 1 p is well defined on and has a Dirac mass at p if and only if ν(ϕ,p) > 0 (see [D2]). It follows from the proof of Proposition 1.2 that if ν(ϕ,p) > 0, then ωϕ n has a Dirac mass at p, hence ϕ / E(,ω). If ν(ϕ,p) = 0, then we can find a convex increasing function χ : R R such that χ( ) = and ( χ) p ω ϕ ω n 1 < +. Using Stokes theorem (in the spirit of the p fundamental inequality in [GZ2]), it follows that ( χ) ϕωϕ n 2 n 1 ( χ) p ω ϕ ω n 1 < +, p hence ϕ E χ (,ω) E(,ω). 4. Sobolev classes 4.1. Weighted gradients. Let χ : R R be a convex increasing function of class C 2. If ϕ PSH(,ω) is smooth then (10) ω + dd c χ ϕ = χ ϕdϕ d c ϕ + χ ϕω ϕ + (1 χ ϕ)ω. So if χ ( 1) 1 and ϕ 1, then χ ϕ PSH(,ω). It is well-known that (ω-)psh functions have gradient in L 2 ε loc for all ε > 0 [Hö], but in general not in L 2 loc. The previous computation indicates that a weighted version of the gradient is in L 2 (). We denote by W 1,2 (,ω) the set of functions ϕ PSH(,ω) whose gradient is square integrable. Since ω is Kähler, we can in fact define, for ϕ PSH(,ω), the function ϕ = ϕ ω a.e. on by ϕ 2 := dϕ d c ϕ ω n 1 /ω n. Note that ϕ W 1,2 (,ω) if and only if ϕ L 2 (,ω n ). Proposition 4.1. Let χ : R R be a convex increasing function of class C 2. Then for every ϕ PSH(,ω), χ ϕ ϕ 2 ω n supχ (ϕ) ω n. In particular, if ϕ 1 and 0 < p < 1/2, then ( ϕ) p W 1,2 (,ω). Proof. Let M = sup χ (ϕ) and ϕ j := max(ϕ, j) for j N. It follows from (10) that χ ϕ j ϕ j 2 ω n = χ ϕ j dϕ j d c ϕ j ω n 1 M ω n.

19 18 DAN COMAN, VINCENT GUEDJ, AND AHMED ZERIAHI This shows that the sequence f j := χ (ϕ j ) ϕ j 2 is bounded in L 1 (,ω n ). Since ϕ j ց ϕ, we use [Hö, Theorem 4.1.8] to conclude, after taking a subsequence, that f j χ (ϕ) ϕ 2 a.e. on. The inequality in the statement now follows from Fatou s lemma. For our second claim, set ϕ a := ( ϕ) a, 0 < a < 1. Applying (10) with χ(t) = ( t) a yields ϕ a PSH(,ω) and ( ϕ) a 2 dϕ d c ϕ ω n 1 1 a(1 a) If p = a/2 then dϕ p d c ϕ p = p 2 ( ϕ) 2p 2 dϕ d c ϕ, hence dϕ p d c ϕ p ω n 1 p ω n. 2(1 2p) Note that using χ(t) = t/log(m t), for M large enough, one can improve the previous weighted L 2 -bound to ( ϕ) 1 [log(m ϕ)] 2 ϕ 2 ω n < +. Observe also that this cannot be improved much farther: in the case when ω ϕ is the current of integration along a hypersurface, ( ϕ) 1/2 does not have square integrable gradient. Next, we show that functions from the class DMA loc (,ω) satisfy stronger weighted gradient boundedness conditions. Proposition 4.2. If ϕ DMA loc (,ω), ϕ 1, then ( ϕ) n 2 dϕ d c ϕ ω n 1 1 ( ϕ) n 1 ω ϕ ω n 1. n 1 In particular, DMA loc (,ω) W 1,2 (,ω). Proof. If ϕ j := max(ϕ, j), j N, then integrating by parts we get (n 1) ( ϕ j ) n 2 dϕ j d c ϕ j ω n 1 ( ϕ j ) n 1 ω ϕj ω n 1. The sequence of functions f j := ( ϕ j ) n 2 ϕ j 2 is thus uniformly bounded in L 1 (,ω n ). Since ϕ j ց ϕ, the conclusion follows as in the previous proof by [Hö, Theorem 4.1.8] and by Fatou s lemma Blowing up and down. We saw in Section 2 that there are many functions ϕ DMA(,ω) whose gradient does not belong to L 2 (). This condition, although the best possible in the local two-dimensional theory, is thus not the right one from the global point of view. We show here that this condition does not behave well under a birational change of coordinates. For simplicity, and without loss of generality, we restrict ourselves to the two-dimensional local setting. Let π : Y B be the blow up at the origin of a ball B C 2, and let E be the exceptional divisor. ( ) Lemma 4.3. If δ > 0 then π L 2+δ loc (B) L 1 loc (Y ), but ( π L 2 loc (B)) L 1 loc (Y ). ω n.

20 DEFINITION OF THE COMPLE MONGE-AMPÈRE OPERATOR 19 Proof. Let z = (x, y) B, and fix a coordinate chart in Y given by (s, t) (s,st,[1 : t]) Y. Then π(s,t) = (x,y), x = s, y = st. For fixed positive constants M and a, let = {(s,t) : s < a, t < M}, K = π( ) = {(x,y) : x < a, y < M x }. Let f L 2+δ loc (B), and f = π f, so f(s,t) = f(s,st). Note that the function z x 2 is in L 2 ǫ (K) for every ǫ > 0. We take γ = δ/(1 + δ) and apply Hölder s inequality with conjugate exponents 2 + δ and 2 γ: f = K f x 2 ( K f 2+δ ) 1/(2+δ) ( K 1 x 4 2γ ) 1/(2 γ). Hence f L 1 loc (Y ). The function f(z) = 1/( z 2 log z ) is in L 2 loc (B), but f(s,t) C s 2 ( log s + 1) 1 on, for some C > 0. So f L 1 ( ). ( ) In dimension n, a similar proof shows that π L n+δ loc (B) L 1 loc (Y ). Example 4.4. π ( PSH W 1,4 loc (B) ) W 1,2 loc (Y ). Indeed, let u α (z) = ( log z ) α, 0 < α < 1. One checks easily that u α W 1,4 (B) if α < 3/4. Let ũ α = π u α. With the notation from the proof of Lemma 4.3, we have for s small ( ũ α (s,t) = log s log ) α 1 + t 2 ũ α, s (s,t) C s ( log s ) 1 α, for some constant C > 0. So ũ α W 1,2 loc ( ) if α 1/2. Note that if u PSH W 1,4+δ loc (B), δ > 0, then, by the Sobolev embedding theorem, u is continuous and so is π u. Hence π u W 1,2 loc (Y ) (see [Bl1]). The previous example shows that the condition ϕ L 2 () does not behave well under blow-up. We show it behaves well under blowing down. Proposition 4.5. Let T be a positive closed current of bidegree (1,1) on B, and let R = π T ν[e], where ν = ν(t,0). If R has psh potentials in W 1,2 1,2 loc (Y ), then T has psh potentials in Wloc (B). Recall that any positive closed (1,1)-current in the blow up of B at the origin writes R = π T ν[e] + λ[e], where T is a positive closed current in B, ν = ν(t,0) is the Lelong number of T at the origin, and λ 0. Clearly R cannot have potentials in W 1,2 loc if λ > 0. Proof. Let u be a psh potential of T on B. We only have to check that the gradient of u is L 2 -integrable in a neighborhood of the origin. Using the notation from the proof of Lemma 4.3, a psh potential for R on is v(s,t) = u(s,st) ν log s. Hence on K we have u(x,y) = v(x,y/x) + ν log x. Therefore for almost all (x,y) K u v (x,y) = x s (x,y/x) y v x 2 t (x,y/x) + ν 2x, u y (s,t) = 1 v x t (x,y/x).

21 20 DAN COMAN, VINCENT GUEDJ, AND AHMED ZERIAHI Now K K y 2 x 4 1 x 2 v t (x,y/x) v t (x,y/x) 2 2 = = t 2 v t (s,t) v t (s,t) 2. Since the function (x,y) 1/x is in L 2 (K), we conclude that the partial derivatives of u belong to L 2 (K). A similar argument shows that the partial derivatives of u are in L 2 (K ), where K = {(x,y) : y < a, x < M y } Compact singularities. We give here an important class of functions in DMA loc (,ω). Let D be a divisor on and set L D (,ω) = {ϕ PSH(,ω)/ϕ is bounded near D}. Thus the singularities of ϕ L D (,ω) are constrained to a compact subset of \ D. When D = H is a hyperplane of the complex projective space = P n, the set L D (,ω) is in one-to-one correspondence with the Lelong class L + (C n ) of psh functions u in C n such that u(z) log z is bounded near infinity. So these are the ω-psh analogues of the psh functions with compact singularities introduced by Sibony in [Sib] (see also [D2]). Proposition 4.6. If D is an ample divisor then L D (,ω) DMA loc(,ω). Proof. Fix ϕ L D (,ω) and let V be a small neighborhood of D, so that ϕ is bounded in V. We can assume that ϕ 0 and ωn = 1. Let ω be a smooth semi-positive closed (1,1) form in the cohomology class of D, such that ω 0 in \V. Since D is ample, ω is cohomologous to a Kähler form ω 0. For simplicity, we assume ω 0 = ω (otherwise we bound ω Cω 0 in all arguments below). Hence ω = ω +dd c χ, where χ is a smooth function on, chosen to be either negative or positive, as we like. We assume here χ 0, and we first observe that ϕ L 1 (ω ϕ ω n 1 ): ( ϕ)ω ϕ ω n 1 = ( ϕ)ω ϕ ω ω n 2 + ( ϕ)ω ϕ dd c χ ω n 2 ϕ L (V ) ω ϕ ω ω n 2 + χω ϕ ( dd c ϕ) ω n 2 ϕ L (V ) + χ L () < +, since dd c ϕ ω, χω ϕ 0 and ω ϕ ω ω n 2 = ωn = 1. It follows that the positive current ωϕ 2 := ω ϕ ω+dd c (ϕω ϕ ) is well defined. We can thus show by a similar argument that ϕ L 1 (ωϕ 2 ωn 2 ), so that ωϕ 3 is also well defined, and so on. At last, we show that ϕ L 1 (ωϕ n 1 ω). We now prove that ϕ 2 L 1 (ωϕ n 2 ω 2 ). We assume here that χ 0. Observe that dd c ϕ 2 = 2dϕ d c ϕ 2ϕdd c ϕ 2( ϕ)ω ϕ, therefore ϕ 2 ω n 2 ϕ ω 2 ϕ 2 ω n 2 ϕ ω ω + 2 χ L () 2, ( ϕ)ω n 1 ϕ ω. The integrals are finite because ω has support in V, where ϕ is bounded, and because ϕ L 1 (ωϕ n 1 ω). Thus ϕ 2 L 1 (ωϕ n 2 ω 2 ). Similar integration by parts allows us to show that ϕ 3 L 1 (ωϕ n 3 ω 3 ), by using ϕ 2 L 1 (ωϕ n 2 ω 2 ). Continuing like this, we see that ϕ DMA loc (,ω).

22 DEFINITION OF THE COMPLE MONGE-AMPÈRE OPERATOR 21 Proposition 4.6 shows that PSH(,ω) L loc ( \ {p}) DMA loc(,ω). Indeed, if is projective, one can find for each p a divisor D p. In the general case, it follows from the proof that one only needs to construct a smooth semi-positive form ω cohomologous to a Kähler form, and such that ω 0 near p. This can be achieved on any Kähler manifold. 5. Concluding remarks In this section we restrict our attention to the two-dimensional case. In the local setting of an open subset U C 2, a psh function u on U belongs to the domain of definition D(U) if and only if the gradient of u is locally square integrable (W 1,2 ) on U [Bl1]. Such characterization allows to prove important properties of D(U), such as convexity and stability under taking the maximum of elements of D(U) with arbitrary psh functions (see [Bl1]). Let be a compact Kähler surface and ω be a Kähler form on. In order to prove further properties of DMA(,ω) it would be useful to obtain equivalent characterizations for this domain. We present here the connection in certain cases between DMA(,ω) and certain energy classes Direct sums. Let = P 1 P 1 and ω = ω 1 + ω 2, where ω i = π i ω is the pull-back of the Fubini-Study form ω of P 1 by the projection onto the i th factor, i = 1,2. Proposition 5.1. Let ϕ P SH(, ω) be of the form ϕ(x,y) = u(x) + v(y), where u,v PSH(P 1,ω ). Then (i) ϕ E 1 (ω,ω) u,v E 1 (P 1,ω ) ϕ E 1 (,ω); (ii) ϕ E(ω,ω) u,v E(P 1,ω ) ϕ E(,ω); (iii) ϕ DMA(,ω) ϕ E(ω,ω). Proof. (i) Note that ϕω ϕ ω = uω 1,u ω 2 + uω 1 ω 2,v + v ω 1 ω 2,v + v ω 1,u ω 2, where the integrals uω 1 ω 2,v and v ω 1,u ω 2 are always finite by Fubini s theorem. We use here the obvious notations ω 1,u := (ω 1 + dd c u)(x) and ω 2,v := (ω 2 + dd c v)(y). This shows that ϕ E 1 (ω,ω) if and only if u,v E 1 (P 1,ω ). If u,v E 1 (P 1,ω ) then ϕ W 1,2, hence ϕ DMA(,ω) and ωϕ 2 = 2ω 1,u ω 2,v. By Fubini s theorem ϕωϕ 2 = 2 uω 1,u + 2 v ω 2,v, P 1 P 1 so ω 2 ϕ ({ϕ = }) = 0. We conclude that ϕ E(,ω), hence ϕ E1 (,ω). This formula also shows that ϕ E 1 (,ω) implies that u,v E 1 (P 1,ω ).

23 22 DAN COMAN, VINCENT GUEDJ, AND AHMED ZERIAHI (ii) and (iii). The equivalence ϕ E(ω,ω) u,v E(P 1,ω ) is a direct consequence of the following equality: ω ϕ ω = ω 1,u + ω 2,v. {v= } {ϕ= } {u= } We show next that u,v E(P 1,ω ) = ϕ E(,ω). Let ϕ j,u j,v j be the canonical approximants and set E j = {u > j} {v > j}. Since the bounded ω-psh functions ϕ 2j and u j + v j coincide on the plurifine open set E j, we have by [BT4] that 1 Ej ω 2 ϕ 2j = 2 1 Ej ω uj ω vj = 2 1 {u> j} ω uj 1 {v> j} ω vj. Note that the product measure ω u ω v puts full mass 2 on the set {u > } {v > }, and that 1 {ϕ> 2j} ωϕ 2 2j 1 Ej ωϕ 2 2j. This shows that the sequence of measures 1 {ϕ> 2j} ωϕ 2 2j increases to a measure with total mass 2, hence ϕ E(,ω). We conclude the proof by showing that ϕ DMA(,ω) implies that u,v E(P 1,ω ). Assume for a contradiction that v E(P 1,ω ). We may assume that there exists a compact K {v = } {[1 : w] : w C} so that dd c V = a > 0, V (w) := log 1 + w 2 + v([1 : w]). K We use here (bi)homogeneous coordinates [z 0 : z 1 ],[w 0 : w 1 ] on. Let u j be smooth ω -psh functions decreasing to u on P 1, and set U j (z) = log 1 + z 2 + u j ([1 : z]), Φ j (z,w) = max(u j (z) + V (w),log z ζ j), where ζ C. The functions Φ j yield functions ϕ j DMA(,ω) decreasing to ϕ, hence (dd c Φ j ) 2 ωϕ 2 on C 2. We will show that ωϕ 2 a. {z=ζ} K Since ζ is arbitrary, we get a contradiction. For r > 0 let χ 1 0 be a smooth function such that χ 1 (z) = 1 in the closed disc E r of radius r centered at ζ and χ 1 is supported in the disc D r of radius 2r centered at ζ. For fixed j let N be an open neighborhood of K so that V (w) < log r j max Dr U j for w N, and let χ 2 0 be a smooth function supported in N such that χ 2 (w) = 1 on K. Let χ(z,w) = χ 1 (z)χ 2 (w). Since dd c χ dd c Φ j is supported on the open set {U j (z) + V (w) < log z ζ j} it follows that χ(dd c Φ j ) 2 = (U j + V )dd c χ dd c Φ j χdd c V dd c Φ j = Φ j χ 2 dd c χ 1 (z) dd c V (w). Note that U j (z) + V (w) < log r j < log z ζ j

24 DEFINITION OF THE COMPLE MONGE-AMPÈRE OPERATOR 23 on the support of χ 2 dd c χ 1, thus χ(dd c Φ j ) 2 (log z ζ j)χ 2 dd c χ 1 dd c V = χdd c log z ζ dd c V dd c V = a. We conclude that ωϕ 2 E r K lim sup (dd c Φ j ) 2 a, j E r K K and as r 0, that {z=ζ} K ω2 ϕ a The case = P 2. We produce now similar examples in the case of = P 2 with ω the Fubini-Study form. Let [t : z : w] denote the homogeneous coordinates and ϕ be a ω-psh function with Lelong number 1 at point p = [1 : 0 : 0]. It is easy to see that ϕ can be written as (11) ϕ[t : z : w] = 1 2 log z 2 + w 2 t 2 + z 2 + u[z : w] + w 2 where u is a ω -psh function on {t = 0} P 1. Proposition 5.2. If u / E(P 1,ω) then ϕ / DMA(P 2,ω). Proof. Suppose ϕ DMA(P 2,ω) and let ϕ j be functions defined by (11) with u replaced by u j, where u j are bounded ω-psh on P 1 decreasing to u. Then ϕ j decreases to ϕ and ωϕ 2 j = δ p is the Dirac mass at p, hence ωϕ 2 = δ p. On the other other hand, we are going to construct another sequence of functions j DMA(P 2,ω) decreasing to ϕ such that ω 2 j does not converge to δ p. Let K {t = 0} be a compact so that u = on K and ω u (K) = a > 0, and let j ([t : z : w]) = max ( ϕ([t : z : w]),log t 1 ) 2 log( t 2 + z 2 + w 2 ) j. Then j DMA loc (P 2,ω) by Proposition 4.6. One can show, as in the proof of Proposition 5.1, that ω 2 j (K) a, for all j. This contradicts that ωϕ 2 = δ p. For functions ϕ as in (11), it is easy to show that ϕ DMA loc (P 2,ω) if and only if u W 1,2 (P 1 ). It is an interesting question whether ϕ DMA(P 2,ω) if u E(P 1,ω ). A concrete example is ϕ α [t : z : w] := 1 2 log z 2 + w 2 t 2 + z 2 + w 2 [1 12 log z 2 z 2 + w 2 where 0 < α < 1. Then ϕ α / E(P 2,ω) since it has positive Lelong number at p, and ϕ α / DMA loc (P 2,ω) if α 1/2. It would be of interest to know if ϕ α DMA(P 2,ω) for some α [1/2,1]. ] α,