Meromorphically normal families and a meromorphic Montel-Carath\'eodory theorem

TL;DR

This paper establishes new sufficient conditions for meromorphic normality of families of mappings into projective space and proves a meromorphic version of the Montel-Carathéodory theorem, extending classical results to the meromorphic setting.

Contribution

It introduces a general criterion for meromorphic normality influenced by Fujimoto's work and develops meromorphic analogues of recent results on normal families and the Montel-Carathéodory theorem.

Findings

Provided sufficient conditions for meromorphic normality.

Established a meromorphic version of the Montel-Carathéodory theorem.

Extended classical normal family results to meromorphic mappings.

Abstract

In this paper, we present various sufficient conditions for a family of meromorphic mappings on a domain $D\subset \mathbb{C}^m$ into $\mathbb{P}^n$ to be meromorphically normal. Meromorphic normality is a notion of sequential compactness in the meromorphic category introduced by Fujimoto. We give a general condition for meromorphic normality that is influenced by Fujimoto's work. The approach to proving this result allows us to establish meromorphic analogues of several recent results on normal families of $\mathbb{P}^n$-valued holomorphic mappings. We also establish a meromorphic version of the Montel-Carath\'eodory theorem.

Equations

$$H:=\left\{[w_{0}:w_{1}:\cdots:w_{n}]\in\mathbb{P}^{n}\,\big{|}\sum\nolimits_{l=0}^{n}a_{l}w_{l}=0\right\},$$

$$H_{k}:=\left\{[w_{0}:w_{1}:\cdots:w_{n}]\in\mathbb{P}^{n}\,\big{|}\sum\nolimits_{l=0}^{n}a_{l}^{(k)}w_{l}=0\right\}.$$

$$\mathcal{D}(H_1,\dots ,H_{n+1}):=\left|\det \left[\begin{array}{c}{\alpha }_1\\ ⋮\\ {\alpha }_{n+1}\end{array}\right]\right|.$$

$$D(H_1,\dots ,H_q):=\prod _{1\le k_1<\cdots <k_{n+1}\le q}\mathcal{D}(H_{k_1},\dots ,H_{k_{n+1}}).$$

$${\Gamma }_f:=\overline{\{(z,f(z)):z\in D\setminus I(f)\}}$$

$${\Gamma }_f\text{ is not a subset of }D\times H.$$

$${\nu }_f(a):=\min \{j|P_j\not\equiv 0\}$$

$$\mathsf{supp}\nu :=\{z\in D|\nu (z)\ne 0\}.$$

$$\underset{j\to \infty }{\mathrm{limsup}}{S}_j:=\bigcap _{k=1}^{\infty }\bigcup _{j\ge k}{S}_j=\bigcup _{k=1}^{\infty }\bigcap _{j\ge k}{S}_j=:\underset{j\to \infty }{\mathrm{liminf}}{S}_j.$$

$${\nu (f,H)|}_U(z):={\nu }_{f^{\ast }H}(z),z\in U,$$

$$H_{j}:=\left\{[w_{0}:w_{1}:\cdots:w_{n}]\in\mathbb{P}^{n}\,\big{|}\sum\nolimits_{l=0}^{n}a_{j,\,l}w_{l}=0\right\},$$

$$H:=\left\{[w_{0}:w_{1}:\cdots:w_{n}]\in\mathbb{P}^{n}\,\big{|}\sum\nolimits_{l=0}^{n}a_{l}w_{l}=0\right\}.$$

$$\widetilde{f}_{j}(z):=\big{(}f_{j,\,0}(z),f_{j,\,1}(z),\dots,f_{j,\,n}(z)\big{)}$$

$$f(z):=[f_0(z):f_1(z):\cdots :f_n(z)]\forall z\in U\setminus {\bigcap }_{l=0}^n{f}_l^{-1}\{0\},$$

$$f_j(z):=[\cos jz:\sin jz].$$

$$H_1$$

$$H_2$$

$$H_3$$

$$f_j(z):=[1:e^{jz}].$$

$$f_j(z):=[i\cos jz:\sin jz:\sin jz].$$

$$g_j(z):=\left[1:{\prod }_{1\le t\le k_j}\frac{z-{\alpha }_{j,t}}{1-{\bar{\alpha }}_{j,t}z}:{\prod }_{1\le t\le k_j}\frac{z-{\alpha }_{j,t}}{1-{\bar{\alpha }}_{j,t}z}\right].$$

$$H_1$$

$$H_2$$

$$H_3$$

$$H_4$$

$$H_5$$

$$H_6$$

$$X\cap\Big{(}\bigcap\nolimits_{k\in I}H_{k}\Big{)}=\emptyset\,\,\,\text{ for every }I\subset\{1,\dots,2N+1\}\text{ such that }|I|=N+1,$$

$$f_j(D)\not\subset H_{k,j}\text{ and }{g}_j(D)\not\subset H_{k,j};D(H_{1,j},\dots ,H_{2n+1,j})>0;\text{ and }$$

$$\mathsf{supp}\nu (f_j,H_{k,j})\subseteq \mathsf{supp}\nu (g_j,H_{k,j});$$

$$S_k:=\mathsf{supp}\nu (g,H_k),k=1,\dots ,2n+1,$$

$$E:=\bigcup _{k=1}^{2n+1}{S}_k.$$

$$\widetilde{h}_{j}=\big{(}h_{j,\,0},h_{j,\,1},\ldots,h_{j,\,n}\big{)}\text{ and }\widetilde{h}=\big{(}h_{0},h_{1},\ldots,h_{n}\big{)}$$

$$Z:=\{\begin{array}{ll}{\bigcap }_{k\in J}{H}_k,& \text{if J=∅},\\ {\mathbb{P}}^n,& \text{if J=∅}.\end{array}$$

Full text

1. Introduction and Main Results

The work in this paper is influenced, chiefly, by two different types of results on families of holomorphic or meromorphic mappings into $n$-dimensional complex projective space, $\mathbb{P}^{n}$, $n\geq 2$. The first influence is the work of Fujimoto, who introduced the notion of meromorphic convergence (or $m$-convergence) for a sequence of meromorphic mappings from a domain $D\subset\mathbb{C}^{m}$ into $\mathbb{P}^{n}$. A family $\mathcal{F}$ of meromorphic mappings from $D\subset\mathbb{C}^{m}$ into $\mathbb{P}^{n}$ is said to be meromorphically normal if, roughly speaking, any sequence in $\mathcal{F}$ has a subsequence $\{f_{j}\}$ with the property that any point $a\in D$ admits a neighborhood $U\ni a$ such that — fixing homogeneous coordinates on $\mathbb{P}^{n}$ — for each $f_{j}$ there is a reduced representation $[f_{j,\,0}:f_{j,\,1}:\cdots:f_{j,\,n}]$ for $\left.f_{j}\right|_{U}$ and such that each $\{f_{j,\,l}\}$, $0\leq l\leq n$, converges compactly on $U$ to a holomorphic function $f_{l}$ and $f_{l}\not\equiv 0$ for at least one $l\in\{0,1,\dots,n\}$. Fujimoto called the latter notion of convergence $m$-convergence. We shall discuss this notion, and what it means, more rigorously in Section 2. At this juncture, we merely remark that $m$-convergence is a more well-behaved mode of convergence than the notion of “quasi-regular convergence” introduced by Rutishauser [8]. Indeed, Fujimoto presented the following sufficient condition for meromorphic normality (among other objectives) as an improvement of Rutishauser’s result:

Result 1.1** ([5, Theorem 4.3]).**

Let $D$ be a domain in $\mathbb{C}^{m}$ and let $\mathcal{F}$ be a family of meromorphic mappings from $D$ into $\mathbb{P}^{n}$. Let $H_{1},\dots,H_{2n+1}$ be $2n+1$ hyperplanes in general position in $\mathbb{P}^{n}$ such that for each $f\in\mathcal{F}$ and each $k\in\{1,\dots,2n+1\}$, $f(D)\not\subset H_{k}$. Suppose that for each closed ball $\overline{B}\subset D$, the volumes of $f^{-1}(H_{k})\cap\overline{B}$, viewing $f^{-1}(H_{k})$ as divisors, are uniformly bounded for $k=1,\dots,2n+1$, and $f\in\mathcal{F}$. Then the family $\mathcal{F}$ is meromoprhically normal.

In its general appearance — and in the use of the Kobayashi hyperbolicity of \mathbb{P}^{n}\setminus\big{(}\cup_{k=1}^{2n+1}~{}H_{k}\big{)} ($H_{1},\dots,H_{2n+1}$ are $2n+1$ hyperplanes in $\mathbb{P}^{n}$ as above) in its proof — the above result suggests the following natural question: Is there a version of the Montel–Carathéodory theorem for families of meromorphic mappings from $D\subset\mathbb{C}^{m}$ into $\mathbb{P}^{n}$? To elaborate: the classical Montel–Carathéodory theorem states that a family of holomorphic $\mathbb{P}^{1}$-valued mappings on a planar domain is normal if this family omits three fixed, distinct points in $\mathbb{P}^{1}$. This generalizes to higher dimensions: a family of holomorphic $\mathbb{P}^{n}$-valued mappings on a domain $D\subset\mathbb{C}^{m}$ is normal if this family omits $2n+1$ hyperplanes in $\mathbb{P}^{n}$ located in general position. This result is due to Dufresnoy [3] (also see Kiernan–Kobayashi [6, Section 4]). It is natural to ask whether, under the hypothesis of the latter result, a family of meromorphic mappings into $\mathbb{P}^{n}$ is meromorphically normal. We answer this question in the affirmative. This (see Corollary 1.6) follows from a rather general criterion for meromorphic normality.

To motivate the criterion just alluded to, we must introduce the other result that influences our work. To state this result, we need to introduce an essential quantity which — given a collection of hyperplanes $H_{1},\dots,H_{q}$ in $\mathbb{P}^{n}$, $q\geq n+1$, in general position — quantifies in a canonical way to what extent this collection is in general position. To this end, we fix a system of homogeneous coordinates $w=[w_{0}:w_{1}:\cdots:w_{n}]$ on $\mathbb{P}^{n}$, whence any hyperplane $H$ in $\mathbb{P}^{n}$ can be given by

[TABLE]

where $(a_{0},a_{1},\dots,a_{n})=:\alpha\in\mathbb{C}^{n+1}$ is a non-zero vector. In particular, we can take $\alpha\in\mathbb{C}^{n+1}$ such that $\|\alpha\|=1$. Let $H_{1},\dots,H_{n+1}$ be hyperplanes in $\mathbb{P}^{n}$, and let \alpha_{k}:=\big{(}a_{0}^{(k)},\dots,a_{n}^{(k)}\big{)} be non-zero unit vectors in $\mathbb{C}^{n+1}$ such that for each $k\in\{1,\dots,n+1\}$, $H_{k}$ is given by

[TABLE]

We define

[TABLE]

It is absolutely elementary to see — since $\alpha_{1},\dots,\alpha_{n+1}$ are unit vectors — that $\mathcal{D}(H_{1},\dots,H_{n+1})$ depends only on $\{H_{k}:1\leq k\leq n+1\}$ and is independent of the choice of $\alpha_{k}$. Now let $H_{1},\dots,H_{q}$ be hyperplanes in $\mathbb{P}^{n}$, where $q\geq n+1$. Set

[TABLE]

Recall that the hyperplanes $H_{1},\dots,H_{q}$ are in general position if $D(H_{1},\dots,H_{q})>0$.

We are now in a position to state the second result that influences the present work.

Result 1.2** ([11, Theorem 2.10]).**

Let $D$ be a planar domain and $q\geq 3n+1$ be an integer. Let $\mathcal{F}$ and $\mathcal{G}$ be two distinct families of holomorphic mappings from $D$ into $\mathbb{P}^{n}$. Suppose that the following conditions are satisfied:

1. $(a)$

For each $f\in\mathcal{F}$, there exist a $g\in\mathcal{G}$ and hyperplanes $H_{1,\,f},\dots,H_{q,\,f}$ (which may depend on $f$) such that $f$ and $g$ share $H_{k,\,f}$ on $D$, i.e., $f^{-1}(H_{k,\,f})=g^{-1}(H_{k,\,f})$ for $k=1,\dots,q$; 2. $(b)$

$\inf\{D(H_{1,\,f},\dots,H_{q,\,f}):f\in\mathcal{F}\}>0$; and 3. $(c)$

The family $\mathcal{G}$ is normal.

Then the family $\mathcal{F}$ is normal.

Two natural questions arise immediately in connection with the above result:

• •

Given the close connection between the normality of a family of holomorphic mappings from a domain $D$ into some subset of $\mathbb{P}^{n}$ and the Kobayashi hyperbolicity of the latter subset, could the number of hyperplanes featured in Result 1.2 be reduced and yet yield the same conclusion?

• •

Does a version of Result 1.2 hold true for the domain $D\subset\mathbb{C}^{m}$, $m\geq 2$?

It turns out that the first question is slightly naive since the hypothesis of Result 1.2 is absolutely lacking in information on the extent to which $f(D)$ avoids $H_{k,\,f}$, $f\in\mathcal{F}$, $1\leq k\leq q$. Moreover, Yang et al. show that the number of hyperplanes featured in Result 1.2 cannot, in general, be taken to be less than $3n+1$: see [11, Example 1]. As for the second question: Yang, Liu and Pang in [12] present several improvements to Result 1.2, one of which implies a version of the latter result for mappings in $m$ variables $m\in\mathbb{N}$.

The last two observations led us to look for a variant of Result 1.2 wherein one considers a family $\mathcal{F}$ of meromorphic maps from $D\subset\mathbb{C}^{m}$, $m\in\mathbb{N}$, into $\mathbb{P}^{n}$ and establishes the meromorphic normality of $\mathcal{F}$. In this enterprise, we were influenced by Result 1.1. Loosely speaking, if we augment the hypothesis of Result 1.2 by requiring that $f(D)\not\subset H_{k,\,f}$, $f\in\mathcal{F}$ — as in Result 1.1 — then one only needs to consider $2n+1$ hyperplanes in $\mathbb{P}^{n}$ to play the role analogous to the above result. These are the motivations for our first theorem: i.e., Theorem 1.3 below. Of course, given that $f\in\mathcal{F}$ may possess a non-empty indeterminacy locus, it is not immediately clear what $f(D)\not\subset H_{k,\,f}$ means. We shall defer the explanation to the end of this section. It turns out that the proof of our first theorem opens up an approach to other new results — with simpler, perhaps more attractive hypotheses — which we shall present right after we state:

Theorem 1.3**.**

Let $D$ be a domain in $\mathbb{C}^{m}$. Let $\mathcal{F}$ be a family of meromorphic mappings from $D$ into $\mathbb{P}^{n}$, $\mathcal{G}$ a family of holomorphic mappings from $D$ into $\mathbb{P}^{n}$, and $\mathcal{H}$ a collection of hyperplanes in $\mathbb{P}^{n}$. Suppose that, for each $f\in\mathcal{F}$, there exist a $g_{f}\in\mathcal{G}$ and a subset $\{H_{1,\,f},\ldots,\,H_{2n+1,\,f}\}\subset\mathcal{H}$ such that

1. $(a)$

$f(D)\not\subset H_{k,\,f}$* for $k=\,1,\ldots,\,2n+1$;* 2. $(b)$

$\inf\{D(H_{1,\,f},\dots,H_{2n+1,\,f}):f\in\mathcal{F}\}>0$; and 3. $(c)$

$\mathsf{supp}\,\nu(f,H_{k,\,f})\subseteq g_{f}^{-1}(H_{k,\,f})$* for $k=\,1,\ldots,\,2n+1$.*

Furthermore, assume that for any mapping in the closure of $\mathcal{G}$, its range is not a subset of any hyperplane in the closure of $\mathcal{H}$. If the family $\mathcal{G}$ is normal, then $\mathcal{F}$ is meromorphically normal.

The object $\mathsf{supp}\,\nu(f,H_{k,\,f})$ refers to the support of a certain non-negative divisor, whose precise definition is given in Section 2. If $f$ were holomorphic, then the set $\mathsf{supp}\,\nu(f,H_{k,\,f})$ would equal $f^{-1}(H_{k,\,f})$. Condition $(c)$ should be thought of as “$g_{f}^{-1}(H_{k,\,f})\supseteq f^{-1}(H_{k,\,f})$”, but as $f\in\mathcal{F}$, in general, has a non-empty indeterminacy locus, our condition needs to be stated with care. The collections $\mathcal{G}$ and $\mathcal{H}$ above are understood to be large. Indeed, the larger these collections are, the less restrictive are the constraints that they impose on $\mathcal{F}$. We shall see that the condition on the pair $(\mathcal{G},\mathcal{H})$ towards the end of the statement of Theorem 1.3 is needed due to $\mathcal{G}$ and $\mathcal{H}$ being large — see Theorem 1.4 below. The reader may ask whether one could do without this condition. However, the condition on the pair $(\mathcal{G},\mathcal{H})$ stated above is essential, as we show via Example 3.3 below.

If we replace the family $\mathcal{G}$ in the above theorem by a finite collection and let the collection of hyperplanes be discrete, then it turns out that — just following the approach of the proof of Theorem 1.3 — we may now allow $\mathcal{G}$ to comprise meromorphic mappings. This leads to a theorem in a similar spirit as Theorem 1.3, but with a more attractive statement — in that it involves simpler auxiliary objects and requires fewer conditions to be checked.

Theorem 1.4**.**

Let $D$ be a domain in $\mathbb{C}^{m}$. Let $\mathcal{F}$ be a family of meromorphic mappings from $D$ into $\mathbb{P}^{n}$ and $\mathcal{G}$ a finite family of meromorphic mappings from $D$ into $\mathbb{P}^{n}$. Let $\mathcal{H}$ be a discrete collection of hyperplanes in $\mathbb{P}^{n}$. Suppose that for each $f\in\mathcal{F}$, there exist a $g_{f}\in\mathcal{G}$ and a subset $\{H_{1,\,f},\ldots,\,H_{2n+1,\,f}\}\subset\mathcal{H}$ such that

1. $(a)$

$f(D)\not\subset H_{k,\,f}$* for $k=\,1,\ldots,\,2n+1$;* 2. $(b)$

The hyperplanes $H_{1,\,f},\ldots,\,H_{2n+1,\,f}$ are in general position; and 3. $(c)$

$\mathsf{supp}\,\nu(f,H_{k,\,f})\subseteq\mathsf{supp}\,\nu(g_{f},H_{k,\,f})$* for $k=\,1,\ldots,\,2n+1$.*

If $g(D)\not\subset H$ for every $g\in\mathcal{G}$ and $H\in\mathcal{H}$, then $\mathcal{F}$ is meromorphically normal.

We ought to stress that Theorem 1.4 is not merely a variant of Theorem 1.3. It also forms a step towards establishing a version of the Montel–Carathéodory theorem for families of meromorphic mappings from $D\subset\mathbb{C}^{m}$ into $\mathbb{P}^{n}$. But first, we state the following result, which is a meromorphic analogue of [11, Theorem 2.8] by Yang–Fang–Pang:

Corollary 1.5**.**

Let $D$ be a domain in $\mathbb{C}^{m}$. Let $\mathcal{F}$ be a family of meromorphic mappings from $D$ into $\mathbb{P}^{n}$, and let $H_{1},\dots,H_{2n+1}$ be $2n+1$ hyperplanes in general position in $\mathbb{P}^{n}$. Suppose that for each pair of mappings $f,g\in\mathcal{F}$ and each $k\in\{1,\dots,2n+1\}$, $\mathsf{supp}\,\nu(f,H_{k})=\mathsf{supp}\,\nu(g,H_{k})$. Then the family $\mathcal{F}$ is meromorphically normal.

The next result is the meromorphic analogue of the Montel–Carathéodory theorem that we had referred to at the beginning of this section. It is a corollary of the last two results.

Corollary 1.6** (Meromorphic Montel–Carathéodory Theorem).**

Let $D$ be a domain in $\mathbb{C}^{m}$. Let $\mathcal{F}$ be a family of meromorphic mappings from $D$ into $\mathbb{P}^{n}$. Suppose that $H_{1},\dots,H_{2n+1}$ are $2n+1$ hyperplanes in general position in $\mathbb{P}^{n}$. If for each $k\in\{1,\dots,2n+1\}$ and each $f\in\mathcal{F}$, $\nu(f,H_{k})\equiv 0$, then $\mathcal{F}$ is meromorphically normal.

Our last theorem shows that one has a version of Result 1.2 wherein one infers that a family of meromorphic mappings $\mathcal{F}$ satisfying conditions analogous to those in Result 1.2 is meromorphically normal. Moreover, one can state such a result wherein the domain of all the maps in $\mathcal{F}$ is a domain in $\mathbb{C}^{m}$ for any $m\geq 1$. Our theorem is as follows:

Theorem 1.7**.**

Let $D$ be a domain in $\mathbb{C}^{m}$. Let $\mathcal{F}$ be a family of meromorphic mappings from $D$ into $\mathbb{P}^{n}$, $\mathcal{G}$ a family of holomorphic mappings from $D$ into $\mathbb{P}^{n}$, and $\mathcal{H}$ a collection of hyperplanes in $\mathbb{P}^{n}$. Suppose that, for each $f\in\mathcal{F}$, there exist a $g_{f}\in\mathcal{G}$ and a subset $\{H_{1,\,f},\ldots,\,H_{3n+1,\,f}\}\subset\mathcal{H}$ such that:

1. $(a)$

$f(D)\not\subset H_{k,\,f}$* and $g_{f}(D)\not\subset H_{k,\,f}$ for $k=\,1,\ldots,\,3n+1$;* 2. $(b)$

$\inf\{D(H_{1,\,f},\dots,H_{3n+1,\,f}):f\in\mathcal{F}\}>0$; and 3. $(c)$

$\mathsf{supp}\,\nu(f,H_{k,\,f})\subseteq g_{f}^{-1}(H_{k,\,f})$* for $k=\,1,\ldots,\,3n+1$.*

If the family $\mathcal{G}$ is normal, then $\mathcal{F}$ is meromorphically normal.

It turns out — for reasons analogous to those that apply to Result 1.2 — that the number $3n+1$ in the theorem above is sharp: see Example 3.3.

The proof of Theorem 1.3 will be presented in Section 5. As the discussion right after the statement of this theorem suggests, there are a few basic notion that need to be elaborated upon. This will be the focus of the next section. The proofs of Theorem 1.4 and its corollaries will be presented in Section 6, while the proof of Theorem 1.7 will be presented in Section 7.

We end this section with a brief explanation of some common notations.

1.1. Some notations

We fix the following notation, which we shall use without any further clarification.

1. (1)

As in the discussion above, $\|\boldsymbol{\cdot}\|$ will denote the Euclidean norm. Expressions like “unit vector” will be with reference to this norm. 2. (2)

Let $f$ be a meromorphic mapping of a domain $D\subset\mathbb{C}^{m}$ into $\mathbb{P}^{n}$ and let $I(f)$ denote the indeterminacy locus of $f$ (which would be the empty set if $m=1$). We write

[TABLE]

(which would be precisely the graph of $f$ if it were holomorphic). Let $H$ be a hyperplane in $\mathbb{P}^{n}$. The notation $f(D)\not\subset H$ is our shorthand for the condition

[TABLE]

2. Basic notions

This section is devoted to elaborating upon concepts and terminology that made an appearance in Section 1, and to introducing certain basic notions that we shall need in our proofs.

In this section $D$ will always denote a domain in $\mathbb{C}^{m}$.

Let $f\not\equiv 0$ be a holomorphic function on $D$. For a point $a\in D$, let $f(z)=\sum_{j=0}^{\infty}P_{j}(z-a)$ denote the power-series representation of $f$ in a neighborhood of $a$, where $P_{j}$ is either identically zero or a homogeneous polynomial of degree $j$. The number

[TABLE]

is said to be the zero-multiplicity of $f$ at $a$. An integer-valued function $\nu:D\rightarrow\mathbb{Z}$ is called a divisor on $D$ if for each point $a\in D$ there exist holomorphic functions $g\not\equiv 0$ and $h\not\equiv 0$ in a neighborhood $U$ of $a$ such that $\nu(z)=\nu_{g}(z)-\nu_{h}(z)$ for all $z\in U$. A divisor $\nu$ on $D$ is said to be non-negative if $\nu(z)\geq 0$ for all $z\in D$. We define the support $\mathsf{supp}\,\nu$ of a non-negative divisor $\nu$ on $D$ by

[TABLE]

Let $\{\nu_{j}\,|\,j\in\mathbb{N}\}$ be a sequence of non-negative divisors on a domain $D\subset\mathbb{C}^{m}$. We say that the sequence $\{\nu_{j}\}$ converges to a non-negative divisor $\nu$ on $D$ if each $a\in D$ has a neighborhood $U$ such that there exist holomorphic functions $h_{j}(\not\equiv 0)$ and $h(\not\equiv 0)$ in $U$ with $\nu_{j}(z)=\nu_{h_{j}}(z)$ and $\nu(z)=\nu_{h}(z)$ for all $z\in U$, and $\{h_{j}\}$ converges compactly to $h$ on $U$.

The following result, due to Stoll, confirms that the support behaves continuously as a function on the space of non-negative divisors into the space of closed sets.

Result 2.1** (paraphrasing [9, Theorem 4.10]).**

Let $\{\nu_{j}\}$ be a sequence of non-negative divisors on the domain $D$. If $\{\nu_{j}\}$ converges to a non-negative divisor $\nu$, then the sequence $\{\mathsf{supp}\,\nu_{j}\}$ converges to $\mathsf{supp}\,\nu$.

We must clarify here that, given a sequence of closed sets $\{S_{j}\}$ of $D$, we say that $S_{j}$ converges to $S$ if

[TABLE]

Let $S$ be an analytic set of codimension at least $2$ in $D$. By the Thullen–Remmert–Stein theorem, any non-negative divisor $\nu$ on $D\setminus S$ can uniquely be extended to a non-negative divisor $\hat{\nu}$ on $D$. Moreover, we have the following result given by Fujimoto.

Result 2.2** ([5, page 26, (2.9)]).**

With $D$ and $S\varsubsetneq D$ as above, if a sequence of non-negative divisors $\{\nu_{j}\,|\,j\in\mathbb{N}\}$ on $D\setminus S$ converges to a divisor $\nu$ on $D\setminus S$, then $\{\hat{\nu}_{j}\}$ converges to $\hat{\nu}$ on $D$, where $\hat{\nu}_{j}$ and $\hat{\nu}$ are the extensions of $\nu_{j}$ and $\nu$ respectively.

Let us now consider a meromorphic mapping $f:D\rightarrow\mathbb{P}^{n}$. Fixing a system of homogeneous coordinates on $\mathbb{P}^{n}$, for each $a\in D$, we have a holomorphic map $\widetilde{f}(z):=(f_{0}(z),f_{1}(z),\dots,f_{n}(z))$ on some neighborhood $U$ of $a$ such that, with $I(f)$ denoting the indeterminacy set of $f$:

• •

$f(z)=[f_{0}(z):f_{1}(z):\cdots:f_{n}(z)]$ for each $z\in U\setminus I(f)$; and

• •

$I(f)\cap U=\{z\in U\,|\,f_{0}(z)=f_{1}(z)=\cdots=f_{n}(z)=0\}$.

We shall call any such holomorphic map $\widetilde{f}:U\rightarrow\mathbb{C}^{n+1}$ an admissible representation (or reduced representation) of $f$ on $U$. Note that the set $\{z\in U\,|\,f_{0}(z)=f_{1}(z)=\cdots=f_{n}(z)=0\}$ is of codimension at least $2$.

Let $H$ be a hyperplane as defined in (1.1), whence it is a divisor in $\mathbb{P}^{n}$. Let $f:D\rightarrow\mathbb{P}^{n}$ be a meromorphic mapping such that $f(D)\not\subset H$. Under this condition — see subsection 1.1 for what this means — it is possible to define the pullback of $H$ under $f$ as a divisor in $D$, which we shall denote by $\nu(f,H)$. To briefly see why this is so, consider any $a\in D$, take an admissible representation $\widetilde{f}:=(f_{0},f_{1},\dots,f_{n})$ of $f$ on a neighborhood $U$ of $a$, and consider the holomorphic function $f^{*}H:=a_{0}f_{0}+a_{1}f_{1}+\dots+a_{n}f_{n}$. It follows from the definition of an admissible representation that, in a neighborhood of $a$, the values of the divisor $\nu_{f^{*}H}$ do not depend on the choice of admissible representations. It is now easy to check that if one defines $\nu(f,H)$ by

[TABLE]

then $\nu(f,H)$ is well defined globally to give a divisor on $D$.

Let $\{H_{j}\,|\,j\in\mathbb{N}\}$ be a sequence of hyperplanes in $\mathbb{P}^{n}$. Then, a limiting hyperplane of $\{H_{j}\}$ is the limit of any convergent subsequence — viewing $H_{j}$, $j=1,2,3,\dots$, as points in the dual projective space — of $\{H_{j}\,|\,j\in\mathbb{N}\}$. It is useful, in view of our proofs below, to describe a limiting hyperplane quantitatively as well. To this end: note that each $H_{j}$ has the representation

[TABLE]

where $(a_{j,\,0},\dots,a_{j,\,n})=:\alpha_{j}\in\mathbb{C}^{n+1}$ such that $\|\alpha_{j}\|=1$. Since the sphere $\mathcal{S}:=\{z\in\mathbb{C}^{n+1}\,:\,\|z\|=1\}$ is compact, there exist a subsequence $\{\alpha_{j_{\mu}}\}$ of $\{\alpha_{j}\}$ and $\alpha:=(a_{0},\dots,a_{n})\in\mathcal{S}$ such that $\alpha_{j_{\mu}}\rightarrow\alpha$ as $\mu\rightarrow\infty$. The hyperplane

[TABLE]

is the limit of the sequence $\{H_{j_{\mu}}\,|\,\mu\in\mathbb{N}\}$. Conversely, any limiting hyperplane of $\{H_{j}\}$ arises in this manner. (Related to the last statement is the following observation: if $\{H_{j}\}$ is a convergent sequence in the dual projective space, then, since the vectors $\alpha_{j}$ associated to $H_{j}$, $j=1,2,3,\dots$, as described above are not uniquely determined, the auxiliary sequence $\{\alpha_{j}\}$ need not be convergent. However, if $\{H_{j}\}$ is convergent, then each subsequential limit of $\{\alpha_{j}\}$ would determine the same hyperplane.) Let $\mathcal{H}$ be a collection of hyperplanes in $\mathbb{P}^{n}$. Given the structure of the space of all hyperplanes in $\mathbb{P}^{n}$, the closure of $\mathcal{H}$ — which appears in the statement of Theorem 1.3 — is just the union of $\mathcal{H}$ and the set of all limiting hyperplanes of $\mathcal{H}$.

Let $M$ be a compact connected Hermitian manifold. The space $\operatorname{Hol}(D,M)$ of holomorphic mappings from $D$ into $M$ is endowed with the compact-open topology.

Definition 2.3**.**

A family $\mathcal{F}\subset\operatorname{Hol}(D,M)$ is said to be normal if $\mathcal{F}$ is relatively compact in $\operatorname{Hol}(D,M)$.

Definition 2.4** ([5, Definition 3.1]).**

Let $\{f_{j}\,|\,j\in\mathbb{N}\}$ be a sequence of meromorphic mappings from $D$ into $\mathbb{P}^{n}$. The sequence $\{f_{j}\}$ is said to converge meromorphically on $D$ to a meromorphic mapping $f$ if, for each $a\in D$, there exists an open neighborhood $U$ of $a$ and an admissible representation

[TABLE]

of $f_{j}$ on $U$, $j=1,2,3,\dots$, such that, for each $l\in\{0,1,\dots,n\}$, the sequence $\{f_{j,\,l}\}$ converges uniformly on compact subsets of $U$ to a holomorphic function $f_{l}$ on $U$ with the property that

[TABLE]

where $f_{l_{0}}(z)\not\equiv 0$ on $U$ for some $l_{0}$.

We now have all the terminology needed for the definition that is central to the discussion in Section 1.

Definition 2.5** ([5, Definition 4.1]).**

A family $\mathcal{F}$ of meromorphic mappings from $D$ into $\mathbb{P}^{n}$ is said to be a meromorphically normal family if any sequence in $\mathcal{F}$ has a meromorphically convergent subsequence on $D$.

3. Some Examples

We now provide the examples alluded to in Section 1.

Example 3.1**.**

The condition $(c)$ in the statement of Theorem 1.3 is essential.

Let $D$ be the open unit disc in $\mathbb{C}$, and $\mathcal{F}_{1}=\{f_{j}(z)\,|\,j\in\mathbb{N}\}$, where $f_{j}:D\rightarrow\mathbb{P}^{1}$ is defined by

[TABLE]

Let $\mathcal{G}_{1}$ be the singleton consisting of the map $D\ni z\longmapsto[1:z/2]$. Call this map $g$. Let

[TABLE]

These hyperplanes are in general position. The condition $(c)$ in the statement of Theorem 1.3 fails because $g^{-1}(H_{3})=\{0\}$ whereas $f_{j}^{-1}(H_{3})$ has a large number of points for $j$ large. Thus, all the conditions in the statement of Theorem 1.3 hold true except for $(c)$ (here $\mathsf{supp}\,\nu(f_{j},H_{3})=f_{j}^{-1}(H_{3})$). However, the family $\mathcal{F}_{1}$ fails to be meromorphically normal. $\blacktriangleleft$

Example 3.2**.**

The number $2n+1$ is sharp in the statement of Theorem 1.3.

Let $D$ be the open unit disc in $\mathbb{C}$, and $\mathcal{F}_{2}=\{f_{j}(z)\,|\,j\in\mathbb{N}\}$, where $f_{j}:D\rightarrow\mathbb{P}^{1}$ is defined by

[TABLE]

Let $\mathcal{G}_{2}$ be the singleton consisting of the map ${\sf id}_{D}$. Let $H_{k}:=\{[w_{0}:w_{1}]\,|\,w_{k}=0\}$, $k=0,1$, be two hyperplanes in $\mathbb{P}^{1}$. These hyperplanes are in general position. Clearly, $\mathcal{F}_{2},\,\mathcal{G}_{2}$, and $\{H_{1},H_{2}\}$ satisfy all other conditions — with the understanding that $k$ is limited to $k=1,2$ $(=2n)$ — in the statement of Theorem 1.3. However, $\mathcal{F}_{2}$ is not meromorphically normal. $\blacktriangleleft$

The following example illustrates that we cannot eliminate the condition imposed on the pair $(\mathcal{G},\mathcal{H})$. Observe: this example also confirms that the number $3n+1$ in the statement of Theorem 1.7 is also sharp. This example is taken from [11], where it serves a different purpose (somewhat resembling our previous observation). We repurpose that example as follows:

Example 3.3** (paraphrasing [11, Example 1]).**

The condition imposed on the pair $(\mathcal{G},\mathcal{H})$ in the statement of Theorem 1.3 is essential. Specifically: the conclusion of Theorem 1.3 need not follow if the range of some limit point of $\mathcal{G}$ is contained in some hyperplane in the closure of $\mathcal{H}$. Additionally, the number $3n+1$ in the statement of Theorem 1.7 is also sharp.

Let $D$ be the open unit disc in $\mathbb{C}$, and $\mathcal{F}_{3}=\{f_{j}(z)\,|\,j\in\mathbb{N}\}$, where $f_{j}:D\rightarrow\mathbb{P}^{2}$ is defined by

[TABLE]

Let us denote by $\alpha_{j,\,1},\alpha_{j,\,2},\dots,\alpha_{j,\,k_{j}}$ the zeros of $\sin jz$ in $D$. Let $\mathcal{G}_{3}=\{g_{j}\,|\,j\in\mathbb{N}\}$, where $g_{j}:D\rightarrow\mathbb{P}^{2}$ is defined by

[TABLE]

Let

[TABLE]

It is easy to see that $D(H_{1},\dots,H_{6})>0$. Also, we have $g_{j}(D)\not\subset H_{k}$ for all $j$ and $k=1,\dots,6$. Note that $g_{j}(z)\rightarrow g(z)$, where $g$ is the map $D\ni z\mapsto[1:0:0]$, this clearly shows that $g(D)\subset H_{5}$. It is not hard to see that $f_{j}^{-1}(H_{k})=g_{j}^{-1}(H_{k})$ for all $j$ and $k=1,\dots,6$. Thus, all the conditions in the statement of:

• •

Theorem 1.3 hold true except for the condition imposed on the pair $(\mathcal{G},\mathcal{H})$.

• •

Theorem 1.7 hold true, with the understanding that $k$ is limited to $k=1,\dots,6$ $(=3n)$.

However, $\mathcal{F}_{3}$ fails to be meromorphically normal. $\blacktriangleleft$

4. Essential lemmas

In order to prove our theorems, we need to state certain known results.

One of the well-known tools in the theory of normal families in one complex variable is Zalcman’s lemma. Roughly speaking, it says that the failure of normality implies that a certain kind of infinitesimal convergence must take place. The higher dimensional analogue of Zalcman’s rescaling lemma is as follows:

Lemma 4.1** ([1, Theorem 3.1]).**

Let $M$ be a compact complex space, and $\mathcal{F}$ a family of holomorphic mappings from a domain $D\subset\mathbb{C}^{m}$ into $M$. The family $\mathcal{F}$ is not normal if and only if there exist

1. $(a)$

a point $\xi_{0}\in D$ and a sequence $\{\xi_{j}\}\subset D$ such that $\xi_{j}\rightarrow\xi_{0}$; 2. $(b)$

a sequence $\{f_{j}\}\subset\mathcal{F}$; 3. $(c)$

a sequence $\{r_{j}\}\subset\mathbb{R}$ with $r_{j}>0$ and $r_{j}\rightarrow 0$; and 4. $(d)$

a sequence $\{u_{j}\}$ of unit vectors in $\mathbb{C}^{m}$

such that $h_{j}(\zeta):=f_{j}(\xi_{j}+r_{j}u_{j}\zeta)$, where $\zeta\in\mathbb{C}$ satisfies $\xi_{j}+r_{j}u_{j}\zeta\in D$, converges uniformly on compact subsets of $\mathbb{C}$ to a non-constant holomorphic mapping $h:\mathbb{C}\rightarrow M.$

Remark 4.2*.*

We remark that the result of Aladro–Krantz in [1] has a weaker hypothesis than in Lemma 4.1. In the case where $M$ is non-compact, there is a case missing from their analysis. The arguments needed in this case were provided by [10, Theorem 2.5]. At any rate, Lemma 4.1 is the version of the Aladro–Krantz theorem that we need.

The following result, due to Fujimoto, is about the extension of the domain of meromorphic convergence of a certain sequence of meromorphic mappings.

Lemma 4.3** ([5, Proposition 3.5]).**

Let $D$ be a domain in $\mathbb{C}^{m}$, and $S$ a proper analytic subset of $D$. Let $\{f_{j}\}$ be a sequence of meromorphic mappings from $D$ into $\mathbb{P}^{n}$. Suppose that $\{f_{j}\}$ converges meromorphically to a meromorphic mapping $f$ on $D\setminus S$. If there exists a hyperplane $H$ in $\mathbb{P}^{n}$ such that $f(D\setminus S)\not\subset H$ and the sequence $\{\nu(f_{j},H)\}$ of divisors converges on $D$, then $\{f_{j}\}$ converges meromorphically on $D$.

Eremenko gave the following interesting result wherein every holomorphic mapping $f$ from the complex plane $\mathbb{C}$ into a projective variety in $\mathbb{P}^{n}$ becomes a constant mapping provided $f$ omits a finite number of certain hypersurfaces.

Lemma 4.4** ([4, Theorem 1]).**

Let $X\subset\mathbb{P}^{n}$ be a projective variety, and $N$ a positive integer. Let $H_{1},\dots,H_{2N+1}$ be $2N+1$ hypersurfaces in $\mathbb{P}^{n}$ with the property

[TABLE]

where $|I|$ is the cardinality of the set $I$. Then, every holomorphic mapping from $\mathbb{C}$ into X\setminus\big{(}\cup_{k=1}^{2N+1}H_{k}\big{)} is constant.

5. The proof of Theorem 1.3

Certain parts of our proof of Theorem 1.3 will rely on techniques similar to those in [2, 7]. Since our proof will, at a certain stage, rely upon results involving the convergence of divisors, we shall rephrase Condition $(c)$ in a form that involves the supports of the divisors $g_{f}^{-1}(H_{k})$, $k=1,\dots,2n+1$. With these words, we give the

Proof of Theorem 1.3.

Let $\{f_{j}\}\subset\mathcal{F}$ be an arbitrary sequence. By the hypothesis of the theorem, there exist:

• •

a sequence $\{g_{j}\}\subset\mathcal{G}$;

• •

$2n+1$ sequences of hyperplanes $\left\{H_{k,\,j}\right\}$, $k=1,\ldots,2n+1$; and

• •

$2n+1$ hyperplanes $H_{1},\dots,H_{2n+1}$;

such that for each $j$ and each $k$

[TABLE]

[TABLE]

and such that the hyperplane $H_{k}$ is a limiting hyperplane of $\left\{H_{k,\,j}\right\}$, $k=1,\ldots,2n+1$. To be more precise: it follows from the discussion in Section 2 that there exists an increasing sequence $\{j_{\mu}\}\subset\mathbb{N}$ such that $H_{k}$ is the limit of $\left\{H_{k,\,j_{\mu}}\right\}$, $k=1,\dots,2n+1$. We also record the following, which will be relevant to the proof of Theorem 1.4:

• $(*)$

If $\mathcal{H}$ is a discrete collection of hyperplanes, then the sequences $\{H_{k,\,j_{\mu}}\}$ are constant subsequences.

Owing to the condition $(b)$, the hyperplanes $H_{1},\ldots,H_{2n+1}$ are in general position in $\mathbb{P}^{n}$. This follows from the definition of $D(H_{1},\dots,H_{2n+1})$ and the quantitative discussion on limiting hyperplanes, of a sequence of hyperplanes, in Section 2.

Since the family $\mathcal{G}$ is normal, there exists a subsequence of $\{g_{j_{\mu}}\}$ that converges compactly to a holomorphic mapping $g$. By hypothesis, $g(D)\not\subset H_{k},\ k=1,\ldots,2n+1$. Set

[TABLE]

and

[TABLE]

By Result 2.1, $E$ is either an empty set or an analytic subset of codimension 1 in $D$.

Fix a point $z_{0}\in D\setminus E$ and choose a relatively compact open neighborhood $U_{z_{0}}$ of $z_{0}$ in $D\setminus E$. We now pass to that subsequence of $\{f_{j}\}$ that is indexed by the integer-sequence that indexes the subsequence of $\{g_{j_{\mu}}\}$ introduced in the previous paragraph. At this stage, we may, without loss of generality, relabel the two sequences referred to as $\{g_{j}\}$ and $\{f_{j}\}$. With this relabelling, we conclude — in view of (5.1) and the fact that $U_{z_{0}}\cap S_{k}=\emptyset$ — that $U_{z_{0}}\cap\mathsf{supp}\,\nu\left(f_{j},\,H_{k,\,j}\right)=\emptyset$ for every $j$ and every $k$. Hence \big{\{}\!\left.f_{j}\right|_{U_{z_{0}}}\!\big{\}}\subset\operatorname{Hol}\left(U_{z_{0}},\,\mathbb{P}^{n}\right).

We shall now prove that the sequence \big{\{}\!\left.f_{j}\right|_{U_{z_{0}}}\!\big{\}} has a subsequence that converges compactly on $U_{z_{0}}$. Let us assume that the latter is not true, and aim for a contradiction. Then, by Lemma 4.1, there exist

1. (i)

a subsequence of \big{\{}\!\left.f_{j}\right|_{U_{z_{0}}}\!\big{\}}, which we may label without causing confusion — just for this paragraph — as \big{\{}\!\left.f_{j}\right|_{U_{z_{0}}}\!\big{\}}; 2. (ii)

a point $\xi_{0}\in U_{z_{0}}$ and a sequence $\{\xi_{j}\}\subset U_{z_{0}}$ such that $\xi_{j}\rightarrow\xi_{0}$; 3. (iii)

a sequence $\{r_{j}\}\subset\mathbb{R}$ with $r_{j}>0$ such that $r_{j}\rightarrow 0$; and 4. (iv)

a sequence $\{u_{j}\}$ of unit vectors in $\mathbb{C}^{m}$

such that — defining the maps $h_{j}:\zeta\mapsto f_{j}(\xi_{j}+r_{j}u_{j}\zeta)$ on suitable neighborhoods of $0\in\mathbb{C}$ — $\{h_{j}\}$ converges uniformly on compact subsets of $\mathbb{C}$ to a non-constant holomorphic mapping $h:\mathbb{C}\rightarrow\mathbb{P}^{n}$. Then, there exist admissible representations

[TABLE]

of $h_{j}$ and $h$ respectively such that $\left\{h_{j,\,l}\right\}$ converge uniformly on compact subsets of $\mathbb{C}$ to $h_{l}$, $0\leq l\leq n$. This implies that \big{\{}{h}_{j}^{*}H_{k,\,j}\big{\}} converge uniformly on compact subsets of $\mathbb{C}$ to $h^{*}H_{k}$, $1\leq k\leq 2n+1$. Recall that we defined $f^{*}H$, where $H$ is any hyperplane, in Section 2. By Hurwitz’s theorem, and the fact that for each $k\in\{1,\ldots,2n+1\},$ $U_{z_{0}}\cap\mathsf{supp}\,\nu\left(f_{j},\,H_{k,\,j}\right)=\emptyset$ for every $j$, one of the following holds for each $k$:

1. (1)

$h(\mathbb{C})\subset H_{k}$, or 2. (2)

$h(\mathbb{C})\cap H_{k}=\emptyset$.

Now, let J:=\left\{k\in\{1,\ldots,2n+1\}\,\big{|}\,h(\mathbb{C})\subset H_{k}\right\}, and

[TABLE]

We now make the following claim:

• •

$|J^{c}|\geq 2\dim_{\mathbb{C}}Z+1$; and

• •

for each $I\subset J^{c}$ with $|I|=\dim_{\mathbb{C}}Z+1$, $Z\cap\left(\bigcap_{k\in\,I}H_{k}\right)=\emptyset$.

If $J=\emptyset$, then we are done because in that case $Z=\mathbb{P}^{n}$, and the hyperplanes $H_{1},\ldots,\,H_{2n+1}$ are in general position. Now, we consider the case when $J\neq\emptyset$. Since the hyperplanes $H_{1},\ldots,\,H_{2n+1}$ are in general position, we have:

• For each $1\leq t\leq 2n+1$ and $I\subset\{1,\ldots,2n+1\}$ with $|I|=t$, $\bigcap_{k\in\,I}\!H_{k}$ is of pure dimension equal to \dim_{\mathbb{C}}\big{(}\bigcap_{k\in\,I}\!H_{k}\big{)}=\max\{n-t,\,-1\} (here, it is understood that $\dim_{\mathbb{C}}(\emptyset)=-1)$.

The above follows from Bézout’s theorem. Thus, we get

[TABLE]

Since $h(\mathbb{C})\subset Z$, $\dim_{\mathbb{C}}Z\geq 0$, whence we get

[TABLE]

Hence,

[TABLE]

The inequality above is, again, a consequence of $H_{1},\ldots,\,H_{2n+1}$ being in general position. Moreover, if $I\subset J^{c}$ with $|I|=\operatorname{dim}_{\mathbb{C}}Z+1$, then by (5.2),

[TABLE]

and

[TABLE]

where the last equality holds because the hyperplanes $H_{1},\ldots,H_{2n+1}$ are in general position. This establishes the above claim. At this stage, we can appeal to Lemma 4.4, to conclude that $h$ is constant, which is a contradiction. Thus, it follows that the sequence \big{\{}\!\left.f_{j}\right|_{U_{z_{0}}}\!\big{\}} — by which we now mean the sequence considered at the beginning of this paragraph — has a subsequence that converges compactly on $U_{z_{0}}$. Let us denote the limit of this subsequence by $f^{U_{z_{0}}}$.

Cover $D\setminus E$ by a countable collection of connected open subsets of $D\setminus E$. Note that the point $z_{0}\in D\setminus E$ and the neighborhood $U_{z_{0}}\ni z_{0}$ in the last paragraph were arbitrary (with the proviso $U_{z_{0}}\cap E=\emptyset$). Thus, from the conclusion of the last paragraph and by a standard diagonal argument, we conclude that there exists a subsequence of the original sequence $\{f_{j}\}$ — which we can again denote simply by $\{f_{j}\}$ — and a holomorphic map $f$ on $D\setminus E$ such that $\{f_{j}\}$ converges compactly on $D\setminus E$ to $f$.

We shall now show that there is a subsequence of $\left\{f_{j}\right\}$ that is meromorphically convergent in $D$. Since the hyperplanes $H_{1},\ldots,H_{2n+1}$ are in general position in $\mathbb{P}^{n}$, there exists a subset $L\subset\{1,\ldots,2n+1\}$ with $|L|\geq n+1$ such that $f^{*}H_{k}\not\equiv 0$ on $D\setminus E$ for all $k\in L$. Let $z_{1}\in D$ be an arbitrary point. By the general-position condition, again, there exist an open ball $B(z_{1},\epsilon)\subset D$ and $k_{0}\in L$ such that $\mathsf{supp}\,\nu(g,\,H_{k_{0}})\cap\overline{B(z_{1},\epsilon)}=\emptyset$. This implies that by passing to appropriate subsequences once more — and relabelling them by $j\in\mathbb{N}$ — we have subsequences $\{f_{j}\}$, $\{g_{j}\}$ and $\{H_{k_{0},\,j}\}$ such that, firstly, $\mathsf{supp}\,\nu(g_{j},\,H_{k_{0},\,j})\cap\overline{B(z_{1},\epsilon)}=\emptyset.$ Then, by (5.1), we have

[TABLE]

Now, we define meromorphic mappings $F_{j}:B(z_{1},\epsilon)\rightarrow\mathbb{P}^{n+1}$, $j\in\mathbb{N}$, as follows: for any $z\in B(z_{1},\epsilon)$, if $\widetilde{f}_{j}=(f_{j,\,0},f_{j,\,1},\ldots,f_{j,\,n})$ is an admissible representation of $f_{j}$ on a neighborhood $U_{z}\subset B(z_{1},\epsilon)$, then $F_{j}$ is such that it has the admissible representation $\widetilde{F}_{j}=(f_{j,\,0},f_{j,\,1},\dots,f_{j,\,n},f_{j}^{*}H_{k_{0},\,j})$ on $U_{z}$. Let $Q_{l}$, $l=0,1,\dots,n$, be hyperplanes in $\mathbb{P}^{n}$ defined by

[TABLE]

and let $\overline{Q}_{l}$, $l=0,1,\dots,n+1$, be hyperplanes in $\mathbb{P}^{n+1}$ defined by

[TABLE]

Clearly, $\{F_{j}\}$ converges compactly on $B(z_{1},\epsilon)\setminus E$ to a holomorphic mapping $F$ from $B(z_{1},\epsilon)\setminus E$ into $\mathbb{P}^{n+1}$, and if \widetilde{f}=\big{(}f_{0},\dots,f_{n}\big{)} is an admissible representation of $f$ on an open subset $U\subset B(z_{1},\epsilon)\setminus E$, then $F$ has an admissible representation $\widetilde{F}=(f_{0},\,f_{1},\dots,f_{n},f^{*}H_{k_{0}})$ on $U$. Since $f$ is holomorphic on $B(z_{1},\epsilon)\setminus E$, there exists $l_{0}$, $0\leq l_{0}\leq n$ such that $f^{*}Q_{l_{0}}\not\equiv 0$ on $B(z_{1},\epsilon)\setminus E$. Hence, $F^{*}\overline{Q}_{l_{0}}\not\equiv 0$ on $B(z_{1},\epsilon)\setminus E.$ Therefore, there exists a $j_{0}\in\mathbb{N}$ such that $f_{j}^{*}Q_{l_{0}}\not\equiv 0$ and $F_{j}^{*}\overline{Q}_{l_{0}}\not\equiv 0$ on $B(z_{1},\epsilon)\setminus E$ for all $j\geq j_{0}$. Since $f^{*}H_{k_{0}}\not\equiv 0$ on $B(z_{1},\epsilon)\setminus E$, we have $F^{*}\overline{Q}_{n+1}\not\equiv 0$ on $B(z_{1},\epsilon)\setminus E.$ Also, by (5.3), we have $\nu(F_{j},\overline{Q}_{n+1})=0.$ Therefore by Lemma 4.3, $\{F_{j}\}$ converges meromorphically on $B(z_{1},\epsilon)$. This implies that the sequence of divisors \big{\{}\nu(F_{j},\,\overline{Q}_{l_{0}})\big{\}}_{j\geq\,j_{0}} converges on $B(z_{1},\epsilon),$ hence \big{\{}\nu(f_{j},\,Q_{l_{0}})\big{\}}_{j\geq\,j_{0}} converges on $B(z_{1},\epsilon).$ By Lemma 4.3 again, $\{f_{j}\}_{j\geq\,j_{0}}$ is meromorphically convergent on $B(z_{1},\epsilon).$ By a diagonal argument — with details analogous to the argument made above — we extract a further subsequence that is meromorphically convergent on $D$ to a meromorphic mapping which agrees with $f$ on $D\setminus E$. This completes the proof. ∎

6. The proofs of Theorem 1.4 and its corollaries

Proof of Theorem 1.4.

Let $\{f_{j}\}\subset\mathcal{F}$ be an arbitrary sequence. We can find $2n+1$ sequences of hyperplanes $\left\{H_{k,\,j}\right\}$, $k=1,\ldots,2n+1$, associated to $\{f_{j}\}$ exactly as in the beginning of the proof of Theorem 1.3. The purpose of Condition $(b)$ of Theorem 1.3 was to ensure that the hyperplanes $H_{1},\dots,H_{2n+1}$ presented right at the beginning of the proof of Theorem 1.3 are in general position. As highlighted by $(*)$ in the proof of Theorem 1.3: in our present setting (i.e., $\mathcal{H}$ is discrete) $H_{k}$ is just the single hyperplane repeated ad infinitum that constitutes the constant sequence $\{H_{k,\,j_{\mu}}\}$ referred to by $(*)$, $k=1,\dots,2n+1$. Now, $H_{1},\dots,H_{2n+1}$ being in general position follows from $(b)$ in the statement of Theorem 1.4.

Since the family $\mathcal{G}$ is a finite family, we can find a meromorphic mapping $g\in\mathcal{G}$ such that we can extract a subsequence of $\{f_{j}\}$, which we can continue to label as $\{f_{j}\}$ without loss of generality, such that

[TABLE]

for each $j$ and each $k$. By hypothesis, $g(D)\not\subset H_{k}$ for all $k\in\{1,\dots,2n+1\}$. Now, define the sets $S_{k}$ and $E$ as they were defined in the proof of Theorem 1.3. Thus, we have all of the ingredients to be able to repeat verbatim the argument beginning from the third paragraph of the proof of Theorem 1.3 until the fifth paragraph thereof to extract from $\{f_{j}\}\subset\mathcal{F}$ a subsequence — which we may continue, without loss of generality, to refer to as $\{f_{j}\}$ — that converges compactly on $D\setminus E$ to a map $f$ that is holomorphic on $D\setminus E$.

Since the hyperplanes $H_{1},\ldots,H_{2n+1}$ are in general position in $\mathbb{P}^{n}$, there exists a subset $L\subset\{1,\ldots,2n+1\}$ with $|L|\geq n+1$ such that $f^{*}H_{k}\not\equiv 0$ on $D\setminus E$ for all $k\in L$.

We shall now show that there is a subsequence of $\left\{f_{j}\right\}$ that is meromorphically convergent in $D\setminus\bigcap_{k\in\,L}S_{k}$. Let $z_{1}\in D\setminus\bigcap_{k\in\,L}S_{k}$ be an arbitrary point. There exist an open ball $B(z_{1},\epsilon)\subset D\setminus\bigcap_{k\in\,L}S_{k}$ and $k_{0}\in L$ such that $S_{k_{0}}\cap\overline{B(z_{1},\epsilon)}=\emptyset.$ Hence, $\mathsf{supp}\,\nu(g,\,H_{k_{0}})\cap\overline{B(z_{1},\epsilon)}=\emptyset$. This implies that by passing to appropriate subsequences once more — and relabelling them by $j\in\mathbb{N}$ — we have subsequences $\{f_{j}\}$, $\{g_{j}\}$ and $\{H_{k_{0},\,j}\}$ such that, firstly, $\mathsf{supp}\,\nu(g_{j},\,H_{k_{0},\,j})\cap\overline{B(z_{1},\epsilon)}=\emptyset.$ Then, by (6.1), we have

[TABLE]

We are again in a position to repeat the argument in the final paragraph of the proof of Theorem 1.3, mutatis mutandis, to conclude that $\{f_{j}\}$ has a subsequence — which we may again relabel as $\{f_{j}\}$ — that is meromorphically convergent on $D\setminus\bigcap_{k\in\,L}S_{k}$ to a meromorphic mapping $\varphi$ (which agrees with $f$ on $D\setminus E$).

We now claim that the indeterminacy set $I(g)=\bigcap_{k\in\,L}S_{k}$. We remark here that this claim presents no contradiction when $m=1$, in which case $g$ is holomorphic. This is because, by holomorphicity of $g$, we get

[TABLE]

The latter is a consequence of the general-position condition, which here translates into $\{H_{k}:k\in L\}$ being a set of distinct points. At any rate, the following argument is not much different and holds for any $m$. Clearly, $I(g)\subset\bigcap_{k\in\,L}S_{k}.$ Since $|L|\geq n+1$, and $H_{k}$, $k\in L$, are in general position in $\mathbb{P}^{n}$, we have $\bigcap_{k\in\,L}S_{k}\subset I(g)$ — hence the claim. Now, the meromorphic normality of $\mathcal{F}$ on $D$ follows from the following argument: Take a hyperplane $H$ in $\mathbb{P}^{n}$ such that $\varphi(D\setminus\bigcap_{k\in\,L}S_{k})\not\subset H$. Clearly, $\left\{\nu(f_{j},\,H)\right\}$ converges on $D\setminus\bigcap_{k\in\,L}S_{k}$. Since the codimension of the set $\bigcap_{k\in\,L}S_{k}=I(g)$ is at least 2, we have by Result 2.2 that $\left\{\nu(f_{j},\,H)\right\}$ converges on $D$. Therefore by Lemma 4.3, $\left\{f_{j}\right\}$ converges meromorphically on $D$. This completes the proof. ∎

The proofs of Corollaries 1.5 and 1.6 are absolutely direct applications. For the sake of completeness, we now provide their proofs:

Proof of Corollary 1.5.

Take, and fix, an arbitrary mapping from the family $\mathcal{F}$ and name it $g$. Define $\mathcal{G}:=\{g\}$ and $\mathcal{H}:=\{H_{1},\dots,H_{2n+1}\}$. Now, the meromorphic normality of $\mathcal{F}$ is evident from the Theorem 1.4. ∎

Proof of Corollary 1.6.

Notice that for each $k\in\{1,\dots,2n+1\}$, and each pair of mappings $f,g\in\mathcal{F}$, $\mathsf{supp}\,\nu(f,\,H_{k})=\mathsf{supp}\,\nu(g,\,H_{k})=\emptyset.$ Hence, by Corollary 1.5, $\mathcal{F}$ is meromorphically normal. ∎

7. The proof of Theorem 1.7

Proof of Theorem 1.7.

Let $\{f_{j}\}\subset\mathcal{F}$ be an arbitrary sequence. By hypothesis, there exist

• •

a sequence $\{g_{j}\}\subset\mathcal{G}$; and

• •

$3n+1$ sequences of hyperplanes $\{H_{k,\,j}\}$, $k=1,\dots,3n+1$;

such that for each $j$ and each $k$

[TABLE]

[TABLE]

Let each $H_{k,\,j}$ have the representation

[TABLE]

As in the proof of Theorem 1.3, we get $3n+1$ hyperplanes with the representations

[TABLE]

In view of the condition $(b)$, the hyperplanes $H_{1},\dots,H_{3n+1}$ are in general position in $\mathbb{P}^{n}$.

Again as in the proof of the Theorem 1.3, we can extract a subsequence of $\{g_{j}\}$ that converges compactly to a holomorphic mapping $g$. Since $H_{1},\dots,H_{3n+1}$ are in general position in $\mathbb{P}^{n}$, we get a subset $L\subset\{1,\dots,3n+1\}$ with $|L|\geq 2n+1$ such that $g(D)\not\subset H_{k}$, for all $k\in L$. Set

[TABLE]

and

[TABLE]

By Result 2.1, $E$ is either an empty set or an analytic subset of codimension 1 in $D$. The proof follows along the same lines as the argument beginning from the third paragraph of the proof of Theorem 1.3. ∎

Acknowledgements

I would like to express my sincere gratitude to Gautam Bharali for helpful discussions during the course of this work. I would like to thank him especially for his suggestions towards strengthening some of the results in this paper. This work is supported by the SERB National Post-doctoral Fellowship (N-PDF) (Grant no. PDF/2017/001140) and a UGC CAS-II grant (Grant no. F.510/25/CAS-II/2018(SAP-I)).