NORMAL FAMILIES OF MEROMORPHIC MAPPINGS SHARING HYPERSURFACES

In this paper, we prove some normality criteria for families of meromorphic mappings of a domain D ⊂ C m into the complex projective space CP n under a condition on the inverse images (ignoring counting multiplicities) of moving hypersurfaces

NORMAL FAMILIES OF MEROMORPHIC MAPPINGS SHARING HYPERSURFACES Nguyen Thi Thu Hang[1] and Tran Van Tan[2] Department of Mathematics, Hanoi National University of Education 136- Xuan Thuy street, Cau Giay, Hanoi, Vietnam [1] thuhanghp2003@gmail.com, [2] tranvantanhn@yahoo.com. Abstract In this paper, we prove some normality criteria for families of meromorphic mappings of a domain D ⊂ Cm into the complex projective space CP n under a condition on the inverse images (ignoring counting multiplicities) of moving hypersurfaces. Keywords: Normal family, Meromorphic mappings, Moving hypersurfaces, Nevanlinna theory. Mathematics Subject Classification 2000: 32A10, 32C10, 32H20. 1 Introduction The Little Picard Theorem states that if a meromorphic function on the complex plane C omits three distinct points in C, then it is a constant function; and the classical result of Montel says that the family F of meromorphic functions on a domain D ⊂ C is normal if there are three distinct points a, b, c ∈ C such that each element of F omits each of a, b, and c in D. The Little Picard Theorem was generalized to the case of entire curves in the complement of 2n + 1 hyperplanes in general position in CP n by Fujimomto [5], and to the case of entire curves in the complement of 2n+1 hypersurfaces in general position in CP n by Eremenko [4]. According to Bloch’s principle, to every Picard-type theorem there should belong a corresponding normality criterion. The normality result corresponding to the aforementioned Picardtype theorems was proved by Tu [12], and Tu-Li [13]. In this paper, we examine this problem for the case where the mappings of the family can meet the hyperplanes (and hypersurfaces). Let f be a meromorphic mapping of a domain D in Cm into CP n . Then for each a ∈ D, f has a reduced representation f = (f0 , · · · , fn ) on a neighborhood U of a in D which means that each fi is a holomorphic function on 1 U and f (z) = (f0 (z) : · · · : fn (z)) outside the analytic set (of all points of indetermination of f ) I(f ) := {z : f0 (z) = · · · = fn (z) = 0} of codimension ≥ 2. In 1974, Fujimoto [6] introduced the notion of a meromorphically normal family into the complex projective space. m n A sequence {fk }∞ k=1 of meromorphic mappings of a domain D in C into CP is said to converge meromorphically on D to a meromorphic mapping f of D into CP n if and only if, for any z ∈ D, each fk has a reduced representation fk = (fk0 , · · · , fkn ) on some fixed neighborhood U of z such that {fki }∞ k=1 converges uniformly on compact subsets of U to a holomorphic function fi (0 ≤ i ≤ n) on U with the property that (f0 , · · · , fn ) is a representation of f in U. A family F of meromorphic mappings of a domain D in Cm into CP n is said to be meromorphically normal on D if any sequence in F has a meromorphically convergent subsequence on D. Denote by HD the ring of all holomorphic functions on D. Let Q be a homogeneous polynomial in HD [x0 , . . . , xn ] of degree d ≥ 1. Denote by Q(z) the homogeneous polynomial over C obtained by substituting a specific point z ∈ D into the coefficients of Q. We define a moving hypersurface in CP n to be any homogeneous polynomial Q ∈ HD [x0 , . . . , xn ] such that the coefficients of Q have no common zero point. We say that moving hypersurfaces {Qj }qj=1 (q ≥ t + 1) in CP n are in t−subgeneral position with respect to a subset X ⊂ CP n , if there exists z ∈ D such that for any 1 ≤ j0 < · · · < jt ≤ q the system of equations Qji (z)(w0 , . . . , wn ) = 0 0≤i≤t has no solution (w0 , . . . , wn ) = (0, . . . , 0) satisfying (w0 : · · · : wn ) ∈ X. Let Q be a moving hypersurface in CP n , and let f be a meromorphic mapping of D into CP n . For z ∈ D, take f˜ = (f0 , . . . , fn ) is a reduced representation of f in a neighborhood of U of z. The divisor ν(f,Q) (z) := νQ(f˜) (z)(z ∈ U ) is determined independently of a choice of reduced representations, and hence is well-defined on the totality of D. Here, νQ(f˜) is the zero divisor of holomorphic function Q(f˜). In [6], Fujimoto obtained the following interesting result. 2 Theorem F. Let F be a family of meromorphic mappings of a domain (2n+1) D ⊂ Cm into CP n and let {Hj }j=1 be hyperplanes in CP n in general position such that for each f ∈ F, f (D) ⊂ Hj (j = 1, . . . , 2n + 1) and for any fixed compact subset K of D, the 2(m − 1)-dimensional Lebesgue areas of f −1 (Hj ) ∩ K (j = 1, . . . , 2n + 1) inclusive of multiplicities for all f in F are bounded above by a fixed constant. Then F is a meromorphically normal family on D. We refer the readers to [3,9,11,12,13], for extensions of the above theorem for both cases of fixed and moving hypersurfaces. I would like to remark that in all of these results, the multiplicity of intersections are taken into account. In this paper, we obtain the following results, where multiplicity of intersections are disregarded. Theorem 1.1. Let X ⊂ CP n be a projective variety. Let Q1 , . . . , Q2t+1 be moving hypesurfaces in CP n in t-subgeneral position with respect to X. Let F be a family of meromorphic mappings f of a domain D ⊂ Cm into X, such that Qj (f ) ≡ 0, for all j ∈ {1, . . . , 2t + 1}. Assume that a) f −1 (Qj ) = g −1 (Qj ) (as sets) for all f, g in F, and for all j ∈ {1, . . . , 2t+ 1}, −1 b) dim(∩2t+1 (Qi )) ≤ m − 2 for f ∈ F. j=1 f Then F is a meromorphically normal family on D For each moving hyperplane H in CP n defined by the homogeneous polynomial H := a0 x0 + · · · + an xn ∈ HD [x0 , . . . , xn ] (the coefficients a0 , . . . , an have no common zero point), we define a holomorphic map H ∗ of D into CP n with the reduced representation (a0 : · · · : an ). Let 2t + 1(t ≥ n) moving hyperplanes Hj := aj0 x0 + · · · + ajn xn ∈ HD [x0 , . . . , xn ] (j = 1, . . . , 2t + 1). Define D(H1 , . . . , H2t+1 ) := ( det(aj i )0≤i, ≤n ). L⊂{1,...,2t+1},#L=t+1 {j0 ,...,jn }⊂L The moving hyperplanes {Hj } are said to be in pointwise t−subgeneral position if for every z ∈ D, the fixed hyperplanes {Hj (z)} in CP n in t−subgeneral position. It is clear that H1 , . . . , H2t+1 are in pointwise t−subgeneral position in CP n if and only if D(H1 , . . . , H2t+1 )(z) > 0, for all z ∈ D. For the case of hyperplanes, we obtain the following result. 3 Theorem 1.2. Let F be a family of meromorphic mappings of a domain D ⊂ Cm into CP n . For each f in F, we consider 2t + 1 moving hyperplanes ∗ : f ∈ F} (j = 1, . . . , 2t + 1) are H1f , · · · , H(2t+1)f in CP n such that {Hjf normal families and there exists a positive constant δ0 satisfying D(H1f , . . . , H(2t+1)f )(z) > δ0 , for all z ∈ D, f ∈ F. 1 Let m1 , . . . , m2t+1 be positive integers and may be ∞ such that 2t+1 j=1 mj < 1. Assume that Hjf (f ) ≡ 0 for all j ∈ {1, . . . , 2t + 1}, f ∈ F, and two following conditions are satisfied: a) {z : 1 ≤ ν(f,Hjf ) (z) ≤ mj } = {z : 1 ≤ ν(g,Hjg ) (z) ≤ mj } for all f, g in F, and for all j ∈ {1, . . . , 2t + 1}, b) I(f ) ⊂ ∪2t+1 j=1 {z : 1 ≤ ν(f,Hjf ) (z) ≤ mj }, and Hjf (f ) ≡ 0, for all j ∈ {1, . . . , 2t + 1} and f ∈ F, where I(f ) is the set of all points of indetermination of f. Then F is a meromorphically normal family on D Acknowledgements: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED). The second named author was partially supported by Vietnam Institute for Advanced Study in Mathematics, and by the Abdus Salam International Centre for Theoretical Physics, Italy. We thank Si Duc Quang for helpful discussions. 2 Notations Let ν be a nonnegative divisor on C. For each positive integer (or +∞) p, we define the counting function of ν (where multiplicities are truncated by p) by r [p] nν (t) dt (1 < r < +∞) t [p] N (r, ν) := 1 [p] where nν (t) = |z|≤t min{ν(z), p}. For a holomorphic mapping f of C into CP n , and a homogeneous polynomial [p] Q in CP n , with Q(f˜) ≡ 0, we define N(f,Q) (r) := N [p] (r, ν(f,Q) ). We simply [+∞] write N(f,Q) (r) for N(f,Q) (r). 4 Let f be a holomorphic mapping of C into CP n . For arbitrary fixed homogeneous coordinates (w0 : · · · : wn ) of CP n , we take a reduced representation f = (f0 , · · · , fn ) of f. Set f = max{|f0 |, . . . , |fn |}. The characteristic function of f is defined by 2π 2π 1 log f (re ) dθ − 2π 1 Tf (r) := 2π iθ 0 log f (eiθ ) dθ, 1 < r < +∞. 0 We state the First and the Second Main Theorems in Nevanlinna theory: First Main Theorem. Let f be a holomorphic mapping of C into CP n and Q be a homogeneous polynomial in C[x0 , · · · , xn ] of degree d ≥ 1, such that Q(f ) ≡ 0. Then N(f,Q) (r) d · Tf (r) + O(1) for all r > 1. Second Main Theorem. Let f be a linearly nondegenerate holomorphic mapping of C into CP n and H1 , . . . , Hq be fixed hyperplanes in CP n in N −subgeneral position (q > N ≥ n). Then, q [n] (q − 2N + n − 1)Tf (r) N(f,Hj ) (r) + o Tf (r) j=1 for all r except for a subset E of (1, +∞) of finite Lebesgue measure. 3 Proof of our results In order to prove Theorems 1.1-1.2, we need some preparations. Definition 3.1 ([13], Definition 4.4). Let {νi }∞ i=1 be a sequence of nonnegative divisors on a domain D in Cm . It is said to converge to a nonnegative divisor ν on D if and only if any a ∈ D has a neighborhood U such that there exist nonzero holomorphic functions h and hi on U with νi = νhi and ν = νh on U such that {hi }∞ i=1 converges to h uniformly on compact subsets of U . Let S be an analytic set in D of codimension ≥ 2. By Thullen-RemmertStein’s theorem [14], any non-negative divisor ν on D \ S can be uniquely extended to a divisor ν on D. Moreover, we have 5 Lemma 3.1 ([6], page 26, (2.9)). If a sequence {νk }∞ k=1 of non-negative divisors on D \ S converges to ν on D \ S, then {νk } converges to ν on D. Lemma 3.2 ([6], Proposition 3.5). Let {fi } be a sequence of meromorphic mappings of a domain D in Cm into CP n and let E be a thin analytic subset of D. Suppose that {fi } meromorphically converges on D \ E to a meromorphic mapping f of D \ E into CP n . If there exists a hyperplane H in CP n such that f (D \ E) ⊂ H and {ν(fi ,H) } is a convergent sequence of divisors on D, then {fi } is meromorphically convergent on D. Similarly to Corollary 3.1 in [6], we give the following corollary: Corollary 3.1. Let S be an analytic set of codimension ≥ 2 in a domain D ⊂ Cm . Let {fk }∞ k=1 be a sequence of meromorphic mappings of D into CP n . Assume that {fk |D\S } meromorphically converges to a meromorphic mapping f of D \ S, then {fk } is meromorphically convergent on D. Proof. Let H be a hyperplane in CP n such that f (D \ S) ⊂ H. It is clear that {ν(fk ,H) } converges on D \ S. Then, by Lemma 3.1, {ν(fk ,H) } converges on D. Therefore, by Lemma 3.2, {fk } meromorphically converges on D. Lemma 3.3 ([1], Theorem 3.1). Let F be a family of holomorphic mappings of a domain D in Cm into CP n . The family F is not normal on D if and only if there exist sequences {pj } ⊂ D with {pj } → p0 ∈ D, {fj } ⊂ F, {ρj } ⊂ R with ρj > 0 and {ρj } → 0, and Euclidean unit vectors {uj } ⊂ Cm , such that gj (ξ) := fj (pj + ρj uj ξ), where ξ ∈ C satisfies pj + ρj uj ξ ∈ D, converges uniformly on compact subsets of C to a nonconstant holomorphic mapping g of C into CP n . Lemma 3.4 ([4], Theorem 1). Let X be a closed subset of CP n (with respect to the usual topology of a real manifold of dimension 2n) and let D1 , . . . , D2 +1 be (fixed) hypersurfaces in CP n , in −subgeneral position with respect to X. +1 Then, every holomorphic mapping f of C into X \ (∪2j=1 Dj ) is constant. Proof of theorem 1.1. Assume that X is defined by homogeneous polyd nomials Q2t+2 , . . . , Qs in C[x0 , . . . , xn ]. By replacing Qi by Qj j where dj is a suitable positive integer, we may assume that Qj (j = 1, . . . , s) have the same degree d. Set Td := (i0 , . . . , in ) ∈ Nn+1 : i0 + · · · + in = d . 0 6 Assume that ajI xI Qj = (j = 1, . . . , s) I∈Td where ajI ∈ HD for all I ∈ Td , j ∈ {1, . . . , 2t + 1}, ajI ∈ C for all I ∈ Td , j ∈ {2t + 2, . . . , s}, xI = xi00 · · · xinn for x = (x0 , . . . , xn ) and I = (i0 , . . . , in ). Let T = (. . . , tkI , . . . ) (k ∈ {1, . . . , s}, I ∈ Td ) be a family of variables. Set tjI xI ∈ Z[T, x], Qj = j = 1, . . . , s. I∈Td For each subset L ⊂ {1, · · · , 2t + 1} with |L| = t + 1, take RL ∈ Z[T ] is the resultant of (s−t) homogeneous polynomials Qj , j ∈ {2t+2, . . . , s}∪L. Since Qj j∈L are in t−subgeneral position with respect to X, there exists z0 ∈ D such that (s − t) homogeneous polynomials Q2t+2 , . . . , Qs , and Qj (z0 ), j ∈ L have non-trivial common solutions in Cn+1 (note that X is defined by the polynomials Qi ∈ C[x0 , . . . , xn ], i ∈ {2t + 2, . . . , s}). This means that RL (. . . , akI , . . . )(z0 ) = 0. By the assumption, for each j ∈ {1 . . . , 2t + 1}, the set Aj := Q−1 j (f ) does not depend on the mapping f ∈ F. Set A := {z ∈ D : RL (· · · , akI , · · · )(z) = 0}. L⊂{1,··· ,2t+1},#L=t+1 Then E := (∪2t+1 i=1 Ai ) ∪ A is a thin analytic subset of D. ∞ Let {fk }k=1 ⊂ F be an arbitrary sequence. For any fixed point z0 ∈ D \ E, there exists an open ball B(z0 , ) in D \ E such that fk−1 (Qi ) ∩ B(z0 , ) = ∅, for allk ≥ 1, andi ∈ {1, . . . , 2t + 1}. (3.1) Since Qi (f˜k ) = 0 on B(z0 , ), we get that B(z0 , ) ∩ I(fk ) = ∅. This implies, m {fk |B(z0 , ) }∞ k=1 ⊂ Hol(B(z0 , ), CP ). We now prove that {fk |B(z0 , ) }∞ k=1 is a normal family on B(z0 , ). In∞ deed, suppose that {fk |B(z0 , ) }k=1 is not normal on B(z0 , ), then by Lemma 3.3, there exist a subsequence (again denoted by {fk |B(z0 , ) }∞ k=1 ) and p0 ∈ 7 B(z0 , ), {pk }∞ k=1 ∈ B(z0 , ) with pk → p0 , {ρk } ⊂ (0, +∞) with ρj → 0, Euclidean unit vectors {uk } ∈ Cm such that the sequence of holomorphic maps gk (ξ) := fk (pk + ρk uk ξ) : ∆rk → CP n , (rk → ∞) converges uniformly on compact subsets of C to a nonconstant holomorphic map g : C → CP n . Then, there exist reduced representations g˜k = (gk0 , · · · , gkn ) of gk and a representation g˜ = (g0 , · · · , gn ) of g such that {˜ gki } converges uniformly on compact subsets of C to g˜i . This implies that Qj (pk + ρk uk ξ)(˜ gk (ξ)) converges uniformly on compact subsets of C to Qj (p0 )(˜ g (ξ)). By (3.1) and Hurwitz’s theorem, for each j ∈ {1, . . . , 2t + 1} we have Img ∩ Qj (p0 ) = ∅, or Img ⊂ Qj (p0 ). Here, we identify the polynomial Qj (p0 ) ∈ C[x0 , · · · , xn ] with the hypersurface in CP n defined by Qj (p0 ). It is clear that Q1 (z0 ), . . . , Q2t+1 (z0 ) are in t−subgeneral position with respect to X, sicne p0 ∈ A. Since, Imfk ⊂ X, for all k ≥ 1, we get that Img ⊂ X. Without loss of generality, we may assume that Img ⊂ Qj (p0 ) for 1 ≤ j ≤ v and Img ∩ Q(z0 ) = ∅ for j ∈ {v + 1, . . . , 2t + 1} (we take v = 0 for the case where Img ∩ Qj (z0 ) = ∅ for all j ∈ {1, . . . , 2t + 1}). We have Img ⊂ M := X ∩ (∩vj=1 Qj (p0 )). Case 1: v is even, v = 2 . We consider 2(t − ) + 1 hypersurfaces Qv+1 (z0 ), . . . , Q2t+1 (z0 ). For any subset T ⊂ {v + 1, . . . , 2t + 1}, with #T = (t − ) + 1, we have M ∩ (∩j∈T Qj (z0 )) = X ∩ (∩j∈T Qj (z0 )) ∩ (∩vi=1 Qi (z0 )) = ∅ (note that #(T ∪ {1, . . . , v}) = t − + 1 + v = t + 1 + ≥ t + 1). This means that 2(t − ) + 1 hypersurfaces Qv+1 (z0 ), . . . , Q2t+1 (z0 ) are in (t− )− subgeneral position with respect to M. On the other hand, Img ⊂ M \ (∪2t+1 j=v+1 Qj (z0 )). Hence, by Lemma 3.4, g is constant; this is a contradiction. Therefore, {fk |Uz0 }∞ k=1 is a normal family on B(z0 , ). Case 2: v is old, v = 2 + 1. We consider 2(t − − 1) + 1 hypersurfaces Qv+1 (z0 ), . . . , Q2t (z0 ). For any subset T ⊂ {v + 1, . . . , 2t}, with #T = t − , we have M ∩ (∩j∈T Qj (z0 )) = X ∩ (∩j∈T Qj (z0 )) ∩ (∩vi=1 Qi (z0 )) = ∅ (note that #(T ∪ {1, . . . , v}) = (t − ) + v = t + 1 + ≥ t + 1). This means that 2(t − − 1) + 1 hypersurfaces Qv+1 (z0 ), . . . , Q2t (z0 ) are in (t − )−subgeneral position with respect to M. 8 On the other hand, Img ⊂ M \ (∪2t j=v+1 Qj (z0 )). Hence, by Lemma 3.4, g is constant; this is a contradiction. Hence, {fk |Uz0 }∞ k=1 is a normal family on B(z0 , ). By the usual diagonal argument, we can find a subsequence (again denoted by {fk }∞ k=1 ) which converges uniformly on compact subsets of D \ E to a holomorphic map f, Imf ⊂ X. By the assumption, ∩2t+1 j=1 Sj is an analytic set of codimension ≥ 2. It is clear that I(fk ) ⊂ Sj for all k ≥ 1 and j ∈ {1, . . . , 2t + 1}. Therefore, I(fk ) ⊂ ∩2t+1 j=1 Sj for all k ≥ 1 and j ∈ {1, . . . , 2t + 1}. This means that {fk }k≥1 are holomorphic on D \ (∩2t+1 j=1 Sj ). 2t+1 ∗ For any fixed point z ∈ D \(∩j=1 Sj ), there exist an open ball B(z ∗ , ρ) ⊂ ∗ D \ (∩2t+1 j=1 Sj ) and an index j0 ∈ {1, . . . , 2t + 1} such that Sj0 ∩ B(z , ρ) = ∅. This means that ∗ Q−1 j0 (fk ) ∩ B(z , ρ) = ∅. (3.2) ∗ n+1 We define holomorphic mappings {Fk }∞ as follows: k=1 of B(z , ρ) into CP for any z ∈ B(z ∗ , ρ), if fk has a reduced representation f˜k = (fk0 , · · · , fkn ) on a neighborhood Uz ⊂ B(z ∗ , ρ) then Fk has a reduced representation F˜k = d d (fk0 , · · · , fkn , Qj0 (f˜k )) on Uz . Let Hi (i = 0, · · · , n) be hyperplanes in CP n defined by Hi = {(w0 : · · · wn )|wi = 0} and let H i (i = 0, · · · , n + 1) be hyperplanes in CP n+1 defined by H i = {(w0 : · · · wn+1 )|wi = 0}. It is easy to see that {Fk } converges uniformly on compact subset of B(z ∗ , ρ)\ E to a holomorphic map F of B(z ∗ , ρ) \ E into CP n+1 , and if f has a reduced representation f˜ = (f0 , · · · , fn ) on an open subset U ⊂ B(z ∗ , ρ) \ E then F has reduced representation F˜ = (f0d , · · · , fnd , Qj0 (f˜)) on U . Since f is holomorphic on B(z ∗ , ρ) \ E, there exists i0 (0 ≤ i0 ≤ n) such that H i0 (F ) ≡ Hi0 (f ) ≡ 0 on B(z ∗ , ρ) \ E. Then there exists k0 > 0 such that H i0 (Fk ) ≡ Hi0 (fk ) ≡ 0 on B(z ∗ , ρ) \ E for all k k0 . Since Qj0 (f ) ≡ 0 on B(z ∗ , ρ) \ E, we have H n+1 (F ) ≡ 0 on B(z ∗ , ρ) \ E. On the other hand, by (3.2), for all k ≥ 1, we have Fk−1 (H n+1 ) = fk−1 (Qj0 ) ∩ B(z ∗ , ρ) = ∅. 9 Therefore, by Lemma 3.2, {Fk } is meromorphically convergent on B(z ∗ , ρ). This implies that the sequence of divisors {ν(Fk ,H i ) } converges on B(z ∗ , ρ), 0 and hence {ν(fk ,Hi0 ) } converges on B(z ∗ , ρ). By again Lemma 3.2, {fk } meromorphically converges on B(z ∗ , ρ), for any z ∗ ∈ D \ (∩2t+1 j=1 Sj ). Hence, {fk } 2t+1 meromorphically converges on D \ (∩j=1 Sj ). On the other hand, ∩2t+1 j=1 Sj is an analytic set of codimension ≥ 2. Hence, by Corollary 3.1, {fk } meromorphically converges on D. Then F is a meromorphically normal family on D. We have completed the proof of Theorem 1.1. ✷ Proof of theorem 1.2. Let {fk }∞ k=1 ⊂ F be an arbitrary sequence. Since ∗ {Hjf , f ∈ F} is a (holomorphically) normal family, without loss of generality, ∗ }∞ we may assume that, for each j ∈ {1, . . . , 2t + 1}, the sequence {Hjf k k=1 converges uniformly on every compact subset of D to a holomorphic map L∗j with the reduced representation (bj0 : · · · : bjn ). Let Lj (j = 1, . . . , 2t + 1) be moving hyperplanes defined by homogeneous polynomials bj0 x0 + · · · + bjn xn . Since D(H1fk , . . . , H(2t+1)fk )(z) ≥ δ0 for all z ∈ D, k ≥ 1, we have D(L1 , . . . , L2t+1 )(z) ≥ δ0 for all z ∈ D. This means that L1 , . . . , L2t+1 are in pointwise t−subgenral position in CP n . By the assumption, for each j ∈ {1 . . . , 2t + 1}, the set Sj := {z : 1 ≤ ν(f,Hjf ) ≤ mj } does not depend on the mapping f ∈ F. We now prove that: ∞ Claim 1: There exists a subsequence of {fk }∞ k=1 (again denoted by {fk }k=1 ) that converges uniformly on compact subset of D \(∪2t+1 j=1 S j ) to a holomorphic 2t+1 map f on D \ (∪j=1 S j ). Here S j is the colosure of Sj . For any fixed point z0 ∈ D \ (∪2t+1 j=1 S j ), we take an open ball B(z0 , ) ⊂ 2t+1 D \ (∪j=1 S j ). Then, for all j ∈ {1, . . . , 2t + 1}, Sj ∩ B(z0 , ) = ∅. (3.3) This implies, Hjfk (fk ) has no zeros of multiplicity ≥ mj on B(z0 , ) for all ∞ k ≥ 1 and j ∈ {1, . . . , 2t + 1}. Since I(fk ) ⊂ ∪2t+1 j=1 Sj , we have that {fk }k=1 are holomorphic on B(z0 , ). We now prove that {fk |B(z0 , ) }∞ k=1 is a normal family on B(z0 , ). Indeed, suppose that {fk |B(z0 , ) }∞ is not normal on B(z0 , ), then by Lemma k=1 3.3, there exist a subsequence (again denoted by {fk |B(z0 , ) }∞ k=1 ) and p0 ∈ B(z0 , ), {pk }∞ ∈ B(z , ) with p → p , {ρ } ⊂ (0, +∞) with ρj → 0, 0 k 0 k k=1 10 Euclidean unit vectors {uk } ∈ Cm such that the sequence of holomorphic maps gk (ξ) := fk (pk + ρk uk ξ) : ∆rk → CP n , (rk → ∞) converges uniformly on compact subsets of C to a nonconstant holomorphic map g : C → CP n . Then, there exist reduced representations g˜k = (gk0 , · · · , gkn ) of gk and a reduced representation g˜ = (g0 , · · · , gn ) of g such that {˜ gki } converges uniformly on compact subsets of C to g˜i . This implies that Hj (pk + ρk uk ξ)(˜ gk (z)) converges uniformly on compact subsets of C to Lj (p0 )(˜ g (ξ)). By (3.3) and Hurwitz’s theorem, for each j ∈ {1, . . . , 2t + 1} we have that Img ⊂ Lj (p0 ), or Lj (p0 )(˜ g ) have no zeros of multiplicity ≤ mj . Since L1 , . . . , L2t+1 are in pointwise t−subgeneral position in CP n , we have that fixed hyperplanes L1 (p0 ), . . . , L2t+1 (p0 ) are in t−subgeneral position in CP n . Case 1: Img ⊂ Lj (p0 ), for all j ∈ {1, . . . , 2t + 1}. Let CP ⊂ CP n be the small least projective subspace satisfying Img ⊂ CP . Then CP ⊂ Lj (p0 ) for all j ∈ {1, . . . , 2t + 1}, and (fixed) hyperplanes L1 (z0 ) ∩ CP , . . . , L2t+1 (z0 ) ∩ CP are in t−subgeneral position in CP . Since g is nonconstant, we have ≥ 1. It is clear that g is linearly non-degenerate in CP . By the First and Second Main Theorems, we have · Tg (r) = (2t + 1 − 2t + − 1)Tg (r) 2t+1 [ ] ≤ N(g,Lj (p0 )) (r) + o(Tg (r)) j=1 2t+1 ≤ j=1 mj N(g,Lj (p0 )) (r) + o(Tg (r)) 2t+1 ≤ j=1 mj Tg (r) + o(Tg (r)), for all r > 1 except for a subset of (1, +∞) of finite Lebesgue measure. 1 Therefore, 2t+1 j=1 mj ≥ 1; this is a contradiction. Case 2: There exists j ∈ {1, . . . , 2t+1} such that Img ⊂ Lj (p0 ). Without loss of generality, we may assume that Img ⊂ Lj (p0 ) for j ≤ k (k ≥ 1), and Img ⊂ Lj (p0 ) for j ∈ {k + 1, . . . , 2t + 1}. Set P := (∩kj=1 Lj (p0 )). Let CP s ⊂ P be the small least projective subspace containing Img. Since g is nonconstant, we get that s ≥ 1. Since, 11 L1 (z0 ), . . . , L2t+1 (z0 ) are in t−subgeneral position in CP n , we have that (2t − k + 1) (fixed) hyperplanes Lk+1 (z0 ) ∩ CP s , . . . , L2t+1 (z0 ) ∩ CP s are in (t − k) subgeneral position in CP s (note that CP s ⊂ Lj (z0 ) for all j ∈ {k + 1, . . . , 2t + 1}). Therefore, by the First and Second Main Theorem, for any > 0, we have (k + s)Tg (r) = (2t − k + 1 − 2(t − k) + s − 1)Tg (r) 2t+1 [s] ≤ N(g,Lj (p0 )) (r) + o(Tg (r)) j=k+1 2t+1 ≤ s N(g,Lj (p0 )) (r) + o(Tg (r)) mj j=k+1 2t+1 ≤ s Tg (r) + o(Tg (r)), m j j=k+1 for all r > 1 except for a subset of (1, +∞) of finite Lebesgue measure. 1 k+s Therefore, 2t+1 j=k+1 mj ≥ s > 1; this is a contradiction. Hence, {fk |B(z0 , ) }∞ k=k0 is a normal family on B(z0 , ). By the usual diagonal argument, we can find a subsequence (again denoted by {fk }∞ k=1 ) which 2t+1 converges uniformly on compact subset of D \ (∪j=1 S j ) to a holomorphic map f. The proof of Claim 1 has been completed. ✷ Take a fixed mapping f∗ ∈ F with a reduced representation f˜∗ . Set Sj◦ := {z ∈ Sj : z is a regular point of Hjf∗ (f˜∗ )}, j = 1, . . . , 2t + 1. Under the notation as in Claim 1, we now prove that: ◦ ∗ Claim 2: For each z ∗ ∈ ∪2t+1 j=1 Sj , there exists an open ball B(z , ) such that {fk } meromorphically converges on B(z ∗ , ). Let an arbitrary point z ∗ ∈ Sj◦0 , for some j0 . We have m∗j0 := ν(f∗ ,Hj0 f∗ ) (z ∗ ) ≤ mj0 (note that z ∗ ∈ Sj0 ). Since z ∗ is a regular point of the function Hj0 f∗ (f˜∗ ), there exists a ball B(z ∗ , ) ⊂ D such that all the zero points of Hj0 f∗ (f˜∗ ) in B(z ∗ , ) have the same multiplicity m∗j0 . Therefore, Hj0 f (f˜) (f ∈ F have the same zero set in B(z ∗ , ) and all the zero points of Hj0 f (f˜) in B(z ∗ , ) have multiplicity m∗j0 . Therefore, {ν(fk ,Hj0 fk ) |B(z∗ , ) }∞ k=1 is a constant sequence of divisors. 12 (3.4) ∗ n+1 We define meromorphic mappings {gk }∞ as follows: k=1 of B(z , ) into CP ∗ for any z ∈ B(z , ), if fk has a reduced representation f˜k = (fk0 , · · · , fkn ) on a neighborhood Uz ⊂ B(z ∗ , ) then gk has a reduced representation g˜k = d d , Hj0 fk (f˜k )) on Uz . Let Hi (i = 0, · · · , n) be hyperplanes in CP n , · · · , fkn (fk0 defined by Hi = {(w0 : · · · wn )|wi = 0} and let H i (i = 0, · · · , n + 1) be hyperplanes in CP n+1 defined by H i = {(w0 : · · · wn+1 )|wi = 0}. 2t+1 −1 Set E := ∪2t+1 j=1 f∗ (Hjf ∗ ). Then E is a thin analytic subset of D and ∪j=1 Sj ⊂ −1 ∪2t+1 j=1 f∗ (Hjf ∗ ). By Claim 1, it is easy to see that {gk } converges uniformly on compact subsets of B(z ∗ , ρ) \ E to a holomorphic map g of B(z ∗ , ) \ E into CP n+1 , and if f has a reduced representation f˜ = (f0 , · · · , fn ) on an open subset U ⊂ B(z ∗ , ρ) \ E then g has reduced representation g˜ = (f0d , · · · , fnd , Lj0 (f˜)) on U . Since f is holomorphic on B(z ∗ , ) \ E, there exists i0 (0 ≤ i0 ≤ n) such that Hi0 (f ) ≡ 0 on B(z ∗ , )\E, and hence H i0 (g) ≡ 0 on B(z ∗ , ) \ E. Then, there exists k0 > 0 such that Hi0 (fk ) ≡ H i0 (gk ) ≡ 0 on B(z ∗ , ) \ E for all k k0 . We have ν(gk ,H n+1 ) ≡ ν(fk ,Hj0 fk ) on B(z ∗ , ). Therefore, by (3.3) and Lemma 3.3, {gk } meromorphically converges on B(z ∗ , ). It implies that {ν(gk ,H i ) } 0 converges on B(z ∗ , ρ), and hence {ν(fk ,Hi0 ) } converges on B(z ∗ , ρ). By again Lemma 3.3, we get that {fk }k≥k0 meromorphically converges on B(z ∗ , ). This completes the proof of Claim 2. ✷ We have Sj \Sj◦ ⊂ singf∗−1 (Hjf∗ ), where singf∗−1 (Hjf∗ ) means the singular locus of the (reduction of the) analytic set f∗−1 (Hjf∗ ). Indeed, otherwise there existed a ∈ (Sj \ Sj◦ ) ∩ regf∗−1 (Hjf∗ ). Then m∗ := ν(f∗ ,Hjf∗ ) (a) > mj . Since a is a regular point of (f∗−1 (Hjf∗ ), by R¨ uckert Nullstellensatz, there exists nonzero holomorphic function h, u on a neighborhood U of a such that dh and u have no zero point and Hjf∗ (f∗ ) = hm∗ ·u on U. Since a ∈ Sj , there exists b ∈ Sj ∩ U. Hence, mj ≥ ν(f∗ ,Hjf∗ ) (b) = m∗ ; this is a contradtiction. Thus, −1 Sj \ Sj◦ ⊂ singf∗−1 (Hjf∗ ). Set S := ∪2t+1 j=1 singf∗ (Hjf∗ ). Then S is an analytic ◦ set in D of codimension ≥ 2. We have ∪2t+1 j=1 (Sj \ Sj ) ⊂ S. Therefore, from ∞ Claims 1-2, there exists a subsequence of {fk }∞ k=1 (again denoted by {fk }k=1 ) which meromorphically converges on D \ S. Hence, by Corollary 3.1, {fk }∞ k=1 meromorphically converges on D. Hence, F is a meromorphically normal family on D. We have completed the proof of Theorem 1.2. ✷ 13 References [1] G. Aladro and S.G Krantz, A criterion for normality in Cn , J Math. Anal. Appl., 161 (1991), 1-8. [2] G. Dethloff and T. V. Tan, A second main theorem for moving hypersurface targets, Houston J. Math. 37(2011), 79-111. [3] Gerd Dethloff , D. D. Thai, and P. N. T. Trang, Normal families of meromorphic mappings of several complex variables for moving hypersurfaces in a complex projective space, Preprint math. CV/arXiv:13010687. [4] A. Eremenko, A Picard type theorem for holomorphic curves, Period. Math. Hung. 38 (1999), 39-42. [5] H. Fujimoto, Extensions of the big Picard’s theorem, Tohoku. Math. J., 24 (1972), 415 - 422. [6] H. Fujimoto, On families of meromorphic maps into the complex projective space, Nagoya Math. J., 54 (1974), 21- 51. [7] J. Joseph and M. H. Kwack, Extension and convergence theorems for families of normal maps in several complex variables, Proc. Amer. Math. Soc., 125 (1997), 1675-1684. [8] J. Joseph and M. H. Kwack, Some classical theorems and families of normal maps in several complex variables, Complex Variables., 29 (1996), 343-362. [9] P. N. Mai, D. D. Thai and P. N. T. Trang, Normal families of meromorphic mappings of several complex variables into CP n , Nagoya Math. J., 180 (2005), 91-110. [10] W. Stoll, Normal families of non-negative divisors, Math. Z., 84 (1964), 154-218. [11] S. D. Quang and T. V. Tan, Normal families of meromorphic mappings of several complex variables into CP n for moving hypersurfaces, Ann. Polon. Math., 94, 97-110. [12] Z. Tu, Normal criteria for families of holomorphic mappings of several complex variables into CP n , Proc. Amer. Math. Soc., 127 (2005), 10391049. [13] Z. Tu and P. Li, Normal families of meromorphic mappings of several complex variables into CP n for moving targets, Sci. China. Ser. A Math., 48 (2005), 355 - 368. 14 ¨ [14] P. Thullen, Uber die wesentlichen Singularit¨aten analytischer Functionen und Fl´achen in Raum von n komplexen Ver¨anderlichen, Math. Ann., 111 (1935), 137-157. [15] J. Wang, A generalization of Picard’s theorem with moving targets, Complex Variables, 44(2001), 39-45. 15 [...]... Trang, Normal families of meromorphic mappings of several complex variables into CP n , Nagoya Math J., 180 (2005), 91-110 [10] W Stoll, Normal families of non-negative divisors, Math Z., 84 (1964), 154-218 [11] S D Quang and T V Tan, Normal families of meromorphic mappings of several complex variables into CP n for moving hypersurfaces, Ann Polon Math., 94, 97-110 [12] Z Tu, Normal criteria for families. .. Dethloff , D D Thai, and P N T Trang, Normal families of meromorphic mappings of several complex variables for moving hypersurfaces in a complex projective space, Preprint math CV/arXiv:13010687 [4] A Eremenko, A Picard type theorem for holomorphic curves, Period Math Hung 38 (1999), 39-42 [5] H Fujimoto, Extensions of the big Picard’s theorem, Tohoku Math J., 24 (1972), 415 - 422 [6] H Fujimoto, On families. .. exists a subsequence of {fk }∞ k=1 (again denoted by {fk }k=1 ) which meromorphically converges on D \ S Hence, by Corollary 3.1, {fk }∞ k=1 meromorphically converges on D Hence, F is a meromorphically normal family on D We have completed the proof of Theorem 1.2 ✷ 13 References [1] G Aladro and S.G Krantz, A criterion for normality in Cn , J Math Anal Appl., 161 (1991), 1-8 [2] G Dethloff and T V Tan,... complex variables into CP n for moving hypersurfaces, Ann Polon Math., 94, 97-110 [12] Z Tu, Normal criteria for families of holomorphic mappings of several complex variables into CP n , Proc Amer Math Soc., 127 (2005), 10391049 [13] Z Tu and P Li, Normal families of meromorphic mappings of several complex variables into CP n for moving targets, Sci China Ser A Math., 48 (2005), 355 - 368 14 ¨ [14] P Thullen,... (1972), 415 - 422 [6] H Fujimoto, On families of meromorphic maps into the complex projective space, Nagoya Math J., 54 (1974), 21- 51 [7] J Joseph and M H Kwack, Extension and convergence theorems for families of normal maps in several complex variables, Proc Amer Math Soc., 125 (1997), 1675-1684 [8] J Joseph and M H Kwack, Some classical theorems and families of normal maps in several complex variables,... Lemma 3.3, {gk } meromorphically converges on B(z ∗ , ) It implies that {ν(gk ,H i ) } 0 converges on B(z ∗ , ρ), and hence {ν(fk ,Hi0 ) } converges on B(z ∗ , ρ) By again Lemma 3.3, we get that {fk }k≥k0 meromorphically converges on B(z ∗ , ) This completes the proof of Claim 2 ✷ We have Sj \Sj◦ ⊂ singf∗−1 (Hjf∗ ), where singf∗−1 (Hjf∗ ) means the singular locus of the (reduction of the) analytic... > 1 except for a subset of (1, +∞) of finite Lebesgue measure 1 k+s Therefore, 2t+1 j=k+1 mj ≥ s > 1; this is a contradiction Hence, {fk |B(z0 , ) }∞ k=k0 is a normal family on B(z0 , ) By the usual diagonal argument, we can find a subsequence (again denoted by {fk }∞ k=1 ) which 2t+1 converges uniformly on compact subset of D \ (∪j=1 S j ) to a holomorphic map f The proof of Claim 1 has been completed... that all the zero points of Hj0 f∗ (f˜∗ ) in B(z ∗ , ) have the same multiplicity m∗j0 Therefore, Hj0 f (f˜) (f ∈ F have the same zero set in B(z ∗ , ) and all the zero points of Hj0 f (f˜) in B(z ∗ , ) have multiplicity m∗j0 Therefore, {ν(fk ,Hj0 fk ) |B(z∗ , ) }∞ k=1 is a constant sequence of divisors 12 (3.4) ∗ n+1 We define meromorphic mappings {gk }∞ as follows: k=1 of B(z , ) into CP ∗ for... such that the sequence of holomorphic maps gk (ξ) := fk (pk + ρk uk ξ) : ∆rk → CP n , (rk → ∞) converges uniformly on compact subsets of C to a nonconstant holomorphic map g : C → CP n Then, there exist reduced representations g˜k = (gk0 , · · · , gkn ) of gk and a reduced representation g˜ = (g0 , · · · , gn ) of g such that {˜ gki } converges uniformly on compact subsets of C to g˜i This implies... n+1 defined by H i = {(w0 : · · · wn+1 )|wi = 0} 2t+1 −1 Set E := ∪2t+1 j=1 f∗ (Hjf ∗ ) Then E is a thin analytic subset of D and ∪j=1 Sj ⊂ −1 ∪2t+1 j=1 f∗ (Hjf ∗ ) By Claim 1, it is easy to see that {gk } converges uniformly on compact subsets of B(z ∗ , ρ) \ E to a holomorphic map g of B(z ∗ , ) \ E into CP n+1 , and if f has a reduced representation f˜ = (f0 , · · · , fn ) on an open subset U ⊂ B(z ... Tan, Normal families of meromorphic mappings of several complex variables into CP n for moving hypersurfaces, Ann Polon Math., 94, 97-110 [12] Z Tu, Normal criteria for families of holomorphic mappings. .. Thai and P N T Trang, Normal families of meromorphic mappings of several complex variables into CP n , Nagoya Math J., 180 (2005), 91-110 [10] W Stoll, Normal families of non-negative divisors,... sequence of meromorphic mappings of D into CP n Assume that {fk |DS } meromorphically converges to a meromorphic mapping f of D S, then {fk } is meromorphically convergent on D Proof Let H