VALUE DISTRIBUTION FOR SEQUENCES OF RATIONAL MAPPINGS AND COMPLEX DYNAMICS

ALEXANDER RUSSAKOVSKII AND BERNARD SHIFFMAN

Abstract. We obtain results on the asymptotic equidistribution of the pre-images of linear subspaces for sequences of rational mappings between projective spaces. As an application to complex dynamics, we consider the iterates P_{k} of a rational mapping P of ℙ^{n}. We show, assuming a condition on the topological degree λ of P, that there is a probability measure μ on ℙ^{n} such that the discrete measures λ^{−k}P_{k}^{∗}δ_{w} converge to μ for all w ∈ ℙ^{n} outside a pluripolar set.

1. Introduction

The study of value distribution for sequences of mappings may be regarded as an analogue of Nevanlinna theory; instead of studying the asymptotic behavior of the area of a pre-image of an analytic set in a ball when its radius tends to infinity, one investigates the asymptotics of the pre-images of an analytic set under a sequence of mappings. One of the main reasons for studying this subject is its applications to complex dynamics, although value distribution theory for sequences is also of independent interest.

Many investigations have the Brolin-Lyubich Theorem (see [Br, Ly, FLM]) as their starting point. This theorem can be formulated as follows:

Let R(z) be a rational function of degree d ≥ 2, and let R_{k} be its k-th iterate. Then there is an invariant probability measure μ on ℙ^{1} = ℂ ∪ {∞} such that for all w outside an exceptional set E ⊂ ℙ^{1} containing at most 2 points,

\frac{1}{d^{k}} (R_{k})^{∗}δ_{w} → μ.

Here (R_{k})^{∗}δ_{w} is a discrete measure in ℙ^{1}, counting the number of roots of the equation R_{k}(z) = w with multiplicities; convergence is in the weak sense.

Date: March 28, 1996.

Shiffman’s research was partially supported by National Science Foundation Grant DMS-9500491.

The research of both authors at MSRI is partially supported by National Science Foundation Grant DMS-9022140.

Many of the recent papers on holomorphic dynamics in several complex variables (see [BS1, BS2, BS3, BLS1, BLS2, HP, FS1, FS2, FS3, FS4]) have dealt with various extensions of this theorem to n variables in particular cases. This result was also extended by Sodin [So] to the case where {R_{n}} is an arbitrary sequence of rational functions of one variable with rapidly increasing degrees. Sodin proved that if one agrees to omit a larger set E of exceptional values, then the pre-images of the remaining values w ∈ ℙ^{1} \ E are equidistributed in a certain sense.

In a recent paper [RS], the phenomenon of equidistribution for sequences of polynomial mappings ℂ^{n} → ℙ^{m} was studied. The phenomenon and the corresponding results may be described as follows. We let ω = ω_{n} denote the Kähler form of the Fubini-Study metric on complex projective n-space ℙ^{n} (normalized so that ∫_{ℙ^{n}} ω^{n} = 1).

Suppose that {P_{k}} is a sequence of polynomial mappings ℂ^{n} → ℙ^{m}, such that ∑ \frac{1}{δ_{1}(P_{k})} < ∞, where δ_{1}(P_{k}) denotes the maximal degree of the components of P_{k}. Then

(i) pre-images of all but an exceptional pluripolar set of complex hyperplanes in ℙ^{m} are equidistributed with the pull-backs P_{k}^{∗}ω.

(ii) if all P_{k} are non-degenerate, then pre-images of all but a pluripolar set of points in ℙ^{m} are equidistributed with P_{k}^{∗}ω^{m} (m ≤ n).

The exceptional sets in (i) and (ii) above are described in terms of a “proximity sequence.”

In [RS], only the cases of codimension 1 and m were treated. The present paper deals with all intermediate codimensions 1 ≤ ℓ ≤ min(n, m) and rational mappings ℙ^{n} → ℙ^{m} (Theorem 1.1) as well as giving a higher dimensional version (with pluripolar exceptional set) of the Brolin-Lyubich theorem (Theorem 1.3).

In order to state our results on the asymptotic equidistribution of pre-images of linear subspaces of intermediate dimension, we let 𝔾(ℓ, m) denote the Grassmannian of projective linear subspaces of codimension ℓ in ℙ^{m}. Note that 𝔾(m, m) = ℙ^{m}. If P : ℙ^{n}−→ℙ^{m} is a meromorphic (i.e., rational) map, we let δ_{ℓ}(P) denote the degree of P^{−1}(W) for generic W ∈ 𝔾(ℓ, m) (1 ≤ ℓ ≤ min(n, m)). (For generic W, P^{−1}(W) has pure codimension ℓ.) One easily sees that δ_{1}(P) is the degree of the polynomials in a representation of P (using homogeneous coordinates); if P is holomorphic (i.e., regular), then δ_{ℓ}(P) = δ_{1}(P)^{ℓ}. In Section 4, we give an analytic description of the “intermediate degrees” δ_{ℓ}(P) and we show that δ_{k+ℓ}(P) ≤ δ_{k}(P)δ_{ℓ}(P). Intermediate degrees have been considered also in [Fr].

Our first result on the equidistribution of pre-images under an arbitrary sequence of rational maps of projective spaces is as follows:

Theorem 1.1. Let {P_{k}} be a sequence of rational mappings from ℙ^{n} to ℙ^{m}. Let 1 ≤ ℓ ≤ min(n, m), and let {a_{k}} be a sequence of positive numbers such that

∑_{k=1}^{∞} \frac{δ_{ℓ−1}(P_{k})}{a_{k}} < +∞ .

Then there exists a pluripolar subset E of 𝔾(ℓ, m) such that

\frac{1}{a_{k}} (P_{k}^{∗}[W] − P_{k}^{∗}ω^{ℓ}) → 0

as k → ∞ for all W ∈ 𝔾(ℓ, m) \ E.

Here, convergence means weak convergence in D′^{ℓ,ℓ}(ℙ^{n}). In Section 6, we give a description of the exceptional set E in terms of the “proximity sequence.” The pull-back P_{k}^{∗}ω^{ℓ} is smooth off the indeterminacy locus of P and has locally integrable coefficients. For generic W, P_{k}^{∗}[W] is given by integration over P_{k}^{−1}(W). Precise definitions of the pull-back currents P_{k}^{∗}[W], P_{k}^{∗}ω^{ℓ} ∈ D′^{ℓ,ℓ}(ℙ^{n}) are given in Section 3.

As an application of Theorem 1.1 to complex dynamics, we have the following result on the equidistribution of iterated pre-images: Corollary 1.2. Let P : ℙ^{n}−→ℙ^{n} be a rational mapping, and let P_{k} denote the k-th iterate of P. If a > δ_{ℓ−1}(P), then

\frac{1}{a^{k}} (P_{k}^{∗}[W] − P_{k}^{∗}ω^{ℓ}) → 0

for all W ∈ 𝔾(ℓ, m) outside a pluripolar set.

To obtain Corollary 1.2 from Theorem 1.1, we use the fact that δ_{ℓ−1}(P_{k}) ≤ δ_{ℓ−1}(P)^{k}, which we verify in Section 4 (see Lemma 4.6).

For holomorphic maps P : ℙ^{n} → ℙ^{n} of degree d ≥ 2 it was shown in [HP] and [FS4] that \frac{1}{d^{k}} log |P_{k}|^{2} converges uniformly to a Green’s function G (where P_{k} is the k-th iterate of P), and hence by Bedford-Taylor [BT],

\frac{1}{d^{nk}} P_{k}^{∗}ω^{n} → (dd^{c}G)^{n} = μ ,

where μ is a probability measure on ℙ^{n} which is invariant in the sense that P_{∗}μ = μ.

However, Fornaess and Sibony [FS4] showed that for a non-holomorphic map P : ℙ^{2}−→ℙ^{2} of degree d, the number of points in the generic fibre of P is strictly less than d^{2}. (We give a generalization of this fact as Lemma 4.4.) Thus, for such P it follows that

\frac{1}{d^{2k}} P_{k}^{∗}ω^{2} → 0 .

So the question of existence of a nontrivial limit measure in the meromorphic case remained open. The following result gives a limit measure and equidistribution for the iterates of meromorphic maps on ℙ^{n}:

Theorem 1.3. Let P : ℙ^{n}−→ℙ^{n} be a rational map, let λ = δ_{n}(P) denote the topological degree of P, and write

μ_{k} = \frac{1}{λ^{k}} P_{k}^{∗}ω^{n} .

If λ > δ_{n−1}(P), then the sequence {μ_{k}} converges to a probability measure μ on ℙ^{n} and

\frac{1}{λ^{k}} P_{k}^{∗}[W] → μ

for all points W ∈ ℙ^{n} outside a pluripolar set.

Remark: We have for any test function φ,

(P_{∗}μ_{k+1}, φ) = \frac{1}{λ^{k+1}} ∫ (φ ∘ P)P_{k+1}^{∗}ω^{n} = \frac{1}{λ^{k+1}} ∫ P^{∗}(φP_{k}^{∗}ω^{n}) = \frac{1}{λ^{k}} ∫ φP_{k}^{∗}ω^{n} = (μ_{k}, φ) .

Thus

P_{∗}μ_{k+1} = μ_{k} .  (1)

If μ(I_{P}) = 0, then P_{∗}μ would be well-defined and hence we could let k → ∞ in (1) to conclude that μ is an invariant measure, i.e., P_{∗}μ = μ.

Further results will be given in a subsequent paper.

2. Examples and open questions

We begin with several elementary examples of sequences of iterated mappings illustrating our results. These examples are the extensions to ℙ^{2} of proper polynomial mappings of ℂ^{2}. Besides equidistribution, we are going to look at invariance properties of the limit currents and measures. Example 1. (This example was discussed in [RS].) Let P_{k}(z, w) be the k-th iterate of the mapping P = (z^{δ}, w^{δ}) : ℂ^{2} → ℂ^{2}. We first restrict our attention to ℂ^{2}. It is easy

to see that the uniform convergence \frac{1}{δ^{k}} log(1 + |P_{k}(z, w)|^{2}) → G(z, w) takes place, where

G(z, w) = sup(log^{+} |z|^{2}, log^{+} |w|^{2})

is the plurisubharmonic Green function of the unit polydisk with logarithmic growth at infinity. It follows that

dd^{c}G = lim_{k→∞} \frac{1}{δ^{k}} P_{k}^{∗}ω.

The current T = dd^{c}G is concentrated on the boundary of the unit polydisk and on the set {|z| = |w| > 1}. This current is invariant under the mapping P in the sense that

P^{∗}T = δT .

(The pull-backs of closed positive (1, 1)-currents by holomorphic and meromorphic maps are defined in Section 3.) According to the results of [RS], this current is the limit of pre-images of all nonexceptional hyperplanes (lines).

Every hyperplane of the form z = c or w = c has pre-images of similar form. These pre-images tend towards the cylinder |z| = 1 or |w| = 1, respectively. Thus all such hyperplanes are exceptional. A third family of exceptional hyperplanes consists of those passing through the origin. Pre-images of these hyperplanes tend towards the “cone” {|z| = |w|}. We now consider the complex projective plane ℙ^{2} = ℂ^{2} ∪H_{∞}, identifying (z, w) ∈ ℂ^{2} with (1, z, w) ∈ ℙ^{2}. We let ℙ^{2∗} ≈ ℙ^{2} denote the parameter space of hyperplanes in ℙ^{2}; the point (ζ_{0}, ζ_{1}, ζ_{2}) ∈ ℙ^{2∗} represents the hyperplane {(z, w) ∈ ℂ^{2} : ζ_{0} + ζ_{1}z + ζ_{2}w = 0}. (The point (1, 0, 0) represents the hyperplane at infinity, H_{∞}.) Thus we see that the set of exceptional hyperplanes consists of the three pencils in ℙ^{2∗} : {ζ_{0} = 0}, {ζ_{1} = 0}, {ζ_{2} = 0}. We note that, besides T, there are at least five more (linearly independent) invariant closed (1, 1)-currents (on ℂ^{2}), T_{1} = dd^{c} log^{+} |z|^{2}, T_{2} = dd^{c} log^{+} |w|^{2}, T_{3} = dd^{c} log max(|z|^{2}, |w|^{2}), T_{4} = [H_{1}] = dd^{c} log |z|^{2}, T_{5} = [H_{2}] = dd^{c} log |w|^{2} (where H_{1} = {z = 0}, H_{2} = {w = 0}) with the same property

P^{∗}T_{i} = δT_{i}.

In fact, they are limits of pre-images of the corresponding exceptional hyperplanes. (These currents are invariant on ℙ^{2} as well as on ℂ^{2}, and we have the additional invariant current T_{6} = [H_{∞}] on ℙ^{2}.)

Now consider pre-images of points. The measure μ = T^{2} = (dd^{c}G)^{2}, being

the limit of \frac{1}{δ^{2k}} P_{k}^{∗}ω^{2}, is concentrated on the distinguished boundary of the unit polydisk; hence pre-images of most points must tend to the torus by Theorem 2 of [RS]. However, all points of the form (0, c) or (c, 0) have pre-images of the form (0, c′) (respectively (c′, 0)) and are definitely exceptional (as well as the points of H_{∞}). So in this case the exceptional set is the same union of 3 hyperplanes in ℙ^{2∗}.

The measure μ possesses invariance properties, P^{∗}μ = δ^{2}μ. Note that we also have μ = T_{1} ∧ T_{2}.

The map P extends to a rational mapping: Q(t, z, w) = (t^{δ}, z^{δ}, w^{δ}) : ℙ^{2} → ℙ^{2}, which we call the projectivization of P. (For this example, Q is holomorphic.) The situation then becomes very symmetric with respect to all variables. To simplify the terminology, we will call the corresponding currents (measures) “projectivizations” of the ones defined on ℂ^{n}. Instead of, say, dd^{c} log^{+} |z|^{2} = dd^{c} log(|z|^{2} ∨ 1) one has to consider dd^{c} log(|z|^{2} ∨|t|^{2}), where ∨ stands for maximum, and so on. The same points and hyperplanes are exceptional, and because of the absence of the indeterminacy set, no difficulty occurs.

Example 2. The situation changes if we consider the iterates of the mapping P = (z^{d_{1}}, w^{d_{2}}) : ℂ^{2} → ℂ^{2} with d_{1} ≠ d_{2}. For simplicity, let P = (z^{2}, w^{3}). It is easy to see

that in this case \frac{1}{3^{k}} log(1 + |P_{k}(z, w)|^{2}) → G(z, w), where now G(z, w) = log^{+} |w|^{2}. It follows as before that

dd^{c}G = lim_{k→∞} \frac{1}{3^{k}} P_{k}^{∗}ω;

however this time the current T = dd^{c}G is concentrated on the cylinder {|w| = 1}. It is invariant for the mapping P:

P^{∗}T = 3T .

Applying the results of [RS], we see that this current is the limit of the pre-images of all nonexceptional hyperplanes.

The set of exceptional hyperplanes in ℂ^{2} consists of the two pencils in ℙ^{2∗} : {ζ_{0} = 0}, {ζ_{2} = 0}. Besides T, there are at least three more invariant currents: T_{1} = dd^{c} log^{+} |z|^{2}, T_{2} = dd^{c} log |z|^{2}, T_{3} = dd^{c} log |w|^{2}. However this time

P^{∗}T_{1} = 2T_{1} .

(Note that for any smooth form S on ℂ^{2}, we have P_{∗}P^{∗}S = 6S; by smoothing and taking limits, one sees that this identity is also valid if S is a positive (1, 1)-current on ℂ^{2}. Hence P_{∗}T_{1} = 3T_{1}, P_{∗}T = 2T.) The currents T_{i} are the limits of pre-images of the corresponding families of exceptional hyperplanes. The coefficient 2 for T_{1} makes the situation somewhat different as we shall see.

Now consider pre-images of points. The results of [RS] do not provide useful information since now T^{2} = (dd^{c}G)^{2} ≡ 0. According to Theorem 1.3, this is not a surprise, since δ_{2}(P) = 6, not 9, and we have a limit

μ = lim_{k→∞} \frac{1}{6^{k}} P_{k}^{∗}ω^{2},

which is the same measure (concentrated on the distinguished boundary of the unit polydisk) as in Example 1. So pre-images of most points must concentrate there. Again, the exceptional set for points is the same union of 3 hyperplanes in ℙ^{2}. The measure μ possesses the invariance property P^{∗}μ = 6μ. Note that we also have μ = T ∧ T_{1}.

Consider the projectivization of this mapping: Q(t, z, w) = (t^{3}, tz^{2}, w^{3}) : ℙ^{2} → ℙ^{2} and the sequence Q_{k} of its iterations. Note that although P is holomorphic, Q is only meromorphic and has one indeterminacy point (0, 1, 0) which is the reason for all “anomalies.” Since H_{∞} is contracted by Q to the fixed point (0, 0, 1), the map Q is “generic” in the sense of Fornaess and Sibony [FS3, FS4]. Also, the graph G of the mapping Q is singular. So it is really necessary to resolve the singularities of G to define all our currents correctly (see Section 3). Note that we have δ_{1}(Q_{k}) = 3^{k}, δ_{2}(Q_{k}) = 6^{k} < δ_{1}^{2}(Q_{k}). In accordance with the results of [FS3, FS4], there is an invariant (1, 1)-current T = dd^{c} log(|w|^{2} ∨ |t|^{2}) which is the projectivization of the above mentioned current on ℂ^{2}. The projectivization of the other invariant current, T_{1}, is no longer invariant, since now Q^{∗}T_{1} = 2T_{1} + H_{∞}.

The situation is even more complicated for the point case. The results of [FS3, FS4] do not provide a nontrivial invariant measure since T^{2} = 0. According to Theorem 1.3, there is a limit measure μ′ in this case. In fact, μ′ = [dd^{c} log(|z|^{2}∨|w|^{2}∨|t|^{2})]^{2} is the projectivization of μ and is invariant on ℙ^{2}. We have μ′ = T ∧ T_{1} as before, although now T_{1} is not invariant itself. The measure μ′ has the property Q^{∗}μ′ = 6μ′.

Example 3. Finally, consider the sequence of iterations of P (z, w) = (w^{3}, z^{2}). This example is not generic in the sense of Fornaess and Sibony [FS3, FS4], since H_{∞} is contracted by (the projectivization of) P to the indeterminacy point (0, 1, 0), so their results cannot be applied. Since δ_{1}(P_{k}) ≠ δ_{1}^{k}(P) (which is the case for non-generic maps) we cannot conclude that the sequence of currents \frac{1}{δ_{1}(P_{k})}P_{k}^{∗}ω converges. Instead we have two subsequences

{P_{2k} = (z^{6^{k}}, w^{6^{k}})} , {P_{2k+1} = (w^{3·6^{k}}, z^{2·6^{k}})} .

The first subsequence is the same as in Example 1 with δ = 6, and the corresponding current subsequence thus converges to

T_{1} = dd^{c} log^{+}(|z|^{2} ∨ |w|^{2}).

The second subsequence of currents converges to

T_{2} = dd^{c} log^{+}(|w|^{2} ∨ |z|^{4/3}) .

These currents are responsible for the distribution of pre-images of hyperplanes. As for the invariance properties, there seems to be no invariant current since we have P^{∗}T_{1} = 3T_{2}, P^{∗}T_{2} = 2T_{1}.

For the codimension two case, we have δ_{2}(P_{k}) = δ_{2}^{k}(P) = 6^{k}. So there is a limit measure which is the same as in the two previous examples and is invariant.

We state here some open problems. In the Brolin-Lyubich Theorem, the exceptional set consists of two points. In the above examples, the exceptional sets are unions of at most 3 hyperplanes in ℙ^{2}. This leads to the questions:

Can we further describe the exceptional set in Theorem 1.3? Is the exceptional set in fact algebraic?

Another question is:

Does the measure μ in Theorem 1.3 charge the indeterminacy set of P?

If the answer to this question is “no”, then by the remark following Theorem 1.3, P_{∗}μ would be well-defined and μ would be an invariant measure.

3. Notation and terminology

We let E^{p,q}(X), D^{p,q}(X), D′^{p,q}(X) denote the spaces of (complex-valued) C^{∞} forms, compactly supported C^{∞} forms, and currents, respectively, of bidegree (p, q) on a complex manifold X and we use the standard differentials d = ∂ + \overline{∂}, d^{c} = (4π\sqrt{−1})^{−1}(∂ − \overline{∂}). Points in complex projective n-space ℙ^{n} are identified with their representations z = (z_{0}, z_{1}, …, z_{n}) in homogeneous coordinates. We shall regard the Grassmannian 𝔾(ℓ, m) of projective linear subspaces of codimension ℓ in ℙ^{m} as a subvariety of ℙ(^{⋀m+1−ℓ}ℂ^{m+1}).

If P : ℙ^{n}−→ℙ^{m} is a non-constant meromorphic map, it is a well-known consequence of Chow’s theorem that P must be rational, i.e., P can be written in the form P = (P_{0}, …, P_{n}) where P_{j} ∈ ℂ[z_{0}, …, z_{n}] and deg P_{0} = ⋯ = deg P_{m} = d. We can assume that the P_{j} have no common factors; we then say that deg P = d. (This notion of degree should not be confused with the topological degree of an equidimensional rational map, which is the number of points in the pre-image of a generic point in the range.) We let I_{P} ⊂ ℙ^{n} denote the indeterminacy locus of P (the points where P is not holomorphic); I_{P} is an algebraic subvariety of codimension ≥ 2.

Suppose f : Y → X is a holomorphic mapping of complex manifolds. If α is a current on X, f^{∗}α is not always defined. However, we shall define f^{∗}α in two special cases: First, we suppose α = uγ where γ ∈ E^{p,q}(X) is a smooth form and u is the difference of plurisubharmonic functions. Assume further that f(Y) is not contained in the ±∞ locus of u. Then u ∘ f is the difference of plurisubharmonic functions on Y and hence is in L_{loc}^{1}(Y). We define f^{∗}α = (u ∘ f)f^{∗}γ, which is clearly independent of the representation α = uγ. The second case we consider is that of a current of the form [D] ∧ γ, where [D] is the current of integration over a divisor D on X and γ is a smooth form as before. We assume also that f(Y) ⊄ Supp D so that f^{∗}D is a divisor on Y. We then define f^{∗}([D] ∧ γ) = [f^{∗}D] ∧ f^{∗}γ. These two definitions are consistent in the following way. Suppose α = log |g|^{2} · γ where g is a meromorphic function on X such that neither the zeroes nor the poles of g contain f(Y) and γ is a closed (p, q)-form on X. Then dd^{c}α = [D] ∧ γ where D = Div(g). Hence

f^{∗}dd^{c}α = [f^{∗}D] ∧ f^{∗}γ = dd^{c}(log |g ∘ f|^{2} · f^{∗}γ) = dd^{c}f^{∗}α .  (2)

Let P : ℙ^{n}−→ℙ^{m} be a rational map. For a smooth (p, q)-form η ∈ E^{p,q}(ℙ^{m}) we define the pull-back current P^{∗}η ∈ D′^{p,q}(ℙ^{m}) as follows: We let G_{P} ⊂ ℙ^{n} × ℙ^{m} denote the graph of P (which is an irreducible algebraic subvariety of ℙ^{n} × ℙ^{m}) and we consider a desingularization G̃ →^{ρ} G_{P}. We have the commutative diagram:

G̃ ; π_{1} ↙  ↘ π_{2} ; ℙ^{n}  −→^{P} ℙ^{m}  (3)

We then define

P^{∗}η = π_{1∗}π_{2}^{∗}η .

Note that P^{∗}η has coefficients in L_{loc}^{1} and has singular support contained in the indeterminacy locus I_{P} of P. In fact, (P |_{ℙ^{n}\I_{P}})^{∗}η is the usual pull-back of the form η.

We consider the current of integration [W] ∈ D′^{ℓ,ℓ}(ℙ^{m}) and define the pull-back π_{2}^{∗}[W] to be the current of integration over the algebraic (n − ℓ)-cycle π_{2}^{∗}W on G̃ (using the diagram (3)), whenever dim π_{2}^{−1}(W) = n − ℓ. If we represent W as the intersection of hyperplanes H_{1}, …, H_{ℓ} in ℙ^{m}, then π_{2}^{∗}W is the intersection of divisors π_{2}^{∗}H_{1} ∩⋯∩π_{2}^{∗}H_{ℓ}. For a definition of this intersection, which is a formal sum of the irreducible components of π_{2}^{−1}(W) with positive integer coefficients, see [Ha, Appendix A] or Definitions 2.3 and 2.4.2 (or Example 7.1.10) in [Fu]. This pull-back, or intersection, can also be defined analytically as follows. Let g_{j} be a local defining function for π_{2}^{∗}H_{j}, for 1 ≤ j ≤ ℓ, and write g = (g_{1}, …, g_{ℓ}). Then by Griffiths and King [GK, 1.10] (see also [Sh, I.12, Th. 3]), we have the local formula

π_{2}^{∗}[W] = dd^{c} (log |g|^{2}(dd^{c} log |g|^{2})^{ℓ−1}) .

(Alternately, π_{2}^{∗}[W] = dd^{c} log |g_{1}|^{2} ∧ … ∧ dd^{c} log |g_{ℓ}|^{2}, where the existence of this product of currents is guaranteed by Demailly [De].) One way to verify that these definitions are all equivalent is to first consider generic hyperplanes H_{1}, …, H_{ℓ} so that the divisors π_{2}^{∗}H_{1}, …, π_{2}^{∗}H_{ℓ} are smooth hypersurfaces (of multiplicity 1) in G̃ intersecting transversely. (This is possible by Lemma 4.1 in the next section.) Then for this case, π_{2}^{∗}[W] is the current of integration over a smooth submanifold. In the general case, the current π_{2}^{∗}[W] is the weak limit of the pull-backs of generic intersections W^{ν} = H_{1}^{ν} ∩ ⋯ ∩ H_{ℓ}^{ν} converging to W. The existence of the limit follows, for example, from [Fu, Cor. 11.1] for the algebraic definition and from [De] or [FS5, Cor. 3.6] for the analytic definition.

We now state the Poincaré-Lelong formula for linear subspaces of ℙ^{m} and describe its pull-backs by a rational map P : ℙ^{n}−→ℙ^{m}. Let W ∈ 𝔾(ℓ, m) be an (m−ℓ)-plane in ℙ^{m}. For each W ∈ 𝔾(ℓ, m), we define the current

Λ_{W} = log \frac{|ζ|^{2}|W |^{2}}{|ζ ∧ W |^{2}} ∑_{j=0}^{ℓ−1}(d_{ζ}d_{ζ}^{c} log |ζ ∧ W |^{2})^{j} ∧ ω_{ζ}^{ℓ−1−j} ∈ D′^{ℓ−1,ℓ−1}(ℙ_{ζ}^{m}) ,

which (by definition) has locally integrable coefficients. We have the generalized Poincaré-Lelong formula for W [GK, 1.15] (see also [Sh, II.6, pp. 68-69]),

dd^{c}Λ_{W} = ω^{ℓ} − [W] .  (4)

Now let W = H_{1} ∩ ⋯ ∩ H_{ℓ} such that dim π_{2}^{−1}(W) = n − ℓ, where we use the notation of (3). Applying the generalized Poincaré-Lelong formula ([GK, 1.15] or [Sh, II.6, pp. 68-69]), to the divisors π_{2}^{∗}H_{1}, …, π_{2}^{∗}H_{ℓ} of the lifted hyperplane-section bundle π_{2}^{∗}O_{ℙ^{m}}(1) with Chern form π_{2}^{∗}ω, we obtain

dd^{c}π_{2}^{∗}Λ_{W} = π_{2}^{∗}ω^{ℓ} − π_{2}^{∗}[W] ,  (5)

where π_{2}^{∗}Λ_{W} ∈ D′^{ℓ−1,ℓ−1}(G̃) is given by

(π_{2}^{∗}Λ_{W})(z̃) =^{def} log \frac{|π_{2}(z̃)|^{2}|W |^{2}}{|π_{2}(z̃) ∧ W |^{2}} ∑_{j=0}^{ℓ−1}(dd^{c} log |π_{2}(z̃) ∧ W |^{2})^{j} ∧ π_{2}^{∗}ω^{ℓ−1−j} ∈ D′^{ℓ−1,ℓ−1}(G̃).

(6)

In particular, π_{2}^{∗}Λ_{W} has L_{loc}^{1} coefficients and is smooth on G̃\π_{2}^{−1}(W). We define the currents

\begin{aligned}P^{∗}Λ_{W} & = & π_{1∗}(π_{2}^{∗}Λ_{W}) , \\ P^{∗}[W] & = & π_{1∗}[π_{2}^{−1}W] . \end{aligned}  (7)

By applying π_{1∗} to (5), we obtain the current identity on ℙ^{n},

dd^{c}P^{∗}Λ_{W} = P^{∗}ω^{ℓ} − P^{∗}[W].  (8)

Note that for generic W, π_{2}^{∗}[W] has multiplicity identically 1 and contains no components inside the exceptional locus of π_{1}, and thus [P^{∗}W] is the current of integration over the closure of (P |_{ℙ^{n}\I_{P}})^{−1}(W).

4. The intermediate degrees of a rational map

In this section, we give some properties of the intermediate degrees δ_{ℓ}(P) of a rational map P, which we also describe analytically and topologically. We use the following consequence of Bertini’s theorem:

Lemma 4.1. Let Y be a projective algebraic manifold and let f : Y → ℙ^{m} be a nonconstant holomorphic map. Then for a generic hyperplane H ⊂ ℙ^{m}, the divisor f^{∗}H is smooth and has multiplicity 1.

Proof: Apply Bertini’s theorem (see, for example, [GH, p. 137]) to the complete linear system of f^{∗}H. □

For a subvariety V ⊂ ℙ^{m}, we write P^{−1}(V) = π_{1}(π_{2}^{−1}(V)) (using the notation in (3)). We let #(S) denote the cardinality of a set S. We begin with a formula for the integral of certain singular forms on ℙ^{n}:

Lemma 4.2. Let P_{j} : ℙ^{n}−→ℙ^{m_{j}}, 1 ≤ j ≤ n, be rational maps, and let I = I_{P_{1}} ∪ ⋯ ∪ I_{P_{n}}.

i) For generic hyperplanes H_{1} ⊂ ℙ^{m_{1}}, …, H_{n} ⊂ ℙ^{m_{n}},

∫_{ℙ^{n}\I} P_{1}^{∗}ω_{m_{1}} ∧ ⋯ ∧ P_{n}^{∗}ω_{m_{n}} = # ( ⋂_{j=1}^{n} P_{j}^{−1}(H_{j}) \ I) ,

ii)

∫_{ℙ^{n}\I} P_{1}^{∗}ω_{m_{1}} ∧ ⋯ ∧ P_{n}^{∗}ω_{m_{n}} ≤ ∏_{j=1}^{n} deg P_{j} .

To verify Lemma 4.2, we first give a topological description of the integral. Let X = ℙ^{m_{1}} × ⋯ × ℙ^{m_{n}} and write P = (P_{1}, …, P_{n}) : ℙ^{n}−→X; then I = I_{P}. Consider the commutative diagram

G̃
π_{1} ↙  ↘ π_{2}
ℙ^{n}  −→^{P}  X

where G̃ is a desingularization of the graph of P and π_{1}, π_{2} are the projections. Let p_{j} : X → ℙ^{m_{j}} denote the projection to the j-th factor and let P̃_{j} = p_{j} ∘π_{2} : G̃ → ℙ^{m_{j}}, for 1 ≤ j ≤ n. Let t_{m} denote the positive generator of H^{2}(ℙ^{m}, ℤ), and write

t_{m_{1}} × ⋯ × t_{m_{n}} = p_{1}^{∗}t_{m_{1}} • ⋯ • p_{n}^{∗}t_{m_{n}} ∈ H^{2n}(X, ℤ) ,

where • denotes the cup product in the cohomology ring.

Lemma 4.3. Using the notation of Lemma 4.2,

∫_{ℙ^{n}\I} P_{1}^{∗}ω_{m_{1}} ∧ ⋯ ∧ P_{n}^{∗}ω_{m_{n}} = (π_{2}^{∗}(t_{m_{1}} × ⋯ × t_{m_{n}}), G̃)
= (P̃_{1}^{∗}t_{m_{1}} • ⋯ • P̃_{n}^{∗}t_{m_{n}}, G̃) ∈ ℤ .

Proof: Since the de Rham class of the Kähler form ω_{m} on ℙ^{m} equals t_{m}, we have by Section 3,

∫_{ℙ^{n}\I} P_{1}^{∗}ω_{m_{1}} ∧ ⋯ ∧ P_{n}^{∗}ω_{m_{n}} = ∫_{ℙ^{n}\I_{P}} P^{∗}(ω_{m_{1}} × ⋯ × ω_{m_{n}})
= (π_{1∗}π_{2}^{∗}(ω_{m_{1}} × ⋯ × ω_{m_{n}}), 1)
= (π_{2}^{∗}(ω_{m_{1}} × ⋯ × ω_{m_{n}}), 1)
= (π_{2}^{∗}(t_{m_{1}} × ⋯ × t_{m_{n}}), G̃) .

The second equality follows from our definitions. □ Proof of Lemma 4.2: Let E ⊂ G̃ be the exceptional locus of π_{1}. By Lemma 4.1 applied to P̃_{1} : G̃ → ℙ^{m_{1}}, there is a hyperplane H_{1} ⊂ ℙ^{m_{1}} such that the divisor π_{2}^{∗}(H_{1} × ℙ^{m_{2}} × ⋯ × ℙ^{m_{n}}) is a smooth hypersurface Y_{1} ⊂ G̃ of multiplicity 1 with dim Y_{1} ∩ E < n − 1. By Lemma 4.1, we can inductively find hyperplanes H_{2} ⊂ ℙ^{m_{2}}, …, H_{n} ⊂ ℙ^{m_{n}} such that, writing

Y_{j} = π_{2}^{−1}(H_{1} × ⋯ × H_{j} × ℙ^{m_{j+1}} × ⋯ × ℙ^{m_{n}}) ,

Y_{j} is a smooth submanifold of G̃ of dimension n − j, dim Y_{j} ∩ E < n − j, and the divisor (P̃_{j}|_{Y_{j−1}})^{∗}H_{j} (on Y_{j−1}) has multiplicity 1, or equivalently, P̃_{j}^{−1}(H_{j}) intersects Y_{j−1} transversely. In particular, dim Y_{n} = 0 and Y_{n} ∩ E = ∅. Write H′_{j} = p_{j}^{−1}(H_{j}), for 1 ≤ j ≤ n, so that

Y_{j} = π_{2}^{−1}(H′_{1} ∩ ⋯ ∩ H′_{j}) = π_{2}^{−1}(H′_{j}) ∩ Y_{j−1} .

A codimension j submanifold S of a complex manifold Y determines the current of integration [S] ∈ D′^{j,j}(Y) given by ([S], φ) = ∫_{S} φ for a test form φ. We also let [S] denote the de Rham class in H^{2j}(Y, ℝ) containing the current [S]. (If Y is compact of dimension n, then the cohomology class [S] is the Poincaré dual of the (2n−2j)-cycle S.) If two submanifolds S_{1}, S_{2} intersect transversely, then [S_{1} ∩ S_{2}] = [S_{1}] • [S_{2}] in the cohomology ring of Y. In particular, [H_{j}] = t_{j} ∈ H^{2}(ℙ^{m_{j}}, ℤ) ⊂ H^{2}(ℙ^{m_{j}}, ℝ) and

[H_{1} × ⋯ × H_{n}] = [H′_{1} ∩ ⋯ ∩ H′_{n}] = t_{m_{1}} × ⋯ × t_{m_{n}} .

Furthermore, our construction of the H_{j}, Y_{j} implies that

[Y_{j}] = [Y_{j−1}] • π_{2}^{∗}[H′_{j}] = π_{2}^{∗}[H′_{1}] • ⋯ • π_{2}^{∗}[H′_{j}] = π_{2}^{∗}[H′_{1} ∩ ⋯ ∩ H′_{j}] .

Therefore,

π_{2}^{∗}(t_{m_{1}} × ⋯ × t_{m_{n}}) = π_{2}^{∗}[H_{1} × ⋯ × H_{n}] = [Y_{n}]  (9)

(where the points of Y_{n} have multiplicity 1).

Let A_{j} : ℂ^{m_{j}+1} → ℂ be a linear map defining the hyperplane H_{j}, and consider the polynomial

Q_{j} = A_{j}(P_{j0}, …, P_{jn}) ∈ ℂ[z_{0}, …, z_{n}] ,

where P_{j} = (P_{j0}, …, P_{jn}), for 1 ≤ j ≤ n. We then have

{z ∈ ℙ^{n} : Q_{j}(z) = 0} = π_{1} (π_{2}^{−1}(H′_{j})) ⊃ I_{P_{j}} .  (10)

Since Y_{j} ⊂ G̃ \ E and π_{1} maps G̃ \ E bijectively to ℙ^{n} \ I_{P}, we have

π_{1}(Y_{n}) = π_{1} (π_{2}^{−1}(H′_{1} ∩ ⋯ ∩ H′_{n})) \ I_{P}
= ⋂_{j=1}^{n} π_{1} (π_{2}^{−1}(H′_{j})) \ I_{P} = ⋂_{j=1}^{n} P_{j}^{−1}(H_{j}) \ I_{P}
= {z ∈ ℙ^{n} \ I_{P} : Q_{1}(z) = ⋯ = Q_{n}(z) = 0} .

Thus by (9),

(π_{2}^{∗}(t_{m_{1}} × ⋯ × t_{m_{n}}), G̃) = ([Y_{n}], G̃) = #(Y_{n}) .

By Bézout’s Theorem,

#(Y_{n}) = #(π_{1}(Y_{n})) = # ( ⋂_{j=1}^{n} P_{j}^{−1}(H_{j}) \ I_{P} ) ≤ ∏_{j=1}^{n} deg Q_{j} = ∏_{j=1}^{n} deg P_{j} .

The conclusion follows from Lemma 4.3.

Definition: Let P : ℙ^{n}−→ℙ^{m} be a rational map. We define the intermediate degrees δ_{ℓ}(P) of P by the formula

δ_{ℓ}(P) = ∫_{ℙ^{n}\I_{P}} P^{∗}ω_{m}^{ℓ} ∧ ω_{n}^{n−ℓ}

for 1 ≤ ℓ ≤ min(n, m).

We shall show in Lemma 4.4 below that the intermediate degrees are also given by the geometric definition in the introduction; in particular, if m = n, then δ_{n}(P) is the topological degree of P, which is defined as the cardinality of P^{−1}(x), for a generic point x ∈ ℙ^{n}. Clearly, δ_{ℓ}(P) > 0 if and only if rank P ≥ ℓ. It is easy to verify that δ_{1}(P) = deg P, and if P is holomorphic (this can happen only if m ≥ rank P = n), then δ_{ℓ}(P) = (deg P)^{ℓ}, for ℓ ≤ n.

It follows from Lemma 4.2 applied to the maps P_{1} = ⋯ = P_{ℓ} = P , P_{ℓ+1} = ⋯ = P_{n} = Id_{ℙ^{n}} that in general,

δ_{ℓ}(P) ≤ (deg P)^{ℓ} , for 1 ≤ ℓ ≤ n .  (11)

We shall give a more general inequality in Lemma 4.7 below.

Lemma 4.4. Let P : ℙ^{n}−→ℙ^{m} be a rational map. Then

δ_{ℓ}(P) = deg P^{−1}(W) ≤ (deg P)^{ℓ}

for generic W ∈ 𝔾(ℓ, m), with equality if and only if codimI_{P} > ℓ. In particular, δ_{n}(P) ≤ (deg P)^{n}, with equality if and only if P is holomorphic.

Proof: We shall apply Lemma 4.2 with P_{1} = ⋯ = P_{ℓ} = P , P_{ℓ+1} = ⋯ = P_{n} = Id_{ℙ^{n}}. By part (ii) of the lemma, δ_{ℓ}(P) ≤ (deg P)^{ℓ}. For generic W = H_{1} ∩ ⋯ ∩ H_{ℓ}, π_{2}^{−1}(W) is of codimension ℓ and has no components contained in the exceptional locus of π_{1}, and thus P^{−1}(W) has pure dimension n − ℓ and dim P^{−1}(W) ∩ I_{P} < n − ℓ. Hence for generic hyperplanes H_{1}, …, H_{n}, we have

⋂_{j=1}^{n} P_{j}^{−1}(H_{j}) \ I_{P} = P^{−1}(W) ∩ H_{ℓ+1} ∩ ⋯ ∩ H_{n} ,

where W = H_{1} ∩ ⋯ ∩ H_{ℓ}, and thus by part (i),

δ_{ℓ}(P) = # (P^{−1}(W) ∩ H_{ℓ+1} ∩ ⋯ ∩ H_{n}) = deg P^{−1}(W) .

Furthermore, using the notation in the proof of Lemma 4.2,

{z ∈ ℙ^{n} : Q_{1}(z) = ⋯ = Q_{ℓ}(z) = 0} = P^{−1}(W) ∪ I_{P} .

Since deg Q_{1} = ⋯ = deg Q_{ℓ} = deg P, it follows from Bézout’s theorem (see for example [Fu, Example 8.4.6]) that δ_{ℓ}(P) < (deg P)^{ℓ} if dim I_{P} ≥ n − ℓ. If dim I_{P} < n − ℓ, then P^{−1}(W) ⊃ I_{P} (by dimension considerations) and Bézout’s theorem gives equality.

In fact, if n = 2 in Lemma 4.4, then δ_{2}(P) = (deg P)^{2} − q, where q is the number of points of I_{P} counting multiplicity. This is illustrated by the following example.

Example: Let P : ℙ^{2}−→ℙ^{2} be given by

P (z_{0}, z_{1}, z_{2}) = (z_{1}z_{2}, z_{0}z_{2}, z_{0}z_{1}) = (\frac{1}{z_{0}} , \frac{1}{z_{1}} , \frac{1}{z_{2}} ) .

Then δ_{1}(P) = 2, δ_{2}(P) = 1. Note that in this example I_{P} consists of the three points (1, 0, 0), (0, 1, 0), (0, 0, 1).

Lemma 4.5. Suppose that P : ℙ^{n}−→ℙ^{m} is a rational map and L : ℂ^{m+1} → ℂ^{M+1} is a linear map such that ImageP ⊄ ℙ(L^{−1}(0)). Let P_{L} = L̂ ∘ P : ℙ^{n}−→ℙ^{M}, where L̂ : ℙ^{m}−→ℙ^{M} is the map induced from L. Then

∫_{ℙ^{n}\I}(P_{L}^{∗}ω)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ} ≤ δ_{ℓ}(P)

for 1 ≤ k ≤ ℓ ≤ n, where I = I_{P} ∪ I_{P_{L}}.

Proof: We can assume without loss of generality that M = m. We first consider the case where L is nonsingular and thus L̂ is biholomorphic. Therefore

∫_{ℙ^{n}\I}(P_{L}^{∗}ω)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ} = ∫_{ℙ^{n}\I}(P^{∗}ω′)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ}

where ω′ = L̂^{∗}ω. Since ω′ and ω are in the same de Rham class, it follows from Section 3 (or by the proof of Lemma 4.3) that

∫_{ℙ^{n}\I}(P^{∗}ω′)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ} = ∫_{ℙ^{n}\I}(P^{∗}ω)^{ℓ} ∧ ω^{n−ℓ} = δ_{ℓ}(P) .

We now suppose that L is singular. Choose a sequence {L_{ν}} of nonsingular linear operators on ℂ^{m+1} such that L_{ν} → L. We can write

(P_{L_{ν}}^{∗}ω)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ} = f_{ν}ω^{n} , (P_{L}^{∗}ω)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ} = fω^{n}

where f_{ν}, f are non-negative C^{∞} functions on ℙ^{n} \ I. Then f_{ν} → f pointwise on ℙ^{n} \ I, and hence by Fatou’s Lemma,

∫_{ℙ^{n}\I}(P_{L}^{∗}ω)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ} = ∫_{ℙ^{n}\I} fω^{n} ≤ lim inf_{ν→∞} ∫_{ℙ^{n}\I} f_{ν}ω^{n} = δ_{ℓ}(P) .

Lemma 4.6. Let P : ℙ^{n}−→ℙ^{m} , Q : ℙ^{m}−→ℙ^{r} be rational maps. Then

δ_{ℓ}(Q ∘ P) ≤ δ_{ℓ}(P)δ_{ℓ}(Q) .

Proof: Let η = Q^{∗}ω^{ℓ} ∈ D′^{ℓ,ℓ}(ℙ^{n}). We smooth η by an approximate identity {ψ_{ε}}_{ε>0} with respect to a Haar measure h on GL(n + 1, ℂ) to obtain

η_{ε} =^{def} ∫_{GL(n+1,ℂ)}(g^{∗}η)ψ_{ε}(g)dh(g) ∈ E^{ℓ,ℓ}(ℙ^{n}) .

Then η_{ε} → η pointwise as ε → 0, η_{ε} ≥ 0, and we have the identity in de Rham cohomology,

[η_{ε}] = [η] = δ_{ℓ}(Q)[ω^{ℓ}] ∈ H^{2ℓ}(ℙ^{n}, ℤ) .

Using the commutative diagram (3), we then have

∫_{ℙ^{n}} P^{∗}η_{ε} ∧ ω^{n−ℓ} = ∫_{G} π_{2}^{∗}η_{ε} ∧ π_{1}^{∗}ω^{n−ℓ} = ∫_{G} δ_{ℓ}(Q)π_{2}^{∗}ω^{ℓ} ∧ π_{1}^{∗}ω^{n−ℓ} = δ_{ℓ}(Q)δ_{ℓ}(P) .

Therefore, by Fatou’s lemma,

δ_{ℓ}(Q ∘ P) = ∫_{ℙ^{n}\I_{Q∘P}} P^{∗}η ∧ ω^{n−ℓ} ≤ lim inf_{ε→0} ∫_{ℙ^{n}} P^{∗}η_{ε} ∧ ω^{n−ℓ} = δ_{ℓ}(Q)δ_{ℓ}(P) .

Lemma 4.7. Let P : ℙ^{n}−→ℙ^{m} be a rational map. Then

δ_{k+ℓ}(P) ≤ δ_{k}(P)δ_{ℓ}(P) .

Proof: Let η = P^{∗}ω^{ℓ} ∈ D′^{ℓ,ℓ}(ℙ^{m}) and consider the smooth forms η_{ε} as in the above proof. As before,

[η_{ε}] = δ_{ℓ}(P)[ω^{ℓ}] ∈ H^{2ℓ}(ℙ^{n}, ℤ) ,

and

∫_{ℙ^{n}} P^{∗}ω^{k} ∧ η_{ε} ∧ ω^{n−k−ℓ} = ∫_{G} π_{2}^{∗}ω^{k} ∧ π_{1}^{∗}(η_{ε} ∧ ω^{n−k−ℓ})
= δ_{ℓ}(P) ∫_{G} π_{2}^{∗}ω^{k} ∧ π_{1}^{∗}ω^{n−k} = δ_{ℓ}(P)δ_{k}(P) .

The conclusion follows as above by letting ε → 0 and applying Fatou’s lemma.

5. The proximity function

Let P : ℙ^{n}−→ℙ^{m} be a rational map, and let 1 ≤ ℓ ≤ min(n, m). Recall that if codimP^{−1}(W) = ℓ, then equations (6), (7) define a current P^{∗}Λ_{W} ∈ D′^{ℓ−1,ℓ−1}(ℙ^{n}) with locally integrable coefficients. Hence we can define the proximity function m_{P}^{ℓ} : 𝔾(ℓ, m) → [0, +∞] by

m_{P}^{ℓ} (W) = \{ \begin{matrix}(P^{∗}Λ_{W} , ω^{n−ℓ+1}) = ∫_{ℙ^{n}\(I_{P} ∪I_{P,W})} P^{∗}Λ_{W} ∧ ω^{n−ℓ+1} & if codimP^{−1}(W) = ℓ \\ +∞ & if codimP^{−1}(W) < ℓ \end{matrix}

for W ∈ 𝔾(ℓ, m), where I_{P,W} is the indeterminacy locus of the map z ↦ P (z) ∧ W ∈ ℙ(⋀^{m+2−ℓ} ℂ^{m+1}). We give 𝔾(ℓ, m) the Kähler metric ω induced from the natural embedding 𝔾(ℓ, m) ⊂ ℙ(⋀^{m+1−ℓ} ℂ^{m+1}).

The following key estimate is used in our proof of Theorem 1.1.

Lemma 5.1. If P : ℙ^{n}−→ℙ^{m} is a rational map, then m_{P}^{ℓ} ∈ L^{1}(𝔾(ℓ, m)) and

dd^{c}m_{P}^{ℓ} ≤ ℓδ_{ℓ−1}(P)ω ,

for 1 ≤ ℓ ≤ min(n, m).

Proof: (The techniques used for this proof are somewhat similar to those of [Sk1, Sk2].) Assume first that ℓ ≥ 2. (The estimate of Lemma 5.1 is straightforward for the hyperplane case ℓ = 1; we give the argument for this case at the end of this proof. See also [RS] for a complete treatment of pre-images of hyperplanes.) Write E = ⋀^{m+1−ℓ} ℂ^{m+1}, Ẽ = ⋀^{m+2−ℓ} ℂ^{m+1} and let λ_{0} : E → ℂ, λ_{1} : Ẽ → ℂ be linear functions of unit norm. For ζ ∈ ℂ^{m+1}, W ∈ E, we define the augmented exterior product

ζ ˜∧W = (ζ_{0}W_{0}, ζ ∧ W) ∈ ℂ ⊕ Ẽ

where W_{0} = λ_{0}(W). Let

X = ℙ_{z}^{n} × 𝔾(ℓ, m)_{W} × ℙ_{ζ}^{m} × ℙ(Ẽ)_{θ} × ℙ(ℂ ⊕ Ẽ)_{η} .

(The subscripts z, W, ζ, θ, η serve to identify the variables used in this discussion.)

We further write λ_{1}(θ) = θ_{1} for θ ∈ Ẽ. If η = (c, θ) ∈ ℂ ⊕ Ẽ, we write η_{0} = c, η_{1} = θ_{1} = λ_{1}(θ). By making a linear change of coordinates in ℙ_{ζ}^{m} we can assume without loss of generality that Image(P) ⊄ {ζ ∈ ℙ^{m} : ζ_{0} = 0}. We consider the current

Ω = log \frac{|ζ|^{2}|W |^{2}|θ_{1}|^{2}|η_{0}|^{2}}{|ζ_{0}|^{2}|W_{0}|^{2}|θ|^{2}|η_{1}|^{2}} ω_{z}^{n−ℓ+1} ∧ ∑_{j=0}^{ℓ−1} ω_{θ}^{j} ∧ ω_{ζ}^{ℓ−1−j} ∈ D′^{n−1,n−1}(X) .

Let Q : ℙ^{n} × 𝔾(ℓ, m)−→X be the meromorphic (rational) map given by

Q(z, W) = (z, W, P (z), P (z) ∧ W, P (z)˜∧W) .  (12)

(The redundancy in the map Q is needed to facilitate our proof.) We can assume that λ_{1} was chosen so that λ_{1}(P (z) ∧ W) ≢ 0. We shall show (see Lemma 5.2) that

m_{P}^{ℓ} = π_{2∗}Q^{∗}Ω ,

where π_{2} : ℙ^{n}×𝔾(ℓ, m) → 𝔾(ℓ, m) is the projection. Of course, since Q is not regular, we must define the pull-back Q^{∗}Ω, which we do as follows. Let

Y →^{ρ} Image(Q) ⊂ X

be a desingularization of the image of Q. (Note that Image(Q) is an algebraic subvariety of X and can be identified with the graph of Q.) Let π_{1} : X → ℙ^{n} × 𝔾(ℓ, m) be the projection, and write ρ_{1} = π_{1} ∘ ρ, ρ_{2} = π_{2} ∘ ρ_{1} so that we have the commutative diagram:

ℙ^{n}×𝔾(ℓ, m) ⟵^{ρ_{1}} Y

Q ↓^{|} ρ ↙ ↓ ρ_{1} ↘ ρ_{2}

X ⟶^{π_{1}} ℙ^{n}×𝔾(ℓ, m) →^{π_{2}} 𝔾(ℓ, m)

We then define Q^{∗}Ω = ρ_{1∗}ρ^{∗}Ω, where ρ^{∗}Ω is given as in Section 3. Let I_{Q} ⊂ ℙ^{n} × 𝔾(ℓ, m) denote the indeterminacy locus of Q; I_{Q} is an algebraic subvariety of codimension ≥ 2. Write U = ℙ^{n} × 𝔾(ℓ, m) \ I_{Q}, and let Q_{0} = Q|_{U} : U → X. Since ρ_{1} maps ρ_{1}^{−1}(U) biholomorphically onto U, we have Q_{0} = ρ ∘ ρ_{1}^{−1}|_{U}. Hence

(Q^{∗}Ω)|_{U} = (ρ_{1}^{−1}|_{U})^{∗}ρ^{∗}Ω = Q_{0}^{∗}Ω ,

so our definition of Q^{∗} agrees with the usual one at regular points. Since Q^{∗}Ω has coefficients in L_{loc}^{1}, it is the extension to ℙ^{n} × 𝔾(ℓ, m) of Q_{0}^{∗}Ω with zero mass on I_{Q}.

We first note the following:

Lemma 5.2. m_{P}^{ℓ} = ρ_{2∗}ρ^{∗}Ω = π_{2∗}Q^{∗}Ω ∈ L^{1}(𝔾(ℓ, m)).

Proof: By the definition of Q^{∗}Ω we have

ρ_{2∗}ρ^{∗}Ω = π_{2∗}ρ_{1∗}ρ^{∗}Ω = π_{2∗}Q^{∗}Ω .  (13)

Since Q^{∗}Ω has coefficients in L_{loc}^{1} and its degree, 2n, is the fibre dimension of π_{2}, π_{2∗}Q^{∗}Ω ∈ L^{1}(𝔾(ℓ, m)) and

(π_{2∗}Q^{∗}Ω)(W) = ∫_{ℙ_{z}^{n}×{W }\I_{Q}} Q^{∗}Ω < +∞ , for a.a. W ∈ 𝔾(ℓ, m) .  (14)

Since η_{1} ∘ Q = θ_{1} ∘ Q and η_{0} ∘ Q = P_{0}(z)W_{0} = (ζ_{0}W_{0}) ∘ Q, we have

Q^{∗}Ω|_{U} = Q_{0}^{∗}Ω = log \frac{|P (z)|^{2}|W |^{2}}{|P (z) ∧ W |^{2}} ω_{z}^{n−ℓ+1} ∧ ∑_{j=1}^{ℓ−1}(dd^{c} log |P (z) ∧ W |^{2})^{j} ∧ (P^{∗}ω_{ζ})^{ℓ−1−j} .

Thus for a.a. W ∈ 𝔾(ℓ, m), I_{Q} ⊅ ℙ^{n} × {W } and

m_{P}^{ℓ} (W) = ∫_{ℙ^{n}\(I_{P} ∪I_{P,W})} P^{∗}Λ_{W} ∧ ω^{n−ℓ+1} = ∫_{ℙ_{z}^{n}×{W }\I_{Q}} Q^{∗}Ω ≤ +∞ .  (15)

The desired identity follows from (13), (14), and(15). □

Remark: We could use (6), (7) to define m_{P}^{ℓ} (W) for all W ∈ ℙ(⋀^{m+1−ℓ} ℂ^{m+1}); then Lemma 5.1 remains valid on ℙ(⋀^{m+1−ℓ} ℂ^{m+1}).

We are now ready to compute dd^{c}m_{P}^{ℓ}. We let H_{ζ}^{0}, H_{W}^{0} , H_{θ}^{1}, H_{η}^{0}, H_{η}^{1} denote the hyperplanes in X given by the divisors of ζ_{0}, W_{0}, θ_{1}, η_{0}, η_{1} respectively, and we let

D = H_{ζ}^{0} + H_{W}^{0} − H_{θ}^{1} + H_{η}^{1} − H_{η}^{0} = Div (\frac{ζ_{0}W_{0}η_{1}}{θ_{1}η_{0}} ) .

We have

\begin{aligned}dd^{c}Ω & = (ω_{ζ} + ω_{W} − ω_{θ} − D) ∧ ω_{z}^{n−ℓ+1} ∧ ∑_{j=0}^{ℓ−1}ω_{θ}^{j} ∧ ω_{ζ}^{ℓ−1−j} \\ & = [ω_{ζ}^{ℓ} − ω_{θ}^{ℓ} + (ω_{W} − D) ∧ ∑_{j=0}^{ℓ−1}ω_{θ}^{j} ∧ ω_{ζ}^{ℓ−1−j}] ∧ ω_{z}^{n−ℓ+1} . \end{aligned}  (16)

Lemma 5.3. The divisor ρ^{∗}D is effective (i.e., is locally the divisor of a holomorphic function).

Proof: (Our choices of ζ_{0} and θ_{1} guarantee that ρ(Y) ⊄ Supp D so that ρ^{∗}D is defined.) Let y_{0} ∈ Y be arbitrary, and let

X′ = {(z, W, ζ, θ, η) ∈ X : z_{i} ≠ 0, λ(W) ≠ 0, ζ_{j} ≠ 0, λ′(θ) ≠ 0}

where 0 ≤ i ≤ n, 0 ≤ j ≤ m and λ : E → ℂ, λ′ : Ẽ → ℂ are linear functions chosen so that ρ(y_{0}) ∈ X′. Let

g = \frac{ζ̃_{0}W̃_{0}}{θ̃_{1}} \frac{η_{1}}{η_{0}} ∈ Mer(X′)

where ζ̃_{0} = ζ_{0}/ζ_{j}, W̃_{0} = W_{0}/λ(W) = (λ_{0}/λ)(W), θ̃_{1} = θ_{1}/λ′(θ) = (λ_{1}/λ′)(θ). Then Div(g) = D|_{X′}. We must show that g ∘ ρ is holomorphic at y_{0}. Now

\begin{matrix}g ∘ Q = \frac{P_{0}(z)}{P_{j}(z)} \frac{W_{0}}{λ(W)} \frac{λ′(P (z) ∧ W)}{λ_{1}(P (z) ∧ W)} \frac{λ_{1}(P (z) ∧ W)}{P_{0}(z)W_{0}} \\ = \frac{λ′(P (z) ∧ W)}{P_{j}(z)λ(W)} = \frac{λ′(P (z̃) ∧ W̃)}{P_{j}(z̃)} \end{matrix}

where z̃ = z_{i}^{−1}z ∈ ℂ^{n+1}, W̃ = λ(W)^{−1}W ∈ E. Write, for y ∈ Y,

ρ_{1}(y) = (z(y), W (y)) ,

z̃(y) = z_{i}^{−1}(y)z(y), W̃(y) = λ(W (y))^{−1}W (y). Since ρ is holomorphic and ρ(y_{0}) ∉ Div(ζ_{j}), there is a neighborhood Y_{0} of y_{0} so that P_{μ}(z̃(y)) = φ(y)f_{μ}(y) for y ∈ Y_{0} and 0 ≤ μ ≤ m, where φ, f_{0}, …, f_{m} ∈ O(Y_{0}), φ ≢ 0, and f_{j}(y_{0}) ≠ 0. Thus

g ∘ ρ = g ∘ Q ∘ ρ_{1} = \frac{λ′(φF ∧ W̃)}{φf_{j}} = \frac{λ′(F ∧ W̃)}{f_{j}}

on Y_{0}, where F = (f_{0}, … f_{m}). Since f_{j}(y_{0}) ≠ 0, g ∘ ρ is holomorphic at y_{0}. □

We now complete the proof of Lemma 5.1: By Lemma 5.3 and (16), we have

ρ^{∗}dd^{c}Ω ≤ ρ^{∗} [(ω_{ζ}^{ℓ} + ω_{W} ∧ ∑_{0}^{ℓ−1} ω_{θ}^{j} ∧ ω_{ζ}^{ℓ−1−j}) ∧ ω_{z}^{n−ℓ+1}] .

By Section 3, ρ^{∗}dd^{c}Ω = dd^{c}ρ^{∗}Ω. Hence by Lemma 5.2,

\begin{aligned}dd^{c}m_{P}^{ℓ} & = dd^{c}(ρ_{2∗}ρ^{∗}Ω) = ρ_{2∗}ρ^{∗}dd^{c}Ω \\ & ≤ ρ_{2∗}ρ^{∗} [(ω_{ζ}^{ℓ} + ω_{W} ∧ ∑_{0}^{ℓ−1} ω_{θ}^{j} ∧ ω_{ζ}^{ℓ−1−j}) ∧ ω_{z}^{n−ℓ+1}] . \end{aligned}

Since ρ_{2∗}ρ^{∗} = π_{2∗}Q^{∗} and

Q_{0}^{∗}(ω_{ζ}^{ℓ} ∧ ω_{z}^{n−ℓ+1}) = (dd^{c} log |P (z)|^{2})^{ℓ} ∧ ω_{z}^{n−ℓ+1} = 0 ,

we obtain

\begin{aligned}dd^{c}m_{P}^{ℓ} & ≤ ∑_{j=0}^{ℓ−1} π_{2∗}Q^{∗} (ω_{W} ∧ ω_{θ}^{j} ∧ ω_{ζ}^{ℓ−1−j} ∧ ω_{z}^{n−ℓ+1}) \\ & = ∑_{j=0}^{ℓ−1} (∫_{ℙ_{z}^{n}×\{W \}}(dd^{c} log |P (z) ∧ W |^{2})^{j} ∧ (P^{∗}ω_{ζ})^{ℓ−1−j} ∧ ω_{z}^{n−ℓ+1}) ω_{W} . \end{aligned}

By Lemma 4.5 with L : ℂ^{m+1} → Ẽ given by L(ζ) = ζ ∧ W,

∫_{ℙ_{z}^{n}×{W }}(dd^{c} log |P (z) ∧ W |^{2})^{j} ∧ (P^{∗}ω_{ζ})^{ℓ−1−j} ∧ ω_{z}^{n−ℓ+1} ≤ δ_{ℓ−1}(P) ,

and the desired inequality follows.

We now modify (and simplify) the above argument for the case ℓ = 1: Identify

G(1, m) = ℙ(^{⋀m}ℂ^{m+1}) = ℙ(ℂ^{m+1∗}) = ℙ^{m∗} .

Let X = ℙ_{z}^{n} × ℙ_{W}^{m∗} × ℙ_{ζ}^{m} and consider the current

Ω = log \frac{|ζ|^{2}|W |^{2}}{|(W, ζ)|^{2}} ω_{z}^{n} ∈ D′^{n,n}(X) .

Then

dd^{c}Ω = (ω_{ζ} + ω_{W} − D) ∧ ω_{z}^{n} ,

where D = Div(W, ζ). Then m_{P}^{1} = π_{2∗}Q^{∗}Ω, where

Q = (z, W, P (z)) : ℙ_{z}^{n} × ℙ_{W}^{m∗} → X .

We conclude as before that dd^{c}m_{P}^{1} = π_{2∗}Q^{∗}dd^{c}Ω ≤ ω_{W}. □

6. Description of the exceptional set

In this section we prove Theorem 1.1, giving a description of the exceptional set E in terms of the proximity function as follows. Let P = {P_{k}} be a sequence of rational mappings from ℙ^{n} to ℙ^{m} as in Theorem 1.1 and fix a sequence A = {a_{k}} of positive numbers. We define the exceptional set

E_{ℓ}(P, A) = \{W ∈ 𝔾(ℓ, m) : lim sup_{k→∞} \frac{m_{k}(W)}{a_{k}} > 0\} ,  (17)

where m_{k} = m_{P_{k}}^{ℓ}. (Thus by definition, E_{ℓ}(P, A) contains those planes W such that infinitely many of the pre-images P_{k}^{−1}(W) have codimension less than ℓ.) The following two propositions yield Theorem 1.1 with E = E_{ℓ}(P, A).

Proposition 6.1. Let P = {P_{k}} be a sequence of rational mappings from ℙ^{n} to ℙ^{m} and let A = {a_{k}} be a sequence of positive numbers. Let 1 ≤ ℓ ≤ min(n, m). Then for all W ∈ 𝔾(ℓ, m) \ E_{ℓ}(P, A),

\frac{1}{a_{k}} (P_{k}^{∗}[W] − P_{k}^{∗}ω^{ℓ}) → 0

as k → ∞.

Proposition 6.2. Let P = {P_{k}}, A = {a_{k}} be as in Proposition 6.1. If

∑_{k=1}^{∞} \frac{δ_{ℓ−1}(P_{k})}{a_{k}} < +∞ ,

then E_{ℓ}(P, A) is pluripolar in 𝔾(ℓ, m).

Before proving Propositions 6.1 and 6.2, we note a corollary to Theorem 1.1 on the equidistribution of pre-images for subsequences of a given sequence of rational mappings. This corollary uses the following terminology: For a current T ∈ D′^{p,p}(ℙ^{n}) of order 0, we let ∥T ∥ denote the total variation measure of ℙ^{n}, which is the regular measure on ℙ^{n} given by

∥T ∥(U) = sup {|(T, φ)| : φ ∈ D^{n−p,n−p}(U), ∥φ∥ ≤ 1}

for U open in ℙ^{n}. Here, ∥φ∥ denotes the comass norm of a compactly supported form φ (see Federer [Fe, 1.8.1, 4.1.7]). The quantity ∥T ∥(ℙ^{n}) is called the mass of T. If T is positive, then it follows from Wirtinger’s inequality (see for example, [Fe, 1.8.2]) that ∥T ∥(ℙ^{n}) = (T, ω^{n−p}). In particular, ∥ω^{p}∥(ℙ^{n}) = 1.

Corollary 6.3. Let P = {P_{k}} be as in Theorem 1.1, let 1 ≤ ℓ ≤ min(n, m), and suppose that

\frac{δ_{ℓ−1}(P_{k})}{δ_{ℓ}(P_{k})} → 0 .

Let M ⊂ D′^{ℓ,ℓ}(ℙ^{n}) be the set of weak cluster points of \{\frac{1}{δ_{ℓ}(P_{k})}P_{k}^{∗}ω^{ℓ}\}. Then M ≠ ∅, and for every current η ∈ M,

i) η is a positive current ii) η has mass 1 iii) there is a subsequence {P′_{k}} of {P_{k}} and a pluripolar set E such that

\frac{1}{δ_{ℓ}(P′_{k})}P′_{k}^{∗}[W] → η

for all W ∈ 𝔾(ℓ, m) \ E.

VALUE DISTRIBUTION FOR SEQUENCES AND COMPLEX DYNAMICS 21

Proof (assuming Theorem 1.1): Since

∥\frac{1}{δ_{ℓ}(P_{k})}P_{k}^{∗}ω^{ℓ}∥(ℙ^{n}) = \frac{1}{δ_{ℓ}(P_{k})} ∫ P_{k}^{∗}ω^{ℓ} ∧ ω^{n−ℓ} = 1,

it follows that M ≠ ∅. Let

η = lim_{k→∞} \frac{1}{δ_{ℓ}(P′_{k})}P_{k}^{′∗}ω^{ℓ} ∈ M

for some subsequence {P′_{k}} of P. Then (i) is obvious, and (ii) follows from

∥η∥(ℙ^{n}) = (η, ω^{ℓ−1}) = lim_{k→∞} \frac{1}{δ_{ℓ}(P′_{k})}(P_{k}^{′∗}ω^{ℓ}, ω^{n−ℓ}) = 1 .

Finally, choose a subsequence \{P′′_{k}\} of \{P′_{k}\} such that ∑ \frac{δ_{ℓ−1}(P′′_{k})}{δ_{ℓ}(P′′_{k})} < +∞ , and apply Theorem 1.1 with a_{k} = δ_{ℓ}(P′′_{k}) to obtain (iii). Proof of Proposition 6.1: Let φ ∈ D^{n−ℓ,n−ℓ}(ℙ^{n}) be an arbitrary real form and choose a constant c_{φ} such that −c_{φ}ω^{n−ℓ+1} ≤ dd^{c}φ ≤ c_{φ}ω^{n−ℓ+1}. Then for all W ∈ 𝔾(ℓ, m) such that codimP_{k}^{−1}(W) = ℓ, we have by (8),

|(P_{k}^{∗}[W] − P_{k}^{∗}ω^{ℓ}, φ)| = |(P_{k}^{∗}Λ_{W} , dd^{c}φ)| ≤ c_{φ}m_{k}(W) .

The conclusion follows from the definition of E_{ℓ}(P, A). Proof of Proposition 6.2: Let P : ℙ^{n}−→ℙ^{m} be a rational map. We write

S_{P} = {W ∈ 𝔾(ℓ, m) : codimP^{−1}(W) < ℓ} ;

S_{P} is an algebraic subvariety (which is usually empty) of 𝔾(ℓ, m) and thus is pluripolar. By (15) we can write

P^{∗}Λ_{W} ∧ ω_{z}^{n−ℓ+1} = f(z, W)ω_{z}^{n} ,

where i) 0 ≤ f(z, W) ≤ +∞, ii) f ∈ L^{1}(ℙ^{n} × 𝔾(ℓ, m)), iii) f is continuous on the set U =^{def} ℙ^{n} × 𝔾(ℓ, m) \ I_{Q} . Here Q is the map given by equation (12) in Section 5. By the definition of the proximity function,

m_{P}^{ℓ} (W) = ∫_{ℙ_{z}^{n}} f(z, W)ω_{z}^{n} , for W ∈ 𝔾(ℓ, m) \ S_{P} .

Since

I_{Q} ∩ (ℙ^{n} × {W }) = [I_{P} ∪ P^{−1}(W)] × {W } ,

m_{P}^{ℓ} (W) ∈ [0, +∞] is well defined by the above integral.

Let (−m_{P}^{ℓ})^{∗} be the upper-regularization of (−m_{P}^{ℓ}) given by

(−m_{P}^{ℓ})^{∗}(W) = lim sup_{ξ→W}[−m_{P}^{ℓ} (ξ)] ,

for W ∈ 𝔾(ℓ, m), and write

m_{P ∗}^{ℓ}(W) = −(−m_{P}^{ℓ})^{∗}(W) = lim inf_{ξ→W} m_{P}^{ℓ} (ξ) .

We claim that

m_{P}^{ℓ} (W) = m_{P ∗}^{ℓ}(W) for all W ∈ 𝔾(ℓ, m) \ S_{P} .  (18)

To demonstrate (18), let W_{0} ∈ 𝔾(ℓ, m) \ S_{P}, and let {W_{ν}} be a sequence of points converging to W_{0} such that m_{P}^{ℓ} (W_{ν}) → m_{P ∗}^{ℓ}(W_{0}). Since f(z, W_{ν}) → f(z, W_{0}) for almost all z ∈ ℙ^{n} (precisely, for z ∉ I_{P} ∪ P^{−1}(W_{0})), we have by Fatou’s Lemma,

m_{P}^{ℓ} (W_{0}) = ∫_{ℙ_{z}^{n}} f(z, W_{0})ω_{z}^{n} ≤ lim inf_{ν→∞} ∫_{ℙ_{z}^{n}} f(z, W_{ν})ω_{z}^{n} = lim inf_{ν→∞} m_{P}^{ℓ} (W_{ν}) = m_{P ∗}^{ℓ}(W_{0}) .

Since by definition, m_{P}^{ℓ} (W_{0}) ≥ m_{P ∗}^{ℓ}(W_{0}), we obtain (18).

A function u on a complex manifold Y (with values in ℝ ∪ −{∞}) is said to be quasi-plurisubharmonic (quasi-psh for short) if u is locally equal to the sum of a C^{∞} function and of a plurisubharmonic (psh) function, or equivalently, if dd^{c}u is bounded below by a continuous real (1, 1)-form. By Lemma 5.1, −m_{P ∗}^{ℓ} is quasi-psh.

We now let P = {P_{k}}, A = {a_{k}} be as in Proposition 6.2 and suppose that ∑ \frac{δ_{ℓ−1}(P_{k})}{a_{k}} < +∞. It suffices to show that E_{ℓ}(P, A) is pluripolar in an arbitrary affine open set

G′ =^{def} {W ∈ 𝔾(ℓ, m) : λ(W) ≠ 0}

given by a linear function λ : E → ℂ. . We write m_{k} = m_{P_{k}}^{ℓ} and we consider the functions u_{k} : G′ → ℝ ∪ {−∞} given by

u_{k} = v − \frac{m_{k∗}}{ℓδ_{ℓ−1}(P_{k})}

where

v(W) = log \frac{|W |^{2}}{|λ(W)|^{2}} .  (19)

We assume that ∥λ∥ = 1, so v(W) ≥ 0. By Lemma 5.1, u_{k} is plurisubharmonic on G′. Next, construct the series

u = ∑_{k=1}^{∞} \frac{δ_{ℓ−1}(P_{k})}{a_{k}}u_{k} .  (20)

To see that u is plurisubharmonic, represent it as a limit of the sequence

τ_{k} = S · v − \frac{1}{ℓ} ∑_{j=1}^{k} \frac{m_{j∗}}{a_{j}} ,

where

S = ∑_{k=1}^{∞} \frac{δ_{ℓ−1}(P_{k})}{a_{k}} .

Since m_{k∗} ≥ 0, {τ_{k}} is a decreasing sequence of plurisubharmonic functions on G′, so the limit is either plurisubharmonic or identically −∞. To see that the latter case is impossible, we average u(W) over all W with respect to Haar probability measure σ on 𝔾(ℓ, m). It is well known (e.g., see [Sh, Ch. 2, § 4, Th. 7]) that

∫_{𝔾(ℓ,m)} Λ_{W} (z)dσ(W) = c_{ℓ,m}ω^{ℓ−1}(z)  (21)

for some constant c_{ℓ,m}. Therefore, by Fubini’s theorem,

∫_{𝔾(ℓ,m)} m_{P_{k}}(W)dσ(W) = ∫_{𝔾(ℓ,m)} (∫_{ℙ^{n}} P_{k}^{∗}Λ_{W} ∧ ω^{n−l+1}) dσ(W)

= c_{ℓ,m} ∫_{ℙ^{n}} P_{k}^{∗}ω^{l−1} ∧ ω^{n−ℓ+1} = c_{ℓ,m}δ_{ℓ−1}(P_{k}) .

Hence

∫ u(W)dσ(W) ≥ −\frac{1}{ℓ} ∑_{k=1}^{∞} \frac{1}{a_{k}} ∫ m_{k}(W)dσ(W)

= −\frac{1}{ℓ} ∑_{k=1}^{∞} \frac{c_{ℓ,m}δ_{ℓ−1}(P_{k})}{a_{k}}
= −\frac{c_{ℓ,m}}{ℓ} S > −∞ .

Thus u is plurisubharmonic.

Finally, if W ∈ E_{ℓ}(P, A) \ ⋃_{k} S_{P_{k}}, then by definition, for an infinite number of indices k we have

εa_{k} < m_{k}(W) = m_{k∗}(W)

for some ε > 0, and therefore

\frac{δ_{ℓ−1}(P_{k})}{a_{k}}u_{k}(W) ≤ \frac{δ_{ℓ−1}(P_{k})}{a_{k}}v(W) − \frac{ε}{ℓ} ,

so u(W) = −∞. Since the S_{P_{k}} are pluripolar, it follows that the set E_{ℓ}(P, A) is pluripolar.

7. Limit measures for iterates of rational maps

In this section we prove Theorem 1.3. We continue to use the notation from Section 6.

Assume that P : ℙ^{n}−→ℙ^{n} is as in Theorem 1.3 and write λ = δ_{n}(P), δ = δ_{n−1}(P). Let

h = \frac{1}{λ}P^{∗}ω^{n} ∈ D′^{n,n}(ℙ^{n}).

By the definition of the topological degree δ_{n}(P), ∫_{ℙ^{n}} h = 1. Claim: If f is a quasi-plurisubharmonic function on ℙ^{n}, then

∫_{ℙ^{n}} fh > −∞ .  (22)

(Note that h ≥ 0; since f is bounded above, the claim is equivalent to saying that fh is L^{1}.) To verify the claim, we again consider the commutative diagram (3). We then have

∫ fh = \frac{1}{λ} ∫_{G}(f ∘ π_{1})π_{2}^{∗}ω^{n} > −∞

since f ∘ π_{1} is quasi-plurisubharmonic on G̃ and hence is in L^{1}, verifying (22). Choose a > 0 such that δ < a < λ. By Lemma 4.6, δ_{n−1}(P^{k}) ≤ δ^{k} and thus

∑ \frac{δ_{n−1}(P^{k})}{a^{k}} = S < +∞.

Applying Theorem 1.1 with ℓ = n, P = {P_{k}}, where P_{k} is the k-th iterate of P, and A = {a^{k}}, we conclude that

\frac{m_{k}(W)}{a^{k}} → 0  (23)

for points W of ℙ^{n} outside a pluripolar set.

Since pluripolar sets have Lebesgue measure zero, (23) is valid for a.a. W ∈ ℙ^{n}. Let

U =^{def} ∑_{k=1}^{∞} \frac{m_{k}}{a^{k}} = Sv − u,

where v, u are given by (19),(20). By the proof of Proposition 6.2, u is psh and hence −U is quasi-plurisubharmonic. Therefore, by (22),

∫_{ℙ^{n}} Uh < +∞  (24)

Let φ ∈ D^{0}(ℙ^{n}) be arbitrary. Since ∫_{ℙ^{n}} h = 1,

\frac{1}{a^{k}} (P_{k}^{∗}h − P_{k}^{∗}ω^{n}, φ) = \frac{1}{a^{k}} ∫ (P_{k}^{∗}[W] − P_{k}^{∗}ω^{n}, φ)h(W)
= −\frac{1}{a^{k}} ∫ (P_{k}^{∗}Λ_{W} , dd^{c}φ)h(W) .

As in the proof of Proposition 6.1, |(P_{k}^{∗}Λ_{W} , dd^{c}φ)| ≤ c_{φ}m_{k}(W), and we conclude that

|\frac{1}{a^{k}} (P_{k}^{∗}Λ_{W} , dd^{c}φ)| ≤ c_{φ}U(W) .  (25)

By (23), (24) and (25), we can let k → +∞ and apply Lebesgue’s dominated convergence theorem to conclude that

\frac{1}{a^{k}} (P_{k}^{∗}h − P_{k}^{∗}ω^{n}, φ) → 0  (26)

as k → +∞.

We note that

\frac{1}{λ^{k+1}} P_{k+1}^{∗}ω^{n} = \frac{1}{λ^{k}} P_{k}^{∗}h,

as an identity of currents with L_{loc}^{1} coefficients.

Hence by (26)

(\frac{1}{λ^{k+1}} P_{k+1}^{∗}ω^{n} − \frac{1}{λ^{k}} P_{k}^{∗}ω^{n}, φ) = \frac{1}{λ^{k}} (P_{k}^{∗}h − P_{k}^{∗}ω^{n}, φ) ≤ (\frac{a}{λ})^{k} .

for k sufficiently large. Therefore the sequence \{(\frac{1}{λ^{k}} P_{k}^{∗}ω^{n}, φ)\} is Cauchy. Since φ is arbitrary, it follows that \frac{1}{λ^{k}} P_{k}^{∗}ω^{n} converges to a measure μ.

Let μ_{k} = \frac{1}{λ^{k}} P_{k}^{∗}ω^{n}. Since the μ_{k} are probability measures, so is μ. The last conclusion of Theorem 1.3 then follows from Theorem 1.1.

Acknowledgements

Part of the work on this paper was done while both authors were at the Mathematical Sciences Research Institute in Berkeley. This work was started when the first author was visiting Johns Hopkins University. The authors would like to thank these organizations for their hospitality and support.

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Alexander Russakovskii, Theory of Functions Department, Institute for Low Temperature Physics, 310164 Kharkov, Ukraine

    E-mail address: russakov@math.jhu.edu, russakov@msri.org

Bernard Shiffman, Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA

    E-mail address: shiffman@math.jhu.edu, shiffman@msri.org

Current address: Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720, USA (both authors, until June 1996)