Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 24, 1999, 89–104

ELIMINATION OF DEFECTS OF MEROMORPHIC MAPPINGS OF C^{m} INTO P^{n}(C)

Seiki Mori

Yamagata University, Faculty of Science, Department of Mathematical Sciences Yamagata, 990-8590 Japan; mori@kszaoh3.kj.yamagata-u.ac.jp

Abstract. The Nevanlinna defect relation and other results on the Nevanlinna theory assert that each meromorphic mapping f of C^{m} into P^{n}(C) has few deficient hyperplanes in P^{n}(C). However, it seems to me that meromorphic mappings with a deficient hyperplane must be very few. In this paper, we show that for an arbitrary given transcendental meromorphic mapping f (which may be linearly degenerate), we can eliminate all deficient hyperplanes in the sense of Nevanlinna by a small deformation of f.

1. Introduction

For a nondegenerate meromorphic mapping f of C^{m} into the complex projective space P^{n}(C), Nevanlinna’s defect relation

∑_{j=1}^{N} δ_{f}(H_{j}) ≤ n + 1

holds for hyperplanes {H_{j}}_{j=1}^{N} ∈ P^{n}(C)^{∗} in a general position. Hence the set of hyperplanes with a positive Nevanlinna deficiency in a set X ⊂ P^{n}(C)^{∗} of hyperplanes in a general position is at most countable. (Such a hyperplane is called a deficient hyperplane or a defect.) Sadullaev [5] proved that the set of hyperplanes with a positive Valiron deficiency is of capacity zero (or it is a locally pluripolar set). Furthermore, we observe that the set of Valiron deficient hyperplanes has projective logarithmic capacity zero in the sense of Molzon–Shiffman–Sibony [3]. These results assert that defects of a meromorphic mapping are very few, and sometimes the Nevanlinna theory is called the equidistribution theory. However, it seems to me that meromorphic mappings with deficient hyperplanes must be very few.

In this note, we deal with the problem of whether we can eliminate all defects for a given meromorphic mapping of C^{m} into P^{n}(C) by a small deformation of the mapping. In one dimensional case, for any finite-order transcendental entire function f, the inequality

∑_{a∈C} δ_{f}(a) ≤ δ_{f′}(0)

1991 Mathematics Subject Classification: Primary 32H; Secondary 30D.

holds (see Hayman [2, p. 104]), and we also find α ∈ C such that f̃(z) = f(z)+αz satisfies δf̃′(0) = 0. Hence we see

∑_{a∈C} δf̃(a) = 0,

that is, f̃ does not have a finite defect. In Section 3, we shall prove that for any transcendental meromorphic mapping f of C^{m} into P^{n}(C), we can eliminate all Nevanlinna defects by a small deformation f̃ of f. Here a small deformation f̃ of f means that their Nevanlinna order functions satisfy

|T_{f}(r) − Tf̃(r)| ≤ O(log r), r → +∞.

2. Preliminaries

2.1. Notation and terminology. Let z = (z_{1}, …, z_{m}) be the natural coordinate system in C^{m}. Set

⟨z, ξ⟩ = ∑_{j=1}^{m} z_{j}ξ_{j} for  ξ = (ξ_{1}, …, ξ_{m}),  ∥z∥^{2} = ⟨z, \overline{z}⟩,

B(r) = {z ∈ C^{m} | ∥z∥ < r}, ∂B(r) = {z ∈ C^{m} | ∥z∥ = r},
ψ = dd^{c} log ∥z∥^{2}  and  σ = d^{c} log ∥z∥^{2} ∧ ψ^{m−1},

where d^{c} = (\sqrt{−}1/4π)(¯∂ − ∂), and ψ^{k} = ψ ∧ ⋯ ∧ ψ (k -times).

Let f be a meromorphic mapping of C^{m} into P^{n}(C). Then f has a reduced representation (f_{0} : ⋯: f_{n}), where f_{0}, …, f_{n} are holomorphic functions on C^{m} with codim_{C}{z ∈ C^{m} | f_{0}(z) = ⋯ = f_{n}(z) = 0} ≥ 2. We write f = (f_{0}, …, f_{n}) as the same letter of the meromorphic mapping f. Denote D^{α}f = (D^{α}f_{0}, …, D^{α}f_{n}) for a multi-index α, where D^{α}ϕ = ∂^{|α|}ϕ/∂z_{1}^{α_{1}} ⋯ ∂z_{m}^{α_{m}}, α = (α_{1}, …, α_{m}), |α| = α_{1} + ⋯ + α_{m} and a function ϕ.

Definition. (See Fujimoto [4, Section 4].) We define the generalized Wronskian of f by

W_{α^{0},...,α^{n}}(f) = det(D^{α^{k}}f : 0 ≤ k ≤ n),

for n + 1 multi-indices α^{k} = (α_{1}^{k}, …, α_{m}^{k}), 0 ≤ k ≤ n.

By Fujimoto [4, Section 4], for every linearly nondegenerate meromorphic mapping f of C^{m} into P^{n}(C), there are n+1 multi-indices α^{0}, …, α^{n} such that {D^{α^{0}}f, …, D^{α^{n}}f} is an admissible basis with |α^{k}| ≤ n+1. Then W_{α^{0},...,α^{n}}(ϕf) = ϕ^{n+1}W_{α^{0},...,α^{n}}(f) ≢ 0 holds for any nonzero holomorphic function ϕ on C^{m}, where ϕf = (ϕf_{0}, …, ϕf_{n}).

For a nonconstant meromorphic mapping f of C^{m} into P^{n}(C), the proximity function m_{f}(r, H) and the counting function N_{f}(r, H) of a hyperplane H in P^{n}(C) are given by

m_{f}(r, H) := ∫_{∂B(r)} log \frac{∥f∥∥a∥}{|⟨f, a⟩|}σ

and

N_{f}(r, H) := ∫_{r_{0}}^{r} \frac{dt}{t} ∫_{(f^{∗}H)∩B(t)} ψ^{m−1}

for some fixed r_{0} > 0, where

H = {w = (w_{0}, …, w_{n}) ∈ C^{n+1} \ {0} | ∑_{j=0}^{n} a_{j}w_{j} = 0},

a = (a_{0}, …, a_{n}) ∈ C^{n+1} \ {0} and f^{∗}H denotes the pullback of H under f. The Nevanlinna order function T_{f}(r) of f is given by

T_{f}(r) := ∫_{r_{0}}^{r} \frac{dt}{t} ∫_{B(t)} f^{∗}ω∧ψ^{m−1},

where ω = {ω_{α}} = dd^{c} log ∑_{j=0}^{n}(|w_{j}/w_{α}|^{2}), in a neighborhood U_{α} := {w_{α}≠ 0}. We write

N(r, (ϕ)) := ∫_{r_{0}}^{r} \frac{dt}{t} ∫_{(ϕ)∩B(t)} ψ^{m−1},

where (ϕ) denotes the divisor determined by a meromorphic function ϕ on C^{m}. We note that

log ∥f∥ = log \sqrt{|f_{0}|^{2} + ⋯ + |f_{n}|^{2}} = log(|f_{0}| + ⋯ + |f_{n}|) + O(1).

Then we observe that

T_{f}(r) = ∫_{∂B(r)} log(∑_{j=0}^{n} |f_{j}|^{2})^{1/2}σ + O(1) = ∫_{∂B(r)} log ∑_{j=0}^{n} |f_{j}|σ + O(1).

The Nevanlinna deficiency δ_{f}(H) and the Valiron deficiency Δ_{f}(H) of a hyperplane H for f are given by

δ_{f}(H) := lim inf_{r→+∞} \frac{m_{f}(r, H)}{T_{f}(r)} = 1 − lim sup_{r→+∞} \frac{N_{f}(r, H)}{T_{f}(r)}

and

Δ_{f}(H) := lim sup_{r→+∞} \frac{m_{f}(r, H)}{T_{f}(r)} = 1 − lim inf_{r→+∞} \frac{N_{f}(r, H)}{T_{f}(r)}.

We now define the projective logarithmic capacity of a set in the projective space P^{n}(C) (see Molzon–Shiffman–Sibony [3, p. 46]). Let E be a compact subset of P^{n}(C) and 𝒫(E) denote the set of probability measures supported on E. We define

V_{μ}(x) := ∫_{w∈P^{n}(C)} log \frac{∥x∥∥w∥}{|⟨x, w⟩|} dμ(w), μ ∈ 𝒫(E)

and

V (E) := inf_{μ∈𝒫(E)} sup_{x∈P^{n}(C)} V_{μ}(x).

Define the projective logarithmic capacity of E by

C(E) := \frac{1}{V (E)}.

(This is a Frostman type capacity.) If V (E) = +∞, we say that the set E is of projective logarithmic capacity zero. For an arbitrary subset K of P^{n}(C), we put

C(K) = sup_{E⊂K} C(E),

where the supremum is taken over all the compact subset E of K. Note that there is a probability measure μ_{0} ∈ 𝒫 such that V (E) = sup_{x∈P^{n}(C)} V_{μ_{0}}(x).

2.2. Some results. A. Vitter [7] proved the following theorem:

Theorem A (Lemma on logarithmic derivatives). Let f = (f_{0} : f_{1}) be a reduced representation of a meromorphic mapping f: C^{m} → P^{1}(C). Set F = f_{1}/f_{0} . Then there exist positive constants a_{1} , a_{2} , a_{3} such that

∫_{∂B(r)} log^{+} |\frac{F_{z_{j}}}{F} |σ ≤ a_{1} + a_{2} log r + a_{3} log T_{f}(r), j = 1, …, m //.

Here the notation “A(r) ≤ B(r) //” means that the inequality A(r) ≤ B(r) holds for r outside a countable union of intervals I of finite Lebesgue measure.

Molzon–Shiffman–Sibony [3] proved the following result on the projective logarithmic capacity.

Theorem B ([3, p. 47]). Let φ: [0, 1] → P^{n}(C) be a real smooth nondegenerate arc in P^{n}(C) and K a compact subset of [0, 1]. Then the projective logarithmic capacity C(φ(K)) is positive if and only if K has a positive logarithmic capacity in C.

Here “smooth nondegenerate arc φ” means that there exists a lift φ̃: [0, 1] → C^{n+1} \ {0} such that the k -th derivatives {φ^{(k)}(t)}_{k≥0} of φ̃(t) span C^{n+1} for every t ∈ [0, 1].

Furthermore, using a similar argument to Tsuji [6, p. 199, Theorem V.5], we have

Proposition 1. Let f be a meromorphic mapping of C^{m} into P^{n}(C) such that lim_{r→+∞} T_{f}(r) = +∞. Then there exist r_{1} < r_{2} < ⋯ < r_{n} → +∞ and sets E_{n} : E_{n+1} ⊂ E_{n}, n = 1, 2, …, in P^{n}(C)^{∗} with

V (E_{n})≥ 2 log T_{f}(r_{n})

such that, if H does not belong to E_{n},

m_{f}(r, H) ≤ 4\sqrt{T_{f}(r)} log T_{f}(r)

for r > r_{n}. Hence

lim_{r→+∞} \frac{m_{f}(r, H)}{T_{f}(r)} = 0

outside a set E ⊂ P^{n}(C)^{∗} of projective logarithmic capacity zero. Here P^{n}(C)^{∗} denotes the dual projective space of P^{n}(C) which consists of all hyperplanes in P^{n}(C).

This is proved by a method similar to Tsuji [6], using the Frostman type capacity C(E) = 1/V (E) and the de la Vallee Poussin type capacity

C̃(E) := sup_{λ∈ℳ} \{λ(E) | U_{λ}(z) := ∫_{w∈P^{n}(C)} log \frac{∥z∥∥w∥}{|⟨z, w⟩|} dλ(w) ≤ 1 for all z ∈ E\},

where ℳ denotes a set of Borel measures on E. Details are omitted here. The proof (Lemma B) of the inequality (7) in Tsuji [6, p. 199] for the projective logarithmic potential is the following. We include here the proofs of Lemma A and Lemma B, since Ninomiya’s book is written in Japanese.

Lemma A (cf. Ninomiya [4]). Let E be a Borel set in P^{n}(C). Then we have

C(E) = C̃(E).

Proof. For any ε > 0, we take a Borel measure μ ≥ 0 with μ(E) = C̃(E)+ε. Then the inequality

sup_{z∈P^{n}} ∫_{w∈P^{n}} log \frac{∥z∥∥w∥}{|⟨z, w⟩|} dμ(w) > 1

holds, and there is a measure μ_{0} ∈ 𝒫(E) such that

V (E) = sup_{z∈P^{n}} ∫_{E} log \frac{∥z∥∥w∥}{|⟨z, w⟩|} dμ_{0}(w).

We put μ_{1} = (C̃(E) + ε)μ_{0} . Then we have

sup_{z∈P^{n}} ∫_{w∈P^{n}} log \frac{∥z∥∥w∥}{|⟨z, w⟩|} dμ_{1}(w) > 1,

or, equivalently,

sup_{z∈P^{n}} ∫_{E} log \frac{∥z∥∥w∥}{|⟨z, w⟩|} dμ_{0}(w) > \frac{1}{C̃(E) + ε}.

Hence we get V (E) ≥ 1/(C̃(E) + ε), so C̃(E) ≥ C(E). Next we shall show C̃(E) ≤ C(E) under the assumption C̃(E) > 0. For any ε > 0, there is a measure ν such that ν(E) = C̃(E) − ε and

sup_{z∈P^{n}} ∫_{w∈P^{n}} log \frac{∥z∥∥w∥}{|⟨z, w⟩|} dν(w) ≤ 1.

Now we put μ_{0} = ν/ν(E). Then we see μ_{0}(E) = 1, so μ_{0} ∈ 𝒫(E), and we have

\frac{1}{C(E)} = V (E) = inf_{μ∈𝒫} sup_{z∈P^{n}} ∫_{w∈P^{n}} log \frac{∥z∥∥w∥}{|⟨z, w⟩|} dμ(w)

≤ sup_{z∈P^{n}} ∫_{w∈P^{n}} log \frac{∥z∥∥w∥}{|⟨z, w⟩|} dμ_{0}(w)

≤ sup_{z∈P^{n}} ∫_{w∈P^{n}} log \frac{∥z∥∥w∥}{|⟨z, w⟩|} \frac{dν(w)}{ν(E)} ≤ \frac{1}{ν(E)} = \frac{1}{C̃(E) − ε}.

Thus we have C̃(E) − ε ≤ C(E), and hence we have C̃(E) ≤ C(E). Therefore we obtain C(E) = C̃(E).

Lemma B (cf. Ninomiya [4]). Let E_{1}, E_{2}, …, E_{n}, … be Borel sets in P^{n}(C), and E = ⋃_{n=1}^{∞} E_{n}. Then

\frac{1}{V (E)} ≤ \frac{1}{V (E_{1})} + \frac{1}{V (E_{2})} + ⋯ \frac{1}{V (E_{n})} + · · · .

Proof. For any ε > 0, there is a measure μ ∈ 𝒫(E) such that

sup_{z∈P^{n}} ∫_{E} log \frac{∥z∥∥w∥}{|⟨z, w⟩|} dμ(w) < V (E) + ε.

Put μ′ = μ/(V (E) + ε) (≥ 0). Then we have

sup_{z∈P^{n}} ∫_{E} log \frac{∥z∥∥w∥}{|⟨z, w⟩|} dμ′(w) < 1.

Since log(∥z∥∥w∥/|⟨z, w⟩|) ≥ 0, we have

sup_{z∈P^{n}} ∫_{E_{n}} log \frac{∥z∥∥w∥}{|⟨z, w⟩|} dμ′(w) < 1.

Hence μ′(E_{n}) ≤ C̃(E_{n}), and we obtain

\frac{1}{V (E) + ε} = μ′(E) ≤ ∑_{n=1}^{∞} μ′(E_{n}) ≤ ∑_{n=1}^{∞} C̃(E_{n}) = ∑_{n=1}^{∞} \frac{1}{V (E_{n})}.

Proposition 2. Consider the set

(1) \begin{aligned}𝒜 = \{(1, a_{1}, …, a_{n}, a_{1}a_{2}, …, a_{1}a_{n}, a_{2}a_{3}, …, & a_{n−1}a_{n}, a_{1}a_{2}a_{3}, …, ∏_{j=1}^{n}a_{j}) \\ & | a_{j} ∈ C\} ⊂ P^{N}(C)^{∗}, \end{aligned}

where N = 2^{n} − 1. After some rearrangement it contains vectors of the form (1, α, α^{2}, …, α^{N}), α ∈ C.

Proof. Let a_{1} = α^{k_{1}} , a_{2} = α^{k_{2}} , …, a_{l} = α^{k_{l}}, …, a_{n} = α^{k_{n}}, where

k_{1} = 1,  k_{m} = ∑_{l=1}^{m−1} k_{l} + 1,  m = 2, 3, …, n.

We shall prove this by induction. In the case where n = 2, we have a_{1} = α, a_{2} = α^{2} , so a_{1}a_{2} = α^{3} . Hence we obtain (1, a_{1}, a_{2}, a_{1}a_{2}) = (1, α, α^{2}, α^{3}) ∈ C^{2^{2}} \ {0}, so this case is proved. Supposing that it was proved until n, we shall prove the case where n + 1. We set a_{n+1} = α^{k_{n+1}} , where k_{n+1} = ∑_{l=1}^{n} k_{l} + 1. Then we observe that k_{n+1} = 2^{n}, since

k_{n+1} = ∑_{l=1}^{n} k_{l} + 1 = ∑_{l=1}^{n−1} k_{l} + k_{n} + 1 = 2k_{n} = 2 · 2^{n−1} = 2^{n}

when k_{n} = 2^{n−1} . By the assumption of induction, we can get 1, α, α^{2}, …, α^{2^{n}−1} by using a_{1}, …, a_{n}, a_{1}a_{2}, …, ∏_{j=1}^{n}a_{j}. Hence we can get α^{2^{n}}, α^{2^{n}+1}, …, α^{2^{n+1}−1} by using a_{1}a_{n+1}, …, a_{n}a_{n+1}, a_{1}a_{2}a_{n+1}, …, ∏_{j=1}^{n}a_{j}a_{n+1} .

3. Elimination of defects of meromorphic mappings

For a meromorphic mapping g of C^{m} into P^{n}(C), we can eliminate all defects by a small deformation of g. We say that meromorphic mapping g is transcendental if

lim_{r→+∞} \frac{T_{g}(r)}{log r} = +∞.

Note that a meromorphic mapping g is rational if and only if

T_{g}(r) = O(log r), r → +∞.

Lemma 1. There are monomials ζ_{1}, …, ζ_{n} in z_{1}, …, z_{m} such that any n derivatives in {D^{α}ζ := (D^{α}ζ_{1}, …, D^{α}ζ_{n}) | |α| ≤ n + 1} are linearly independent over the field M of meromorphic functions on C^{m}, where α = (α_{1}, …, α_{m}) ∈ Z_{≥0} is a multi-index and D^{α}ζ_{k} = ∂^{|α|}ζ_{k}/∂z_{1}^{α_{1}} ⋯ ∂z_{m}^{α_{m}}.

Proof. We now take positive integers k_{1}, …, k_{nm}, inductively. First, we take some positive integer k_{1} > n, and then an integer k_{j} as large as

k_{j} > ∏_{r=1}^{j−1}(k_{r}!), j = 2, …, nm.

We now put

ζ = (ζ_{1}, ζ_{2}, …, ζ_{n}) = (z_{1}^{k_{1}} ⋯ z_{m}^{k_{m}}, z_{1}^{k_{m+1}} ⋯ z_{m}^{k_{2m}}, …, z_{1}^{k_{(n−1)m+1}} ⋯ z_{m}^{k_{nm}}).

Then, for a multi-index α = (α_{1}, …, α_{m}) with |α| ≤ n + 1, we have

D^{α}(ζ_{s}) = D^{α}(z_{1}^{k_{(s−1)m+1}} ⋯ z_{m}^{k_{sm}})

= \frac{k_{(s−1)m+1}!}{(k_{(s−1)m+1} − α_{1})!} ⋯ \frac{k_{sm}!}{(k_{sm} − α_{1})!}z_{1}^{k_{(s−1)m+1}−α_{1}} ⋯ z_{m}^{k_{sm}−α_{m}}

(= A_{s,α}z_{1}^{k_{(s−1)m+1}−α_{1}} ⋯ z_{m}^{k_{sm}−α_{m}}, say.)

Hence we can write

D^{α}ζ = ϕ_{α}(A_{1,α}ξ_{1}, …, A_{n,α}ξ_{n}),

where ϕ_{α} ∈ M and ξ_{i} is some monomial in z_{1}, …, z_{m}. Then for any n multiindices α^{j_{1}}, …, α^{j_{n}}, we observe that

| \begin{matrix}A_{1,α^{j_{1}}} & … & A_{n,α^{j_{1}}} \\ A_{1,α^{j_{2}}} & … & A_{n,α^{j_{2}}} \\ ⋮ & ... & ⋮ \\ A_{1,α^{jn}} & … & A_{n,α^{jn}} \end{matrix} | ≠ 0.

Thus any n derivatives in {D^{α}ζ | |α| ≤ n + 1} are linearly independent over the field M of meromorphic functions on C^{m}.

We note that we can take k_{1}, …, k_{nm} arbitrarily large.

Lemma 2. Let h = (h_{0} : h_{1} : ⋯: h_{n}) be a reduced representation of a meromorphic mapping of C^{m} into P^{n}(C) and ζ_{1}, …, ζ_{n} linearly independent monomials in z_{1}, …, z_{m} as in Lemma 1. Then there exist (ã_{1}, …, ã_{n}) such that ã_{j} = α^{k_{j}}, j = 1, …, n, with k_{1} = 1, k_{m} = ∑_{l=1}^{m−1} k_{l} + 1, m = 2, 3, …, n, α ∈ C, and

f := (h_{0} : h_{1} + ã_{1}ζ_{1}h_{0} : h_{2} + ã_{2}ζ_{2}h_{0} : ⋯: h_{n} + ã_{n}ζ_{n}h_{0})

is a reduced representation of a linearly nondegenerate meromorphic mapping of C^{m} into P^{n}(C).

Proof. Let

Φ(c_{j}; a_{j}) ≡ Φ(c_{0}, c_{1}, …, c_{n}; a_{1}, …, a_{n})

:= c_{0}h_{0} + c_{1}(h_{1} + a_{1}ζ_{1}h_{0}) + c_{2}(h_{2} + a_{2}ζ_{2}h_{0}) + ⋯ + c_{n}(h_{n} + a_{n}ζ_{n}h_{0})

= (c_{0} + a_{1}c_{1}ζ_{1} + a_{2}c_{2}ζ_{2} + ⋯ + a_{n}c_{n}ζ_{n})h_{0} + c_{1}h_{1} + ⋯ + c_{n}h_{n}.

Suppose that for a_{1}, …, a_{n}, there are c_{0}, …, c_{n} such that Φ(c_{j}; a_{j}) ≡ 0. Then

(2) c_{0}h_{0} + c_{1}h_{1} + ⋯ + c_{n}h_{n} = −(c_{1}a_{1}ζ_{1}h_{0} + c_{2}a_{2}ζ_{2}h_{0} + ⋯ + c_{n}a_{n}ζ_{n}h_{0}).

Now we expand h_{j}, j = 0, …, n, to the Taylor series at the origin, and especially we write

h_{0} = ∑_{|β|=0}^{∞} d_{β}z_{1}^{β_{1}} ⋯ z_{m}^{β_{m}} = ∑_{|β|=0}^{∞} d_{β}z^{β},

where β = (β_{1}, …, β_{m}) is a multi-index. We write β^{0} as an index of one of the lowest terms of h_{0} . Then all z^{β^{0}}, ζ_{1}z^{β^{0}}, …, ζ_{n}z^{β^{0}} are different from each other. Now we compare the coefficients of both sides of (2) on each ζ_{i}z^{β^{0}} , i = 1, …, n, and z^{β^{0}} . Then we have

\{ \begin{matrix}c_{0}d_{n}^{0} + c_{1}d_{n}^{1} + ⋯ + c_{n}d_{n}^{n} = −(c_{1}a_{1}e_{n}^{1} + ⋯ + c_{n}a_{n}e_{n}^{n}), \\ c_{0}d_{n−1}^{0} + c_{1}d_{n−1}^{1} + ⋯ + c_{n}d_{n−1}^{n} = −(c_{1}a_{1}e_{n−1}^{1} + ⋯ + c_{n−1}a_{n−1}e_{n−1}^{n−1}), \\ ⋯ \\ c_{0}d_{1}^{0} + c_{1}d_{1}^{1} + ⋯ + c_{n}d_{1}^{n} = −c_{1}a_{1}e_{1}^{1}, \\ c_{0}d_{0}^{0} + c_{1}d_{0}^{1} + ⋯ + c_{n}d_{0}^{n} = 0. \end{matrix}

Here d_{i}^{j}, e_{i}^{j} are determined by h_{0}, …, h_{n}, ζ_{1}, …, ζ_{n}. Hence we have

(3) \{ \begin{matrix}c_{0}d_{n}^{0} + c_{1}(d_{n}^{1} − a_{1}e_{n}^{1}) + ⋯ + c_{n}(d_{n}^{n} − a_{n}e_{n}^{n}) = 0, \\ c_{0}d_{n−1}^{0} + c_{1}(d_{n−1}^{1} − a_{1}e_{n−1}^{1}) + ⋯ + c_{n−1}(d_{n−1}^{n−1} − a_{n−1}e_{n−1}^{n−1}) + c_{n}d_{n−1}^{n} = 0, \\ ⋯ \\ c_{0}d_{1}^{0} + c_{1}(d_{1}^{1} − a_{1}e_{1}^{1}) + c_{2}d_{1}^{2} + ⋯ + c_{n}d_{1}^{n} = 0, \\ c_{0}d_{0}^{0} + c_{1}d_{0}^{1} + ⋯ + c_{n}d_{0}^{n} = 0, \end{matrix}

where d_{0}^{0} = d_{β^{0}} ≠ 0. Then, if the homogeneous equations (3) have nontrivial solutions c_{0}, …, c_{n}, the determinant of the matrix consisting of these coefficients of (3) must be identically zero. Namely,

(4) | \begin{matrix}d_{n}^{0} & d_{n}^{1} − a_{1}e_{n}^{1} & ∗ & ⋯ & ∗ & d_{n}^{n} − a_{n}e_{n}^{n} \\ d_{n−1}^{0} & d_{n−1}^{1} − a_{1}e_{n−1}^{1} & ∗ & ⋯ & d_{n−1}^{n−1} − a_{n−1}e_{n−1}^{n−1} & d_{n−1}^{n} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ d_{1}^{0} & d_{1}^{1} − a_{1}e_{1}^{1} & d_{1}^{2} & ⋯ & ∗ & d_{1}^{n} \\ d_{0}^{0} & d_{0}^{1} & ∗ & ⋯ & ∗ & d_{0}^{n} \end{matrix} | = 0.

Here,

(d_{n}^{n} − a_{n}e_{n}^{n})(d_{n−1}^{n−1} − a_{n−1}e_{n−1}^{n−1}) ⋯ (d_{1}^{1} − a_{1}e_{1}^{1})d_{0}^{0} = d_{β^{0}}^{n+1}a_{1} ⋯ a_{n} + ⋯ + ∏_{i=0}^{n} d_{i}^{i}

and e_{i}^{i} = d_{β^{0}} = d_{0}^{0} ≠ 0. Then the term including a_{1}a_{2} ⋯ a_{n} in (4) appears only in the above term, so (4) ≢ 0 as a function of a_{1}, …, a_{n}. If we take a_{1} = a, a_{l} = a^{k_{l}}, where k_{l} = ∑_{j=1}^{l−1} k_{j} + 1, l = 2, …, n, then (4) is a polynomial in a of degree ∑_{l=1}^{n} k_{l} + 1 = 2^{n} − 1 (≠ constant). Hence (4) has at most 2^{n} − 1 solutions. Therefore, if we change a ∈ C continuously, there exists an ã ∈ C such that (4) does not equal zero, that is, (4) does not have a nontrivial solution. Thus we can choose ã_{1}, …, ã_{n} such that h_{0}, h_{1} + ã_{1}ζ_{1}h_{0}, …, h_{n} + ã_{n}ζ_{n}h_{0} are linearly independent over C, so f is linearly nondegenerate.

Lemma 3. Let f = (f_{0} : ⋯: f_{n}) and h = (h_{0} : ⋯, h_{n}) be as in Lemma 2. Then we have

|T_{f}(r) − T_{h}(r)| ≤ O(log r), r → ∞.

Proof.

log ∥f∥ = log(|h_{0}| + ∑_{k=1}^{n} |h_{k} + a_{k}ζ_{k}h_{0}|) + O(1)

≤ log(|h_{0}| + ∑_{k=1}^{n}(|h_{k}| + |a_{k}ζ_{k}| |h_{0}|)) + O(1)

≤ log{(1 + ∑_{k=1}^{n}(1 + |a_{k}ζ_{k}|))(|h_{0}| + |h_{k}|)} + O(1)

≤ log{(1 + ∑_{k=1}^{n}(1 + |a_{k}ζ_{k}|))∥h∥} + O(1)

= log 2(1 + ∑_{k=1}^{n} |a_{k}ζ_{k}|) + log ∥h∥ + O(1).

Hence we have

T_{f}(r) = ∫_{∂B(r)} log ∥f∥σ

≤ ∫_{∂B(r)} log ∥h∥σ + ∫_{∂B(r)} log 2(1 + ∑_{k=1}^{n} |a_{k}ζ_{k}|)σ + O(1)

= T_{h}(r) + O(log r), r → ∞.

On the other hand,

log ∥h∥ = log ∑_{k=0}^{n} |h_{k}| + O(1)

≤ log(|h_{0}| + ∑_{k=1}^{n}(|h_{k} + a_{k}ζ_{k}h_{0}| + |a_{k}ζ_{k}h_{0}|)) + O(1)

≤ log(∑_{k=0}^{n} |f_{k}| + ∑_{k=1}^{n} |a_{k}ζ_{k}| |h_{0}|) + O(1)

≤ log(∥f∥ + ∑_{k=1}^{n} |a_{k}ζ_{k}| ∥f∥) + O(1)

≤ log{(1 + ∑_{k=1}^{n} |a_{k}ζ_{k}|)(∥f∥)} + O(1)

≤ log ∥f∥ + log(1 + ∑_{k=1}^{n} |a_{k}ζ_{k}|) + O(1).

Thus we have

T_{h}(r) ≤ ∫_{∂B(r)} log ∥f∥σ + ∫_{∂B(r)} log(1 + ∑_{k=1}^{n} |a_{k}ζ_{k}|)σ + O(1)
= T_{f}(r) + O(log r), r → ∞.

We shall prove the main theorem.

Theorem. Let g: C^{m} → P^{n}(C) be a given transcendental meromorphic mapping. Then there exists a regular matrix L = (l_{ij})_{0≤i,j≤n} of the form l_{ij} = c_{ij}ζ_{i} + d_{ij}, (c_{ij}, d_{ij} ∈ C : 0 ≤ i, j ≤ n), such that det L≠ 0 and f := L · g: C^{m} → P^{n}(C) is a meromorphic mapping without Nevanlinna deficient hyperplanes, where ζ_{1}, …, ζ_{n} are monomials in z_{1}, …, z_{m} and linearly independent over C.

Here the mapping f := L · g: C^{m} → P^{n}(C) means a product of the matrix L = (l_{ij}) and a vector of a reduced representation g̃ =^{t} (g_{0} : …: g_{n}) of g which does not depend on the choice of g̃, and a Nevanlinna deficient hyperplane H for f means a hyperplane with δ_{f}(H) > 0.

Remark 1. For the mappings as in the theorem, the inequality |T_{f}(r) − T_{g}(r)| ≤ O(log r), r → +∞ holds, and also the mapping g may be linearly degenerate or of infinite order.

Remark 2. A rational mapping g always has a Nevanlinna defect hyperplane if m = 1 or there is a regular linear change L_{0} such that L_{0} · g has a reduced representation which consists of polynomials including different degrees. But otherwise g does not have Nevanlinna deficient hyperplanes.

Remark 3. If g is of finite order, we can replace “Nevanlinna deficiency” with “Valiron deficiency” in the conclusion of the theorem.

Remark 4. If m = 1, we can take ζ_{k} = z^{k}, k = 1, …, n.

Proof of the theorem. There is a regular linear change L_{1} such that

h := L_{1} · g = (h_{0} : ⋯: h_{n}): C^{m} → P^{n}(C)

is a reduced representation of the meromorphic mapping h and

(5)  N(r, (h_{j})) = (1 − o(1))T_{h}(r),  r → +∞, j = 0, 1, …, n.

Now we choose ζ_{1}, …, ζ_{n} as in Lemma 1. (We take larger gap powers if necessary.) Set

𝒜 = {(1, a_{1}, …, a_{n}, a_{1}a_{2}, …, a_{1}a_{n}, …, ∏_{j=1}^{n}a_{j}) | a_{j} ∈ C}.

Then the set 𝒜 from Proposition 2 contains E = {(1, α, α^{2}, …, α^{2^{n}−1}) | α ∈ C}. Hence 𝒜 contains a nondegenerate smooth arc σ([0, 1]) of σ: [0, 1] → P^{N}(C)^{∗}. Therefore, by Theorem B, 𝒜 is of positive projective logarithmic capacity. Hence by Lemma 2 the set of vectors a = (a_{1}, …, a_{n}) ∈ C^{n} \ {0} such that f^{a} = (h_{0}, h_{1} + a_{1}ζ_{1}h_{0}, ⋯, h_{n} + a_{n}ζ_{n}h_{0}) is linearly nondegenerate, has a positive projective logarithmic capacity. On the other hand, the number of couples of n + 1 multi-indices α^{j_{0}}, …, α^{j_{n}} with |α^{j_{k}}| ≤ n + 1 is at most finite. Hence, there are multi-indices β^{j_{0}}, …, β^{j_{n}} and a subset ℬ of 𝒜 with a positive projective logarithmic capacity such that for any vector a ∈ ℬ, {D^{β^{j_{0}}} f^{a}, …, D^{β^{jn}} f^{a}} is an admissible basis. We write these multi-indices β^{0}, …, β^{n} instead of β^{j_{0}}, …, β^{j_{n}}. Then for these multi-indices β^{0}, …, β^{n} we consider the generalized Wronskian of f := f^{a}, (a = (a_{1}, …, a_{n}) ∈ ℬ). Thus we can write

W_{β} := W_{β^{0},...,β^{n}}(f) = W_{β^{0},...,β^{n}}(h_{0}, h_{1} + a_{1}ζ_{1}h_{0}, …, h_{n} + a_{n}ζ_{n}h_{0})

= h_{0}^{n+1}W_{β^{0},...,β^{n}}(1, \frac{h_{1}}{h_{0}} + a_{1}ζ_{1}, …, \frac{h_{n}}{h_{0}} + a_{n}ζ_{n})

= h_{0}^{n+1}W_{β^{0},...,β^{n}}(1, h_{1} + a_{1}ζ_{1}, …, h_{n} + a_{n}ζ_{n})

= h_{0}^{n+1}(W_{0} + a_{1}W_{1} + ⋯ + ∏_{i=1}^{n} a_{i}W_{N}) ≢ 0,

where h_{j} = h_{j}/h_{0} , j = 1, …, n, and W_{k} is a generalized Wronskian of some 1, h_{1}, a_{1}ζ_{1}, …, h_{n}, a_{n}ζ_{n}, 0 ≤ k ≤ N = 2^{n} − 1, by the multi-linearity of a determinant. Next we consider the meromorphic mapping F of the form

F := (W_{0}/d : W_{1}/d : ⋯: W_{N}/d): C^{m} → P^{N}(C),

where d = d(z) is a meromorphic function consisting of common factors among W_{0}, …, W_{N} such that W_{0}/d, …, W_{N}/d are holomorphic functions without common factors up to unit. This d(z) exists, because C^{m} is a Cousin domain. Then there is an i_{0} such that h_{i_{0}} is transcendental and W_{β^{0},...,β^{n}}(ζ_{1}, …, h_{i_{0}}, …, ζ_{n}) is not identically zero, since h is transcendental. Hence it is not proportionate to W_{N} = W_{β^{0},...,β^{n}}(ζ_{1}, …, ζ_{n}) ≢ 0. Thus the meromorphic mapping F is not constant. Therefore there exists an a_{0} = (1, ã_{1}, …, ã_{n}, ã_{1}ã_{2}, …, ∏_{j=1}^{n} ã_{j}) ∈ ℬ such that

(6)  lim sup_{r→∞} \frac{m_{F} (r, H_{a_{0}})}{T_{F} (r)} = 0,

since the set of Valiron deficient hyperplanes of a nonconstant meromorphic mapping is of projective logarithmic capacity zero, that is,

N_{F} (r, (⟨F, a_{0}⟩)) = N_{F} (r, H_{a_{0}}) = (1 − o(1))T_{F} (r), r → +∞,

where

H_{a_{0}} = {ξ = (ξ_{0}, …, ξ_{N}) | ⟨ξ, a_{0}⟩ = 0}

and

⟨F, a_{0}⟩ = {W_{0} + ã_{1}W_{1} + ⋯ + ∏_{k=1}^{n} ã_{k}W_{N}}/d.

For this ã_{1}, …, ã_{n}, we consider the meromorphic mapping given by the following reduced representation:

f := L_{2} · h = (f_{0} : ⋯: f_{n}): C^{m} → P^{n}(C),

where

L_{2} = ( \begin{matrix}1 & 0 & ⋯ & 0 \\ ã_{1}ζ_{1} & 1 & ⋯ & 0 \\ ã_{2}ζ_{2} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ ã_{n}ζ_{n} & 0 & ⋯ & 1 \end{matrix} ) , det L_{2} = 1 ≠ 0;

hence

f_{0} = h_{0}  and  f_{k} = h_{k} + ã_{k}ζ_{k}h_{0},  k = 1, …, n.

Then from Lemma 3 we obtain

T_{f}(r) = T_{g}(r) + O(log r) = (1 + o(1))T_{g}(r), r → +∞,

if g is not rational.

Lemma 4. Let F and f be as above. Then there exists a positive number K such that

T_{F} (r) ≤ KT_{f}(r).

Proof.

T_{F} (r) = ∫_{∂B(r)} log(|W_{0}| + |W_{1}| + ⋯ + |W_{N}|)\frac{1}{|d|}σ

= ∫_{∂B(r)} log(|W_{0}| + |W_{1}| + ⋯ + |W_{N}|)σ − ∫_{∂B(r)} log |d|σ

≤ ∑_{j=1}^{N} ∫_{∂B(r)} log^{+} |W_{j}|σ + K_{1}N(r, (h_{0})) + O(1)

≤ ∑_{j=1}^{N}\{∫_{∂B(r)} log^{+} \frac{|W_{j}|}{|h_{j_{1}} ⋯ h_{j_{s}}ζ_{j_{s+1}} ⋯ ζ_{j_{n}}|}σ

+ ∫_{∂B(r)} log^{+} |h_{j_{1}} ⋯ h_{j_{s}}ζ_{j_{s+1}} ⋯ ζ_{j_{n}}|σ} + K_{1}N(r, (h_{0})) + O(1)

= o(T_{h}(r)) + K_{2}T_{h}(r) + O(1) ≤ KT_{f}(r)

for some positive constants K ≥ K_{2} ≥ K_{1} > 0, using Theorem A.

We now continue the proof of the theorem. We take an arbitrary vector b = (b_{0}, …, b_{n}) ∈ C^{n+1} \ {0}, which determines the hyperplane H = {w ∈ C^{n+1} \ {0} | ⟨w, b⟩ = 0} in P^{n}(C). We may assume b_{n} ≠ 0. Set

A = ⟨f, b⟩ = ∑_{k=0}^{n} b_{k}f_{k}.

Elimination of defects of meromorphic mappings 103

We note that

| \begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 0 & 1 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \\ b_{1} & b_{2} & ⋯ & b_{n−1} & b_{n} \end{matrix} | = b_{n} ≠ 0;

hence we observe that f_{0}, f_{1}, …, f_{n−1}, A are linearly independent over C. Thus we have

m_{f}(r, H_{b}) = ∫_{∂B(r)} log \frac{∥f∥}{|A|}σ

= ∫_{∂B(r)} log \frac{|W_{β^{0},...,β^{n}}(f_{0}, …, f_{n})|}{|A| |f_{0}| ⋯ |f_{n−1}|}σ

+ ∫_{∂B(r)} log \frac{∥f∥ |f_{0}| ⋯ |f_{n−1}|}{|W_{β^{0},...,β^{n}}(f_{0}, …, f_{n})|}σ

= ∫_{∂B(r)} log \frac{|b_{n}^{−1}| |W_{β^{0},...,β^{n}}(f_{0}, …, f_{n−1}, A)|}{|A| |f_{0}| ⋯ |f_{n−1}|}σ

+ ∫_{∂B(r)} log \frac{∥f∥ |f_{0}| ⋯ |f_{n−1}|}{|f_{0}|^{n+1}}σ

+ ∫_{∂B(r)} log \frac{|f_{0}|^{n+1}}{|W_{β^{0},...,β^{n}}(f_{0}, …, f_{n})|}σ + O(1),

≤ ∫_{∂B(r)} log^{+} \frac{|W_{β^{0},...,β^{n}}(f_{0}, …, f_{n−1}, A)|}{|A| |f_{0}| ⋯ |f_{n−1}|}σ + ∫_{∂B(r)} log \frac{∥f∥^{n+1}}{|f_{0}|^{n+1}} σ

+ ∫_{∂B(r)} log \frac{1}{|W_{0} + a_{1}W_{1} + ⋯ + ∏_{j=1}^{n} a_{j}W_{N} |}σ + O(1)

≤ o(T_{f}(r)) + (n + 1)m_{f}(r, H_{(1,0,...,0)})

+ ∫_{∂B(r)} log \frac{1}{|W_{0} + a_{1}W_{1} + ⋯ + ∏_{j=1}^{n} a_{j}W_{N} |}σ

+ ∫_{∂B(r)} log(|W_{0}| + |W_{1}| + ⋯ + |W_{N}|)σ + O(1) ≤ o(T_{f}(r)) + (n + 1)m_{f}(r, H_{(1,0,...,0)})

+ ∫_{∂B(r)} log \frac{(|W_{0}| + |W_{1}| + ⋯ + |W_{N}|)(1/|d|)}{|W_{0} + a_{1}W_{1} + ⋯ + ∏_{j=1}^{n} a_{j}W_{N} |(1/|d|)}σ + O(1)

= o(T_{f}(r)) + o(T_{f}(r)) + ∫_{∂B(r)} log \frac{∥F ∥}{|⟨F, a_{0}⟩|}σ + O(1) (by (5))

= o(T_{f}(r)) + o(T_{F} (r)) = o(T_{f}(r)), // (by (6)), 104 Seiki Mori

since T_{F} (r) ≤ KT_{f}(r), for some K > 0 by Lemma 4, and using Theorem A. Therefore we obtain

δ_{f}(H_{b}) = lim inf_{r→+∞} \frac{m_{f}(r, H_{b})}{T_{f}(r)} = 0,

that is, δ_{f}(H) = 0 for any H ∈ P^{n}(C)^{∗}. This completes the proof of the theorem.

Problem 1. Is the conclusion of the theorem true if “Nevanlinna deficiency” is replaced with “Valiron deficiency”?

It seems to me that the set N of meromorphic mappings with Nevanlinna defects is small in the space M of all meromorphic mappings of C^{m} into P^{n}(C).

Problem 2. Something can probably be said in terms of Baire categories. What if one considers plurisubharmonic functions in M? Is N then pluripolar?

Acknowledgements. The author wishes to thank the referee for his valuable advice.

References

[1] Fujimoto, H.: Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into P^{N_{1}}(C) × ⋯ × P^{N_{k}}(C). - Japanese J. Math. 11, 1985, 233–264.

[2] Hayman, W.K.: Meromorphic Functions. - Oxford, Clarendon, 1964.

[3] Molzon, R.E., B. Shiffman, and N. Sibony: Average growth of estimates for hyperplane sections of entire analytic sets. - Math. Ann. 257, 1981, 43–59.

[4] Ninomiya, N.: Potential Theory. - Kyoritsu-Syuppan, 1975, (Japanese).

[5] Sadullaev, A.: Deficient divisors in Valiron sense. - Math. USSR. Sb. 36(4), 1980, 535– 547.

[6] Tsuji, M.: Potential Theory in Modern Function Theory. - Maruzen, Tokyo, 1959.

[7] Vitter, A.: The lemma of the logarithmic derivative in several complex variables. - Duke Math. J. 44, 1977, 89–104.

Received 28 April 1997