Annales Polonici Mathematici 60:2, 173–197 Reçu par la Rédaction 11.1.1994

Attenuating the singularities of plurisubharmonic functions

by Christer O. Kiselman (Uppsala)

Contents:

1. Introduction

2. Spherical means and spherical suprema

3. The Hörmander–Bombieri theorem and the integrability index 4. Analyticity theorems for sets of plurisubharmonic functions 5. A refined Lelong number

6. Plurisubharmonic functions with attenuated singularities

7. Analyticity theorems for the refined Lelong number References

Resumo: Malfortigi la malregulejojn de plursubharmonaj funkcioj

Ni studas la malregulejojn de la plursubharmonaj funkcioj per metodoj de la konveksa analitiko. Teoremoj pri analitikeco de rafinita nombro de Lelong estas pruvitaj.

Abstract: We study the singularities of plurisubharmonic functions using methods from convexity theory. Analyticity theorems for a refined Lelong number are proved.

1. Introduction

The plurisubharmonic functions, which appear in complex analysis first as logarithms of moduli of holomorphic functions and as analogues of potentials, are useful for many constructions since they are easier to manipulate than holomorphic functions—this is why Lelong [1985] includes them among “les objets souples de l’analyse complexe.” The present paper is written in this spirit: we show how to construct in a straightforward way new plurisubharmonic functions from old ones using standard methods of convex analysis. These new functions can then be used to find analytic varieties that are connected with the original function, or rather with its singularities. We shall therefore first describe how one can measure the singularity of a plurisubharmonic function: this is done using the Lelong number and the integrability index.

The Lelong number measures how big (or “heavy”) the singularities of a plurisubharmonic function are. It generalizes the notion of multiplicity of a zero of a

1991 Mathematics Subject Classification: Primary 32F05, 32B15. Secondary 52A41. Key words and phrases: plurisubharmonic function, analytic variety, convex function, Fenchel transformation. Research partially supported by the Swedish Natural Science Research Council. holomorphic function. To define it, we first form the measure μ = (2π)^{−1}Δf, where Δ is the Laplacian in all 2n real variables Re z_{j}, Im z_{j}. Note that when f = log |h| is the logarithm of the absolute value of a holomorphic function of one variable, then μ is a sum of point masses, one at each zero of h and with weight equal to the multiplicity of the zero. The Lelong number of f at a point x is by definition the (2n − 2)-dimensional density of the measure μ at x. More explicitly, it is the limit as r → 0 of the mean density of μ in the ball of center x and radius r:

(1.1)  ν_{f}(x) = lim_{r→0} \frac{μ(x + rB)}{λ_{2n−2}(rB ∩ C^{n−1})},

where λ_{k} denotes k-dimensional Lebesgue measure. Note that we compare the mass of μ in the ball x+rB with the volume of the ball of the same radius in C^{n−1}, i.e., of real dimension 2n − 2. This makes sense, because if f = log |h| with h holomorphic, then μ is a mass distribution on the (2n − 2)-dimensional zero set of h. If n = 1, λ_{2n−2}(rB ∩ C^{n−1}) = λ_{0}({0}) = 1, and ν_{f}(x) is just the mass of μ at x.

One often approximates a plurisubharmonic function f by f_{j} = max(−j, f) or by smooth functions f_{j} = f ∗ ψ_{j} obtained by convolution. However, in these cases the functions f_{j} never take the value −∞, so their Lelong numbers ν_{f_{j}}(x) are zero everywhere; their singularities as measured by the Lelong number do not approach those of f as j → +∞. Here we shall construct functions f_{τ} depending on a non-negative number τ such that f_{0} = f and f_{τ} has Lelong number ν_{f_{τ}} (x) = (ν_{f}(x) − τ)^{+}. It turns out that the family (f_{τ})_{τ} can be used in various constructions. The singularities of f_{τ} are the same as those of f but attenuated in a certain sense. More precisely, the important property is that ν_{f_{τ}} (x) > 0 if τ < ν_{f}(x), whereas the singularity is completely killed, i.e., ν_{f_{τ}} (x) = 0, if τ > ν_{f}(x). In this context it is convenient to define the Lelong number of a family of plurisubharmonic functions. We prove analyticity theorems for the superlevel sets of such numbers; see section 4.

If f is plurisubharmonic and t a positive number, the function exp(−f/t) may or may not be integrable. The set of all t such that this function is locally integrable in the neighborhood of a certain point is an interval, and its endpoint measures how singular f is. This is the reason behind the integrability index ι_{f} to be defined in section 3 (see (3.3)). From the Hörmander–Bombieri theorem we get analyticity theorems for the integrability index (see (3.4)). There is a relation between the integrability index and the Lelong number: ι_{f} ⩽ ν_{f} ⩽ nι_{f}, where n is the complex dimension of the space; see Theorem 3.5. This relation cannot be improved (see Example 3.6), but nevertheless it will suffice to yield analyticity theorems for the Lelong number. The reason for this is roughly speaking that if we subtract the same quantity τ from two numbers like ν_{f}(x) and ν_{f}(x′) with ν_{f}(x′) > ν_{f}(x) > τ, then the quotient between ν_{f}(x′) − τ and ν_{f}(x) − τ can be large, for instance larger than the dimension n. This is why analyticity theorems for sets of plurisubharmonic functions are useful when it comes to proving analyticity theorems for a single function.

We also consider a refined Lelong number, first presented at a seminar in Tunis on March 6, 1986 (see Kiselman [1987]). This number, denoted by ν_{f}(x, y) and defined in section 5, depends on a vector y and is a concave function of that vector. The methods of convex analysis can therefore be applied to study it. For any fixed y, it is a special case of Demailly’s generalized Lelong number [1987]. For related studies of Lelong numbers, see Abrahamsson [1988] and Wang [1991].

2. Spherical means and spherical suprema

Let f and q be two given plurisubharmonic functions in an open set ω in C^{n}; we write this as f, q ∈ PSH(ω). We define an open set Ω_{q} in C^{n} × C as

(2.1)  Ω_{q} = {(x, t) ∈ ω × C; q(x) + Re t < 0},

and we note immediately that Ω_{q} is pseudoconvex if ω is pseudoconvex, for the function (x, t) ↦ q(x) + Re t is plurisubharmonic in ω × C. We shall assume that q(x) ⩾ − log d_{ω}(x) for all x ∈ ω, denoting by d_{ω}(x) the Euclidean distance from x to the complement of ω, and we note that then (x, t) ∈ Ω_{q} implies that the closed ball of center x and radius |e^{t}| is contained in ω. We define two functions u and U in Ω_{q} by putting

(2.2)  u(x, t) = u_{f}(x, t) = u_{f,q}(x, t) = ∫_{}_{z∈S} f(x + e^{t}z),  (x, t) ∈ Ω_{q};

(2.3)  U(x, t) = U_{f}(x, t) = U_{f,q}(x, t) = sup_{z∈S} f(x + e^{t}z),  (x, t) ∈ Ω_{q}.

Here S is the Euclidean unit sphere, and the barred integral sign indicates the mean value: in general

∫_{}_{A} f = ∫_{A} f/ ∫_{A} 1 provided 0 < ∫_{A} 1 < +∞.

So u_{f}(x, t) is the mean value of f over the sphere x+e^{t}S, and U_{f}(x, t) is the supremum of f over the same sphere. Since we usually keep q fixed, the dependence on that function need not always be shown. If ω ≠ C^{n}, the simplest choice of q is just q = − log d_{ω}. Then q > −∞ everywhere. However, if ω = C^{n}, then it is usually not convenient to use q = − log d_{ω} = −∞, because with this choice of q, the behavior of f at infinity would influence the local properties of the functions we construct. In this case it is best just to take q = 0.

Thanks to our assumption exp(−q(x)) ⩽ d_{ω}(x), the functions u_{f,q} and U_{f,q} are well defined and do not take the value +∞ in Ω_{q}. Moreover, by standard results, u_{f,q}, U_{f,q} ∈ PSH(Ω_{q}). We define them to be +∞ outside Ω_{q}.

Clearly u_{f} ⩽ U_{f}, and we shall see that there are inequalities in the opposite direction. We can note quickly that u_{af+bg} = au_{f} + bu_{g} for non-negative a, b, even for real a, b, which implies that the function u_{f} depends linearly of f in the linear space of all Borel measurable functions which are integrable on spheres, thus in particular on the space δPSH(ω) of delta-plurisubharmonic functions, i.e., the vector space spanned by those plurisubharmonic functions which are not identically minus infinity in any component of ω. We shall see that this implies that the Lelong number is a linear function on δPSH(ω). As to the function U_{f} we can only say that U_{af+bg} ⩽ aU_{f} + bU_{g} for a, b ⩾ 0, which implies that U_{f} is a convex function of f, and the Lelong number a concave function of f. But when it comes to the maximum of two functions, we have U_{max(f,g)} = max(U_{f}, U_{g}), which implies that ν_{max(f,g)} = min(ν_{f}, ν_{g}), whereas for the mean we can say only that u_{max(f,g)} ⩾ max(u_{f}, u_{g}), which implies that ν_{max(f,g)} ⩽ min(ν_{f}, ν_{g}). It is therefore useful to know that the Lelong number can be defined by either u_{f} or U_{f}, because this enables us to use the best properties of either one.

We can define the Lelong number as the slope at minus infinity of the function t ↦ u(x, t). As a consequence of the maximum principle, u(x, t) and U(x, t) are increasing in t; by Hadamard’s three-circle theorem, they are convex functions of t. Therefore their slopes at −∞ exist:

(2.4)  ν_{f}(x) = lim_{t→−∞} \frac{u(x, t)}{t} and N_{f}(x) = lim_{t→−∞} \frac{U(x, t)}{t}

both exist. This follows from the fact that the slopes

\frac{u(x, t) − u(x, t_{0})}{t − t_{0}} and \frac{U(x, t) − U(x, t_{0})}{t − t_{0}}

are increasing in t. The first limit ν_{f}(x) is the Lelong number of f at x, and the definition we shall use in this paper. The definition (1.1) of the Lelong number as the density of a measure is equivalent to (2.4) as can be proved without difficulty using Stokes’ theorem; see Kiselman [1979].

Since u_{f} ⩽ U_{f}, we have ν_{f}(x) ⩾ N_{f}(x). We shall now see that the two numbers are equal. To this end we shall use Harnack’s inequality, which takes the form

(2.5)  \frac{1 + |x|/r}{(1 − |x|/r)^{m−1}} h(0) ⩽ h(x) ⩽ \frac{1 − |x|/r}{(1 + |x|/r)^{m−1}} h(0)

for harmonic functions which satisfy h ⩽ 0 in the ball of radius r in R^{m}. If f is subharmonic in a neighborhood of the closed ball e^{s}B in C^{n}, we can consider its harmonic majorant h there, which satisfies f(x) ⩽ h(x) and

h(0) = ∫_{}_{z∈S} h(e^{s}z) = ∫_{}_{z∈S} f(e^{s}z) = u(0, s).

Therefore

U(0, t) = sup_{e^{t}S} f ⩽ sup_{e^{t}S} h ⩽ \frac{1 − e^{t−s}}{(1 + e^{t−s})^{2n−1}} u(0, s), t < s,

provided only f ⩽ 0 in e^{s}B. If we apply this inequality to the function f − U(0, s), which is non-positive in e^{s}B by definition, we get, writing U(t) instead of U(0, t) for simplicity:

U(t) − U(s) ⩽ \frac{1 − e^{t−s}}{(1 + e^{t−s})^{2n−1}} (u(s) − U(s)),

equivalently,

(2.6)  U(t) ⩽ (1 − λ_{t−s})U(s) + λ_{t−s}u(s),  t < s,

where λ_{t} is defined for t < 0 as

λ_{t} = \frac{1 − e^{t}}{(1 + e^{t})^{2n−1}} .

We can now prove that the two limits in (2.4) are equal. As already noted, ν_{f}(x) ⩾ N_{f}(x). In the other direction we can take for instance s = t + 1 in (2.6) to obtain the estimate

U(t) ⩽ (1 − λ_{−1})U(t + 1) + λ_{−1}u(t + 1),

whence

\frac{U(t)}{t} ⩾ (1 − λ_{−1})\frac{U(t + 1)}{t} + λ_{−1} \frac{u(t + 1)}{t}, t < 0.

Letting t tend to −∞ we see that N_{f}(x) ⩾ ν_{f}(x).

To any given f, q ∈ PSH(ω) we define

(2.7)  φ_{τ} (x) = inf_{t} [u_{f}(x, t) − τ Re t],  x ∈ ω, τ ⩾ 0.

In view of our convention that u_{f}(x, t) = +∞ if (x, t) ∉ Ω_{q}, the infimum operation acts effectively only over those t that satisfy Re t < −q(x). The function τ ↦ −φ_{τ} (x) is the Fenchel transform of R ∋ t ↦ u_{f}(x, t). We assume all the time that e^{−q(x)} does not exceed the distance d_{ω}(x) from x to the boundary of ω, so that u_{f} is well defined. The function (x, t) ↦ u_{f}(x, t) − τ Re t is plurisubharmonic in Ω_{q} and independent of the imaginary part of t. Therefore the minimum principle of Kiselman [1978] can be applied and yields that φ_{τ} is plurisubharmonic in ω. Example. Let us look at the simplest example: f(x) = log |x|, x ∈ C^{n}. We choose q = 0 and form U_{f}(x, t) = log(e^{t}+|x|) for t < q(x) = 0. Then φ_{τ} (x) = inf_{t<0}(U_{f}(x, t)−τt) can be calculated explicitly: it is φ_{τ} (x) = (1 − τ) log |x| + C_{τ} for 0 ⩽ τ < 1, where C_{τ} is a constant which depends on the parameter τ, and φ_{τ} (x) = log(1 + |x|) for τ ⩾ 1. Thus the Lelong number of φ_{τ} at the origin is max(1 − τ, 0) for all τ ⩾ 0.

There is no apparent reason why the Lelong number of the plurisubharmonic function φ_{τ} at x should be a function of ν_{f}(x) and τ; it could as well depend in some other way on the behavior of f near x. However, it turns out that the simple formula for the Lelong number of φ_{τ} in the example holds quite generally:

Theorem 2.1 (Kiselman [1979]). Let f, q ∈ PSH(ω) with q ⩾ − log d_{ω}. Define φ_{τ} by (2.2) and (2.7). Then φ_{τ} ∈ PSH(ω). If ν_{q}(x) = 0, then the Lelong number of φ_{τ} is

(2.8)  ν_{φ_{τ}} (x) = max(ν_{f}(x) − τ, 0) = (ν_{f}(x) − τ)^{+},  x ∈ ω, τ ⩾ 0.

We can also use the function U_{f} instead of u_{f} in the construction; the proof is the same. If τ < 0, then of course ν_{φ_{τ}} (x) = +∞.

We shall give a simplified proof of Theorem 2.1 under the slightly stronger hypothesis that q(x) > −∞. This is quite enough for the applications we have in mind. (As soon as ω ≠ C^{n}, we must indeed have q > −∞ everywhere.)

Lemma 2.2. With f and φ_{τ} as in Theorem 2.1 we have

ν_{φ_{τ}} (x) ⩾ ν_{f}(x) − τ.

Proof. We first note that by the definition of φ_{τ}, we have for any t′

(2.9)  φ_{τ} (y) ⩽ u_{f}(y, t′) − t′τ.

We now take the mean value of both sides of (2.9) when y varies over the sphere x + e^{t}S, which gives

(2.10)  u_{φ_{τ}} (x, t) = ∫_{}_{z∈S} φ_{τ} (x + ze^{t}) ⩽ ∫_{}_{z∈S} ∫_{}_{w∈S} f(x + ze^{t} + we^{t′}) − t′τ

for all real numbers t, t′. Here the double integral defines a measure of total mass 1 which is invariant under rotations around x and has its support in the set

x + e^{t}S + e^{t′}S = {x + z; |e^{t} − e^{t′}| ⩽ |z| ⩽ e^{t} + e^{t′}}.

When we apply this measure to f, the result is a weighted mean of f, which, since it is rotation invariant, is also a mean value of u_{f}(x, s) for various choices of s. The values of s that can occur are those which satisfy

|e^{t} − e^{t′}| ⩽ e^{s} ⩽ e^{t} + e^{t′} = e^{t′′},

where the last equality serves to define a number t′′. Since u_{f}(x, s) is increasing in s, the mean value cannot exceed u_{f}(x, t′′). Thus (2.10) yields

u_{φ_{τ}} (x, t) ⩽ ∫_{}_{z∈S} ∫_{}_{w∈S} f(x + ze^{t} + we^{t′}) − t′τ ⩽ u_{f}(x, t′′) − t′τ.

(The corresponding formula for U_{φ_{τ}} is even simpler to prove.) The only interesting choice is t = t′, so that t′′ = t + log 2. Thus

\frac{u_{φ_{τ}} (x, t)}{t} ⩾ \frac{u_{f}(x, t + log 2)}{t} − τ, t < 0,

and letting t tend to −∞ we get the desired conclusion.

Lemma 2.3. With f and φ_{τ} as in Theorem 2.1, take a number τ > ν_{f}(x). Assume that q(x) > −∞. Then φ_{τ} (x) > −∞. In particular ν_{φ_{τ}} (x) = 0.

Proof. Since ν_{f}(x) < τ < +∞, f is not equal to −∞ identically in a neighborhood of x. The value φ_{τ} (x) is the infimum of the convex function u_{f}(x, t) − tτ of the real variable t when t varies in the interval ]−∞, −q(x)[. This interval is by hypothesis bounded from the right. Moreover by the choice of τ, the function is strictly decreasing when t ≪ 0. Thus its infimum is finite. Proof of Theorem 2.1, assuming that q(x) is finite. The proof consists of the following steps (cf. Kiselman [1992]). First we note that φ_{τ} is a concave function of τ with φ_{0} = f. Therefore ν_{φ_{τ}} (x) is a convex function of τ taking the value ν_{f}(x) for τ = 0, for the Lelong number is as we have seen a linear function of φ, the limit of u_{φ}(x, t)/t. Second we see from Lemma 2.2 that ν_{φ_{τ}} (x) ⩾ ν_{f}(x)−τ. Third we know from Lemma 2.3 (if ν_{f}(x) is finite) that ν_{φ_{τ}} (x) = 0 if τ > ν_{f}(x). The only convex function of τ which has these properties is τ ↦ (ν_{f}(x) − τ)^{+}.

3. The Hörmander–Bombieri theorem and the integrability index

The fundamental tool which will provide us with analytic sets is the Hörmander– Bombieri theorem:

Theorem 3.1. Let ω be a pseudoconvex open set in C^{n}, and let φ ∈ PSH(ω). For every a ∈ ω such that e^{−φ} ∈ Lloc^{2}(a) there exists a holomorphic function h ∈ 𝒪(ω) such that h(a) = 1 and

(3.1)  ∫_{ω} |h|^{2}e^{−2φ}(1 + |z|^{2})^{−3n}dλ_{2n}(z) < +∞.

Here Lloc^{2}(a) denotes the set of all functions that are square integrable in some neighborhood of the point a.

For the proof see Hörmander [1990: Theorem 4.4.4]. Let us denote by 𝒪(ω, φ) the set of all holomorphic functions h in ω which satisfy condition (3.1) for a given function φ. The intersection

(3.2)  V (φ) = ⋂_{h} (h^{−1}(0); h ∈ 𝒪(ω, φ))

is an intersection of zero sets of holomorphic functions, and therefore itself an analytic set. Let us define

I(φ) = {a ∈ ω; e^{−φ} ∉ Lloc^{2}(a)}.

With this notation the theorem says that V (φ) ⊂ I(φ). It is however obvious that I(φ) ⊂ V (φ).

In view of this theorem it is natural to measure the singularity of a plurisubharmonic function φ at a point a by its integrability index ι_{φ}(a):

(3.3)  ι_{φ}(a) = inf_{t>0} [t; e^{−φ/t} ∈ Lloc^{2}(a)].

Let Φ be an arbitrary subset of PSH(ω) and κ a functional on Φ in the sense that there is given a function κ_{φ}: ω → [0, +∞] for every φ ∈ Φ. We introduce a notation for the superlevel sets of such functionals:

E_{c}^{κ}(φ) = {a ∈ ω; κ_{φ}(a) ⩾ c}, c ⩾ 0.

The superlevel sets of the integrability index are analytic varieties. In fact, by the definition of ι,

E_{c}^{ι}(φ) ⊂ I(φ/t) ⊂ E_{t}^{ι}(φ), 0 < t < c.

Using the Hörmander–Bombieri theorem we see that

E_{c}^{ι}(φ) ⊂ V (φ/t) ⊂ E_{t}^{ι}(φ), 0 < t < c.

We now note that by the definition of the superlevel set, the intersection of all E_{t}^{ι}(φ) when t varies in the interval 0 < t < c is just E_{c}^{ι}(φ), so that

(3.4)  E_{c}^{ι}(φ) = ⋂_{0<t<c} V (φ/t),  c ⩾ 0.

Suppose that κ is a functional which is comparable to the integrability index in the sense that the inequality

(3.5)  sι_{φ}(x) ⩽ κ_{φ}(x) ⩽ tι_{φ}(x),  φ ∈ Φ, x ∈ ω,

holds for some positive constants s and t. Then there is of course a relation between the superlevel sets of the two functionals:

(3.6)  E_{tc}^{κ}(φ) ⊂ E_{c}^{ι}(φ) ⊂ E_{sc}^{κ}(φ),  φ ∈ Φ.

If we know that E_{tc}^{κ}(φ) = E_{sc}^{κ}(φ), then E_{tc}^{κ}(φ) equals E_{c}^{ι}(φ) and so is an analytic variety. Of course functions which admit such an interval of constancy in their superlevel sets are very special. But we shall see in the next section that for a set of plurisubharmonic functions such intervals of constancy can appear quite naturally.

We now ask whether the Lelong number is comparable to the integrability index in the sense of (3.5). The answer is well-known, but will be quoted here for convenience.

Theorem 3.2. If φ ∈ PSH(ω), where ω ⊂ C^{n}, and ν_{φ}(a) ⩾ n, then e^{−φ} ∉ Lloc^{2}(a). Thus E_{n}^{ν}(φ) ⊂ I(φ) ⊂ V (φ). In terms of the integrability index we have ν_{φ}(x) ⩽ nι_{φ}(x).

Proof. This result is contained in Skoda [1972, Proposition 7.1], but it is easy to give a proof using the function U = U_{φ} defined by (2.3). If ν_{φ}(a) ⩾ n, then the slope of t ↦ U(a, t) at minus infinity is at least n, and we have

U(a, t) ⩽ U(a, t_{0}) + n(t − t_{0}), t ⩽ t_{0},

for some t_{0}. Rewriting this in terms of φ we see that

φ(z) ⩽ φ(z_{0}) + log (\frac{|z − a|^{n}}{|z_{0} − a|^{n}} ), |z − a| ⩽ |z_{0} − a|,

for a suitable point z_{0} on the sphere |z − a| = e^{t_{0}}, or equivalently

e^{−2φ(z)} ⩾ e^{−2φ(z_{0})} \frac{|z_{0} − a|^{2n}}{|z − a|^{2n}} ,

where the right-hand side is a non-integrable function near a.

In the other direction we have:

Theorem 3.3. If φ ∈ PSH(ω) has a finite value at a point a ∈ ω, then e^{−φ} ∈

Lloc^{2}(a).

For the proof of this result, see Hörmander [1990: Theorem 4.4.5]. The theorem says that I(φ) is contained in the polar set P (φ) = φ^{−1}(−∞) of φ, thus I(φ) ⊂ P (φ).

Combining Theorems 3.1, 3.2 and 3.3 we see that

(3.7)  E_{nc}^{ν}(φ) ⊂ V (φ/c) ⊂ P (φ),  c > 0.

A stronger result in the same direction is

Theorem 3.4. If ν_{φ}(a) < 1, then e^{−φ} ∈ Lloc^{2}(a). Thus I(φ) ⊂ E_{1}^{ν}(φ). Also ι_{φ}(x) ⩽ ν_{φ}(x).

For the proof see Skoda [1972: Proposition 7.1].

Combining Theorems 3.1, 3.2 and 3.4 we get

Theorem 3.5. The Lelong number ν is comparable to the integrability index ι in the sense of (3.5), more precisely,

(3.8)  ι_{φ}(x) ⩽ ν_{φ}(x) ⩽ nι_{φ}(x),  φ ∈ PSH(ω), x ∈ ω ⊂ C^{n},

and

(3.9)  E_{nc}^{ν}(φ) ⊂ V (φ/c) ⊂ E_{c}^{ν}(φ) ⊂ V (nφ/c),  φ ∈ PSH(ω), c > 0.

These inequalities are sharp.

This result is the basis for the analyticity theorems that we shall state. However, the weaker result (3.7) is often sufficient.

That the comparison in (3.8) between the Lelong number and the integrability index cannot be improved follows from simple examples: Example 3.6. The function

f(z) = max(log |z_{1}|^{a}, log |z_{2}|^{b}), z ∈ C^{2},

has integrability index ι_{f}(0) = ab/(a+b) and Lelong number ν_{f}(0) = min(a, b). Thus

\frac{ν_{f}(0)}{ι_{f}(0)} = \frac{a + b}{max(a, b)} = \frac{a^{−1} + b^{−1}}{max(a^{−1}, b^{−1})} ∈ ]1, 2] .

A little more generally, if we take 1 ⩽ k ⩽ n and positive numbers a_{1}, ..., a_{k} and define

f(z) = max_{1⩽j⩽k} (log |z_{j}|^{a_{j}}) , z ∈ C^{n},

then ι_{f}(0) = (∑ a_{j}^{−1})^{−1} (cf. Lemma 5.5 below) and ν_{f}(0) = min a_{j}, so that

\frac{ν_{f}(0)}{ι_{f}(0)} = \frac{∑ a_{j}^{−1}}{max a_{j}^{−1}} ∈ [1, n] .

(These formulas hold if we define a_{j}^{−1} = 0 for j = k + 1, ..., n.) Clearly the quotient ν_{f}/ι_{f} can assume all values in the closed interval [1, n] (we allow k = 1 and k = n). Thus (3.8) is sharp.

4. Analyticity theorems for sets of plurisubharmonic functions

Let ω be an open set in C^{n} and Φ an arbitrary subset of PSH(ω). Let κ be a functional on Φ which is comparable to the integrability index in the sense that (3.5) holds for some positive constants s and t. Then we get from (3.6) and (3.7):

(4.1)  ⋂_{φ∈Φ} E_{tc}^{κ}(φ) ⊂ ⋂_{φ∈Φ} E_{c}^{ι}(φ) ⊂ ⋂_{φ∈Φ} E_{sc}^{κ}(φ) ⊂ ⋂_{φ∈Φ} P (φ).

It is convenient to introduce a notation for these sets:

E_{c}^{κ}(Φ) = ⋂_{φ∈Φ} E_{c}^{κ}(φ)

for any functional κ. If we define the value of the functional on the whole set Φ as

κ_{Φ}(x) = inf_{φ∈Φ} κ_{φ}(x),

then E_{c}^{κ}(Φ) is just the superlevel set of κ_{Φ}. We can also define the polar set of Φ as

P (Φ) = ⋂_{φ∈Φ} P (φ).

With this notation we can write (4.1) as

(4.2)  E_{tc}^{κ}(Φ) ⊂ E_{c}^{ι}(Φ) ⊂ E_{sc}^{κ}(Φ) ⊂ P (Φ).

Theorem 4.1. Let κ be a functional on a subset Φ of PSH(ω) which is comparable to the integrability index in the sense that (3.5) holds for some positive constants s and t. If the superlevel sets E_{c}^{κ}(Φ) are independent of c over an interval of sufficiently large logarithmic length, viz. if E_{sc}^{κ}(Φ) = E_{tc}^{κ}(Φ), then E_{sc}^{κ}(Φ) is an analytic variety. A little more generally, if Y is an analytic subset of ω and Y ∩ E_{sc}^{κ}(Φ) = Y ∩ E_{tc}^{κ}(Φ), then Y ∩ E_{sc}^{κ}(Φ) is analytic.

Proof. Since the result is local, we can assume ω to be pseudoconvex. The inclusions (4.2) then show that E_{sc}^{κ}(Φ) = E_{c}^{ι}(Φ), which is the set of common zeros of a family of holomorphic functions in ω; see (3.4). Similarly, Y ∩ E_{sc}^{κ}(Φ) = Y ∩ E_{c}^{ι}(Φ).

Theorem 4.2. Let ω, κ and Φ be as in Theorem 4.1, and assume in addition that κ is positively homogeneous, i.e., φ ∈ Φ and t > 0 implies tφ ∈ Φ and κ_{tφ}(a) = tκ_{φ}(a). Let Y be an analytic subset of ω, and let X be a subset of Y. Assume that

(4.3)  inf_{x∈X} κ_{φ}(x) > 0 for all φ ∈ Φ;

and

(4.4) for every x ∈ Y ∖ X there exists a function φ ∈ Φ such that κ_{φ}(x) = 0.

Then X is an analytic set.

Proof. Define ψ = φ/ε_{φ}, where ε_{φ} = inf_{x∈X} κ_{φ}(x) > 0, and let Ψ be the set of all functions ψ obtained in this way. Then κ_{ψ}(x) ⩾ 1 for every ψ ∈ Ψ and every x ∈ X. Using the notation for superlevel sets we can write this as

X ⊂ E_{1}^{κ}(Ψ) ⊂ E_{c}^{κ}(Ψ)

for all c with 0 ⩽ c ⩽ 1.

On the other hand, if x ∈ Y ∖ X, then by (4.4) there is a φ ∈ Φ such that κ_{φ}(x) = 0. Thus ψ = φ/ε_{φ} is in Ψ and κ_{ψ}(x) = κ_{φ}(x)/ε_{φ} = 0. So x ∉ E_{c}^{κ}(ψ), c being any positive number. Thus

x ∉ E_{c}^{κ}(Ψ) = ⋂_{ψ∈Ψ} E_{c}^{κ}(ψ).

Therefore E_{c}^{κ}(Ψ) ∩ Y ⊂ X for every c > 0. Combining this with the first part of the proof we see that E_{c}^{κ}(Ψ) ∩ Y = X for all c satisfying 0 < c ⩽ 1. Hence E_{c}^{κ}(Ψ) ∩ Y is constant for these c and Theorem 4.1 yields that X is analytic.

A particular case of Theorem 4.2 is when we can associate with a given function or current a family of plurisubharmonic functions on which our functional takes values that we can control. The following result is of this character. It holds also for functionals which have only a loose connection to the integrability index or the Lelong number; more precisely functionals which are zero at the same time as the integrability index in a semiuniform way:

Theorem 4.3. Let Φ = {φ_{α}; α ∈ A} be a set of plurisubharmonic functions in an open set ω, and let κ be a functional on Φ which is weakly comparable to the integrability index in the sense that

for every ε > 0 there is a δ > 0 such that φ ∈ Φ, ι_{φ}(x) < δ implies κ_{φ}(x) < ε,

and

φ ∈ Φ, κ_{φ}(x) = 0 implies ι_{φ}(x) = 0.

Suppose that the values κ_{φ_{α}}(x) are given by a formula κ_{φ_{α}}(x) = G(H(x), α) for some functions G: [0, +∞] × A → [0, +∞] and H: ω → [0, +∞]. We assume that c ↦ G(c, α) is increasing, and that there exists a number c_{0} such that G(c_{0}, α) > 0 for all α ∈ A. Finally we suppose that for every c < c_{0} there is an α ∈ A such that G(c, α) = 0. Then the superlevel set {x; H(x) ⩾ c_{0}} is analytic.

Proof. We shall apply Theorem 4.2 to X = {x; H(x) ⩾ c_{0}}. First we note that

inf_{x∈X} κ_{φ_{α}}(x) = inf_{x∈X} G(H(x), α) ⩾ G(c_{0}, α) = ε_{α} > 0

for any α ∈ A. Hence ι_{φ_{α}}(x) ⩾ δ_{α} > 0, which means that (4.3) holds for ι. Next, if x ∉ X, then c = H(x) < c_{0} and there is an α such that G(c, α) = 0. We get

κ_{φ_{α}}(x) = G(H(x), α) = G(c, α) = 0;

hence also ι_{φ_{α}}(x) = 0, so that (4.4) holds for ι. Thus Theorem 4.2, applied to ι, shows that X is analytic.

This theorem contains the classical theorem of Siu [1974]. For if we let κ_{φ}(x) = ν_{φ}(x), A = [0, c_{0}[, G(c, α) = (c − α)^{+}, H(x) = ν_{f}(x), and define φ_{α} as

φ_{α}(x) = inf_{t} [u_{f}(x, t) − tα; (x, t) ∈ Ω_{q}], x ∈ ω, α ∈ [0, c_{0}[ ,

then by Theorem 2.1,

ν_{φ_{α}}(x) = (ν_{f}(x) − α)^{+} = G(H(x), α).

The function G(c, α) = (c − α)^{+} satisfies the hypotheses of Theorem 4.3, so it follows that the superlevel set {x; ν_{f}(x) ⩾ c_{0}} is an analytic variety. The singularities of the φ_{α} are the same as those of f, but attenuated to some degree as shown by the formula. This attenuation is the reason behind their usefulness in proving Siu’s theorem and the explanation for the title of this paper.

5. A refined Lelong number

In analogy with the mean value and supremum over a sphere, we can define the mean value and supremum over a polycircle. Let us define, given a function f ∈ PSH(ω),

(5.1)  \begin{aligned}v(x, y) & = v_{f}(x, y) = ∫_{}_{|z_{j}|=|e^{y_{j}} |} f(x + z) \\ & = (2π)^{−n} ∫_{0}^{2π} ds_{1} ⋯ ∫_{0}^{2π} ds_{n}f(x_{1} + e^{y_{1}+is_{1}}, ..., x_{n} + e^{y_{n}+is_{n}}); \end{aligned}

(5.2) V (x, y) = V_{f}(x, y) = sup_{|z_{j}|=|e^{y_{j}} |} f(x + z) = sup_{|z_{j}|=1} f(x_{1} + z_{1}e^{y_{1}}, ..., x_{n} + z_{n}e^{y_{n}}).

Using well-known properties of plurisubharmonic functions we see that v and V are plurisubharmonic functions of the variables (x, y) ∈ ω × C^{n} if Re y_{j} ≪ 0. To be precise, they are plurisubharmonic in the open set

Ω_{0} = {(x, y) ∈ ω × C^{n}; x + z ∈ ω for all z with |z_{j}| ⩽ |e^{y_{j}}|}.

If ω is pseudoconvex, then so is Ω_{0}. Indeed, if d_{ω} now denotes the distance to C^{n} ∖ω measured by the norm ∥z∥_{∞} = max_{j} |z_{j}|, then Ω_{0} is the set of all (x, y) ∈ ω × C^{n} such that Re y_{j} < log d_{ω}(x). We note that (x, y) ↦ − log d_{ω}(x) + max_{j} Re y_{j} is plurisubharmonic in ω × C^{n}, which implies that Ω_{0}, as the subset of ω × C^{n} where this function is negative, is pseudoconvex.

We fix the domain of definition of v and V to be an open pseudoconvex set Ω contained in ω × C^{n} such that

(5.3)  (x, y) ∈ Ω, |z_{j}| ⩽ |e^{y_{j}}| implies x + z ∈ ω,

(this amounts to Ω ⊂ Ω_{0}) and

(5.4)  (x, y) ∈ Ω, Re y′_{j} ⩽ Re y_{j} implies (x, y′) ∈ Ω.

Then we know that v, V ∈ PSH(Ω). We define them as +∞ outside Ω. Moreover y ↦ v(x, y), V (x, y) are convex and coordinatewise increasing.

Sometimes it is necessary to assume also

(5.5)  \{ \begin{matrix}For every x ∈ ω and every y ∈ R_{+}^{n} there is a number t_{x,y} < +∞ \\ such that (x, ty) ∈ Ω implies Re t < t_{x,y}. \end{matrix}

Here R_{+}^{n} is the set of all vectors y ∈ R^{n} with y_{j} > 0 for every j. If ω is bounded, (5.5) of course follows from (5.3).

Definition 5.1. Let f ∈ PSH(ω) and define v by (5.1). Then the function

(5.6)  R_{+}^{n} ∋ y ↦ ν_{f}(x, y) = lim_{t→−∞} \frac{v(x, ty)}{t},  x ∈ ω,

shall be called the refined Lelong number of f at the point x. In the closed cone defined by y_{j} ⩾ 0 we define it by continuity: ν_{f}(x, y) = lim_{z→y,z_{j}>0} ν_{f}(x, z), and we set ν_{f}(x, y) = −∞ if some y_{j} is negative.

It is clear that ν_{f}(x, y) is positively homogeneous in y: ν_{f}(x, ty) = tν_{f}(x, y) for all y ∈ R^{n} and all t > 0. Moreover it is concave as a function of y ∈ R^{n} for fixed x. In fact, y ↦ ν_{f}(x, y) is the asymptotic function of the concave function y ↦ −v_{f}(x, −y) on R_{+}^{n}. Its values are finite on R_{+}^{n} except when f is equal to minus infinity in a whole neighborhood of x. Example. If f(z) = max_{j} (log |z_{j}|^{a_{j}}) with positive numbers a_{j}, then v(0, y) = max_{j}(a_{j}y_{j}) for y ∈ R_{+}^{n}, and the refined Lelong number at the origin is ν_{f}(0, y) = min_{j}(a_{j}y_{j}), y ∈ R_{+}^{n}, a simple concave function of y.

Comparing v and V is analogous to comparing u and U. We use Harnack’s inequality (2.5) with m = 2 repeatedly to get

f(z) ⩽ ∏_{1}^{n} \frac{1 − |z_{j}|/r_{j}}{1 + |z_{j}|/r_{j}} ∫_{}_{|y_{k}|=r_{k}} f(y),

provided f ⩽ 0 is plurisubharmonic in a neighborhood of the polydisk with radii r_{1}, ..., r_{n}. This gives

V (x, z) ⩽ ∏_{1}^{n} \frac{1 − e^{z_{j}−y_{j}}}{1 + e^{z_{j}−y_{j}}} v(x, y), z_{j} < y_{j} ⩽ log r_{j}.

If now f is only plurisubharmonic, we can apply this inequality to the function f − V ( · , y) ⩽ 0. In complete analogy with (2.6) we can write

V (x, z) ⩽ (1 − λ_{z−y})V (x, y) + λ_{z−y}v(x, y), z_{j} < y_{j},

where now

λ_{z} = ∏_{1}^{n} \frac{1 − e^{z_{j}}}{1 + e^{z_{j}}} , z_{j} < 0,

and it follows that the refined Lelong number can be defined using V as well as v.

One motivation for this variant of the Lelong number is in the study of blow-ups of plurisubharmonic functions. To give an idea of how blow-ups work, let us look at a simple example. Consider the mapping T (x) = (x_{1}, x_{1}x_{2}) from C^{2} into C^{2}. The pull-back under this mapping of the algebraic set

Y = {y ∈ C^{2}; y_{2} = 0} ∪ {y ∈ C^{2}; y_{2} = y_{1}^{2}}

is

X = T^{−1}(Y) = {x ∈ C^{2}; x_{1} = 0} ∪ {x ∈ C^{2}; x_{2} = 0} ∪ {x ∈ C^{2}; x_{1} = x_{2}}.

We see that in Y the two branches are tangent to each other at the origin, whereas the irreducible branches of X intersect transversally.

Now use the mapping T to blow up a plurisubharmonic function at the origin. The blow-up is g = T^{∗}f, g(x_{1}, x_{2}) = f(T (x)) = f(x_{1}, x_{1}x_{2}), and it turns out that the usual Lelong number of g at the origin is equal to ν_{f}(0, y) with y = (1, 2).

More generally, a blow-up transforms the refined Lelong number by a linear transformation: if

g(x) = T^{∗}f(x) = f(T (x)) = f(x_{1}^{m_{11}}x_{2}^{m_{12}} ⋯ x_{n}^{m_{1n}}, ..., x_{1}^{m_{n1}}x_{2}^{m_{n2}} ⋯ x_{n}^{m_{nn}})

with det(m_{jk}) ≠ 0, then

ν_{g}(0, y) = ν_{f}(0, z), where z_{j} = ∑ m_{jk}y_{k}.

In fact,

V_{g}(0, y) = sup_{|x_{j}|=e^{y_{j}}} f(x_{1}^{m_{11}}x_{2}^{m_{12}} ⋯ x_{n}^{m_{1n}}, ..., x_{1}^{m_{n1}}x_{2}^{m_{n2}} ⋯ x_{n}^{m_{nn}})

= sup_{t_{j}} f(e^{z_{1}+it_{1}}, ..., e^{z_{n}+it_{n}}) = V_{f}(0, z),

putting

x_{1}^{m_{j1}} ⋯ x_{n}^{m_{jn}} = exp (∑ m_{jk}y_{k} + i ∑ m_{jk}s_{k}) = exp(z_{j} + it_{j}).

Not only is |x_{1}^{m_{j1}} ⋯ x_{n}^{m_{jn}}| = exp z_{j}, but also all arguments of x_{1}^{m_{j1}} ⋯ x_{n}^{m_{jn}} occur independently if det(m_{jk}) ≠ 0.

Proposition 5.2. Let f ∈ PSH(ω), x ∈ ω, and put 1 = (1, 1, ..., 1) ∈ R^{n}. Then ν_{f}(x, 1) = ν_{f}(x).

Proof. The ball of radius e^{t} is contained in the polydisk with all radii equal to e^{t}, which in turn is contained in the ball of radius e^{t}\sqrt{n}. Therefore

U(x, t) ⩽ V (x, t, ..., t) ⩽ U(x, t + \frac{1}{2} log n).

Dividing by t and letting t tend to −∞ we get the result.

Proposition 5.3. If f ∈ PSH(ω), x is a point in ω, and y ∈ R_{+}^{n}, then

(min y_{j})ν_{f}(x) ⩽ ν_{f}(x, y) ⩽ (max y_{j})ν_{f}(x).

Proof. As already noted, the function y ↦ V (x, y) is increasing in each variable, so the result follows from the simple fact that the polydisk with radii y_{j} contains the polydisk with all radii equal to min y_{j} and is contained in the polydisk with all radii equal to max y_{j}. By homogeneity ν_{f}(x, (max y_{j})1) = (max y_{j})ν_{f}(x, 1) and similarly for the minimum. Then we use Proposition 5.2 to replace ν_{f}(x, 1) by ν_{f}(x).

Proposition 5.4. Fix a ∈ C^{n} and let f be any plurisubharmonic function in a neighborhood of a. If there exists a y ∈ R^{n} with all y_{j} ⩾ 0 such that ν_{f}(a, y) ⩾ ∑ y_{j},

then e^{−f} ∉ Lloc^{2}(a).

Proof. Without loss of generality we can assume that a = 0, that f is defined in a neighborhood of the unit polydisk and that V (0, 0) = 0. If there is a y ∈ R_{+}^{n} such that ν_{f}(0, y) ⩾ q ∑ y_{j} for some positive number q, then

sup_{|z_{k}|⩽exp(ty_{k})} f(z) = V (0, ty) = V (0, ty) − V (0, 0) ⩽ qt ∑ y_{j}, for all t ⩽ 0,

which implies that

f(z) ⩽ (max_{k} \frac{1}{y_{k}} log |z_{k}|) q ∑ y_{j}, |z_{j}| ⩽ 1,

or equivalently

e^{−f(z)} ⩾ min_{k} |z_{k}|^{−q ∑_{j} y_{j}/y_{k}} = min_{k} |z_{k}|^{−1/c_{k}}, |z_{j}| ⩽ 1,

putting c_{k} = y_{k}/q ∑ y_{j}. Then ∑ c_{k} = 1/q, and if q ⩾ 1 it follows from Lemma 5.5 below that e^{−f} does not belong to Lloc^{2}(0).

Now in general we know only that there is a y with y_{j} ⩾ 0 satisfying the condition. Then by continuity there are vectors y′ ∈ R_{+}^{n} satisfying ν_{f}(0, y′) ⩾ q ∑ y′_{j} with q arbitrarily close to 1. We can now let q tend to 1 and again conclude that e^{−2f} is not integrable; see the next lemma.

Lemma 5.5. Let c_{1}, ..., c_{n} be positive numbers and define q = 1/ ∑ c_{k}. The function

C^{n} ∈ z ↦ min_{k} |z_{k}|^{−1/c_{k}}

is in Lloc^{2}(0) if and only if q < 1. When q < 1, the square of its L^{2} norm over the unit polydisk is π^{n}/(1 − q), which tends to +∞ as q ↗ 1.

Proof. A direct calculation.

The converse of Proposition 5.4 does not hold in general, as can be seen from the simple example

f(z) = \frac{3}{2} log |z_{1} − z_{2}|, z ∈ C^{2}.

Here

ν_{f}(0, y) = \frac{3}{2} min y_{j} ⩽ \frac{3}{4}(y_{1} + y_{2}), y ∈ R^{2}.

But exp(−f) = |z_{1} − z_{2}|^{−3/2} ∉ Lloc^{2}(0).

However, for functions of (|z_{1}|, ..., |z_{n}|) there is a converse:

Proposition 5.6. Let f be a plurisubharmonic function of (|z_{1}|, ..., |z_{n}|) in a neighborhood of the origin, and assume that there is a number q satisfying 0 < q < 1 such that ν_{f}(0, y) ⩽ q ∑ y_{j} for all y ∈ R_{+}^{n}. Then e^{−f} ∈ Lloc^{2}(0).

For the proof we need a lemma on the behavior of the asymptotic function of a convex function.

Lemma 5.7. Let f be plurisubharmonic in a neighborhood of the closed unit polydisk. Suppose q is a positive number such that ν_{f}(0, y) ⩽ q ∑ y_{j} for all y ∈ R_{+}^{n}, and take any p > q. Then there exists a constant C such that v(0, y) ⩾ p ∑ y_{j} − C for all y ∈ R^{n} with y_{j} ⩽ 0.

Proof. We may assume that v(0, 0) = 0; then v(0, y) ⩽ 0 for all y with y_{j} ⩽ 0. Let Γ denote the simplex

Γ = {y ∈ R^{n}; y_{j} ⩽ 0, ∑ y_{j} = −1}.

Take a number s with p > s > q. The hypothesis that ν_{f}(0, y) ⩽ q ∑ y_{j} implies that there is a constant C_{y} for every y ∈ Γ such that v(ty) ⩾ −st − C_{y}; the question is whether this can be done in a uniform way. Choose finitely many points y^{k} ∈ Γ such that for every y ∈ Γ there is a k and a point z ∈ Γ such that

y^{k} = (1 − \frac{s}{p})z + \frac{s}{p}y.

Then by the convexity of y ↦ v(0, y),

v(0, ty^{k}) ⩽ (1 − \frac{s}{p})v(0, tz) + \frac{s}{p}v(0, ty) ⩽ \frac{s}{p}v(0, ty) t ⩾ 0.

There is a constant C such that v(ty^{k}) ⩾ −st − C for all k and all t ⩾ 0. Hence

v(0, ty) ⩾ \frac{p}{s}v(0, ty^{k}) ⩾ −pt − C, t ⩾ 0, y ∈ Γ,

which is just what we wanted to prove. Proof of Proposition 5.6. It is enough to consider f satisfying the conditions of Lemma 5.7. By the lemma there exists a number p < 1 and a finite constant C such that v(0, y) ⩾ p ∑ y_{j} −C for all y with y_{j} ⩽ 0. This means that f(z) ⩾ p log |z_{1} ⋯ z_{n}|−C and hence

e^{−2f(z)} ⩽ |z_{1} ⋯ z_{n}|^{−2p}e^{2C}, |z_{j}| ⩽ 1.

The right hand side is integrable.

The results proved so far imply that we can determine the integrability index from the refined Lelong number for functions of (|z_{1}|, ..., |z_{n}|):

Theorem 5.8. Let f be a plurisubharmonic function defined in a neighborhood of the origin in C^{n}, and let g be the function of (|z_{1}|, ..., |z_{n}|) whose mean values over any polycircle with center at the origin agree with those of f (thus v_{g}(x, y) = v_{f}(x, y) and ν_{g}(0, y) = ν_{f}(0, y) for all y). Then the integrability index of g is

ι_{g}(0) = sup [ν_{f}(0, y); y_{j} > 0, ∑ y_{j} = 1] ⩽ ι_{f}(0).

Proof. Combine Propositions 5.4 and 5.6.

6. Plurisubharmonic functions with attenuated singularities

With Ω satisfying (5.3) and (5.4) we define

(6.1)  f_{y,α}(x) = inf_{t} [V (x, ty) − αt],  y ∈ R^{n}, α ∈ R, x ∈ ω.

(Note that V (x, ty) = +∞ if (x, ty) ∉ Ω.) This function is plurisubharmonic in view of the minimum principle (Kiselman [1978]) since (x, t) ↦ V (x, ty) − α Re t is plurisubharmonic and independent of the imaginary part of t. We note that the set {y; (x, y) ∈ Ω} is convex in view of (5.4), and that the set {t ∈ R; (x, ty) ∈ Ω} is an interval. We shall study the Lelong number of f_{y,α}. The first result is a generalization of Lemma 2.2.

Lemma 6.1. Let a, b ∈ R_{+}^{n} and let s > 0. Define a ∧ b by (a ∧ b)_{j} = min(a_{j}, b_{j}). Then ν_{f_{b,α}}(x, a) ⩾ ν_{f}(x, a ∧ sb) − αs. In particular ν_{f_{a,α}}(x, a) ⩾ ν_{f}(x, a) − α.

Proof. By the definition of f_{b,α} we always have f_{b,α}(x) ⩽ V (x, sb) − sα for any s ∈ R. We now take the supremum over the polycircle {x + z; |z_{j}| = exp a_{j}}, with an arbitrary a ∈ R^{n}:

V_{f_{b,α}}(x, a) = sup_{|z_{j}|=e^{a_{j}}} f_{b,α}(x + z) ⩽ sup_{|z_{j}|=e^{a_{j}}} [V (x + z, sb) − sα]

= sup_{|z_{j}|=e^{a_{j}}} sup_{|w_{j}|=e^{sb_{j}}} [f(x + z + w) − sα]

= sup_{|z_{j}|=e^{y_{j}}} [f(x + z) − sα] = V_{f}(x, y) − sα,

where y ∈ R^{n} is defined by e^{y_{j}} = e^{a_{j}} + e^{sb_{j}}. Thus

V_{f_{b,α}}(x, a) ⩽ V_{f}(x, y) − sα

for every a and every sb, with y determined from the first two vectors. We now replace a by ta and s by ts with t < 0. Then

(6.2)  V_{f_{b,α}}(x, ta) ⩽ V_{f}(x, ty(t)) − tsα,

where now

y_{j}(t) = \frac{1}{t} log(e^{ta_{j}} + e^{tsb_{j}}),

and we can estimate y_{j}(t) from above and below, writing c = a ∧ sb,

c_{j} ⩾ y_{j}(t) ⩾ \frac{1}{t} log 2 + c_{j}.

Dividing (6.2) by the negative number t and letting t tend to −∞ we get the desired inequality, at least at points of continuity of y ↦ ν_{f}(x, y). If f equals minus infinity near x, all Lelong numbers are equal to +∞. If not, V_{f}(x, ·) is a convex function which is real-valued, hence continuous, in the open set y_{j} > 0, and V_{f}(x, ty(t))/t tends to its asymptotic function ν_{f}(x, c) as t → −∞.

Lemma 6.2. Assume Ω satisfies (5.3), (5.4) and (5.5). If α > ν_{f}(x, y), then f_{y,α}(x) > −∞, in particular ν_{f_{y,α}}(x, z) = 0 for all z ∈ R_{+}^{n}.

Proof. The number f_{y,α}(x) is the infimum of V (x, ty) − αt over all real t such that (x, ty) ∈ Ω. The expression is a convex function of t which is decreasing for t ≪ 0. By (5.5), t < t_{x,y} < +∞. Hence the infimum is finite.

Theorem 6.3. Let ω be open in C^{n} and let Ω be an open set in ω × C^{n} satisfying (5.3), (5.4) and (5.5). Let f ∈ PSH(ω) and define f_{y,α} by (6.1). Then ν_{f_{y,α}}(x, y) is determined by ν_{f}(x, y) and α:

(6.3)  ν_{f_{y,α}}(x, y) = (ν_{f}(x, y) − α)^{+},  x ∈ ω, y ∈ R_{+}^{n}, α ⩾ 0.

Proof. The proof is analogous to that of Theorem 2.1. The function α ↦ ν_{φ_{α}}(x, y) is convex, takes the value ν_{f}(x, y) for α = 0, and has the properties explained in Lemmas 6.1 and 6.2. This determines it uniquely.

7. Analyticity theorems for the refined Lelong number

Theorem 7.1 (Analyticity theorem). For any f ∈ PSH(ω), where ω is open in C^{n}, any y ∈ R^{n} with y_{j} > 0, and any real number c, the superlevel set

E_{y,c}^{ν}(f) = {x ∈ ω; ν_{f}(x, y) ⩾ c}

of the refined Lelong number is an analytic variety in ω.

This theorem was proved in April, 1986, and a proof was presented in June, 1986, at the conference Geometric and Quantitative Complex Analysis in Wuppertal. It is stated without proof in Kiselman [1987]. A proof was published in Kiselman [1992]. The theorem generalizes Siu’s theorem [1974] and is a special case of Demailly’s analyticity theorem [1987]. Proof. The functional κ_{φ}(x) = ν_{φ}(x, y) is comparable to the integrability index (Theorem 3.5 and Proposition 5.3) and is positively homogeneous in φ, so we can apply Theorem 4.3, letting Φ denote the set of all functions f_{y,α} with α ⩾ 0 and using the information in Theorem 6.3.

It is tempting to try to prove an analyticity theorem for numbers that are defined by varying only some of the variables, thus defining

V_{f,k}(x, y_{1}, ..., y_{k}) = sup_{|z_{j}|=e^{y_{j}}} f(x_{1} + z_{1}, ..., x_{k} + z_{k}, x_{k+1}, ..., x_{n})

and

ν_{f,k}(x, y_{1}, ..., y_{k}) = lim_{t→−∞} \frac{V_{f,k}(x, ty_{1}, ..., ty_{k})}{t}, x ∈ ω, (y_{1}, ..., y_{k}) ∈ R_{+}^{k}.

We could call ν_{f,k} a partial Lelong number, because it is the refined Lelong number of a restriction of f to a k-dimensional subspace of C^{n}. Evidently

V_{f,k}(x, y_{1}, ..., y_{k}) ⩽ V_{f}(x, y_{1}, ..., y_{n})

for all choices of y_{k+1}, ..., y_{n} ∈ R, so that

ν_{f,k}(x, y_{1}, ..., y_{k}) ⩾ ν_{f}(x, y_{1}, ..., y_{n}), x ∈ ω, y ∈ R_{+}^{n},

and

ν_{f,k}(x, y_{1}, ..., y_{k}) ⩾ lim ν_{f}(x, y_{1}, ..., y_{n}), x ∈ ω, y ∈ R_{+}^{k},

where the limit is for y_{k+1}, ..., y_{n} → +∞. It is natural to ask whether

(7.1)  ν_{f,k}(x, y_{1}, ..., y_{k}) = lim ν_{f}(x, y_{1}, ..., y_{n}),  x ∈ ω, y ∈ R_{+}^{k}.

Clearly

(7.2)  V_{f,k}(x, y_{1}, ..., y_{k}) = lim V_{f}(x, y_{1}, ..., y_{n}),

where the limit is now taken as y_{k+1}, ..., y_{n} tend to −∞, but this is not enough to guarantee equality in (7.1). In general there is no analyticity theorem for these partial Lelong numbers; see Wang [1991, Examples 4.2 and 4.3]. Under certain hypotheses, however, an analyticity theorem does hold; see Wang [1991, Theorems 4.10 and 4.11]. We shall now consider a very simple case of this kind, which moreover is of some geometric significance. We let L(C^{m}, C^{n}) denote the space of all linear mappings from C^{m} into C^{n}.

Theorem 7.2. Let f ∈ PSH(ω). Define

(7.3)  V′_{f}(x, h, z, y) = sup_{\begin{matrix}|x′|=e^{y_{0}} \\ |z′_{j}|=e^{y_{j}} \end{matrix}} f(x + x′ + h(z + z′))

for (x, h, z, y) ∈ ω × L(C^{m}, C^{n}) × C^{m} × R^{1+m} and

ν′_{f}(x, h, z, y) = lim_{t→−∞} \frac{V′_{f}(x, h, z, ty)}{t}

for (x, h, z, y) ∈ ω × L(C^{m}, C^{n}) × C^{m} × R_{+}^{1+m}. Then the superlevel set

E_{y,c}^{ν′}(f) = {(x, h, z) ∈ ω × L(C^{m}, C^{n}) × C^{m}; ν′_{f}(x, h, z, y) ⩾ c}

is an analytic subvariety of ω × L(C^{m}, C^{n}) × C^{m}.

Proof. For f ∈ PSH(ω) the function F (x, h, z) = f(x + h(z)) is plurisubharmonic in

Ω = {(x, h, z) ∈ ω × L(C^{m}, C^{n}) × C^{m}; x + h(z) ∈ ω}.

The refined Lelong number of this function is then well defined: we first take the supremum over a polycylinder

(7.4)  \begin{aligned}V_{F} & (X, Y) = V_{F} ((x, h, z), Y) = sup [F (X + X′); |X′_{j}| = e^{Y_{j}}] \\ & = sup [f(x + x′ + (h + h′)(z + z′)); |x′| = e^{y_{0}}, |h′_{jk}| = e^{y_{jk}}, |z′_{k}| = e^{y_{k}}] \end{aligned}

for

X = (x, h, z) ∈ Ω, Y = (y_{0}, (y_{jk}), (y_{k})) ∈ R^{1+mn+m},

and then define the refined Lelong number as

ν_{F} (X, Y) = lim_{t→−∞} \frac{V_{F} (X, tY)}{t}, X ∈ Ω, Y ∈ R_{+}^{1+mn+m}.

Here X = (x, h, z) is a point in Ω, and Y = (y_{0}, (y_{jk}), (y_{k})) denotes a direction in R × R^{mn} × R^{m}. The y_{jk} variables are those that correspond to the variation in h; the linear mapping h′ is given by h′(z)_{j} = ∑ h′_{jk}z_{k} and we let the h′_{jk} vary under the constraints |h′_{jk}| = e^{y_{jk}}. We shall see that if the y_{jk} are large compared with y_{0}, then

(7.5)  ν_{F} ((x, h, z), Y) = ν′_{f}(x, h, z, y).

Admitting this for a moment, we see that E_{y,c}^{ν′}(f) = E_{Y,c}^{ν}(F) if the vector y = (y_{0}, y_{1}, ..., y_{m}) ∈ R^{1+m} is completed in a suitable way to Y. By Theorem 7.1 the set E_{Y,c}^{ν}(F) is analytic, which proves the theorem. The passage to the limit in (7.1) is therefore particularly simple: the value is constant for all large values of those variables that tend to plus infinity. In Wang [1991, section 5] it is shown that there are other cases where the limit (7.1) gives an analyticity theorem for the partial Lelong number. It is however necessary to use polydisks with very small radii in some directions to get the result, which shows that the idea of using polydisks rather than balls is essential.

To prove (7.5) we shall compare the supremum in (7.3) with that in (7.4): in the first the mapping h is fixed, in the second it varies. Obviously V_{F} ((x, h, z), Y) ⩾ V′_{f}(x, h, z, y) for all choices of (y_{jk}), which gives ν_{F} ((x, h, z), Y) ⩽ ν′_{f}(x, h, z, y).

On the other hand, if we write

F (X + X′) = f(x + x′ + (h + h′)(z + z′)) = f(x + x′′ + h(z + z′))

with x′′ = x′ + h′(z + z′), then in the definition of V_{F} ((x, h, z), tY) (cf. (7.4)) we have

|x′′| ⩽ e^{ty_{0}} + ∑ e^{ty_{jk}}(|z_{k}| + e^{ty_{k}}) = e^{ty_{0}(t)},

where the last equality is a definition of y_{0}(t). We get

V_{F} ((x, h, z), tY) ⩽ V′_{f}(x, h, z, ty_{0}(t), ty_{1}, ..., ty_{m})

and

y_{0}(t) = \frac{1}{t} log (e^{ty_{0}} + ∑ e^{ty_{jk}}(|z_{k}| + e^{ty_{k}})) → y_{0} as t → −∞,

provided 0 < y_{0} < y_{jk} and y_{k} > 0 for all j and k, yielding

ν_{F} ((x, h, z), Y) = lim_{t→−∞} \frac{V_{F} ((x, h, z), tY)}{t} ⩾ lim_{t→−∞} \frac{V′_{f}(x, h, z, ty_{0}(t), ty_{1}, ..., ty_{m})}{t}
= ν′_{f}(x, h, z, y).

Given positive numbers y_{0} and y_{1}, ..., y_{m} it is therefore enough to choose y_{jk} such that y_{jk} > y_{0} > 0. This means that the change in the variable h is absorbed by the change in the variable x, and we can conclude that (7.5) holds if y_{jk} > y_{0} > 0 and y_{k} > 0.

Theorem 7.3. Define for f ∈ PSH(ω),

(7.6)  V′′_{f}(x, h, y) = sup_{|z_{j}|⩽e^{y_{j}}} f(x + h(z))

for (x, h, y) ∈ ω × GL(n, C) × R^{n} and

ν′′_{f}(x, h, y) = lim_{t→−∞} \frac{V′′_{f}(x, h, ty)}{t}

for (x, h) ∈ ω × GL(n, C) and y ∈ R_{+}^{n}. Then every superlevel set

E_{y,c}^{ν′′}(f) = {(x, h) ∈ ω × GL(n, C); ν′′_{f}(x, h, y) ⩾ c}

of ν′′_{f}(x, h, y) is an analytic subset of ω × GL(n, C).

Proof. We compare the supremum in (7.3) with that in (7.6). In the first, x varies, in the second x is fixed. Since h is now invertible, the variation in x can be absorbed by the variation in z, so that V′_{f}(x, h, 0, y) can be compared with V′′_{f}(x, h, y) as in the previous proof. Explicitly, we first note that we have V′′_{f}(x, h, y_{1}, ..., y_{n}) ⩽ V′_{f}(x, h, 0, y_{0}, ..., y_{n}) for all real y_{0}, y_{1}, ..., y_{n}, which gives ν′′_{f}(x, h, y) ⩾ ν′_{f}(x, h, 0, y_{0}, y) for all y ∈ R_{+}^{n}, y_{0} ∈ R_{+}. For the other direction we look at the supremum defining V′_{f}(x, h, 0, ty_{0}, ..., ty_{n}) (cf. (7.3)); we can write

x + x′ + h(z′) = x + h(h^{−1}(x′) + z′) = x + h(z′′),

where

|z′′_{j}| = |h^{−1}(x′)_{j} + z′_{j}| ⩽ ∥h^{−1}∥e^{ty_{0}} + e^{ty_{j}} = e^{ty_{j}(t)},

if we define

y_{j}(t) = \frac{1}{t} log (∥h^{−1}∥e^{ty_{0}} + e^{ty_{j}}) .

If y_{0} > y_{j}, then y_{j}(t) tends to y_{j} as t → −∞. Therefore

V′_{f}(x, h, 0, ty_{0}, ..., ty_{n}) ⩽ V′′_{f}(x, h, ty_{1}(t), ..., ty_{n}(t)),

which in turn implies

ν′_{f}(x, h, 0, y_{0}, ..., y_{n}) ⩾ ν′′_{f}(x, h, y_{1}, ..., y_{n}), y_{j} > 0, y_{0} > max(y_{1}, ..., y_{n}).

Given any y ∈ R_{+}^{n} we choose y_{0} > max y_{j} and then have E_{y,c}^{ν′′}(f) = E_{(y_{0},y),c}^{ν′}(f). By Theorem 7.2 this is an analytic set.

Finally we shall study another kind of attenuation of the singularities. We define

(7.7)  g_{η}(x) = inf_{y} [V (x, y) − Re η · y; (x, y) ∈ Ω],  x ∈ ω, η ∈ R^{n},

where η · y = η_{1}y_{1} + ⋯ + η_{n}y_{n}. Question. What is the refined Lelong number ν_{g_{η}}(x, y) of g_{η}? Is there a formula like (6.3)?

We shall collect a few results concerning the functions g_{η} which are not enough to answer this question, but which suffice nevertheless to prove an analyticity theorem using them.

Lemma 7.4. Let f ∈ PSH(ω) and define g_{η} by (7.7). Then

ν_{g_{η}}(x, a) ⩾ ν_{f}(x, a ∧ b) − η · b,  η ∈ R^{n}, x ∈ ω, a, b ∈ R^{n};
ν_{g_{η}}(x, a) ⩾ sup_{b} (ν_{f}(x, a ∧ b) − η · b),  η ∈ R^{n}, x ∈ ω, a ∈ R^{n}.

Proof. We always have g_{η} ⩽ f_{b,α} if η · b = α. Using Lemma 6.1 we see that

ν_{g_{η}}(x, a) ⩾ ν_{f_{b,α}}(x, a) ⩾ ν_{f}(x, a ∧ b) − α = ν_{f}(x, a ∧ b) − η · b.

Lemma 7.5. Assume Ω that satisfies (5.3) and (5.4). Let f ∈ PSH(ω) and define g_{η} by (7.7). Take a point x ∈ ω. If a ∈ R_{+}^{n} and ν_{f}(x, a) < c, then there exists a vector η ∈ R^{n} such that η · a < c and g_{η}(x) > −∞. In particular ν_{g_{η}}(x, z) = 0 for all z ∈ R_{+}^{n}.

Proof. If ν_{f}(x, a) < c, there exists an affine minorant of V on the straight line defined by a: if we take a number t_{0} with −t_{0} large enough, there is a number b such that ν_{f}(x, a) ⩽ b < c and

−∞ < V (x, t_{0}a) + b(t − t_{0}) ⩽ V (x, ta)

for every t ∈ R. By the Hahn-Banach theorem, we can find an affine minorant in the whole space extending this one, which means that we can find η ∈ R^{n} such that η · a = b < c and

V (x, t_{0}a) + η · (y − t_{0}a) ⩽ V (x, y)

for every y ∈ R^{n}. Consequently

−∞ < C = V (x, t_{0}a) − η · (t_{0}a) ⩽ V (x, y) − η · y

and −∞ < C ⩽ inf_{y}(V (x, y) − η · y) = g_{η}(x).

Theorem 7.6. Let ω be an open set in C^{n} and Ω an open set in ω × C^{n} satisfying (5.3) and (5.4). Let g_{η} be given by (7.7). Define G_{x,z}(η) = ν_{g_{η}}(x, z). This is a convex function of η ∈ R^{n}, and its Fenchel transform

G̃_{x,z}(y) = sup_{η} [η · y − G_{x,z}(η)] = sup_{η} [η · y − ν_{g_{η}}(x, z)], y ∈ R^{n},

satisfies

(7.8)  −ν_{f}(x, y) ⩽ G̃_{x,z}(−y) ⩽ −ν_{f}(x, y ∧ z),  y, z ∈ R_{+}^{n}.

In particular G̃_{x,z}(−y) = −ν_{f}(x, y) when y ⩽ z. The function (y, z) ↦ G̃_{x,z}(−y) is convex in R^{n} × R^{n}.

Proof. From Lemma 7.4 we see that −ν_{g_{η}}(x, z) ⩽ −ν_{f}(x, y ∧ z) + η · y, which immediately gives the inequality to the right in (7.8).

If ν_{f}(x, y) < c, then by Lemma 7.5 there is an η ∈ R^{n} such that η · y < c and ν_{g_{η}}(x, z) = 0 for all z ∈ R^{n}. Hence by definition G̃_{x,z}(−y) ⩾ −c. Letting c tend to ν_{f}(x, y), we get the inequality to the left in (7.8). This proves the theorem.

Can Theorem 4.3 be applied to prove Theorem 7.1 using instead the g_{η}? We do not know enough about the Lelong numbers of g_{η} to do that—Theorem 4.3 requires an exact formula for the values of the functional. But the more flexible Theorem 4.2 will suffice. To see this, let Φ be the set of all functions g_{η} with η · y < c, where we define g_{η} by (7.7). We shall first check the hypothesis (4.3) of Theorem 4.2. Lemma 7.4 yields

κ_{g_{η}}(x) = ν_{g_{η}}(x, y) ⩾ ν_{f}(x, y) − η · y ⩾ c − η · y > 0

for any point x ∈ X = E_{y,c}^{ν}(f). Thus (4.3) holds. Next, if ν_{f}(x, y) < c, then there is an η such that η · y < c, thus g_{η} ∈ Φ, and κ_{g_{η}}(x) = 0 (Lemma 7.5). This means that also (4.4) holds, and we can conclude again, now using Theorem 4.2, that X = E_{y,c}^{ν}(f) is analytic.

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Author’s address: Uppsala University, Department of Mathematics, P. O. Box 480, SE-751 06 Uppsala, Sweden.