Non-pluripolar energy and the complex Monge–Ampère operator

1 Introduction

Let Ω be a domain in ℂ n , and let u ∈ PSH ⁡ ( Ω ) , i.e. let u be a plurisubharmonic (psh) function on Ω. If u is C 2 , then d ⁢ d c ⁢ u is a positive form, and the associated Monge–Ampère measure is defined as the top wedge power of this form with itself. This positive measure plays a fundamental role in pluripotential theory akin to the role played by the Laplacian in ordinary potential theory. If u is not C 2 , then d ⁢ d c ⁢ u is no longer a form but a current. As is well known the wedge product of currents is typically not well-defined, which raises the question whether it is still possible to define a Monge–Ampère measure for more general psh functions.

Bedford and Taylor [6, 7] solved this problem when u is (locally) bounded. Their idea was to define the Monge–Ampère measure ( d ⁢ d c ⁢ u ) n recursively. Assume that T is a closed positive current of bidegree ( j , j ) . Then T has measure coefficients and since u is bounded, uT is a well-defined current and thus so is d ⁢ d c ⁢ ( u ⁢ T ) . Bedford and Taylor proved that this current is closed and positive. They could then recursively define closed positive currents

The Monge–Ampère operators

have some essential continuity properties. Bedford and Taylor proved that if u ℓ is any sequence of psh functions decreasing to u, then ( d ⁢ d c ⁢ u ℓ ) p converges weakly to ( d ⁢ d c ⁢ u ) p .

We are interested in the situation when u is not locally bounded. Demailly [16, 17] showed that it is possible to extend the Bedford–Taylor Monge–Ampère operators to psh functions that are bounded outside “small” sets. Moreover, Błocki [10] and Cegrell [14] characterized the largest class 𝒟 ⁢ ( Ω ) of psh functions on which there is a Monge–Ampère operator u ↦ ( d ⁢ d c ⁢ u ) n that is continuous under decreasing sequences. For instance, functions in PSH ⁡ ( Ω ) that are bounded outside a compact set in Ω are in 𝒟 ⁢ ( Ω ) , see, e.g., [10]. On the other hand psh functions with analytic singularities, i.e., locally of the form u = c ⁢ log ⁡ | f | 2 + b , where c > 0 , f is a tuple of holomorphic functions, and b is locally bounded, are not in 𝒟 ⁢ ( Ω ) unless their unbounded locus is discrete, see [11] or Proposition 4.2.

To handle more singular psh functions Bedford and Taylor [8] introduced the notion of non-pluripolar Monge–Ampère currents. The idea is to capture the Monge–Ampère currents of the “bounded part” of u ∈ PSH ⁡ ( Ω ) . Note that for any ℓ , max ⁡ ( u , - ℓ ) is psh and locally bounded, and thus ( d ⁢ d c ⁢ max ⁡ ( u , - ℓ ) ) p is well-defined for any p. For each p ≤ n ,

is a form with measure coefficients. The existence of the limit follows from the fact that the Monge–Ampère operators on bounded psh functions are local in the plurifine topology, i.e., if u = v on a plurifine open set, then ( d ⁢ d c ⁢ u ) p = ( d ⁢ d c ⁢ v ) p on that set. One serious issue is that the measure coefficients of 〈 d ⁢ d c ⁢ u 〉 p might be not locally finite, as an example due to Kiselman, [20], shows. If they are locally finite, however, by [12], 〈 d ⁢ d c ⁢ u 〉 p is a closed positive ( p , p ) -current. For instance, this is the case when u has analytic singularities. We refer to the currents 〈 d ⁢ d c ⁢ u 〉 p as non-pluripolar Monge–Ampère currents.

As the name suggests, the non-pluripolar Monge–Ampère currents do not charge pluripolar sets. Thus, since 〈 d ⁢ d c ⁢ u 〉 p cannot capture the behavior on the singular set of u, they do not coincide with Demailly’s extensions of ( d ⁢ d c ⁢ u ) p in general. In particular, it follows that the non-pluripolar Monge–Ampère operators u ↦ 〈 d ⁢ d c ⁢ u 〉 p are far from being continuous under decreasing sequences in general. For instance, if u = log ⁡ | f | 2 , where f is a holomorphic function, then 〈 d ⁢ d c ⁢ u 〉 = 0 , whereas for any sequence u ℓ decreasing to u, d ⁢ d c ⁢ u ℓ converges weakly to d ⁢ d c ⁢ u , which by the Poincaré–Lelong formula is the current of integration [ f = 0 ] along the divisor of f.

The purpose of this paper is to introduce a new class of psh functions together with an extension of the Demailly–Bedford–Taylor Monge–Ampère operators that capture the singular behavior.

If u ∈ 𝒢 ⁢ ( Ω ) and j ≤ n - 1 , then u ⁢ 〈 d ⁢ d c ⁢ u 〉 j is a well-defined current and thus by mimicking the original construction by Bedford and Taylor we can define generalized Monge–Ampère currents.

By using the locality of the non-pluripolar Monge–Ampère operators and the integrability, it follows that [ d ⁢ d c ⁢ u ] p and S p ⁢ ( u ) are closed positive currents. In particular, [ d ⁢ d c ⁢ u ] p dominates 〈 d ⁢ d c ⁢ u 〉 p .

Definitions 1.1 and 1.2 are inspired by the construction of Monge–Ampère currents in [1, 5]. From [5, Proposition 4.1] it follows that psh functions with analytic singularities have locally finite non-pluripolar energy. Thus there are functions in 𝒢 ⁢ ( Ω ) that are not in 𝒟 ⁢ ( Ω ) . If u ∈ PSH ⁡ ( Ω ) has analytic singularities, then the currents [ d ⁢ d c ⁢ u ] p coincide with the Monge–Ampère currents ( d ⁢ d c ⁢ u ) p introduced in [1, 5]. In this case S p ⁢ ( u ) = 𝟏 Z ⁢ [ d ⁢ d c ⁢ u ] p , where Z is the unbounded locus of u. In [4, 3] these Monge–Ampère currents are used to understand non-proper intersection theory in terms of currents. In particular, the Lelong numbers of the currents [ d ⁢ d c ⁢ log ⁡ | f | 2 ] p are certain local intersection numbers, so-called Segre numbers, associated with the ideal generated by f.

In Section 4 we provide other examples of functions in 𝒢 ⁢ ( Ω ) and also psh functions that are not in 𝒢 ⁢ ( Ω ) . For instance, Example 4.3 shows that there are psh functions u ≤ v such that u ∈ 𝒢 ⁢ ( Ω ) but v ∉ 𝒢 ⁢ ( Ω ) .

Given u ∈ 𝒢 ⁢ ( Ω ) , note that max ⁡ ( u , - ℓ ) is a natural sequence of locally bounded psh functions decreasing to u, cf. (1.1). Our first main theorem states that our new Monge–Ampère currents [ d ⁢ d c ⁢ u ] p are the limits of the Monge–Ampère currents of this regularization.

Note that u ℓ = max ⁡ ( u , - ℓ ) corresponds to χ ℓ ⁢ ( t ) = max ⁡ ( t , - ℓ ) . Also note that Theorem 1.3 implies that [ d ⁢ d c ⁢ u ] p coincides with the Bedford–Taylor–Demailly Monge–Ampère current when this is defined.

The Monge–Ampère currents of the natural regularizations max ⁡ ( u , - ℓ ) do not always converge, see Example 5.9, and thus not all psh functions are in 𝒢 ⁢ ( Ω ) .

Since there are functions in 𝒢 ⁢ ( Ω ) that are not in 𝒟 ⁢ ( Ω ) , we cannot expect continuity for all decreasing sequences. Our next result is a twisted version of Theorem 1.3 that illustrates that failure of continuity. Let v be a locally bounded psh function on Ω. Then max ⁡ ( u , v - ℓ ) is another natural sequence of locally bounded psh functions decreasing to u. Moreover, if χ ℓ is as in Theorem 1.3, then also χ ℓ ∘ ( u - v ) + v is a sequence of locally bounded psh functions decreasing to u.

Note that the lower degree Monge–Ampère currents [ d ⁢ d c ⁢ u ] j come into play. Also note that Theorem 1.3 follows from Theorem 1.5 by setting v = 0 .

When u has analytic singularities, Theorem 1.3 first appeared in [2, Theorem 1.1], and Theorem 1.5 appeared in [11, Theorem 1], although formulated slightly differently, cf. Remark 5.7 below. In those papers, using a Hironaka desingularization, the results are reduced to the case with divisorial singularities. Such a reduction is not available in the general case, and in this paper we instead rely on properties from [12] of the non-pluripolar Monge–Ampère operator. In particular, we get new proofs of the results in [2] and [11].

Let us now turn to the global setting. Assume that ( X , ω ) is a compact Kähler manifold of dimension n. Recall that a function φ is said to be ω-psh, φ ∈ PSH ⁡ ( X , ω ) , if whenever h is a local potential for ω, i.e., d ⁢ d c ⁢ h = ω , φ + h is psh. Then d ⁢ d c ⁢ φ + ω is a closed positive current in [ ω ] , and by the d ⁢ d c -lemma any closed positive current in [ ω ] can be written as d ⁢ d c ⁢ φ + ω for some ω-psh φ, and this φ is unique up to adding of constants. Thus studying ω-psh functions is the same as studying closed positive currents in [ ω ] .

If φ ∈ PSH ⁡ ( X , ω ) is bounded, then there are well-defined Monge–Ampère currents ( d ⁢ d c ⁢ φ + ω ) p , locally defined as ( d ⁢ d c ⁢ ( φ + h ) ) p , where h is a local potential for ω. It turns out, [12, Proposition 1.6], that for an unbounded φ the non-pluripolar Monge–Ampère currents 〈 d ⁢ d c ⁢ φ + ω 〉 p are always well-defined. Moreover, [12, Proposition 1.20] showed that

When φ is bounded we have equality

but in general the inequality can be strict.

Our definitions of 𝒢 ⁢ ( X ) and Monge–Ampère currents naturally lend themselves to the global setting.

Since two local potentials differ by a pluriharmonic function, it follows that [ d ⁢ d c ⁢ φ + ω ] p and S p ω ⁢ ( φ ) are well-defined global positive closed currents on X. Note that whether an ω-psh function φ is in 𝒢 ⁢ ( X , ω ) only depends on the current d ⁢ d c ⁢ φ + ω and not on the choice of ω as a Kähler representative in the class [ ω ] . Also the currents [ d ⁢ d c ⁢ φ + ω ] p and S p ω ⁢ ( φ ) only depend on the current d ⁢ d c ⁢ φ + ω .

From Theorem 1.5 we get global regularization results. Given φ ∈ PSH ⁡ ( X , ω ) , note that max ⁡ ( φ , - ℓ ) is a natural sequence of bounded ω-psh functions decreasing to φ.

If η is another Kähler form in [ ω ] , then η = ω + d ⁢ d c ⁢ g for some smooth function g. There is an associated regularization of φ, namely φ ℓ := max ⁡ ( φ - g , - ℓ ) + g , which corresponds to the max-regularization of the current d ⁢ d c ⁢ φ + ω with respect to the alternative decomposition d ⁢ d c ⁢ ( φ - g ) + η .

Note that Theorem 1.8 follows immediately from Theorem 1.9 by setting g = 0 . As in the local case, for φ with analytic singularities Theorems 1.8 and 1.9 follow from [11, Theorem 1], cf. Remark 6.9.

From (1.3) and Theorem 1.8 we get the following mass formula.

In fact, Theorem 1.10 is a cohomological consequence of the definition of [ d ⁢ d c ⁢ φ + ω ] p ; in Section 6.1 we provide a direct proof that does not rely on Theorem 1.8. For φ with analytic singularities this theorem appeared in [2, Theorem 1.2 and Proposition 5.2]. Note that, for j ≤ p , the current S j ω ⁢ ( φ ) captures the mass that “escapes” from 〈 d ⁢ d c ⁢ φ + ω 〉 p at codimension j.

From the local case it follows that ω-psh functions with analytic singularities are in 𝒢 ⁢ ( X , ω ) , and in Section 11 we provide other examples. However, in the global setting we know more about the structure of the class 𝒢 ⁢ ( X , ω ) . In particular, it contains the Błocki–Cegrell class. Note that being in the Błocki–Cegrell class is a local statement, cf. Proposition 10.1 below. We say that φ ∈ PSH ⁡ ( X , ω ) is in 𝒟 ⁢ ( X , ω ) if whenever g is a local d ⁢ d c -potential of ω in an open set 𝒰 ⊂ X , then φ + g ∈ 𝒟 ⁢ ( 𝒰 ) .

Next, as the name suggests, the class 𝒢 ⁢ ( X , ω ) of ω-psh functions with finite non-pluripolar energy can be understood as a finite energy class. Recall that the Monge–Ampère energy of φ ∈ PSH ⁡ ( X , ω ) , introduced in [19], inspired by earlier work [13] in the local setting, is defined as

if φ is bounded and by

in general. The corresponding finite energy class

is convex. Recall that, if φ , ψ ∈ PSH ⁡ ( X , ω ) , then φ is said to be less singular than ψ, φ ⪰ ψ , if φ ≥ ψ + O ⁢ ( 1 ) . If φ ⪰ ψ and ψ ⪰ φ , we say that φ and ψ have the same singularity type and write φ ∼ ψ . The class ℰ ⁢ ( X , ω ) is closed under finite perturbations in the sense that if φ ∈ ℰ ⁢ ( X , ω ) and ψ ∼ φ , then ψ ∈ ℰ ⁢ ( X , ω ) . Moreover, ℰ ⁢ ( X , ω ) is contained in the full mass class

We introduce an alternative energy for φ ∈ PSH ⁡ ( X , ω ) .

Note that

so that 𝒢 ⁢ ( X , ω ) can be thought of as an finite energy class, cf. (1.7).

Although 𝒢 ⁢ ( X , ω ) contains the convex subclass ℰ ⁢ ( X , ω ) it is not convex itself. However, it contains certain other convex energy classes.

Note that if ψ ∈ PSH ⁡ ( X , ω ) ∩ L ∞ ⁢ ( X ) , then ℰ ψ ⁢ ( X , ω ) ⊃ ℰ ⁢ ( X , ω ) . The classes ℰ ψ ⁢ ( X , ω ) have the following properties, similar to ℰ ⁢ ( X , ω ) . Following [12], we say that φ ∈ PSH ⁡ ( X , ω ) has small unbounded locus (sul) if there exists a complete pluripolar closed subset A ⊂ X such that φ is locally bounded outside A, cf. Section 2.2 below.

The paper is organized as follows. In Section 2 we provide some background on the classical and the non-pluripolar Monge–Ampère operators in the local setting. In Section 3 we introduce the class 𝒢 ⁢ ( Ω ) , and more generally classes 𝒢 k ⁢ ( Ω ) of psh functions of locally finite non-pluripolar energy of order k , and our Monge–Ampère operators, and in Section 4 we provide various examples of functions in 𝒢 ⁢ ( Ω ) . In Section 5 we prove the regularity result Theorem 1.5. In fact, we prove a slightly more general version formulated in terms of 𝒢 k ⁢ ( Ω ) .

In Section 6 we extend our definitions and regularity results to the global setting. In Section 7 we recall the classical Monge–Ampère energy and in Sections 8 and 9 we study the non-pluripolar energy and the relative energy, respectively; in particular, we prove Theorems 1.13 and 1.15. As in the local case we introduce more generally non-pluripolar and relative energies of order k and corresponding finite energy classes 𝒢 k ⁢ ( X , ω ) and ℰ k ψ ⁢ ( X , ω ) , and we prove versions of our results formulated in terms of these. In Section 10 we discuss the Błocki–Cegrell class and prove Theorem 1.11. Finally, in Section 11 we give various examples of functions with finite non-pluripolar energy.

2 The complex Monge–Ampère product

Throughout this paper X is a domain in ℂ n or more generally a complex manifold of dimension n. All measures are assumed to be Borel measures. We let d c = 1 4 ⁢ π ⁢ i ⁢ ( ∂ - ∂ ¯ ) , so that d d c log | z 1 | 2 = [ z 1 = 0 ] .

In this section we recall some basic facts about the (non-pluripolar) Monge–Ampère products. We refer to, e.g., [18, Chapter III] for the classical Bedford–Taylor–Demailly theory, see also [7, 8, 12].

First, the plurifine topology is the coarsest topology such that all psh functions on all open subsets of X are continuous. A basis for this topology is given by all sets of the form V ∩ { u > 0 } , where V is open and u is psh in V.

Let T be a closed positive current and let u be a locally bounded psh function on X. Then uT is a well-defined current and

is again a closed positive current. In particular, if u 1 , … , u p are locally bounded psh functions, then the product ∧ ⁡ ∧ ⁡ d ⁢ d c ⁢ u p ⋯ d ⁢ d c ⁢ u 1 T is defined inductively as

It turns out that this product is commutative in the factors d ⁢ d c ⁢ u j and multilinear in the factors d ⁢ d c ⁢ u j and it does not charge pluripolar sets. Moreover, ∧ ⁡ d ⁢ d c ⁢ u p ⋯ d ⁢ d c ⁢ u 1 is local in the plurifine topology, i.e., if O is a plurifine open set and u j = v j pointwise on O, then

The products (2.1) satisfy the following continuity property.

We will use the following result.

3 Local Monge–Ampère currents – The classes 𝒢 k ⁢ ( X )

We slightly extend the definition of 𝒢 ⁢ ( X ) from the introduction, cf. Definition 1.1.

Note that if ω is a smooth strictly positive ( 1 , 1 ) -form, then u ∈ 𝒢 k ⁢ ( X ) if and only if the measure

is locally finite and u is locally integrable with respect to this measure. Clearly

where 𝒢 ⁢ ( X ) is as in Definition 1.1.

If u ∈ 𝒢 k ⁢ ( X ) , then the Monge–Ampère currents

from Definition 1.2 are well-defined ( p , p ) -currents for p = 1 , … , k + 1 .