Let $f(z)$ be a meromorphic function on the complex plane. By the second main theorem of value distribution theory [6], we have the following classical Picard's theorem.

Theorem A  If there exist three mutually distinct points $a_1, \ a_1$ and $a_3$ on the Riemann sphere such that $f(z)-a_i\ (i=1, 2, 3)$ has no zero on the complex plane, then $f$ is a constant.

Let $F$ be a family of meromorphic functions defined on a domain $D$ of the complex plane. $F$ is said to be normal on $D$ if every sequence of functions of $F$ has a subsequence which converges uniformly on every compact subset of $D$ with respect to the spherical metric to a meromorphic function or identically $\infty$ on $D$. Perhaps the most celebrated criteria for normality in one complex variable is the following Montel-type theorem, see ref. [2].

Theorem B  Let $F$ be a family of meromorphic functions on a domain $D$ of the complex plane. Suppose that there exist three mutually distinct points $a_1, \ a_1$ and $a_3$ on the Riemann sphere such that $f(z)-a_i\ (i=1, 2, 3)$ has no zero on $D$ for each $f\in F$. Then F is a normal family on $D$.

The fact that Picard's theorem and normality criteria were so intimately related led Bloch to hypothesise that a family of meromorphic functions which have a property $P$ in common on a domain $D$ is normal on $D$ if the property $P$ forces a meromorphic function on the complex plane to be a constant. This hypothesize is called Bloch's heuristic principle in complex function theory [1, 13]. Although the principle is false in general, many authors proved normality criteria for families of meromorphic functions by starting from Picard-type theorems. Hence an interesting topic is to make the principle rigorous and to find its applications. There are many investigations in this field for one complex variable (see, e.g., [13, 14]).

In the case of higher dimension, the notion of normal family has proved its importance in geometric function theory in several complex variables(see, e.g., [8, 12]). Many Picard-type theorems have established for holomorphic mappings of $\mathbb{C}^n$ into $P^N(\mathbb{C})$(e.g. [4, 5]). Afterwards, the related normality criteria in several complex variables have proved by using various methods (see, e.g., [3, 9]). In particular, Aladro and Krantz [1] proved a criterion for normality in several complex variables and for the first time implemented a Zalcman type heuristic principle in this more general content. Using the heuristic principle obtained by Aladro and Krantz [1], Tu [11]extended Theorem B to the case of families of holomorphic mappings of a domain $D$ in $\mathbb{C}^n$ into $P^N(\mathbb{C})$ related to Fujimoto-Green's and Nochka's Picard-type theorems.

In this paper, inspired by the idea in Tu [11], we shall prove some normality criteria for families of holomorphic mappings of several complex variables into $P^n(\mathbb{C})$, related to Shirosahi's [10] Picard-type theorems.

For the general reference of this paper, see references [10, 11].

Let $P^n(\mathbb{C})$ be the complex projective space of dimension n, and $\rho : \mathbb{C}^{n+1}\setminus\{0\} \rightarrow P^n(C)$ be the standard projective mapping. Let f be a holomorphic mapping of a domain $D$ in $\mathbb{C}^m$ into $P^n(\mathbb{C})$. Then for any $z\in D$, $f$ always has a reduced representation

$ \tilde{f}(z)=(f_0(z), \cdots, f_n(z)) $

on some neighborhood $U$ of $z$ in $D$, that is, $f_0(z), \cdots, f_n(z)$ are holomorphic functions on $U$ without common zeroes, such that $f(z) = \rho(\tilde{f}(z))$ on $U$.

Definition 2.1  A sequence $\{f^{(p)}(z)\}$ of holomorphic mappings from a domain $D$ in $\mathbb{C}^m$ into $P^n(\mathbb{C})$ is said to converge uniformly on compact subsets of $D$ to a holomorphic mapping $f(z)$ of $D$ into $P^n(\mathbb{C})$ if and only if, for any $z\in D$, each $f^{(p)}(z)$ has a reduced representation

$ \tilde{f}^{(p)}(z) = (f^{(p)}_0(z), \cdots, f^{(p)}_{n}(z)) $

on some fixed neighborhood $U$ of $z$ such that $\{f^{(p)}_i (z)\}^\infty_{p=1}$ converges uniformly on compact subsets of $U$ to a holomorphic function $f_i(z) (i = 0, \cdots, n)$ on $U$ with the property that

$ \tilde{f}(z) := (f_0(z), \cdots, f_{n}(z)) $

is a representation of $f(z)$ on $U$, where $f_{i_0}(z)$ is not identically equal to zero on $U$ for some $i_0$.

Let F be a family of holomorphic mappings of a domain $D$ in $\mathbb{C}^m$ into $P^n(\mathbb{C})$. F is said to be a normal family on $D$ if any sequence in F contains a subsequence which converges uniformly on compact subsets of $D$ to a holomorphic mapping of $D$ into $P^n(\mathbb{C})$.

For the detailed discussion about convergence, see reference [3].

A hypersurface $S$ in $P^n(\mathbb{C})$ is the projection of the set of zeros of a nonconstant homogeneous form $P$ in $n + 1$ variables. Let

$ P(w_0, w_1, \cdots, w_n) := \sum a_{i_0, \cdots, i_n}w_0^{i_0}\cdots w_n^{i_n}\ \ (a_{i_0, \cdots, i_n}\in \mathbb{C}) $

be a homogeneous polynomial in $w_0, \cdots, w_n$. Then

$ S := \rho(\{(w_0, w_1, \cdots, w_n) \in \mathbb{C}^{n+1}\setminus\{0\}: P(w_0, w_1, \cdots, w_n) = 0\}) $

is a hypersurface in $P^n(\mathbb{C})$ given by the homogeneous polynomial $P$.

Let $d$ and $p$ be two positive integers with $d > 2p+8$ and $ p > 2$ which have no common factors. Define homogeneous polynomials $H(w_0, w_1) = w_0^d + w_0^pw_1^{d-p} + w_1^d $ with degree $d$.

We consider hypersurfaces given by homogeneous polynomials $P$ and $P_n$, where

$ \begin{eqnarray*} P(w_0, w_1) &=& P_1(w_0, w_1) = H(w_0, w_1), \nonumber\\ P_n(w_0, \cdots, w_n) &=& P_{n-1}(P(w_0, w_1), \cdots, P(w_{n-1}, w_n)) \end{eqnarray*} $

with degree $d^n$ for $n \geq 2$. In [10], M. Shirosahi gave some interesting Picard-type theorems for the hypersurfaces given by homogeneous polynomials $P$ and $P_n$ as follows.

Theorem C  Let $f$ and $g$ be entire functions at least one of which are not identically equal to zero. If $P(f, g)=\alpha^d$ for some entire function $\alpha$, then $(f:g)$ is constant.

Theorem D  Let $f_0, f_1$and $f_2$ be entire functions at least two of which are not identically equal to zero, and let $C$ be a nonzero constant. If $P(f_0, f_1)=CP(f_1, f_2), $then $(f_0:f_1: f_2)$ is constant.

For more $f_j's$, Shirosahi proved the following results in [10].

Theorem E  Let $n \geq 2$ be an integer and $f_0, \cdots, f_n$ entire functions. If at least two of $P(f_0, f_1), \ P(f_1, f_2), \ \cdots, P(f_{n-1}, f_n)$ are not identically equal to zero and $(P(f_0, f_1): P(f_1, f_2): \cdots: P(f_{n-1}, f_n))$ is constant, then $(f_0:\ f_1: \cdots : f_n)$ is constant.

Theorem F  Let $n$ be a positive integer and $f$ a holomorphic mapping of $\mathbb{C}$ into $P^n(\mathbb{C})$ with a reduced representation $(f_0, \cdots, f_n)$. If $P_n(f_0, \cdots, f_n) = 0$, then $f$ is constant.

Theorem G  Let $f$ be a holomorphic mapping of $\mathbb{C}$ into $P^n(\mathbb{C})$ with a reduced representation $(f_0, \cdots, f_n)$. If $P_n(f_0, \cdots, f_n) = \alpha^{d^n}$, for some entire function $\alpha$ not identically equal to zero, then $f$ is constant.

Using the heuristic principle obtained by Aladro and Krantz [1], we will prove the normality criterions for family of holomorphic mappings of several complex variables into the complex projective space, related to Shirosahi's Picard-type theorems as follows:

Theorem 2.1  Let $F$ be a family of holomorphic mappings from a domain $D$ in $\mathbb{C}^m$ into $P^n(\mathbb{C})$. For any $z\in D$, each $f\in F$ has a reduced representation

$ \tilde{f}(z) = (f_0 (z), f_1(z), \cdots, f_n(z)) $

on some neighborhood $U$ of $z$ in $D$. If $(P(f_0, f_1): P(f_1, f_2): \cdots: P(f_{n-1}, f_n))$ is constant on $U$, then $F$ is normal on $D$.

For $n=1$, We immediately obtain normality criteria related to Theorem D.

Corollary 2.1.1  Let $F$ be a family of holomorphic mappings from a domain $D$ in $\mathbb{C}^m$ into $P^2(C)$. For any $z\in D$, each $f\in F$ has a reduced representation

$ \tilde{f}(z) = (f_0 (z), f_1(z), f_2(z)) $

on some neighborhood $U$ of $z$ in $D$. Let C be a nonzero constant, if $P(f_0, f_1)=CP(f_1, f_2)$ on $U$, then $F$ is normal on $D$.

Theorem 2.2  Let $F$ be a family of holomorphic mappings from a domain $D$ in $\mathbb{C}^m$ into $P^n(\mathbb{C})$. For any $z\in D$, each $f\in F$ has a reduced representation

$ \tilde{f}(z) = (f_0 (z), f_1(z), \cdots, f_n(z)) $

on some neighborhood $U$ of $z$ in $D$. If $P_n(f_0, f_1, \cdots, f_n)= \alpha^{d^n}$ on $U$, for some entire function $\alpha$, then $F$ is normal on $D$.

When $n=1$ in Theorem 2.2, we get the normality criteria related to Theorem C.

Corollary 2.2.1  Let $F$ be a family of holomorphic mappings from a domain $D$ in $\mathbb{C}^m$ into $P^1(C)$. For any $z\in D$, each $f\in F$ has a reduced representation

$ \tilde{f}(z) = (f_0 (z), f_1(z)) $

on some fixed neighborhood $U$ of $z$, such that $P(f_0, f_1)=\alpha^d$ on $U$ for some entire function $\alpha$, then $F$ is normal on $D$.

In order to prove Theorem 2.1 and 2.2, we need the following Lemma.

Lemma 3.1  Let F be a family of holomorphic mappings of a domain $D$ in $\mathbb{C}^m$ into $P^n(\mathbb{C})$. The family F is not normal on $D$ if and only if there exists a compact set $K\subset D$ and sequences $\{f_i\}\subset F$, $\{p_i\}\subset K$, $\{r_i\}$ with $r_i > 0$ and $r_i \rightarrow 0^+$ and $\{u_i\}\subset \mathbb{C}^m$ Euclidean unit vectors such that $g_i(\xi):= f_i(p_i + r_iu_i\xi), $ where $\xi\in \mathbb{C}$ satisfies $p_i+ r_iu_i\xi\in D$, converges uniformly on compact subsets of $\mathbb{C}$ to a nonconstant holomorphic mapping g of $\mathbb{C}$ into $P^n(\mathbb{C})$.

For the proof of Lemma 3.1, see Theorem 3.1 in reference [1], Theorem 6.5 in ref. [7] (Cf. reference [14]).

Proof of Theorem 2.1  Suppose that F is not a normal family on $D$. Then, by Lemma 3.1, there exist a compact set $K \subset D$ and sequences $\{f_k\}\subset F$, $\{p_k\}\subset K$, $\{r_k\}$ with $r_k > 0$ and $r_k \rightarrow 0^+$ and $\{u_k\}\subset \mathbb{C}^m$ Euclidean unit vectors such that

$ g_k(\xi):= f_k(p_k + r_ku_k\xi), $

where $\xi\in \mathbb{C}$ satisfying $p_k+ r_ku_k\xi\in D$, converges uniformly on compact subsets of $\mathbb{C}$ to a nonconstant holomorphic mapping $g$ of $\mathbb{C}$ into $P^n(\mathbb{C})$.

On the other hand, we will prove that $g$ must be a constant holomorphic mapping of $\mathbb{C}$ into $P^n(\mathbb{C})$.

In fact, since $K$ is a compact subset of $D$, without loss of generality, we might assume that $\{p_k\}(\subset K)$ converges to $p_0 (\in K)$.

Now, let $k_0$ be a positive integer, such that for $k \geq k_0$, $d(p_k + r_ku_k\xi, p_0) \leq\min \{ \frac{1}{2} d(\mathbb{C}^m - D, p_0), 1\}$ for all $\xi \in \{\xi \in \mathbb{C}; \mid\xi\mid \leq1\}$, where $d(p, q)$ is the Euclidean distance between $p$ and $q$ in $\mathbb{C}^m$.

Let $g$ have a reduced representation $\tilde{g}(\xi) = (g_0(\xi), g_1(\xi), \cdots, g_n(\xi))$ on $\mathbb{C}$.

Since $g_k(\xi):= f_k(p_k + r_ku_k\xi), $ where $\xi\in \mathbb{C}$ satisfies $p_k+ r_ku_k\xi\in D$, converges uniformly on compact subsets of $\mathbb{C}$ to a nonconstant holomorphic mapping g, we have

$ g_k(\xi):= f_k(p_k + r_ku_k\xi)(k \geq k_0) $

that converges to $g$ uniformly on $\{\xi \in \mathbb{C}; \mid\xi\mid \leq1\}$.

Therefore, there exists a compact subset $K_1\subset\{\xi \in \mathbb{C}; \mid\xi\mid \leq1\}$, such that every $g_k(\xi):= f_k(p_k + r_ku_k\xi)(k \geq k_0)$ has a reduced representation

$ \tilde{g}_k(\xi) = (g_{0k}(\xi), g_{1k}(\xi), \cdots, g_{nk}(\xi)) $

on $K_1$ and $g_k(\xi)$ converges to $g(\xi)$ uniformly on $K_1$ as $k\rightarrow\infty$.

By the assumption of Theorem 2.1, we have $(P(g_{0k}, g_{1k}): P(g_{1k}, g_{2k}): \cdots: P(g_{n-1k}, g_{nk}))$ $(k \geq k_0)$ is constant on $K_1$. Hence $(P(g_0, g_1): P(g_1, g_2): \cdots: P(g_{n-1}, g_n))$ is constant on $K_1$, and thus $(P(g_0, g_1): P(g_1, g_2): \cdots: P(g_{n-1}, g_n))$ is constant on $\mathbb{C}$.

For the case at least two of $P(g_0, g_1), \ P(g_1, g_2), \ \cdots, \ P(g_{n-1}, g_n)$ is not identically equal to zero, by Theorem E, $g$ must be a constant holomorphic mapping of $\mathbb{C}$ into $P^n(\mathbb{C})$. For the other case, it is easy to see that $g$ also is a constant mapping.

Therefore, this is a contradiction. The proof of Theorem 2.1 is completed.

Proof of Theorem 2.2  Suppose that F is not a normal family on $D$. According to the proof of Theorem 2.1, $g_k(\xi):= f_k(p_k + r_ku_k\xi)(k \geq k_0)$ has a reduced representation

$ \tilde{g}_k(\xi) = (g_{0k}(\xi), g_{1k}(\xi), \cdots, g_{nk}(\xi)) $

on $K_1$ and $g_k(\xi)$ converges to $g(\xi)$ uniformly on $K_1$ as $k\rightarrow\infty$.

On the other hand, $P_n(g_{0k}, g_{1k}, \cdots, g_{nk})= \alpha^{d^n}$ on $K_1$, where $\alpha$ is an entire function.

Thus $P_n(g_0, g_1, \cdots, g_n)= \alpha^{d^n}$ on $K_1$, hence $P_n(g_0, g_1, \cdots, g_n)= \alpha^{d^n}$ on $\mathbb{C}$.

Case 1   $\alpha$ is identically equal to zero. Using Theorem F, $g$ must be a constant holomorphic mapping of $\mathbb{C}$ into $P^n(\mathbb{C})$, a contradiction.

Case 2   $\alpha$ is not identically equal to zero. By Theorem G, $g$ must be a constant holomorphic mapping of $\mathbb{C}$ into $P^n(\mathbb{C})$, a contradiction.

Thus, $g$ is a constant. This is a contradiction. Theorem 2.2 is proved.