Normal family of meromorphic mappings and Big Picard's theorem
aa r X i v : . [ m a t h . C V ] F e b NORMAL FAMILY OF MEROMORPHIC MAPPINGS AND BIGPICARD’S THEOREM
NGUYEN VAN THIN AND WEI CHEN
Abstract.
In this paper, we prove some results in normal family of meromor-phic mappings intersecting with moving hypersurfaces. As some applications,we establish some results for normal mapping and extension of holomorphicmappings. A our result is strongly extended the Montel’s normal criterion inthe case several variables which is due to Tu in [Proc. Amer. Math. Soc. 127,1039-1049, 1999]. Our results are also strongly extended the results of Tu-Liin [Sci. China Ser. A. 48, 355-364, 2005] and [J. Math. Anal. Appl. 342,629-638, 2006]. Introduction and main results
One of the most important developments in complex analysis in the 20th cen-tury is the so-called Nevanlinna theory dealing with the value distribution ofentire and meromorphic functions. In 1933, H. Cartan [2] generalized the Nevan-linna’s result to holomorphic curve in projective space which is now called theNevanlinna-Cartan theory. Since that time, this problem has been studied in-tensively by many authors. Nevanlinna-Cartan theory has found various applica-tions in complex analysis and geometry complex such as uniqueness set theory,normal family theory, extension of holomorphic mapping and the property hy-perbolic of algebraic variety. In this work, we continue to study the applicationof Nevanlinna-Cartan theory in normal family, normal mapping and extension ofholomorphic mappings.Let F be a family of meromorphic functions defined on a domain D in thecomplex plane C . F is said to be normal on D if every sequence functions of F has a subsequence which converges uniformly on every compact subset of D withrespect to the spherical metric to a meromorphic function or identically ∞ on D . Perhaps the most celebrated criteria for normality in one complex variable is thefollowing Montel-type theorems related to Picard’s theorem of value distributiontheory.
Corresponding author: Nguyen Van Thin.2000
Mathematics Subject Classification.
Primary 30D45, 32H30.Key words: Holomorphic mapping, Normal family, Normal mapping, Meromorphic mapping. Theorem 1. [12]
Let F be a family of meromorphic functions on a domain D in the complex plane C . Suppose that there exist three distinct points w , w and w on the Riemann sphere such that f ( z ) − w i ( i = 1 , , has no zero on D foreach f ∈ F. Then F is a normal family on D . Theorem 2. [13]
Let F be a family of meromorphic functions on a domain D ofthe complex plane C . Suppose that there exist distinct points w , w , . . . , w q ( q ≥ on the Riemann sphere such that f ( z ) − w i has no zero with multiplicities lessthan m i ( i = 1 , , . . . , q ) on D for each f ∈ F , where m i ( i = 1 , , . . . , q ) are q fixed positive integers and may be ∞ , with P qj =1 m j < q − . Then F is a normalfamily on D . In the case of higher dimension, the notion of normal family has proved its im-portance in geometric function theory of several complex variables. Fujimoto [4],Green [8] and Nochka [9] established some Picard-type theorems of value distribu-tion theory for holomorphic mappings from C m to P n ( C ) , related to intersectionmultiplicity of value distribution theory. By starting from Green’s and Nochka’sPicard-type theorems and using the heuristic principle obtained by Aladro andKrantz [1], Tu [16] extended Theorem 1 and Theorem 2 to the case of families ofholomorphic mappings of a domain D in C m . Let A be a nonempty open subset of a domain D in C m such that S = D \ A is an analytic set in D . Let f : A → P n ( C ) be a holomorphic mapping. Let U be a nonempty connected open subset of D . A holomorphic mapping ˜ f from U into C m +1 is said to be a representation of f on U if f ( z ) = ρ ( ˜ f ( z )) for all z ∈ U ∩ A − ˜ f − (0) , where ρ : C m +1 → P n ( C ) is the canonical projection. Aholomorphic mapping f : A → P n ( C ) is said to be a meromorphic mappingfrom D into P n ( C ) if for each z ∈ D , there exists a representation of f on someneighborhood of z in D . Let f be a meromorphic mapping of a domain D in C m into P n ( C ) . Then forany a ∈ D , f always has an reduced representation ˜ f ( z ) = ( f ( z ) , . . . , f n ( z ))on some neighborhood U of a in D , which means that each f i is a holomorphicfunction on U and that f ( z ) = ( f ( z ) : f ( z ) : · · · : f n ( z )) outside the analyticset I ( f ) := { z ∈ U : f ( z ) = f ( z ) = · · · = f n ( z ) = 0 } of codimension at least 2 . Obviously a meromorphic mapping from D into P n ( C ) is a holomorphic mappingfrom D into P n ( C ) if and only if I ( f ) = ∅ . Let f be a holomorphic mapping from D into P n ( C ) and ˜ f be a reduced of f on U. Thus n + 1 meromorphic mappings˜ f i = (cid:18) f f i , . . . , f n f i (cid:19) : U → C n +1 , i = 0 , . . . , n associated a reduced representation ˜ f of f on U do not depend on the represen-tation ˜ f , which will be used lately. Definition 3. [1] Let D be a domain in C m . Let F be a family of holomorphicmappings from D to a compact complex manifold M . This family F is said to be a(holomorphically) normal family on D if any sequence in F contains a subsequencewhich converges uniformly on compact subsets of D to a holomorphic mappingfrom D into M. Definition 4. [5] A sequence { f p } of meromorphic mappings from D to P n ( C ) issaid to converge meromorphically on D to a meromorphic mapping f if and onlyif, for any z ∈ D , each f p has a reduced representation ˜ f p = ( f ,p , f ,p , . . . , f n,p )on some fixed neighborhood U of z in D such that { f i,p } ∞ p =1 converges uniformlyon compact subsets of U to a holomorphic function f i (0 ≤ i ≤ n ) on U with theproperty that f = ( f : f : · · · : f n ) is a representation of f on U, not necessarilya reduced representation. Definition 5. [5] Let F be a family of meromorphic mappings of D into P n ( C ) . One may call F a meromorphically normal family on D if any sequence in F hasa meromorphically convergent subsequence on D . Definition 6. [5] A sequence { f p } of meromorphic mappings from D into P n ( C )is said to be quasi-regular on D if and only if for any z ∈ D has a neighborhood U with property that { f p } converges compactly on U outside a nowhere denseanalytic subset S of U, i.e., for any domain G ⊂ U \ S such that closure G of G in U \ S is a compact subset of U, there is some p such that I ( f p ) ∩ G = ∅ for all p ≥ p and { f p | G } p ≥ p converges uniformly on G to a holomorphic mapping of G into P n ( C ) . Definition 7. [5] Let F be a family of meromorphic mappings of D into P n ( C ) .F is said to be quasi-normal family on D if any sequence in F has a subsequenceso as to be quasi-regular on D . For the detailed discussion about quasi-normal family, see Fujimoto [5].In 2005, Tu and Li extended the result of Tu [16] for moving target and obtainedthe result as follows:
Theorem 8. [18]
Let F be a family of holomorphic mappings of a domain D in C m into P n ( C ) and let H , . . . , H q be q ( q ≥ n + 1) moving hyperplanes in P n ( C ) located in pointwise general position such that each f in F intersects H j on D with multiplicity at least m j ( j = 1 , . . . , q ) , where m , . . . , m q are fixed positiveintegers and may be ∞ , with P qj =1 m j < q − n − n . Then F is a normal familyon D . In 2006, Tu and Li proved some results as follows for normal mapping andBig’s Picard Theorem:
Theorem 9. [19]
Let S be an analytic subset of a domain D in C m with dim C S ≤ m − . Let f be a holomorphic mapping from D − S into P n ( C ) . Let H , . . . , H q be q ( ≥ n + 1) moving hyperplanes in P n ( C ) located in pointwise general positionsuch that f intersects H j on D − S with multiplicity at least m j ( j = 1 , . . . , q ) , where m , . . . , m q are positive integers and may be ∞ , with P qj =1 m j < q − n − n . Then the holomorphic mapping f from D − S into P n ( C ) extends to a holomorphicmapping from D into P n ( C ) . Theorem 10. [19]
Let S be an analytic subset of a domain D in C m with codi-mension one, whose singularities are normal crossings. Let f be a holomorphicmapping from D − S into P n ( C ) . Let H , . . . , H q be q ( ≥ n + 1) moving hyper-planes in P n ( C ) located in pointwise general position such that f intersects H j on D − S with multiplicity at least m j ( j = 1 , . . . , q ) , where m , . . . , m q are pos-itive integers and may be ∞ , with P qj =1 m j < q − n − n . Then the holomorphicmapping f from D − S into P n ( C ) extends to a holomorphic mapping from D into P n ( C ) . In 2015, Dethloff-Thai-Trang [3] generalized the result of Tu-Li [18] for mero-morphic mapping intersecting a class hypersurfaces in weak general position.Detail, they consider the class hypersurfaces with contruction as follows: Let H D be the ring of holomorphic functions on D and let H D [ x , . . . , x n ] be theset of homogeneous polynomials of variables x = ( x , . . . , x n ) over H D . Take Q ∈ H D [ x , . . . , x n ] \ { } with deg( Q ) = d and write Q ( x ) = Q ( x , . . . , x n ) = n d X k =0 a k x I k = n d X k =0 a k x i k . . . x i kn n , where a k ∈ H D , I k = ( i k , . . . , i kn ) with | I k | = i k + · · · + i kn = d for k =0 , . . . , n d and n d = (cid:0) n + dn (cid:1) −
1. Thus each z ∈ D corresponds to an element Q z in C [ x , . . . , x n ] defined by Q z ( x ) = n d X k =0 a k ( z ) x I k , and hence a hypersurface D z := { x ∈ C n +1 : Q z ( x ) = 0 } is associated to z or Q z . The correspondence D from D into hypersurfaces is calleda moving hypersurface in P n ( C ) defined by Q . Let a = ( a , . . . , a n d ) be the vectorfunction associated to D (or Q ). In this paper, we only consider homogeneouspolynomials Q over H D [ x , . . . , x n ] such that coefficients of Q have no commonzeros, so that the hypersurface D z is well defined for each z ∈ D .Let P , . . . , P n be n + 1 moving homogeneous polynomials of common degree in H D [ x , . . . , x n ] . Denote by S ( { P i } ni =0 ) the set of all homogeneous not identicallyzero polynomials Q = P ni =0 b i P i , b i ∈ H D . Let T , . . . , T n be moving hypersur-faces in P n ( C ) with common degree, where T i is defined by the (not identicallyzero) polynomial P i (0 ≤ i ≤ N ) . Denote by ∼ S ( { T i } ni =0 ) the set of all movinghypersurfaces in P n ( C ) which are defined by Q ∈ S ( { P i } ni =0 ) . Let { Q j = P ni =0 b ij P i } qj =1 be q ( q ≥ n + 1) homogeneous polynomials in S ( { P i } ni =0 ) . We say that { Q j } qj =1 are located pointwise in general position in S ( { P i } ni =0 ) if det( b ij k ) ≤ k,i ≤ n = 0for all 1 ≤ j < · · · < j n ≤ q and all z ∈ D . For homogeneous coordinate x = ( x : · · · : x n ) in P n ( C ) , we denote x =( x , . . . , x n ) . Before state the results of Dethloff-Thai-Trang, we need some defi-nitions as follows:
Definition 11.
Let D j be moving hypersurfaces in P n ( C ) of degree d j which isdefined by polynomial homogeneous Q j ∈ H D [ x , . . . , x n ] , j = 1 , . . . , q. We saythat moving hypersurfaces { D j } qj =1 ( q ≥ n + 1) in P n ( C ) are located in (weakly)general position if there exists z ∈ D such that, for any 1 ≤ j < · · · < j n ≤ q, the system of equations ( Q j i ( z )( x , . . . , x n ) = 00 ≤ j ≤ n has only the trivial solution x = (0 , . . . ,
0) in C n +1 . This is equivalent to D ( Q , . . . , Q q )( z ) = Y ≤ j < ···
Theorem 12. [3]
Let F be a family of holomorphic mappings of a domain D in C m into P n ( C ) . Let q ≥ n + 1 be a positive integer. Let m , . . . , m q be positiveintegers or ∞ such that q X j =1 m j < q − n − n . Suppose that for each f ∈ F, there exist n + 1 moving hypersurfaces T ,f , . . . , T n,f in P n ( C ) of common degree and that there exist q moving hypersurfaces D ,f ,. . . , D q,f in ∼ S ( { T i,f } ni =0 ) such that the following conditions are satisfied: ( i ) For each ≤ i ≤ n, the coefficients of the homogeneous polynomials P i,f which define the T i,f are bounded above uniformly for all f in F on compactsubsets of D for all ≤ j ≤ q, the coefficients b ij ( f ) of the linear combinations ofthe P i,f , i = 0 , . . . , n, which define the homogeneous polynomials Q j,f are boundedabove uniformly for all f in F on compact subsets of D , where Q j,f ∈ S ( { P i,f } ni =0 ) is a homogeneous polynomial defined the D j,f , and for any fixed z ∈ D , inf f ∈ F D ( Q ,f , . . . , Q q,f )( z ) > . ( ii ) Assume that f intersects D j,f with multiplicity at least m j for each ≤ j ≤ q, for all f ∈ F . Then F is a holomorphically normal family on D . Theorem 13. [3]
Let F be a family of meromorphic mappings of a domain D of C m into P n ( C ) . Let q ≥ n + 1 be a positive integer. Suppose that for each f ∈ F , there exist n + 1 moving hypersurfaces T ,f , . . . , T n,f in P n ( C ) of common degreeand that there exist q moving hypersurfaces D ,f , . . . , D q,f in ∼ S ( { T i,f } ni =0 ) suchthat the following conditions are satisfied: ( i ) For each ≤ i ≤ n, the coefficients of the homogeneous polynomials P i,f which define the T i,f are bounded above uniformly for all f in F on compactsubsets of D for all ≤ j ≤ q, the coefficients b ij ( f ) of the linear combinations ofthe P i,f , i = 0 , . . . , n, which define the homogeneous polynomials Q j,f are boundedabove uniformly for all f in F on compact subsets of D , where Q j,f ∈ S ( { P i,f } ni =0 ) is a homogeneous polynomial defined the D j,f , and for any { f p } ⊂ F , there exists z ∈ D (which may depend on sequence) such that inf p ∈ N D ( Q ,f p , . . . , Q q,f p )( z ) > . ( ii ) For any fixed compact K of D , the m − -dimensional Lebesgue areas of f − ( D k,f ) ∩ K (1 ≤ k ≤ n + 1) counting multiplicities are bounded above for all f ∈ F (in particular, f ( D ) D k,f (1 ≤ k ≤ n + 1)) . ( iii ) There exists a closed subset S of D with Λ m − ( S ) = 0 such that for anyfixed compact subset K of D − S, the m − -dimensional Lebesgue areas of { z ∈ suppν Q k,f ( ˜ f ) : ν Q k,f ( ˜ f ) ( z ) < m k } ∩ K ( n + 2 ≤ k ≤ q ) ignoring multiplicities for all f ∈ F are bounded above, where { m k } qk = n +2 arefixed positive integers or ∞ with q X k = n +2 m k < q − n − n . Then F is a meromorphically normal family on D . Let D = { D , . . . , D q } be a collection of arbitrary hypersurfaces and Q j be thehomogeneous polynomial in C [ x , . . . , x n ] of degree d j defining D j for j = 1 , . . . , q .Let m D be the least common multiple of the d j for j = 1 , . . . , q and denote n D = (cid:18) n + m D n (cid:19) − . For j = 1 , . . . , q, we set Q ∗ j = Q m D /d j j and let a ∗ j be the vector associated with Q ∗ j .We recall the lexicographic ordering on the ( n + 1)-tuples of natural numbers:let J = ( j , . . . , j n ) , I = ( i , . . . , i n ) ∈ N n +1 , J < I if and only if for some b ∈ { , . . . , n } we have j l = i l for l < b and j b < i b . With the ( n + 1)-tuples I = ( i , . . . , i n ) of non-negative integers, we denote | I | := P j i j . Let ( x : · · · : x n ) be a homogeneous coordinates in P n ( C ) and let { I , . . . , I n D } be a set of ( n + 1)-tuples such that | I j | = m D , j = 0 , . . . , n D , and I i < I j for i < j ∈ { , . . . , n D } . We denote x = ( x , . . . , x n ) ∈ C n +1 , and write x I as x i . . . x i n n where I = ( i , . . . , i n ) ∈ { I , . . . , I n D } . Then we may order the set ofmonomials of degree m D as { x I , . . . , x I n D } .Denote ̺ m D : P n ( C ) → P n D ( C )be the Veronese embedding of degree m D . Let ( w : · · · : w n D ) be the homogeneouscoordinates in P n D ( C ). Then ̺ m D is given by ̺ m D ( x ) = ( w ( x ) : · · · : w n D ( x )) , where w j ( x ) = x I j , j = 0 , . . . , n D . For any hypersurface D j ∈ { D , . . . , D q } and a j = ( a j , . . . , a jn D ) be the vectorassociated with Q ∗ j , we set L j = Q ∗ j = a j w + · · · + a jn D w n D . Then L j is a linear form in P n D ( C ). Let H ∗ j be a hyperplane in P n D ( C ), which isdefined by L j , we say that the hyperplane H ∗ j associated with Q ∗ j (or D j ). Hencefor the collection of hypersurfaces { D , . . . , D q } in P n ( C ), we have the collection H ∗ = { H ∗ , . . . , H ∗ q } of associated hyperplanes in P n D ( C ).Let q > n D be a positive integer. We say that collection D is in general positionfor Veronese embedding in P n ( C ) if { H ∗ , . . . , H ∗ q } is in general position in P n D ( C ) . Hence, the collection D is in general position for Veronese embedding if for anydistinct i , . . . , i n D ∈ { , . . . , q } , the vectors a ∗ i , . . . , a ∗ i n D have rank n D . It is seenthat, for hyperplanes, general position for Veronese embedding is equivalent tothe usual concept of hyperplanes in general position usual. For hypersurfaces,general position for Veronese embedding implies n D -subgeneral position.Let D = { D , . . . , D q } be a collection of arbitrary moving hypersurfaces and Q j be the homogeneous polynomial in H D [ x , . . . , x n ] of degree d j defining D j for j = 1 , . . . , q . Let m D be the least common multiple of the d j for j = 1 , . . . , q anddenote n D = (cid:18) n + m D n (cid:19) − . For j = 1 , . . . , q, we set Q ∗ j = Q m D /d j j and let a ∗ j be the function vector associatedwith Q ∗ j . We set L j = Q ∗ j = a j w + · · · + a jn D w n D . Then L j is a moving linear form in P n D ( C ). Let H ∗ j be a moving hyperplane in P n D ( C ), which is defined by L j (or Q ∗ j ), we say that the hyperplane H ∗ j associatedwith Q ∗ j (or D j ). Hence for the collection of moving hypersurfaces { D , . . . , D q } in P n ( C ), we have the collection H ∗ = { H ∗ , . . . , H ∗ q } of associated moving hyper-planes in P n D ( C ). Definition 14.
Let { D j } qj =1 be moving hypersurfaces in P n ( C ) which are definedby homogeneous Q j ∈ H D [ x , . . . , x n ] of degree d j ( q ≥ n D + 1) , j = 1 , . . . , q. Wesay that moving hypersurfaces { D j } qj =1 are located in (weakly) general positionfor Veronese embedding in P n ( C ) if there exists z ∈ D such that, the collection H ∗ ( z ) = { H ∗ ( z ) , . . . , H ∗ q ( z ) } of associated hyperplanes in P n D ( C ) at z are ingeneral position in P n D ( C ) . Remark 15.
We see that hypersurfaces { D j } qj =1 are located in (weakly) generalposition for Veronese embedding if there exists z ∈ D such that D ( Q ∗ , . . . , Q ∗ q )( z ) > P n D ( C ) . Our results are given as follows:
Theorem 16.
Let F be a family of holomorphic mappings of a domain D in C m into P n ( C ) . Let q ≥ n D + 1 be a positive integer. Let m , . . . , m q be positiveintegers or ∞ such that q X j =1 d j m j < ( q − n D − m D n D . Suppose that for each f ∈ F , there exist q moving hypersurfaces D ,f , . . . , D q,f which are defined by homogeneous polynomials Q ,f , . . . , Q q,f ∈ H D [ x , . . . , x n ] with degree d , . . . , d q respectively such that the following conditions are satisfiedon each compact subset K ⊂ D :( i ) For each ≤ j ≤ q, the coefficients of the homogeneous polynomials Q j,f arebounded above uniformly for all f in F on compact subset K of D . Set m D = lcm ( d , . . . , d q ) , n D = (cid:0) n + m D n (cid:1) − , and Q ∗ j,f = Q m D /d j j,f . Suppose that Q ∗ j,f = P n D i =0 c ij ( f ) ω i , ( ω i = x I i ) in H D [ ω , . . . , ω n D ] is moving linear form defines theassociated hyperplane H ∗ j,f of D j,f ( j = 1 , . . . , q ) and for any { f p } ⊂ F , and forany fixed z ∈ D , inf p ∈ N D ( Q ∗ ,f p , . . . , Q ∗ q,f p )( z ) > . ( ii ) Assume that for each f ∈ F , and ˜ f = ( f , . . . , f n ) is a reduced representationof f ∈ F on D with f i ( z ) = 0 for some i and z, and when m j ≥ ≤| α |≤ m j − ,z ∈ f − ( D j,f ) ∩ K (cid:12)(cid:12)(cid:12) ∂ α ( Q j,f ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } and all f ∈ F . Then F is a holomorphically normal family on D . Theorem 16 is strongly extended the Montel’s normal criterion in the caseseveral variables which due to Tu [16]. From Theorem 16, we get the result asfollows:
Corollary 1.
Let F be a family of holomorphic mappings of a domain D in C m into P n ( C ) . Let q ≥ n D + 1 be a positive integer. Let m , . . . , m q be positive integers or ∞ such that q X j =1 d j m j < ( q − n D − m D n D . Suppose that for each f ∈ F , there exist q moving hypersurfaces D ,f , . . . , D q,f which are defined by homogeneous polynomials Q ,f , . . . , Q q,f ∈ H D [ x , . . . , x n ] with degree d , . . . , d q respectively such that the following conditions are satisfiedon each compact subset K ⊂ D :( i ) For each ≤ j ≤ q, the coefficients of the homogeneous polynomials Q j,f arebounded above uniformly for all f in F on compact subset K ⊂ D . Set m D = lcm ( d , . . . , d q ) , n D = (cid:0) n + m D n (cid:1) − , and Q ∗ j,f = Q m D /d j j,f . Suppose that Q ∗ j,f = P n D i =0 c ij ( f ) ω i , ( ω i = x I i ) in H D [ ω , . . . , ω n D ] is moving linear form defines theassociated hyperplane H ∗ j,f of D j,f ( j = 1 , . . . , q ) and for any { f p } ⊂ F , and forany fixed z ∈ D , inf p ∈ N D ( Q ∗ ,f p , . . . , Q ∗ q,f p )( z ) > . ( ii ) Assume that for each f ∈ F , f intersects D j,f with multiplicity at least m j for each ≤ j ≤ q. Then F is a holomorphically normal family on D . Theorem 17.
Let F be a family of meromorphic mappings of a domain D in C m into P n ( C ) . Let q ≥ n D + 1 be a positive integer. Suppose that for each f ∈ F , there exist q moving hypersurfaces D ,f , . . . , D q,f which defined by homo-geneous polynomials Q ,f , . . . , Q q,f ∈ H D [ x , . . . , x n ] with degree d , . . . , d q respec-tively such that the following conditions are satisfied: ( i ) For each ≤ j ≤ q, the coefficients of the homogeneous polynomials Q j,f are bounded above uniformly for all f in F on compact subsets of D . Set m D = lcm ( d , . . . , d q ) , n D = (cid:0) n + m D n (cid:1) − , and Q ∗ j,f = Q m D /d j j,f . Suppose that Q ∗ j,f = P n D i =0 c ij ( f ) ω i , ( ω i = x I i ) in H D [ ω , . . . , ω n D ] is moving linear form defines theassociated hyperplane H ∗ j,f of D j,f and for any { f p } ⊂ F , there exists z ∈ D (which may depend on sequence) such that inf p ∈ N D ( Q ∗ ,f p , . . . , Q ∗ q,f p )( z ) > . ( ii ) For any fixed compact K of D , the m − -dimensional Lebesgue areas of f − ( D k,f ) ∩ K (1 ≤ k ≤ n D + 1) counting multiplicities are bounded above for all f ∈ F (in particular, f ( D ) D k,f (1 ≤ k ≤ n D + 1)) . ( iii ) There exists a nowhere dense analytic set S in D such that for any fixedcompact subset K of D − S, the m − -dimensional Lebesgue areas of { z ∈ supp ν Q k,f ( ˜ f ) : ν Q k,f ( ˜ f ) ( z ) < m k } ∩ K ( n D + 2 ≤ k ≤ q ) ignoring multiplicities for all f ∈ F are bounded above, where { m k } qk = n D +2 arefixed positive integers or ∞ with q X k = n D +2 d k m k < ( q − n D − m D n D . Then F is a quasi-normal family on D . Theorem 18.
Let S be an analytic subset of a domain D in C m with dim C S ≤ m − . Let f be a holomorphic mapping from D − S into P n ( C ) . Let D , . . . , D q be q ( ≥ n D + 1) moving hypersurfaces in P n ( C ) . Let Q ∗ j = P n D i =0 d ij ω i , ( ω i = x I i ) in H D [ ω , . . . , ω n D ] be moving linear form defines the associated hyperplane H ∗ j of D j such that for any z ∈ D : D ( Q ∗ , . . . , Q ∗ q )( z ) > . Let ˜ f = ( f , . . . , f n ) be a reduced representation of f on D − S with f i ( z ) = 0 forsome i and z, and when m j ≥ ≤| α |≤ m j − ,z ∈ f − ( D j ) (cid:12)(cid:12)(cid:12) ∂ α ( Q j ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } , where m , . . . , m q are positive integers and may be ∞ , with P qj =1 d j m j < ( q − n D − m D n D . Then the holomorphic mapping f from D − S into P n ( C ) extends to a holomorphic mapping from D into P n ( C ) . Theorem 19.
Let S be an analytic subset of a domain D in C m with codimensionone, whose singularities are normal crossings. Let f be a holomorphic mappingfrom D − S into P n ( C ) . Let D , . . . , D q be q ( ≥ n D + 1) moving hypersurfacesin P n ( C ) . Let Q ∗ j = P n D i =0 d ij ω i , ( ω i = x I i ) in H D [ ω , . . . , ω n D ] be moving linearform defines the associated hyperplane H ∗ j of D j such that for any z ∈ D : D ( Q ∗ , . . . , Q ∗ q )( z ) > . Let ˜ f = ( f , . . . , f n ) be a reduced representation of f on D − S with f i ( z ) = 0 forsome i and z, and when m j ≥ ≤| α |≤ m j − ,z ∈ f − ( D j ) (cid:12)(cid:12)(cid:12) ∂ α ( Q j ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } , where m , . . . , m q are positive integers and may be ∞ , with P qj =1 d j m j < ( q − n D − m D n D . Then the holomorphic mapping f from D − S into P n ( C ) extends to a holomorphic mapping from D into P n ( C ) . Theorem 20.
Let F be a family of holomorphic mappings of a domain D in C m into P n ( C ) . Let q ≥ n + 1 be a positive integer. Let m , . . . , m q be positiveintegers or ∞ such that q X j =1 m j < q − n − n . Suppose that for each f ∈ F , there exist n + 1 moving hypersurfaces T ,f , . . . , T n,f in P n ( C ) of common degree and that there exist q moving hypersurfaces D ,f ,. . . , D q,f in ∼ S ( { T i,f } ni =0 ) such that the following conditions are satisfied on eachcompact subset K ⊂ D :( i ) For each ≤ i ≤ n, the coefficients of the homogeneous polynomials P i,f which define the T i,f are bounded above uniformly for all f in F on compactsubset K of D for all ≤ j ≤ q, the coefficients b ij ( f ) of the linear combina-tions of the P i,f , i = 0 , . . . , n, which define the homogeneous polynomials Q j,f are bounded above uniformly for all f in F on compact subset K of D , where Q j,f ∈ S ( { P i,f } ni =0 ) is a homogeneous polynomial defined the D j,f , and for anyfixed z ∈ D , inf f ∈ F D ( Q ,f , . . . , Q q,f )( z ) > . ( ii ) Let ˜ f = ( f , . . . , f n ) be a reduced representation of f ∈ F on D with f i ( z ) = 0 for some i and z, and when m j ≥ ≤| α |≤ m j − ,z ∈ f − ( D j,f ) ∩ K (cid:12)(cid:12)(cid:12) ∂ α ( Q j,f ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } . Then F is a holomorphically normal family on D . Theorem 21.
Let S be an analytic subset of a domain D in C m with dim C S ≤ m − . Let f be a holomorphic mapping from D − S into P n ( C ) . Suppose that thereexist n + 1 moving hypersurfaces T , . . . , T n in P n ( C ) of common degree which aredefined by homogeneous polynomial P , . . . , P n ∈ H D [ x , . . . , x n ] respectively, andthat there exist q moving hypersurfaces D , . . . , D q in ∼ S ( { T i } ni =0 ) such that thefollowing conditions are satisfied: ( i ) Q j ∈ S ( { P i } ni =0 ) is a homogeneous polynomial defined the D j , and for any z ∈ D , D ( Q , . . . , Q q )( z ) > . ( ii ) Let ˜ f = ( f , . . . , f n ) be a reduced representation of f on D − S with f i ( z ) = 0 for some i and z, and when m j ≥ ≤| α |≤ m j − ,z ∈ f − ( D j ) (cid:12)(cid:12)(cid:12) ∂ α ( Q j ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } , where m , . . . , m q are positive integers and may be ∞ , with q X j =1 m j < q − n − n . Then the holomorphic mapping f from D − S into P n ( C ) extends to a holomorphicmapping from D into P n ( C ) . Theorem 22.
Let S be an analytic subset of a domain D in C m with codi-mension one, whose singularities are normal crossings. Let f be a holomorphicmapping from D − S into P n ( C ) . Suppose that there exist n + 1 moving hypersur-faces T , . . . , T n in P n ( C ) of common degree which are defined by homogeneouspolynomials P , . . . , P n ∈ H D [ x , . . . , x n ] respectively, and that there exist q mov-ing hypersurfaces D , . . . , D q in ∼ S ( { T i } ni =0 ) such that the following conditions aresatisfied: ( i ) Q j ∈ S ( { P i } ni =0 ) is a homogeneous polynomial defined the D j , and for any z ∈ D , D ( Q , . . . , Q q )( z ) > . ( ii ) Let ˜ f = ( f , . . . , f n ) be a reduced representation of f on D − S with f i ( z ) = 0 for some i and z, and when m j ≥ ≤| α |≤ m j − ,z ∈ f − ( D j ) (cid:12)(cid:12)(cid:12) ∂ α ( Q j ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } , where m , . . . , m q are positive integers and may be ∞ , with q X j =1 m j < q − n − n . Then the holomorphic mapping f from D − S into P n ( C ) extends to a holomorphicmapping from D into P n ( C ) . Remark 23. ( i ) Our results cover the results due to Tu-Li [18, 19].( ii ) Theorem 20 is a extension the result due to Dethloff-Thai-Trang (Theorem12).Finaly, we consider the case moving hypersurfaces in weakly general positionin P n ( C ) . We prove the results as follows:
Theorem 24.
Let F be a family of holomorphic mappings of a domain D in C m into P n ( C ) . Let q ≥ nn D + 2 n + 1 be a positive integer. Let m , . . . , m q be positiveintegers or ∞ such that q X j =1 m j < q − n − ( n − n D + 1) n D ( n D + 2) . Suppose that for each f ∈ F , there exist q moving hypersurfaces D ,f , . . . , D q,f which are defined by homogeneous polynomials Q ,f , . . . , Q q,f ∈ H D [ x , . . . , x n ] with degree d , . . . , d q respectively, such that the following conditions are satisfiedon each compact subset K ⊂ D :( i ) For each ≤ j ≤ q, the coefficients of the homogeneous polynomials Q j,f arebounded above uniformly for all f in F on compact subset K of D , and for any { f p } ⊂ F , and for any fixed z ∈ D , inf p ∈ N D ( Q ,f p , . . . , Q q,f p )( z ) > . ( ii ) Assume that for each f ∈ F , and ˜ f = ( f , . . . , f n ) is a reduced representationof f ∈ F on D with f i ( z ) = 0 for some i and z, and when m j ≥ ≤| α |≤ m j − ,z ∈ f − ( D j,f ) ∩ K (cid:12)(cid:12)(cid:12) ∂ α ( Q j,f ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } and all f ∈ F . Then F is a holomorphically normal family on D . From Theorem 24, we get the result as follows:
Corollary 2.
Let F be a family of holomorphic mappings of a domain D in C m into P n ( C ) . Let q ≥ nn D + 2 n + 1 be a positive integer. Let m , . . . , m q be positiveintegers or ∞ such that q X j =1 m j < q − n − ( n − n D + 1) n D ( n D + 2) . Suppose that for each f ∈ F , there exist q moving hypersurfaces D ,f , . . . , D q,f which are defined by homogeneous polynomials Q ,f , . . . , Q q,f ∈ H D [ x , . . . , x n ] with degree d , . . . , d q respectively, such that the following conditions are satisfiedon each compact subset K ⊂ D :( i ) For each ≤ j ≤ q, the coefficients of the homogeneous polynomials Q j,f arebounded above uniformly for all f in F on compact subset K of D , and for any { f p } ⊂ F , and for any fixed z ∈ D , inf p ∈ N D ( Q ,f p , . . . , Q q,f p )( z ) > . ( ii ) Assume that f intersects D j,f with multiplicity at least m j for each ≤ j ≤ q, for all f ∈ F . Then F is a holomorphically normal family on D . Let Λ l ( S ) denote the real l -dimensional Hausdorff measure of S ⊂ C m . Wehave the result as follows:
Theorem 25.
Let F be a family of meromorphic mappings of a domain D in C m into P n ( C ) . Let q ≥ nn D + 2 n + 1 be a positive integer. Suppose that for each f ∈ F , there exist q moving hypersurfaces D ,f , . . . , D q,f which are defined byhomogeneous polynomials Q ,f , . . . , Q q,f ∈ H D [ x , . . . , x n ] with degree d , . . . , d q respectively, such that the following conditions are satisfied: ( i ) For each ≤ j ≤ q, the coefficients of the homogeneous polynomials Q j,f arebounded above uniformly for all f in F on compact subsets of D , and for any { f p } ⊂ F , there exists z ∈ D (which may depend on sequence) such that inf p ∈ N D ( Q ,f p , . . . , Q q,f p )( z ) > . ( ii ) For any fixed compact K of D , the m − -dimensional Lebesgue areas of f − ( D k,f ) ∩ K (1 ≤ k ≤ n + 1) counting multiplicities are bounded above for all f ∈ F (in particular, f ( D ) D k,f (1 ≤ k ≤ n + 1)) . ( iii ) There exists a closed subset S in D with Λ m − ( S ) = 0 such that for anyfixed compact subset K of D − S, the m − -dimensional Lebesgue areas of { z ∈ supp ν Q k,f ( ˜ f ) : ν Q k,f ( ˜ f ) ( z ) < m k } ∩ K ( n + 2 ≤ k ≤ q ) ignoring multiplicities for all f ∈ F are bounded above, where { m k } qk = n +2 arefixed positive integers or ∞ with q X k = n +2 m k < q − n − ( n − n D + 1) n D ( n D + 2) . Then F is a meromorphicaly normal family on D . Theorem 26.
Let S be an analytic subset of a domain D in C m with dim C S ≤ m − . Let f be a holomorphic mapping from D − S into P n ( C ) . Let D , . . . , D q be q ( ≥ nn D +2 n +1) moving hypersurfaces in P n ( C ) which are defined by homogeneouspolynomials Q , . . . , Q q ∈ H D [ x , . . . , x n ] with degree d , . . . , d q respectively suchthat such that for any z ∈ D : D ( Q , . . . , Q q )( z ) > . Let ˜ f = ( f , . . . , f n ) be a reduced representation of f on D − S with f i ( z ) = 0 forsome i and z, and when m j ≥ ≤| α |≤ m j − ,z ∈ f − ( D j ) (cid:12)(cid:12)(cid:12) ∂ α ( Q j ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } , where m , . . . , m q are positive integers and may be ∞ , with q X j =1 m j < q − n − ( n − n D + 1) n D ( n D + 2) . Then the holomorphic mapping f from D − S into P n ( C ) extends to a holomorphicmapping from D into P n ( C ) . Theorem 27.
Let S be an analytic subset of a domain D in C m with codi-mension one, whose singularities are normal crossings. Let f be a holomor-phic mapping from D − S into P n ( C ) . Let D , . . . , D q be q ( ≥ nn D + 2 n + 1) moving hypersurfaces in P n ( C ) which are defined by homogeneous polynomials Q , . . . , Q q ∈ H D [ x , . . . , x n ] with degree d , . . . , d q respectively such that for any z ∈ D : D ( Q , . . . , Q q )( z ) > . Let ˜ f = ( f , . . . , f n ) be a reduced representation of f on D − S with f i ( z ) = 0 forsome i and z, and when m j ≥ ≤| α |≤ m j − ,z ∈ f − ( D j ) (cid:12)(cid:12)(cid:12) ∂ α ( Q j ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } , where m , . . . , m q are positive integers and may be ∞ , with q X j =1 m j < q − n − ( n − n D + 1) n D ( n D + 2) . Then the holomorphic mapping f from D − S into P n ( C ) extends to a holomorphicmapping from D into P n ( C ) . Some lemmas
In order to prove our results, we need some lemmas as follows:
Definition 28. [15] Let { A i } be a sequence of closed subsets of D . It is said toconverge to a closed subset A of D if and only if A coincides with the set of all z such that every neighborhood U of z intersects A i for all but finitely many i and,simultaneously, with the set of all z such that every U intersects A i for infinitelymany i. Lemma 1. [15]
Let { N i } be a sequence of pure ( m − -dimensional analyticsubsets of a domain D in C m . If the m − - dimensional Lebesgue areas of N i ∩ K ignoring multiplicities ( i = 1 , , . . . ) are bounded above for any fixedcompact subset K of D , then { N i } is normal in the sense of the convergence ofclosed subsets in D . Lemma 2. [15]
Let { N i } be a sequence of pure ( m − -dimensional analyticsubsets of a domain D in C m . Assume that the m − -dimensional Lebesgueareas of N i ∩ K ignoring multiplicities ( i = 1 , , . . . ) are bounded above for anyfixed compact subset K of D and that { N i } converges to N as a sequence of closedsubsets of D . Then N is either empty or a pure ( m − -dimensional analyticsubset of D . Lemma 3. [1]
Let F be a family of holomorphic mappings of a domain D in C m into P n ( C ) . Then the family F is not normal on D if and only if there exista compact subset K ⊂ D and sequences { f p } ⊂ F, { y p } ⊂ K , { r p } ⊂ R with r p > and r p → + , and { u p } ⊂ C m which are unit vectors such that g p ( ξ ) := f p ( y p + r p u p ξ ) , where ξ ∈ C such that y p + r p u p ξ ∈ D , converges uniformly on compact subsets of C to a nonconstant holomorphic map g of C to P n ( C ) . Definition 29.
Let Ω ⊂ C m be a hyperbolic domain and let M be a completecomplex Hermittian manifold with metric ds M . A holomorphic mapping f ( z )from Ω into M is said to be a normal holomorphic mapping from Ω into M if andonly if there exists a positive constant c such that for all z ∈ Ω and all ξ ∈ T z (Ω) , | ds M ( f ( z ) , df ( z )( ξ )) | ≤ cK Ω ( z, ξ ) , where df ( z ) is the mapping from T z (Ω) into T f ( z ) ( M ) induced by f and K Ω denotethe infinitesimal Kobayashi metric on Ω . Lemma 4. [19]
Let f be a holomorphic mapping of a hyperbolic domain D in C m into P n ( C ) . Then f is not normal on D if and only if there exist { y p } ⊂ D , { r p } with r p > and r p → + and { u p } ⊂ C m Euclidean unit vectors such that g p ( ξ ) := f ( y p + r p u p ξ ) , ξ ∈ C , where lim p →∞ r p d ( y p , C m \ D ) = 0 (where d ( p, q ) is the Euclidean distance between p and q in C m ), converges uniformly on compact subsets of C to a non-constantholomorphic mapping g of C to P n ( C ) . Lemma 5. [3]
Let natural numbers n and q ≥ n + 1 be fixed. Let D kp (1 ≤ k ≤ q, p ≥ be moving hypersurfaces in P n ( C ) such that the following conditions aresatisfied: ( i ) For each ≤ k ≤ q, p ≥ , the coefficients of the homogeneous polynomials Q kp which define the D kp are bounded above uniformly for all p ≥ on compactsubsets of D . ( ii ) There exists z ∈ D such that inf p ∈ N { D ( Q p , . . . , Q qp )( z ) } > δ > Then, we have the following: ( a ) There exists a subsequence { j p } ⊂ N such that for ≤ k ≤ q, Q kj p con-verge uniformly on compact subsets of D to not identically zero homogeneouspolynomials Q k (meaning that the Q kj p and Q k are homogeneous polynomials in H D [ x , . . . , x N ] of the same degree, and all their coefficients converge uniformlyon compact subsets of D ). Moreover, we have that D ( Q , . . . , Q q )( z ) > δ > , thehypersurfaces Q ( z ) , . . . , Q q ( z ) are located in general position, and the movinghypersurfaces Q ( z ) , . . . , Q q ( z ) are located in (weakly) general position. ( b ) There exist a subsequence { j p } ⊂ N and r = r ( δ ) > such that inf p ∈ N D ( Q j p , . . . , Q qj p )( z ) > δ , ∀ z ∈ B ( z , r ) . Lemma 6. [3]
Let { f p } be a sequence of meromorphic mappings of a domain D in C m into P n ( C ) , and let S be a closed subset of D with Λ m − ( S ) = 0 . Supposethat { f p } meromorphically converges on D − S to a meromorphic mapping f of D − S into P n ( C ) . Suppose that, for each f p , there exist n +1 moving hypersurfaces D ,f p , . . . , D n +1 ,f p in P n ( C ) , where the moving hypersurfaces D i,f p may depend on f p , such that the following three conditions are satisfied: ( i ) For each ≤ k ≤ n + 1 , the coefficients of homogeneous polynomial Q k,f p which define D k,f p for all f p are bounded above uniformly for all p ≥ on compactsubsets of D . ( ii ) There exists z ∈ D such that inf p ≥ D ( Q ,f p , . . . , Q n +1 ,f p ( z )) > iii ) The m − -dimensional Lebesgue areas of ( f p ) − ( D k,f p ) ∩ K (1 ≤ k ≤ n + 1 , p ≥ counting multiplicities are bounded above for any fixed compactsubset K of D . Then we have the following: ( a ) { f p } has a meromorphically convergent subsequence on D , and ( b ) if, moreover, { f p } is a sequence of holomorphic mappings of a domain D in C m into P n ( C ) and condition ( iii ) is sharpened to f p ( D ) ∩ D k,f p = ∅ (1 ≤ k ≤ n + 1 , p ≥ , then { f p } has a subsequence which converges uniformly on compact subsets of D to a holomorphic mapping of D to P n ( C ) . The following lemma is Picard type theorem in Nevanlinna-Cartan theory dueto Nochka:
Lemma 7. [9]
Suppose that q ≥ n + 1 hyperplanes H , . . . , H q are given ingeneral position in P n ( C ) and that q positive integers (may be ∞ ) m , . . . , m q aregiven such that q X j =1 m j < q − n − n . Then there does not exist a nonconstant holomorphic mapping f : C → P n ( C ) such that f intersects H j with multiplicity at least m j (1 ≤ j ≤ q ) . Lemma 8. [3]
Let f = ( f : · · · : f n ) : C → P n ( C ) be a holomorphic mapping,and let { P i } n +1 i =0 be n + 1 homogeneous polynomials in general position of commondegree in C [ x , . . . , x n ] . Assume that F = ( F : · · · : F n ) : P n ( C ) → P n ( C ) is the mapping defined by F i ( x ) = P i ( x ) (0 ≤ i ≤ n ) . Then, F ( f ) is a constant map if and only if f is a constant map. Lemma 9. [3]
Let P , . . . , P n be n + 1 homogeneous polynomials of commondegree in C [ x , . . . , x n ] . Also let { Q j } qj =1 ( q ≥ n + 1) be homogeneous polynomials in S ( { P i } ni =0 ) such that D ( Q , . . . , Q q )( z ) = Y ≤ j < ···
Let f be a meromorphic nonconstant mappings from C m into P n ( C ) . Let D i ( i = 1 , . . . , q ) , q ≥ nn D + n + 1 be slowly (with respective f )moving hypersurfaces of P n ( C ) in weakly general position which is defined byhomogeneous polynomials Q i ∈ H D [ x , . . . , x n ] with deg Q i = d i ( i = 1 , . . . , q ) . Set m D = lcm ( d , . . . , d q ) and n D = (cid:0) n + m D n (cid:1) − . Let m , . . . , m q be positiveintegers or ∞ such that q X j =1 m j < q − ( n − n D + 1) n D ( n D + 2) . Assume that Q j ( ˜ f ) ≤ j ≤ q ) and f intersects D j with multiplicity at least m j for each ≤ j ≤ q, then f must be constant mapping. Proof of Theorems
Proof of Theorem 16.
Without loss of generality, we may assume that D is theunit disc. Suppose that F is not normal on D . Then, by Lemma 3, there existsa subsequence denoted by { f p } ⊂ F and y ∈ K , { y p } ∞ p =1 ⊂ K with y p → y , { r p } ⊂ (0 , + ∞ ) with r p → + , and { u p } ⊂ C m , which are unit vectors, suchthat g p ( ξ ) := f p ( y p + r p u p ξ ) converges uniformly on compact subsets of C to anonconstant holomorphic map g of C into P n ( C ) . Let ̺ m D : P n ( C ) → P n D ( C )be the Veronese embedding of degree m D which is given by ̺ m D ( x ) = ( w ( x ) : · · · : w n D ( x )) , where w j ( x ) = x I j , j = 0 , . . . , n D . For w = ( w : · · · : w n D ) , we denote w = ( w , . . . , w n D ) . We see that G p := ρ m D ( g p ) converges uniformly on compact subsets of C to G := ρ m D ( g ) . By Lemma5, Q ∗ jp = Q m D /d j jp := Q m D /d j j,f p converge uniformaly on compact subset of D to Q ∗ j , ≤ j ≤ q, and D ( Q ∗ , . . . , Q ∗ q )( z ) > δ ( z ) > for any fixed z ∈ D . In particular, the moving hyperplanes H ∗ , . . . , H ∗ q definingby moving linear forms Q ∗ , . . . , Q ∗ q (respectively) are located in pointwise generalposition in P n D ( C ) . We see that y p + r p u p ξ belong to a closed disc K = (cid:26) z ∈ C m : || z || ≤ || y || (cid:27) of D for all ξ in a subset compact of C if p is large enough. Thus for j ∈ { , ..., q } ,there exist reduced representations ˜ f p = ( f p , ..., f np ) , ˜ g = ( g , . . . , g n ) of f p , g respectively such that as p → ∞ , ˜ g p ( ξ ) = ˜ f p ( y p + r p u p ξ ) → ˜ g ( ξ ) and hence Q jp,y p + r p u p ξ (˜ g p ( ξ )) → Q j,y (˜ g ( ξ ))(3.1)converge uniformly on compact subsets of C . Claim 1 . g ( C ) intersects with D j,y with multiplicity at least m j . Indeed, for ξ ∈ C with Q j,y (˜ g ( ξ )) = 0, there exist a disc B ( ξ , r ) in C and an index i ∈ { , . . . , n } satisfying g ( B ( ξ , r )) ⊂ { [ x : ... : x n ] ∈ P n ( C ) : x i = 0 } . Since g i ( ξ ) = 0 on B ( ξ , r ), Hurwitz’s theorem shows that g ip ( ξ ) = f ip ( y p + r p u p ξ ) = 0 on B ( ξ , r ) when p is large enough. Thus ˜ g ip ( ξ ) = (cid:18) g p g ip , . . . , g np g ip (cid:19) → ˜ g i ( ξ ), and hence Q jp,y p + r p u p ξ (˜ g ip ( ξ )) → Q j,y (˜ g i ( ξ )) convergeuniformly on B ( ξ , r ), which further implies that( Q jp,y p + r p u p ξ (˜ g ip ( ξ ))) ( k ) → ( Q j,y (˜ g i ( ξ ))) ( k ) (3.2)converge uniformly on B ( ξ , r ) for all k ≥ m j ≥ , sup ≤| α |≤ m j − ,z ∈ f − p ( D jp ) ∩ K | ∂ α ( Q jp ( ˜ f ip )) ∂z α ( z ) | < ∞ hold for all j ∈ { , . . . , q } and all p ≥ , where f ip ( z ) = 0 and D jp is the movinghypersurface associated to Q jp . Since Q j,y (˜ g ( ξ )) = 0, then Hurwitz’s theoremshows that there exits ξ p → ξ such that Q jp,y p + r p u p ξ p (˜ g p ( ξ p )) = 0 , which implies y p + r p u p ξ p ∈ f − p ( D jp ) . Hence there exists
M > m j ≥ ≤ j ≤ q ) and p is large enough | ∂ α ( Q jp ( ˜ f ip )) ∂z α ( y p + r p u p ξ p ) | ≤ M for all α = ( α , . . . , α m ) : 1 ≤ | α | ≤ m j − i ∈ { , . . . , n } , which means (cid:12)(cid:12)(cid:12) ( Q jp,y p + r p u p ξ (˜ g ip ( ξ ))) ( | α | ) (cid:12)(cid:12)(cid:12) ξ = ξ p = | X α c α r | α | p ∂ α ( Q jp ( ˜ f ip )) ∂z α . . . ∂z α m m ( y p + r p u p ξ p ) | ≤ C.r | α | p M, (3.3) where C, c α are suitable constants. Combine (3.8) and (3.9), we get( Q j,y (˜ g i ( ξ ))) ( | α | ) | ξ = ξ = lim p →∞ ( Q jp,y p + r p u p ξ (˜ g ip ( ξ ))) ( | α | ) ξ = ξ p = 0for 1 ≤ | α | ≤ m j − . Hence ξ is a zero of Q j,y (˜ g ) with multiplicity at least m j .Therefore, Claim 1 is proved. We know that G p = ρ m D ( g p ) converges uniformly oncompact subsets of B ( ξ , r ) to G = ρ m D ( g ) . Denote ˜ G by a reduced representaionof G. Note that
Q m D d jj,y (˜ g ( ξ )) = Q ∗ j ( y )( ˜ G ( ξ )) , then G intersects H ∗ j ( y ) withmultiplicity at least m D d j m j for each 1 ≤ j ≤ q, where H ∗ j ( y ) is a associatedhyperplane with linear form Q ∗ j ( y ) in P n D ( C ) . Since H ∗ ( y ) , . . . , H ∗ q ( y ) are ingeneral position in P n D ( C ) . From assumption, q X j =1 d j m j < ( q − n D − m D n D and apply Lemma 7, we get G is a constant holomorphic map. Then g is aconstant holomorphic map from C into P n ( C ) . This is a contradiction. Therefore, F is holomorphically normal family on D . (cid:3) Proof of Theorem 17.
By Lemma 5, Q ∗ jp = Q m D /d j jp := Q m D /d j j,f p converge unifor-maly on compact subset of D to Q ∗ j , ≤ j ≤ q, and D ( Q ∗ , . . . , Q ∗ q )( z ) > δ ( z ) > z ∈ D . In particular, the moving hyperplanes H ∗ , . . . , H ∗ q defining bymoving linear forms Q ∗ , . . . , Q ∗ q (respectively) are located in weak general positionin P n D ( C ) . We recall that with writing both variables z ∈ D and w ∈ P n D ( C ) wehave Q ∗ jp ( z )( w ) → Q ∗ j ( z )( w )uniformly on compact subsets in the variable z ∈ D . By Lemma 1 and Lemma 2,after passing to a subsequence, we may assume that the sequence { f p } satisfies f − p ( D k,f p ) = S k (1 ≤ k ≤ n D + 1)as a sequence of closed subsets of D , where S k are either empty or pure ( m − D , and satisfieslim p →∞ { z ∈ supp ν Q k,fp ( ˜ f p ) | ν Q k,fp ( ˜ f p ) ( z ) < m k } − S = S k ( n D + 2 ≤ k ≤ q )(3.4)as a sequence of closed subsets of D − S, where S k are either empty or pure( m − D − S. Let T = ( . . . , t j , . . . ) (1 ≤ j ≤ q ) be a family of variables. Set ∼ Q ∗ j = P n D j =0 t j w j ∈ Z [ T, w ] . For each L ⊂ { , . . . , q } with | L | = n D + 1 , take ∼ R L isthe resultant of ∼ Q ∗ j , j ∈ L. Since { Q ∗ j } j ∈ L are in weakly general position, then ∼ R L ( . . . , c ij , . . . ) . We set ∼ S := { z ∈ D | ∼ R L ( . . . , c ij , . . . ) = 0 for some L ⊂ { , . . . , q } with | L | = n D + 1 } . Let E = ( ∪ qk =1 S k ∪ ∼ S ) − S. (3.5)Then E is either empty or a pure ( m − D − S. Fixany point z ∈ ( D − S ) − E. (3.6)Choose a relatively compact neighborhood U z of z in ( D − S ) − E. Then { f p | U z } ⊂ Hol ( U z , P n ( C )) . We now prove that the family { f p | U z } is a holomor-phically normal family. For this it is sufficient to show that the family { f p | U z } satisfies all conditions of Corollary 1. Indeed, there exists N such that for all p ≥ N , { f p | U z } does not intersect with D j,f p , ≤ j ≤ n D +1 . From (3.4)-(3.6), wehave { f p | U z } intersect with D j,f p with multiplicity at least m j ( n D + 2 ≤ j ≤ q ) . For all z ∈ U z , we have D ( Q ∗ , . . . , Q ∗ q )( z ) > , if we take m j = ∞ for all1 ≤ j ≤ n D + 1 , then all conditions of Corollary 1 are satisfied. Thus, { f p | U z } isa holomorphically normal family. By the usual diagonal argument, we can finda subsequence (again denoted by { f p } ) which converges uniformly on compactsubsets of ( D − S ) − E to a holomorphic mapping f of ( D − S ) − E into P n ( C ) . Therefore { f p } is quasi-regular on D . Hence, F is a quasi-normal family. Theproof of Theorem 17 is completed. (cid:3) Proof Theorem 18 and Theorem 19.
In order to prove these theorems, we needsome lemmas as follows:
Lemma 11.
Let f be a holomorphic mapping from a bounded domain U in C m into P n ( C ) . Let D , . . . , D q be q ( ≥ n D + 1) moving hypersurfaces in P n ( C ) . Let Q ∗ j = P n D i =0 d ij ω i , ( ω i = x I i ) in H D [ ω , . . . , ω n D ] be a homogeneous linearlydefines the associated hyperplane H ∗ j of D j such that for any z ∈ U : D ( Q ∗ , . . . , Q ∗ q )( z ) > . Let ˜ f = ( f , . . . , f n ) be a reduced representation of f on U with f i ( z ) = 0 for some i and z, and when m j ≥ ≤| α |≤ m j − ,z ∈ f − ( D j ) (cid:12)(cid:12)(cid:12) ∂ α ( Q j ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } , where m , . . . , m q are positive integers and may be ∞ , with P qj =1 d j m j < ( q − n D − m D n D . Then f is a normal holomorphic mappingfrom U into P n ( C ) . Proof.
Suppose that f is not normal on U . Then, by Lemma 4, there exist { y p } ⊂ U , { r p } with r p > r p → + and { u p } ⊂ C m are Euclidean unit vectors suchthat g p ( ξ ) := f ( y p + r p u p ξ ) , ξ ∈ C , where lim p →∞ r p d ( y p , C m \ U ) = 0 , converges uniformly on compact subsets of C to a non-constant holomorphic mapping g of C to P n ( C ) . Since U compact, thenwe may assume that y p → y ∈ U . We have G p := ρ m D ( g p ) converges uniformlyon compact subsets of C to G := ρ m D ( g ) . By assumption, we have D ( Q ∗ , . . . , Q ∗ q )( z ) > z ∈ U . In particular, the moving hyperplanes H ∗ , . . . , H ∗ q defining bymoving linear forms Q ∗ , . . . , Q ∗ q (respectively) are located in pointwise generalposition in P n D ( C ) . There exist the reduced representations ˜ f = ( f , ..., f n ) and ˜ g = ( g , . . . , g n )of f and g respectively such that as p → ∞ , ˜ g p ( ξ ) = ˜ f ( y p + r p u p ξ ) → ˜ g ( ξ ) andhence Q jp,y p + r p u p ξ (˜ g p ( ξ )) → Q j,y (˜ g ( ξ ))(3.7)converge uniformly on compact subsets of C . Claim . g ( C ) intersects with D j,y with multiplicity at least m j . Indeed, for ξ ∈ C with Q j,y (˜ g ( ξ )) = 0, there exist a disc B ( ξ , r ) in C and an index i ∈ { , . . . , n } satisfying g ( B ( ξ , r )) ⊂ { [ x : ... : x n ] ∈ P n ( C ) : x i = 0 } . Since g i ( ξ ) = 0 on B ( ξ , r ), Hurwitz’s theorem shows that g ip ( ξ ) = f i ( y p + r p u p ξ ) = 0 on B ( ξ , r ) when p is large enough. Thus ˜ g ip ( ξ ) = (cid:18) g p g ip , . . . , g np g ip (cid:19) → ˜ g i ( ξ ), and hence Q jp,y p + r p u p ξ (˜ g ip ( ξ )) → Q j,y (˜ g i ( ξ )) convergeuniformly on B ( ξ , r ), which further implies that( Q jp,y p + r p u p ξ (˜ g ip ( ξ ))) ( k ) → ( Q j,y (˜ g i ( ξ ))) ( k ) (3.8) converge uniformly on B ( ξ , r ) for all k ≥ m j ≥ , sup ≤| α |≤ m j − ,z ∈ f − ( D j ) (cid:12)(cid:12)(cid:12) ∂ α ( Q j ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } and all p ≥ , where f i ( z ) = 0 and D j is the movinghypersurface associated to Q j . Since Q j,y (˜ g ( ξ )) = 0, then Hurwitz’s theoremshows that there exits ξ p → ξ such that Q j,y p + r p u p ξ p (˜ g p ( ξ p )) = 0 , which implies y p + r p u p ξ p ∈ f − ( D jp ) . Hence there exists
M > m j ≥ ≤ j ≤ q ) and p is large enough | ∂ α ( Q j ( ˜ f i )) ∂z α ( y p + r p u p ξ p ) | ≤ M for all α = ( α , . . . , α m ) : 1 ≤ | α | ≤ m j − i ∈ { , . . . , n } , which means (cid:12)(cid:12)(cid:12) ( Q j,y p + r p u p ξ (˜ g ip ( ξ ))) ( | α | ) (cid:12)(cid:12)(cid:12) ξ = ξ p = | X α c α r | α | p ∂ α ( Q j ( ˜ f i )) ∂z α . . . ∂z α m m ( y p + r p u p ξ p ) | ≤ C.r | α | p M, (3.9)where C, c α are suitable constants. Combine (3.8) and (3.9), we get( Q j,y (˜ g i ( ξ ))) ( | α | ) | ξ = ξ = lim p →∞ ( Q j,y p + r p u p ξ (˜ g ip ( ξ ))) ( | α | ) ξ = ξ p = 0for 1 ≤ | α | ≤ m j − . Hence ξ is a zero of Q j,y (˜ g ) with multiplicity at least m j .Therefore, Claim is proved. We know that G p = ρ m D ( g p ) converges uniformly oncompact subsets of B ( ξ , r ) to G = ρ m D ( g ) . Denote ˜ G by a reduced representaionof G. Note that
Q m D d jj,y (˜ g ( ξ )) = Q ∗ j ( y )( ˜ G ( ξ )) , then G intersects H ∗ j ( y ) withmultiplicity at least m D d j m j for each 1 ≤ j ≤ q, where H ∗ j ( y ) is a associatedhyperplane with linear form Q ∗ j ( y ) in P n D ( C ) . Since H ∗ ( y ) , . . . , H ∗ q ( y ) are ingeneral position in P n D ( C ) . From assumption, q X j =1 d j m j < ( q − n D − m D n D and apply Lemma 7, we get G is a constant holomorphic map. Then g is aconstant holomorphic map from C into P n ( C ) . This is a contradiction. Therefore, f is normal function on U . (cid:3) Lemma 12. [11]
Let M be a complex manifold and let S be a complex analyticsubset of M with codim S ≥ . Then K M − S = K M on M − S (i.e., the infinitesimalKobayashi metric K M − S is the restriction of K M to M − S ). Now we prove Theorem 18. Fix a point P ∈ S, take a bounded neighborhood U of P in C m (i.e., U is hyperbolic) with U ⊂ D . By Lemma 11, f is normalholomorphic mapping from U − S into P n ( C ) . Thus by Defintion 29 and defintionof the integral distance, there exists a positive constant c such that d P n ( C ) ( f ( z ) , f ( w )) ≤ cd KU − S ( z, w )for all z, w ∈ U − S, where d KU − S and d P n ( C ) denote the Kobayashi distance on U − S and the Fubini-Study distance on P n ( C ) , resepctively. For any z ∈ U ∩ S, let { z i } ∞ i =1 be a sequence of points of U − S so as to converge to z . By Lemma12, we have d P n ( C ) ( f ( z i ) , f ( z j )) ≤ cd KU − S ( z i , z j ) = cd KU ( z i , z j ) . Then { f ( z i ) } ∞ i =1 is a Cauchy sequence of P n ( C ) and hence { f ( z i ) } ∞ i =1 converges toa point a ∈ P n ( C ) . Then f ( z ) has an extension h ( z ) on U so as to be holomorphicon U − S and continuous on U and hence h ( z ) is holomorphic on U by Riemann’sextension theorem. We have competed the proof of Theorem 18.Next, we prove Theorem 19. Fix a point P ∈ S, take a bounded neighborhood U of P in C m with U ⊂ D . By Lemma 11, f is normal holomorphic mappingfrom U − S into P n ( C ) . By assumption of Theorem 19, S is an analytic subset of adomain U in C m with codimension one, whose singularities are normal crossings.Hence f ( z ) extends to a holomorphic mapping from U into P n ( C ) by Theorem2.3 in Joseph and Kwack [7]. We have finished the proof of Theorem 18. (cid:3) Proof Theorem 20, Theorem 21 and Theorem 22.
The proof of Theorem 20 is sim-ilarly to Theorem 16 by using Lemma 3 and Lemma 8. We omit it at here. Inorder to prove two next theorems, we need a lemma as follows:
Lemma 13.
Let f be a holomorphic mapping from U into P n ( C ) . Suppose thatthere exist n +1 moving hypersurfaces T , . . . , T n in P n ( C ) of common degree whichare defined by homogeneous polynomials P , . . . , P n ∈ H D [ x , . . . , x n ] respectively,and that there exist q moving hypersurfaces D , . . . , D q in ∼ S ( { T i } ni =0 ) such thatthe following conditions are satisfied: ( i ) Q j ∈ S ( { P i } ni =0 ) is a homogeneous polynomial defined the D j , and for any z ∈ U , D ( Q , . . . , Q q )( z ) > . ( ii ) Let ˜ f = ( f , . . . , f n ) be a reduced representation of f on U with f i ( z ) = 0 forsome i and z, and when m j ≥ ≤| α |≤ m j − ,z ∈ f − ( D j ) (cid:12)(cid:12)(cid:12) ∂ α ( Q j ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } , where m , . . . , m q are positive integers and may be ∞ , with q X j =1 m j < q − n − n . Then f is a normal holomorphic mapping from U into P n ( C ) . Proof.
The proof of Lemma 13 is similarly to Lemma 11. For convenience thereader, we show it detail. Suppose that f is not normal on U . Then, by Lemma4, there exist { y p } ⊂ U , { r p } with r p > r p → + and { u p } ⊂ C m areEuclidean unit vectors such that g p ( ξ ) := f ( y p + r p u p ξ ) , ξ ∈ C , where lim p →∞ r p d ( y p , C m \ U ) = 0 , converges uniformly on compact subsets of C to a non-constant holomorphic mapping g of C to P n ( C ) . Since U compact, thenwe may assume that y p → y ∈ U . We have G p := F ( g p ) converges uniformly oncompact subsets of C to G := F ( g ) , where F : P n ( C ) → P n ( C )is the mapping defined by F i ( x ) = P i ( x ) (0 ≤ i ≤ n ) . Furthermore, we can write Q j as follows Q j = n X i =0 b ij P i , b ij ∈ H U , j = 1 , . . . , q. Set H j = { x ∈ P n ( C ) | n X i =0 b ij x j = 0 } . By Lemma 9, from D ( Q , . . . , Q q )( z ) > z ∈ D, we have { P i } ni =0 are in general position and { Q j } qj =1 are locatedin general position in S ( { P i } ni =0 ) . This means that the hyperplanes {H j } qj =1 arelocated in general pointwise position in P n ( C ) . By arguments as Lemma 11, G intersects H j ( y ) with multiplicity at least m j for each 1 ≤ j ≤ q. From assumption, q X j =1 m j < q − n − n and apply Lemma 7, we get G = F ( g ) is a constant holomorphic map. Then g is a constant holomorphic map from C into P n ( C ) by Lemma 8. This is acontradiction. Therefore, f is a normal holomorphic mapping on U . (cid:3) Apply Lemma 13, we can prove Theorem 21 and Theorem 22 similarly asTheorem 18 and Theorem 19. Hence, we omit them here. (cid:3)
Proof of Theorem 24.
Suppose that F is not normal on D . Then, by Lemma 3,there exists a subsequence denoted by { f p } ⊂ F and y ∈ K , { y p } ∞ p =1 ⊂ K with y p → y , { r p } ⊂ (0 , + ∞ ) with r p → + , and { u p } ⊂ C m , which are unit vectors,such that g p ( ξ ) := f p ( y p + r p u p ξ ) converges uniformly on compact subsets of C to a nonconstant holomorphic map g of C into P n ( C ) . By Lemma 5, Q jp := Q j,f p converge uniformaly on compact subset of D to Q j , ≤ j ≤ q, and D ( Q , . . . , Q q )( z ) > δ ( z ) > z ∈ D . In particular, the moving hypersurfaces D , . . . , D q defin-ing by moving homogeneous polynomial Q , . . . , Q q (respectively) are located inpointwise general position in P n ( C ) . We recall that with writing both variables z ∈ D and x ∈ P n ( C ) we have Q jp ( z )( x ) → Q j ( z )( x )uniformly on compact subsets in the variable z ∈ D . By arguments as the proof of Theorem 16, the map g intersects D j ( y ) withmultiplicity at least m j . Now, we show that g is a constant mapping. Since Q ( y ) , . . . , Q q ( y ) are in general position in P n ( C ) , then there are at most n hypersurfaces in the family D ( y ) , . . . , D q ( y ) containing image of g. We denote I = { i ∈ { , . . . , q } : Q i ( y )(˜ g ) ≡ } . Therefore, we must have 0 ≤ |I| ≤ n. Notethat q ≥ nn D + 2 n + 1 , then q − |I| ≥ nn D + n + 1 . We see that g intersects with { D j ( y ) } j with multiplicity at least m j , and X j m j ≤ q X j =1 m j < q − n − ( n − n D + 1) n D ( n D + 2) ≤ q − |I| − ( n − n D + 1) n D ( n D + 2) . Apply Lemma 10 for map g and hypersurfaces { D j ( y ) } j I , we get g is a constantmap. This is a contradiction. Therefore, F is holomorphically normal family on D . (cid:3) Proof of Theorem 25.
By Lemma 5, Q jp := Q j,f p converge uniformaly on compactsubset of D to Q j , ≤ j ≤ q, and D ( Q , . . . , Q q )( z ) > δ ( z ) > z ∈ D , where z belongs a neighborhood of z . In particular, the movinghypersurfaces D , . . . , D q defining by moving homogeneous polynomial Q , . . . , Q q (respectively) are located in weak general position in P n ( C ) . We recall that withwriting both variables z ∈ D and x ∈ P n ( C ) we have Q jp ( z )( x ) → Q j ( z )( x )uniformly on compact subsets in the variable z ∈ D . By Lemma 1 and Lemma 2,after passing to a subsequence, we may assume that the sequence { f p } satisfies f − p ( D k,f p ) = S k (1 ≤ k ≤ n + 1)as a sequence of closed subsets of D , where S k are either empty or pure ( m − D , and satisfieslim p →∞ { z ∈ supp ν Q k,fp ( ˜ f p ) | ν Q k,fp ( ˜ f p ) ( z ) < m k } − S = S k ( n + 2 ≤ k ≤ q )(3.10)as a sequence of closed subsets of D − S, where S k are either empty or pure( m − D − S. Let T = ( . . . , t jI , . . . ) (1 ≤ j ≤ q, | I | = d j ) be a family of variables. Set ∼ Q j = P | I j | = d j t jI x I j ∈ Z [ T, x ] . Suppose that Q j ( x ) = Q j ( x , . . . , x n ) = n dj X k =0 e kI j,k x i kj, . . . x i kj,n n , where I j,k = ( i kj, , . . . , i kj,n ) , | I j,k | = d j for k = 0 , . . . , n d j and n d j = (cid:0) n + d j n (cid:1) − . For each L ⊂ { , . . . , q } with | L | = n + 1 , take ∼ R L is the resultant of ∼ Q j , j ∈ L. Since { Q j } j ∈ L are in weakly general position, then ∼ R L ( . . . , e kI j,k , . . . ) ∼ S := { z ∈ D | ∼ R L ( . . . , e kI j,k , . . . ) = 0 for some L ⊂ { , . . . , q } with | L | = n + 1 } . Let E = ( ∪ qk =1 S k ∪ ∼ S ) − S. (3.11)Then E is either empty or a pure ( m − D − S. Fixany point z ∈ ( D − S ) − E. (3.12)Choose a relatively compact neighborhood U z of z in ( D − S ) − E. Then { f p | U z } ⊂ Hol ( U z , P n ( C )) . We now prove that the family { f p | U z } is a holomor-phically normal family. For this it is sufficient to show that the family { f p | U z } satisfies all conditions of Corollary 2. Indeed, there exists N such that for all p ≥ N , { f p | U z } does not intersect with D j,f p , ≤ j ≤ n + 1 . From (3.10)-(3.12),we have { f p | U z } intersect with D j,f p with multiplicity at least m j ( n +2 ≤ j ≤ q ) . For all z ∈ U z , we have D ( Q , . . . , Q q )( z ) > , if we take m j = ∞ for all1 ≤ j ≤ n + 1 , then all conditions of Corollary 2 are satisfied. Thus, { f p | U z } isa holomorphically normal family. By the usual diagonal argument, we can finda subsequence (again denoted by { f p } ) which converges uniformly on compactsubsets of ( D − S ) − E to a holomorphic mapping f of ( D − S ) − E into P n ( C ) . ByLemma 6, { f p } is meromorphically convergent on D . Hence, F is meromorphicalynormal family on D . The proof of Theorem 25 is completed. (cid:3)
Proof Theorem 26 and Theorem 27.
In order to prove these theorems, we needsome lemmas as follows:
Lemma 14.
Let f be a holomorphic mapping from a bounded domain U in C m into P n ( C ) . Let D = { D , . . . , D q } be q ( ≥ nn D + 2 n + 1) moving hypersurfacesin P n ( C ) . Let Q j ∈ H D [ x , . . . , x n ] be a homogeneous polynomial with degree d j which is defined D j such that for any z ∈ U : D ( Q , . . . , Q q )( z ) > . Let ˜ f = ( f , . . . , f n ) be a reduced representation of f on U with f i ( z ) = 0 for some i , then when m j ≥ ≤| α |≤ m j − ,z ∈ f − ( D j ) (cid:12)(cid:12)(cid:12) ∂ α ( Q j ( ˜ f i )) ∂z α ( z ) (cid:12)(cid:12)(cid:12) < ∞ hold for all j ∈ { , . . . , q } , where m , . . . , m q are positive integers and may be ∞ , with q X j =1 m j < q − n − ( n − n D + 1) n D ( n D + 2) . Then f is a normal holomorphic mapping from U into P n ( C ) . Lemma 14 is proved similarly Lemma 13 and Theorem 24. We omit it at here.Theorem 26 and Theorem 27 are proved similarly Theorem 18 and Theorem 19,we omit them at here. (cid:3)
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