Algebraic dependence and finiteness problems of differentiably nondegenerate meromorphic mappings on Kähler manifolds
aa r X i v : . [ m a t h . C V ] A p r NON-INTEGRATED DEFECT RELATION AND FINITENESSPROBLEM OF DIFFERENTIABLY NONDEGENERATEMEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS
SI DUC QUANG , Abstract.
Let M be a complete K¨ahler manifold, whose universal covering is biholo-morphic to a ball B m ( R ) in C m (0 < R ≤ + ∞ ). The first purpose of this article isto establish a non-integerated defect relation with truncation level 1 for differentiablynondegenerate meromorphic mappings with only one hypersurface, which is an union ofsubgeneral position hyperplanes. Our second aim is to study the algebraic dependenceproblem of differentiably meromorphic mappings. We will show that if k differentibilitynondegenerate meromorphic mappings f , . . . , f k of M into P n ( C ) ( n ≥
2) satisfyingthe condition ( C ρ ) and sharing few hyperplanes in subgeneral position regardless ofmultiplicity then f ∧ · · · ∧ f k . For the last aim, we show that there are at most twodifferent differentiably nondegenerate meromorphic mappings of M into P n ( C ) sharing q ( q ∼ N − n + 3 + O ( ρ )) hyperplanes in N − subgeneral position regardless of multiplic-ity. Our results generalize previous finiteness and uniqueness theorems for differentiablymeromorphic mappings of C m and extend some previous results for the case of mappingson K¨ahler manifold. Introduction
In 1985, H. Fujimoto [3] introduced the notion of the non-integrated defect for mero-morphic mappings from a complete K¨ahler manifold into a projective space intersectinghypersurfaces. We recall this definition as follows.Let M be a complete K¨ahler manifold of dimension m with the K¨ahler form ω and let f : M −→ P n ( C ) be a meromorphic mapping. Let µ be a positive integer and D be ahypersurface of degree d in P n ( C ) with f ( M ) D . Denote by Ω f the pull-back of thenormalized Fubini-Study form Ω on P n ( C ), by ν ( f,D ) ( p ) the intersection multiplicity of theimage of f and D at f ( p ) and set ν [ µ ]( f,D ) = min { ν f ( D ) , µ } . Then ν [ µ ]( f,D ) is also consideredas a divisor on M . The non-integrated defect of f with respect to D truncated by level µ is defined by δ [ µ ] f := 1 − inf { η ≥ η satisfies condition ( ∗ ) } . Here, the condition (*) means that there exists a bounded non-negative continuous func-tion h on M with zeros of order not less than ν [ µ ] such that dη Ω f + √− π ∂ ¯ ∂ log h ≥ [ ν [ µ ]( f,D ) ] . Here, for a divisor ν , the notation [ ν ] stands for its generating current. , In the case where the universal covering of the complete K¨ahler manifold M is requiredto be biholomorphic to a ball of C m , H. Fujimoto [3] obtained a result analogous to theNevanlinna-Cartan defect relation as follows.
Theorem A (see [3, Theorem 1.1])
Let M be an m -dimensional complete K¨ahler manifoldand ω be a K¨ahler form of M. Assume that the universal covering of M is biholomorphic toa ball in C m . Let f : M → P n ( C ) be a linearly nondegenerate meromorphic mapping (i.e.,its image is not contained in any hyperplane of P n ( C ) ). Let H , · · · , H q be hyperplanes of P n ( C ) in general position. For some ρ ≥ , if there exists a bounded continuous function h ≥ on M such that ρ Ω f + dd c log h ≥ Ric ω, then q X i =1 δ fn ( H i ) ≤ n + 1 + ρn ( n + 1) . After that this theorem is generalized to the case of hypersurfaces by many authorswith a large number of papers published recently, for instance [10, 13, 14, 15, 16, 18] andothers. However, in all these results the truncation levels of the defect are very largeor lost. Recently, Chen-Han [1] and Thai-Quang [17] independently established somenon-integrated defect relations with better truncation levels. Espectially, in [17], theauthors gave a non-integrated defect relation with the truncation level 1 for differentiablynondegenerate meromorphic mappings intersecting a family of hyperplanes in subgeneralposition of P n ( C ). Here, a mapping f is said to be differentiably nondegenerate if itsdifferential mapping df is surjective at some points. We state here that result of Thai-Quang. Theorem B (see [17, Theorem 4.1])
Let M be an m -dimensional complete K¨ahler mani-fold and ω be a K¨ahler form of M. Assume that the universal covering of M is biholomor-phic to a ball in C m . Let f : M → P n ( C ) be a differentiably non-degenerate meromorphicmapping. Let H , · · · , H q be hyperplanes of P n ( C ) in N − subgeneral position q ≥ N ≥ n .For some ρ ≥ , if there exists a bounded continuous function h ≥ on M such that ρ Ω f + dd c log h ≥ Ric ω, then q X i =1 δ [1] f ( H i ) ≤ N − n + 1 + 2 ρ (2 N − n + 1) . Here, the family of hyperplanes { H , · · · , H q } is said to be in N − subgeneral position iffor any 1 ≤ i < i < · · · < i N , T Nj =0 H i j = ∅ .Now, we consider D = H + · · · + H q be a hypersurfaces of degree q . Then we see that ν [1]( f,D ) ≤ q X i =1 ν [1]( f,H i ) , IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 3 and the inequality becomes to equality if and only if the intersections f − ( H i ) ∩ f − ( H j )has codimension at least two for all 1 ≤ i < j ≤ q . Then, we have δ [1]( f,D ) ≥ q q X i =1 δ [1]( f,H i ) and the equality only occurs in very restricted cases. Therefore, our first aim in this paperis to extend Theorem B to the case where the defect δ [1]( f,D ) is considered instead of δ [1]( f,H i ) .We also show that Theorem B can be made sharper (see Theorem 3.12). Our first resultis stated as follows. Theorem 1.1.
Let { H i } qi =1 be hyperplanes of P n ( C ) in N -subgeneral position. Let M be a complete K¨ahler manifold of dimension m ( m ≥ n ) with a K¨ahler form ω , whoseuniversal covering is biholomorphic to C m or a ball B ( R ) ⊂ C m . Let f be a differentiablynondegenerate meromorphic mapping from M into P n ( C ) . Assume that there is a boundednon negative continuous function h on M , a positive constant ρ such that ρf ∗ Ω + √− π ∂ ¯ ∂ log h ≥ Ric ω, where Ω is the Fubini-Study form on P n ( C ) . Then, we have δ [1] f ( D ) ≤ N − n )2 N − n + 1 + n + 1 q + 2 nρq , where D = H + · · · + H q be a hypersurface of degree q . Hence, in the above theorem, if the family of hyperplanes is assumed to be in generalposition, the defect relations can be given by δ [1] f ( D ) ≤ n + 1 q + 2 nρq . Beside developing the values distribution theory of meromorphic mappings on K¨ahlermanifold, Fujimoto also investigated the uniqueness problem for meromorphic. In [4],Fujimoto proved the following theorem, which is the first uniqueness theorem for mero-morphic mappings on K¨ahler manifold.
Theorem C (see [4, Main Theorem]).
Let M be an m -dimensional connected K¨ahlermanifold whose universal covering is biholomorphic to is biholomorphic to a ball B m ( R ) in C m ( < R ≤ + ∞ ), and let f, g be a linearly non-degenerate meromorphic mappingsof M into P n ( C ) ( m ≥ n ) satisfying the condition ( C ρ ) . Let H , . . . , H q be q hyperplanesof P n ( C ) in general possition. Assume thati) f = g on S qi =1 ( f − ( H i ) ∪ g − ( H i )) ,ii) If q > n + 1 + 2 ρ ( l f + l g ) + m f + m g .Then f = g . Here, l f , l g , m f , m g are positive numbers estimated in an explicit way. For the case where f and g is differentiably non-degenetate, we can take m f = m g = 1 and l f = l g = n .Our second purpose in this paper is to extend the above theorem to the case where thefamily of hyperplanes is in N − subgeneral position. To state our result, we need to recallsome following. SI DUC QUANG , Let M be an m -dimensional connected K¨ahler manifold whose universal covering isbiholomorphic to a ball B m ( R ) in C m (0 < R ≤ + ∞ ). Let f be a non-constantmeromorphic mapping of B m ( R ) into P n ( C ) with a reduced representation f = ( f : · · · : f n ), and H be a hyperplane in P n ( C ) given by H = { a ω + · · · + a n ω n = 0 } , where( a , . . . , a n ) = (0 , . . . , f, H ) = P ni =0 a i f i . We see that ν ( f,H ) is the pull-backdivisor of H by f and is also the divisor generated by the function ( f, H i ).Similarly, if we have a hypersurface D defined by a homogeneous polynomial of de-gree d in n + 1 variables ( x , . . . , x n ), denoted again by D , then we set ( f, D )( z ) = D ( f , · · · , f n )( z ). Then we also see that ν ( f,D ) is the pull-back divisor of D by f and isalso the divisor generated by the function ( f, D ).Let H , . . . , H q be q hyperplanes of P n ( C ) in N − general position. Let d be a positiveinteger. We consider the set D ( f, { H i } qi =1 , d ) of all meromorphic mappings g : M → P n ( C )satisfying the following conditions:(a) ν [ d ]( f,H i ) = ν [ d ]( g,H i ) (1 ≤ i ≤ q ) , (b) f ( z ) = g ( z ) on S qi =1 f − ( H i ).Here, ν [ d ] = min { ν, d } for each divisor ν .Then, our second result in this paper is stated as follows. Theorem 1.2.
Let M be an m -dimensional connected K¨ahler manifold whose universalcovering is biholomorphic to a ball B m ( R ) in C m ( < R ≤ + ∞ ), and let f be adifferentiably nondegenerate meromorphic mapping of M into P n ( C ) ( m ≥ n ) satisfyingthe condition ( C ρ ) . Let H , . . . , H q be q hyperplanes of P n ( C ) in general possition. Let f , . . . , f k (2 ≤ k ≤ n + 1) be differentiably nondegenerate meromorphic mappings in D ( f, { H i } qi =1 , satisfying the condition ( C ρ ) .a) If q > N − n + 1 + k (2 N − n + 1)( k − n + 1) + knρ then f ∧ · · · ∧ f k ≡ .b) If dim f − ( H i ) ∩ f − ( H j ) ≤ m − ≤ i < j ≤ q ) and q > N − n + 1 + kn (2 N − n + 1)( k − N ( n + 1) + kn ρN then f ∧ · · · ∧ f k ≡ . Letting k = 2, we immediately get the following uniqueness theorem. Corollary 1.3.
Let
M, f, H , . . . , H q be as in Theorem 1.2.a) If q > N − n + 1 + 2(2 N − n + 1)( n + 1) + 2 nρ then ♯ D ( f, { H i } qi =1 ,
1) = 1 .b) Assume further that dim f − ( H i ) ∩ f − ( H j ) ≤ m − ≤ i < j ≤ q ) . If q > N − n + 1 + 2 n (2 N − n + 1) N ( n + 1) + 2 n ρN then ♯ D ( f, { H i } qi =1 ,
1) = 1 . Here, by ♯S we denote the cardinality of the set S . IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 5
Remark: Suppose that q = n + 4 and { H i } n +4 i =1 is in general position, i.e., N = n .Then the assumption of the above corollary is fulfilled with ρ < n . Then this result isan extension of the uniqueness theorem for differentiably non-degenerate meromorphicmappings into P n ( C ) sharing a normal crossing divisor of degree n + 4 given firstly byDrouilhet [2, Theorem 4.2].We would like to emphasize here that, in order to study the finiteness problem ofmeromorphic mappings for the case of mappings from C m , almost all authors use Cartan’sauxialiary functions (see Definition 2.2) and compare the counting functions of theseauxialiary functions with the characteristic functions of the mappings. However, in thegeneral case of K¨ahler manifold, this method may do not work since this comparation doesnot make sense if the growth of the characteristic functions do not increase quickly enough.In order to overcome this difficulty, in [12], we introduced the notions of “functions ofsmall integration” and “functions of bounded integration”. Using this notions, we willextend the finiteness theorems for differentiably non-degenerate meromorphic mappingsof C m into P n ( C ) sharing n + 3 hyperplanes (see [9]) to the case of K¨ahler manifolds. Ourlast result is stated as follows. Theorem 1.4.
Let M be an m -dimensional connected K¨ahler manifold whose universalcovering is biholomorphic to a ball B m ( R ) in C m ( < R ≤ + ∞ ), and let f be adifferentiably non-degenerate meromorphic mapping of M into P n ( C ) ( m ≥ n ) satisfyingthe condition ( C ρ ) . Let H , . . . , H q be q hyperplanes of P n ( C ) in N − general possition suchthat dim f − ( H i ) ∩ f − ( H j ) ≤ m − ≤ i < j ≤ q ) . Assume that q > N − n + 1 + 9 n (2 N − n + 1)5 N ( n + 1) + ρ (cid:18) n + 9 n N (cid:19) . Then ♯ D ( f, { H i } qi =1 , ≤ . Remark: Suppose that q = n + 3 and { H i } n +3 i =1 is in general position. Then the assump-tion of the above theorem is fulfilled with ρ < n + 9 . Then this result is an extensionof the finiteness theorems for differentiably non-degenerate meromorphic mappings into P n ( C ) sharing n + 3 hyperplanes in general position of Quang [9, Theorems 1.1,1.2,1.3].2. Basic notions and auxiliary results from the distribution theory
In this section, we recall some notations from the distribution value theory of meromor-phic mappings on a ball B m ( C ) in C m due to [11, 12]. We set k z k = (cid:0) | z | + · · · + | z m | (cid:1) / for z = ( z , . . . , z m ) ∈ C m and define B m ( R ) := { z ∈ C m : k z k < R } (0 < R ≤ ∞ ) ,S ( R ) := { z ∈ C m : k z k = R } (0 < R < ∞ ) . Define v m − ( z ) := (cid:0) dd c k z k (cid:1) m − and σ m ( z ) := d c log k z k ∧ (cid:0) dd c log k z k (cid:1) m − on C m \ { } . SI DUC QUANG , For a divisor ν on a ball B m ( R ) of C m , and for a positive integer p or p = ∞ , we definethe truncated counting function of ν by n ( t, ν ) = R | ν | ∩ B ( t ) ν ( z ) v m − if m ≥ , P | z |≤ t ν ( z ) if m = 1 . and define n [ p ] ( t ) := n ( t, ν [ p ] ) , where ν [ p ] = min { p, ν } . Define N ( r, r , ν ) = r Z r n ( t ) t m − dt (0 < r < r < R ) . Similarly, define N ( r, r , ν [ p ] ) and denote it by N [ p ] ( r, r , ν ).Let ϕ : B m ( R ) −→ C be a meromorphic function. Denote by ν ϕ (res. ν ϕ ) the divisor(resp. the zero divisor) of ϕ . Define N ϕ ( r, r ) = N ( r, r , ν ϕ ) , N [ p ] ϕ ( r, r ) = N [ p ] ( r, r , ( ν ϕ ) [ p ] ) . For brevity, we will omit the character [ p ] if p = ∞ . Throughout this paper, we fix a homogeneous coordi-nates system ( x : · · · : x n ) on P n ( C ). Let f : B m ( R ) −→ P n ( C ) be a meromorphicmapping with a reduced representation f = ( f , . . . , f n ), which means that each f i is aholomorphic function on B m ( R ) and f ( z ) = (cid:0) f ( z ) : · · · : f n ( z ) (cid:1) outside the indetermi-nancy locus I ( f ) of f . Set k f k = (cid:0) | f | + · · · + | f n | (cid:1) / .The characteristic function of f is defined by T f ( r, r ) = Z rr dtt m − Z B ( t ) f ∗ Ω ∧ v m − , (0 < r < r < R ) . By Jensen’s formula, we have T f ( r, r ) = Z S ( r ) log k f k σ m − Z S ( r ) log k f k σ m + O (1) , (as r → R ) . If R = + ∞ , we always choose r = 1 and write N ϕ ( r ) , N [ p ] ϕ ( r ) , T f ( r ) for N ϕ ( r, ,N [ p ] ϕ ( r, , T f ( r,
1) as usual.
Repeating the argument in [3, Proposition 4.5], we have thefollowing.
Proposition 2.1.
Let F , . . . , F l − be meromorphic functions on the ball B m ( R ) in C m such that { F , . . . , F l − } are linearly independent over C . Then there exists an admissibleset { α i = ( α i , . . . , α im ) } l − i =0 ⊂ N m , which is chosen uniquely in an explicit way, with | α i | = P mj =1 | α ij | ≤ i (0 ≤ i ≤ l − such that: IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 7 (i) W α ,...,α l − ( F , . . . , F l − ) Def := det ( D α i F j ) ≤ i,j ≤ l − . (ii) W α ,...,α l − ( hF , . . . , hF l − ) = h l +1 W α ,...,α l − ( F , . . . , F l − ) for any nonzero mero-morphic function h on B m ( R ) . The function W α ,...,α l − ( F , . . . , F l − ) is called the general Wronskian of the mapping F = ( F , . . . , F l − ). Definition 2.2 (Cartan’s auxialiary function [5, Definition 3.1]) . For meromorphic func-tions
F, G, H on B m ( R ) and α = ( α , . . . , α m ) ∈ Z m + , we define the Cartan’s auxiliaryfunction as follows: Φ α ( F, G, H ) := F · G · H · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F G H D α ( F ) D α ( G ) D α ( H ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Lemma 2.3 (see [5, Proposition 3.4]) . If Φ α ( F, G, H ) = 0 and Φ α ( F , G , H ) = 0 for all α with | α | ≤ , then one of the following assertions holds:(i) F = G, G = H or H = F ,(ii) FG , GH and HF are all constant. Let f , f , . . . , f k be m meromorphic mappings from the complete K¨ahler manifold B m ( R ) into P n ( C ), whichsatisfies the condition ( C ρ ) for a non-negative number ρ . For each 1 ≤ u ≤ k , we fix areduced representation f u = ( f u : · · · : f un ) of f u and set k f u k = ( | f u | + · · · + | f u | n ) / .We denote by C ( B m ( R )) the set of all non-negative functions g : B m ( R ) → [0 , + ∞ ]which are continuous (corresponding to the topology of the compactification [0 , + ∞ ]) andonly attain + ∞ in a thin set. Definition 2.4 (see [11, Definition 2.2] and [12, Definition 3.1]) . A function g in C ( B m ( R )) is said to be of small integration with respective to f , . . . , f k at level l if there exist anelement α = ( α , . . . , α m ) ∈ N m with | α | ≤ l , a positive number K , such that for every ≤ tl < p < , Z S ( r ) | z α g | t σ m ≤ K R m − R − r k X u =1 T f u ( r, r ) ! p for all r with < r < r < R < R , where z α = z α · · · z α m m . Remark: In [11], we only give the definition of “function of small integration” for non-negative plurisubharmonic functions. In that way the set of functions of small integrationmay not large enough to contain all neccesary functions. Therefore, here we define thisnotion for all functions in C ( B m ( R )) (as in [12]). However, this does not affect to theproofs in [11].We denote by S ( l ; f , . . . , f k ) the set of all functions in C ( B m ( R )) which are ofsmall integration with respective to f , . . . , f k at level l . We see that, if g belongsto S ( l ; f , . . . , f k ) then g is also belongs to S ( l ; f , . . . , f k ) for every l > l . Moreover, if g is a constant function then g ∈ S (0; f , . . . , f k ). SI DUC QUANG , Proposition 2.5 (see [11, Proposition 2.3] and [12, Proposition 3.2]) . If g i ∈ S ( l i ; f , . . . , f l )(1 ≤ i ≤ s ) then Q si =1 g i ∈ S ( P si =1 l i ; f , . . . , f l ) . Definition 2.6 (see [12, Definition 3.3]) . A meromorphic function h on B m ( R ) is saidto be of bounded integration with bi-degree ( p, l ) for the family { f , . . . , f k } if there exists g ∈ S ( l ; f , . . . , f k ) satisfying | h | ≤ k f k p · · · k f u k p · g, outside a proper analytic subset of B m ( R ) . Denote by B ( p, l ; f , . . . , f k ) the set of all meromorphic functions on B m ( R ) which areof bounded integration of bi-degree ( p, l ) for { f , . . . , f k } . We have the following: • For a meromorphic mapping h , | h | ∈ S ( l ; f , . . . , f k ) iff h ∈ B (0 , l ; f , . . . , f k ). • B ( p, l ; f , . . . , f k ) ⊂ B ( p, l ; f , . . . , f k ) for every 0 ≤ l < l . • If h i ∈ B ( p i , l i ; f , . . . , f k ) (1 ≤ i ≤ s ) then h · · · h m ∈ B ( s X i =1 p i , s X i =1 l i ; f , . . . , f k ) . The following proposition is proved by Fujimoto [7] and reproved by Ru-Sogome [15].
Proposition 2.7 (see [7, Proposition 6.1], also [15, Proposition 3.3]) . Let L , . . . , L l belinear forms of l variables and assume that they are linearly independent. Let F be a mero-morphic mapping from the ball B m ( R ) ⊂ C m into P l − ( C ) with a reduced representation F = ( F , . . . , F l − ) and let ( α , . . . , α l ) be an admissible set of F . Set l = | α | + · · · + | α l | and take t, p with < tl < p < . Then, for < r < R , there exists a positive constant K such that for r < r < R < R , Z S ( r ) (cid:12)(cid:12)(cid:12)(cid:12) z α + ··· + α l W α ,...,α l ( F , . . . , F l − ) L ( F ) . . . L l − ( F ) (cid:12)(cid:12)(cid:12)(cid:12) t σ m ≤ K (cid:18) R m − R − r T F ( R, r ) (cid:19) p . This proposition implies that the function (cid:12)(cid:12)(cid:12)(cid:12) W α ,...,α l ( F , . . . , F l − ) L ( F ) . . . L l − ( F ) (cid:12)(cid:12)(cid:12)(cid:12) belongs to S ( l ; F ).3. Non-integrated defect relation for differentiably nondegeneratemappings intersecting hyperplanes in subgeneral position
In this section we will prove Theorem 1.1. We state here the following lemma onNochka’s weights.
Lemma 3.1 (see also [8, Lemma 3.3 and Lemma 3.4]) . Let H , ..., H q be q hyperplanes in P n ( C ) in N -subgeneral position, where q > N − n + 1 . Then, there are positive rationalconstants ω i (1 ≤ i ≤ q ) satisfying the following:i) < ω i ≤ , ∀ i ∈ { , ..., q } ,ii) Setting ˜ ω = max j ∈ Q ω j , one gets q X j =1 ω j = ˜ ω ( q − N + n −
1) + n + 1 . IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 9 iii) n + 12 N − n + 1 ≤ ˜ ω ≤ nN . iv) Let E i ≥ ≤ i ≤ q ) be arbitrarily given numbers. For R ⊂ { , ..., q } with ♯R = N + 1 , there is a subset R o ⊂ R such that ♯R o = rank { H i } i ∈ R o = n + 1 and Y i ∈ R E ω i i ≤ Y i ∈ R o E i . Next, we will prove the following lemma.
Lemma 3.2.
Let f be a differentiably non-degenerate meromorphic mapping of a ball B m ( R ) in C m into P n ( C ) ( m ≥ n ) with a reduced representation ( f : · · · : f n ) . Let H , . . . , H n be n + 1 hyperplanes of P n ( C ) in general possition. Let α = ( α , . . . , α n ) ∈ ( N m ) n +1 with | α | = 0 , | α i | = 1 (1 ≤ i ≤ n ) such that W := det( D α i f j ; 0 ≤ i, j ≤ n ) .Then we have n X i =0 ν ( f,H i ) − ν W ≤ ν [1] Q ni =0 ( f,H i ) . Proof.
Since W = C det( D α i ( f, H j )) with a nonzero constant C , without loss of generalitywe may suppose that H i = { ω i = 0 } (0 ≤ i ≤ n ) . Then we have ( f, H i ) = f i . Also, wemay assume that α = (1 , , , . . . , , α = (0 , , , . . . , , . . . , α n = (0 , , . . . , , . Let b be a regular point of the analytic set S = { f · · · f n = 0 } and b is not in theindeterminacy locus I ( f ) of f . Then there is a local affine coordinates ( U, x ) around b ,where U is a neighborhood of b in B m ( R ), x = ( x , . . . , x m ) , x ( b ) = (0 , . . . ,
0) such that S ∩ U = { x = 0 } ∩ U .Since b I ( f ), we may suppose that S ∩ U = { f i = 0 } ∩ U (0 ≤ i ≤ l ) and f j ( l + 1 ≤ j ≤ n ) does not vanishes on U . Therefore, we have f i = x t i g j (0 ≤ i ≤ l ) with someholomorphic function g j . We easily see that D α i ( f j /f n ) = ∂ ( f j /f n ) ∂z i = m X s =1 ∂x s ∂z i · ∂∂x s (cid:18) f j f n (cid:19) (0 ≤ j ≤ n − ν ∂∂xs (cid:16) fjfn (cid:17) ( b ) ≥ ( t j − s = 1 t j if s > , ∀ ≤ j ≤ l. On the other hand, we have W = det( D α i f j ; 0 ≤ i, j ≤ n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f f . . . f n∂f ∂z ∂f ∂z . . . ∂f n ∂z ... ... . . . ... ∂f ∂z n ∂f ∂z n . . . ∂f n ∂z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = f n +1 n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ( f /f n ) ∂z ∂ ( f /f n ) ∂z . . . ∂ ( f n − /f n ) ∂z ... ... . . . ... ∂ ( f /f n ) ∂z n ∂ ( f /f n ) ∂z n . . . ∂ ( f n − /f n ) ∂z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . , This implies that ν W ( b ) ≥ min { ν det (cid:18) ∂∂xis (cid:0) fjfn (cid:1) ;0 ≤ j,s ≤ n (cid:19) ( b ); 1 ≤ i < · · · < i n ≤ m }≥ min n X j =0 ν ∂∂xij (cid:0) fjfn (cid:1) ( b ) ≥ t + · · · + t l − n X i =0 ν ( f,H i ) ( b ) − ν [1] Q ni =0 ( f,H i ) ( b ) . Therefore, we have n X i =0 ν ( f,H i ) ( b ) − ν [1] Q ni =0 ( f,H i ) ( b ) ≥ ν W ( b ) . The lemma is proved. (cid:3)
Proof of Theorem 1.1.
By using the universal covering if necessary, it suffices toprove the theorem in the case where M is the ball B m ( R ) of C m .We assume that f has a reduced representation f = ( f , ..., f n ) and each H i is given by H i = { ( x : · · · : x n ) ∈ P n ( C ); a i x + · · · + a in x n = 0 } , where a ij ∈ C and P nj =0 | a ij | = 1. Set( f, H i ) = a i f + · · · + a in f n . Since f is differentiably nondegenerate, df has the rank n at some points outside theindeterminacy locus of f . Hence, there exist indices α = ( α , . . . , α n ) ∈ ( N m ) n +1 with | α | = 0 , | α i | = 1 (1 ≤ i ≤ n ) such that W := det( D α i f j ; 0 ≤ i, j ≤ n )= f n +1 n det( D α i ( f j /f n ); 1 ≤ i ≤ n, ≤ j ≤ n − . (3.3)For each R o = { r o , ..., r on +1 } ⊂ { , ..., q } with rank { H i } i ∈ R o = ♯R o = n + 1, we set W R o ≡ det( D α i ( f, H r j ); 0 ≤ i, j ≤ n ) . Since rank { H r ov (1 ≤ v ≤ n + 1) } = n + 1, there exists a nonzero constant C R o such that W R o = C R o · W .Denote by R o the family of all subsets R o of { , ..., q } with rank { H i } i ∈ R o = ♯R o = n + 1 . Let z be a fixed point. We may suppose that | ( f, H i )( z ) | ≤ | ( f, H i )( z ) | ≤ · · · ≤ | ( f, H i q )( z ) | for a permutation ( i , . . . , i q ) of { , . . . , q } . We set R = { i , . . . , i N +1 } and choose R o ⊂ R such that R o ∈ R o and R o satisfies Lemma 3.1 iv) with respect to numbers (cid:26) || f ( z ) ||| ( f, H i )( z ) | (cid:27) qi =1 .We note that || f ( z ) || ≤ max i ∈ R | ( f, H i )( z ) | ≤ C | ( f, H j )( z ) | ( ∀ j R ) , IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 11 for some positive constant C , which may be chosen not depending on z and R . Then, weget || f ( z ) || ( P qi =1 ω i − n − | W ( z ) || ( f, H )( z ) | ω · · · | ( f, H q )( z ) | ω q ≤ C q − N − | W ( z ) ||| f ( z ) || n +1 Y i ∈ R (cid:18) || f ( z ) ||| ( f, H i )( z ) | (cid:19) ω i ≤ K | W R o ( z ) | · || f ( z ) || ( n +1) || f ( z ) || ( n +1) · Q i ∈ R o | ( f, H i )( z ) | = K | W ( z ) | Q i ∈ R o | ( f, H i )( z ) | , where K is a positive constant, which is chosen not depending on z , R and R o . By notingthat P qi =1 ω i − n − ω ( q − N + n − || f ( z ) || ˜ ω ( q − N + n − | W ( z ) || ( f, H )( z ) | ω · · · | ( f, H q )( z ) | ω q ≤ K X R o ∈R o S R o ( z ) , (3.4)where S R o = | W R o | Q i ∈ R o | ( f, H i ) | . Claim P qi =1 ω i ν ( f,H i ) ( z ) − ν W ( z ) ≤ ν [1]( f,D ) . Indeed, assume that z is a zero of some ( f, H i )( z ) and z is outside the indeterminancylocus I ( f ) of f . Since { H i } qi =1 is in N -subgeneral position, it implies that z is not zero ofmore than N functions ( f, H i ). Without loss of generality, we may assume that z is notzero of ( f, H i ) for each i > N . Put R = { , ..., N + 1 } . Choose R ⊂ R such that ♯R = rank { H i } i ∈ R = n + 1and R satisfies Lemma 3.1 iv) with respect to numbers (cid:8) e ν Qi ( f ) ( z ) (cid:9) qi =1 . Then we have X i ∈ R ω i ν ( f,H i ) ( z ) ≤ X i ∈ R ν ( f,H i ) ( z ) . By Lemma (3.2), this implies that ν W ( z ) = ν W R ( z ) ≥ X i ∈ R ν ( f,H i ) ( z ) − ν [1] Q s ∈ R ( f,H s ) ( z )= q X i =1 ν ( f,H i ) ( z ) − ν [1] Q qs =1 ( f,H s ) ( z ) . Hence, we have q X i =1 ω i ν ( f,H i ) ( z ) − ν W ( z ) ≤ min { , ν Q qs =1 ( f,H i ) ( z ) } = ν [1]( f,D ) . The claim is proved.If M = C m , then by integrating both sides of inequality (3.4) and using the lemma onlogarithmic derivative, we get || ˜ ω ( q − N + n − T f ( r ) + N W ( r ) − q X i =1 ω i N ( f,H i ) ( r ) = o ( T f ( r )) . , Here, the symbol “ || ′′ means the inequality holds for all a ∈ [1 , + ∞ ) outside a set of finiteLebesgue measure. Hence, || ( q − N + n − T f ( r ) ≤ ω q X i =1 ω i N ( f,H i ) ( r ) − N W ( r ) ! + o ( T f ( r )) . From Claim 3.5, we have have the following “second main theorem” : || ( q − N + n − T f ( r ) ≤ ω N [1]( f,D ) ( r ) + o ( T f ( r )) ≤ N − n + 1 n + 1 N [1]( f,D ) ( r ) + o ( T f ( r )) , (3.6)i.e., || (cid:16) qT f ( r ) − N [1]( f,D ) ( r ) (cid:17) ≤ (cid:18) N − n )2 N − n + 1 + n + 1 q (cid:19) qT f ( r ) + O (1) . This implies directly that δ [1] f ( D ) ≤ N − n )2 N − n + 1 + n + 1 q and we have the desired defect relation in this case.Now we consider the case where M = B m ( R ) ( R < + ∞ ). Without loss of generality,we assume that R = 1. We now suppose that δ [1] f ( D ) > − ( q − N + n − ωq + 2 nρq . Then, there exist constants η > u such that e ˜ u | ϕ | ≤ || f || qη , where ϕ is a holomorphic function with ν ϕ = ν [1]( f,D ) and1 − η > − ( q − N + n − ωq + 2 nρq , i.e., ( q − N + n − ω − qη > nρ. Put u = ˜ u + log | ϕ | , then u is a plurisubharmonic and e u ≤ || f || η . Let v ( z ) = log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( n Y i =0 z α i ) | W ( z ) || ( f, H ) ω ( z ) · · · ( f, H q ) ω q ( z ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + u, where z α i = z α i · · · z α im m for α i = ( α i , .., α im ) (0 ≤ i ≤ n ).Therefore, we have the following current inequality2 dd c [ v ] = [ ν W ] − q X j =1 ω i [ ν ( f,H j ) ] + 2 dd c [ u ] ≥ − [ ν [1]( f,D ) ] + [ ν [1]( f,D ) ] = 0 . This implies that v is a plurisubharmonic function on B m (1).On the other hand, by the assumption ρ Ω f + √− π ∂ ¯ ∂ log h ≥ Ric ω, IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 13 there exists a continuous plurisubharmonic function w
6≡ ∞ on B m (1) such that e w dV ≤ || f || ρ v m . Set t = 2 ρ ˜ ω ( q − N + n − − qη > λ ( z ) = ( n Y i =0 z α i ) W ( z ) H ω ( f )( z ) . . . H ω q ( f )( z ) . We see that tn < n · ρ ρn = 1 , and the function u ′ = ω + tv is plurisubharmonic on the K¨ahler manifold M . Choose aposition number p such that 0 < tn < p < . Then, we have e u ′ dV = e w + tv dV ≤ e tv || f || ρ v m = | λ | t e tu || f || ρ v m ≤ | λ | t || f || ρ + tqη v m = | λ | t || f || t ˜ ω ( q − N + n − v m Since P qj =1 ω j = ˜ ω j ( q − N + n −
1) + n + 1 , the above inequality implies that e u dV ≤ | λ | t || f || t ( P qj =1 ω j − n − v m . Integrating both sides of the above inequality over B m (1) , we have Z B m (1) e u dV ≤ Z B m (1) | λ | t || f || t ( P qj =1 ω j − n − v m = 2 m Z r m − (cid:18)Z S ( r ) (cid:0) | λ ||| f || ( P qj =1 ω j − n − (cid:1) t σ m (cid:19) dr ≤ m Z r m − Z S ( r ) X R o ∈R o (cid:12)(cid:12) ( n Y i =0 z α i ) KS R o (cid:12)(cid:12) t σ m ! dr, (3.7)where S R o = | W R o | Q i ∈ R o | ( f, H i ) | . (a) We first consider the case wherelim r → sup T f ( r, r )log 1 / (1 − r ) < ∞ . Choose p > tn < p <
1. Note that P ni =0 | α i | = n . By the lemma onlogarithmic derivative, there exists a positive constant K ′ such that, for every 0 < r 1] with R E dr − r < + ∞ . Choosing r ′ = r + 1 − reT f ( r, r ) , we get T f ( r ′ , r ) ≤ T f ( r, r ) . , Hence, the above inequality implies that X R o ∈R o Z S ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( n Y i =0 z α i ) KS R ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t σ m ≤ K ′′ (1 − r ) p (cid:18) log 11 − r (cid:19) p for all z outside E , where K ′′ is a some positive constant. By choosing K ′′ large enough,we may assume that the above inequality holds for all z ∈ B m (1). Then, the inequality(3.7) yields that Z B m (1) e u dV ≤ m Z r m − K ′′ (1 − r ) p (cid:18) log 11 − r (cid:19) p dr < + ∞ . This contradicts the results of S.T. Yau [19] and L. Karp [6]. Hence, we have δ [1] f ( D ) ≤ − ( q − N + n − ωq + 2 nρq ≤ − ( q − N + n − n + 1)(2 N − n + 1) q + 2 nρq = 2( N − n )2 N − n + 1 + n + 1 q + 2 nρq . Here, the second inequality comes from that 1˜ ω ≤ N − n + 1 n + 1 . The theorem is proved inthis case.(b) We now consider the remaining case wherelim r → sup T ( r, r )log 1 / (1 − r ) = ∞ . It is enough for us to prove the following theorem. Theorem 3.9. With the assumption of Theorem 1.1 and suppose that M = B (1) . Then,we have (cid:16) qT f ( r ) − N [1]( f,D ) ( r ) (cid:17) ≤ (cid:18) N − n )2 N − n + 1 + n + 1 q (cid:19) qT f ( r )+ K ′′ (cid:18) log + − r + log + T f ( r, r ) (cid:19) + O (1) , where K ′′ is a positive constant, for all < r < r < outside a set E ⊂ [0 , with R E dt − t < ∞ . Proof. Repeating the above argument, we have Z S ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( n Y i =0 z α i ) || f || ( P qi =1 ω i − n − | W || ( f, H ) ω · · · ( f, H q ) ω q | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ m ≤ K (cid:18) R m − R − r dT f ( r, r ) (cid:19) p IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 15 for every 0 < r < r < R < 1. Using the concativity of the logarithmic function, we have Z S ( r ) log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( n Y i =0 z α i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ m + ( q X i =1 ω i − n − Z S ( r ) log || f || σ m + Z S ( r ) log | W | σ m − q X j =1 ω j Z S ( r ) log | ( f, H j ) | σ m ≤ K (cid:18) log + R − r + log + T f ( r, r ) (cid:19) (3.10)for some positive constant K .Similarly, we see that q X j =1 ω j N ( H j ,f ) ( r ) − N W ( r ) ≤ N [1]( f,D ) ( r ) + O (1) . Combining these estimates and (2.7), by the Jensen formula, it follows that || ( q X j =1 ω j − n − T f ( r ) ≤ N [1]( f,D ) ( r ) + K (cid:18) log + − r + log + T f ( r, r ) (cid:19) + O (1) . Since P qj =1 ω j = ( q − N + n − ω + n + 1, the above inequality implies that( q − N + n − T f ( r ) ≤ ω N [1]( f,D ) ( r ) + K (cid:18) log + − r + log + T f ( r, r ) (cid:19) + O (1) . On the other hand, since 1˜ ω ≤ N − n + 1 n + 1 , we get( q − N + n − T f ( r ) ≤ N − n + 1 n + 1 N [1]( f,D ) ( r )+ K (cid:18) log + − r + log + T f ( r, r ) (cid:19) + O (1) , i.e., ( qT f ( r ) − N [1]( f,D ) ( r )) ≤ ( q − ( q − N + n − n + 1)2 N − n + 1 ) T f ( r )+ K ′′ (cid:18) log + − r + log + T f ( r, r ) (cid:19) + O (1)= (cid:18) N − n )2 N − n + 1 + n + 1 q (cid:19) qT f ( r )+ K ′′ (cid:18) log + − r + log + T f ( r, r ) (cid:19) + O (1) , where K ′ is a positive constant.The theorem is proved in this case. (cid:3) Remark: The assumption of Theorem 1.1 is the same as that of [17, Theorem 1.4]. Inorder to prove [17, Theorem 1.4], the authors gave the following claim. Claim P qi =1 ω i ν ( f,H i ) ( z ) − ν W ( z ) ≤ P qi =1 ω i ν [1]( f,H i ) ( z ) . , And then, they got the non-integrated defect relation as follows q X i =1 δ [1] f ( H i ) ≤ N − n + 1 + 2 ρ (2 N − n + 1) . However, this inequality is not sharp since the number t in their proof (corresponding tothe number t in the above proof) is not sharply estimated. In fact, from their proof, ifsuppose that q X i =1 δ [1] f ( H i ) > N − n + 1 + 2 nρ ˜ ω . Then, for each j ∈ { , . . . , q } , we may choose constants η j > u j such that e ˜ u j | ϕ j | ≤ || ˜ f || dη j , where ϕ j is a holomorphic function with ν ϕ j = ν [1]( f,H j ) and q − q X j =1 η j > N − n + 1 + 2 nρ ˜ ω . Setting t = 2 ρ ˜ ω ( q − N + n − − P qj =1 η j ) > t | α | = tn = 2 nρ ˜ ω ( q − N + n − − P qj =1 η j ) < . By repeating again the proof of [17, Theorem 1.4], we will get the contradiction. Hencewe have q X i =1 δ [1] f ( H i ) ≤ N − n + 1 + 2 nρ ˜ ω ≤ N − n + 1 + 2 n (2 N − n + 1) ρn + 1 . Then, [17, Theorem 4.1] may be improved to a slightly stronger one as follows. Theorem 3.12. With the assumption of Theorem 1.1, we have q X i =1 δ [1] f ( H i ) ≤ N − n + 1 + 2 n (2 N − n + 1) ρn + 1 . If the family of hyperplanes is assumed to be in general position, the above defectrelations can be given by q X i =1 δ [1] f ( H i ) ≤ n + 1 + 2 ρ n n + 1 . (3.13) 4. Proof of Theorem 1.2 In order to proof Theorem 1.2, we firstly prove the following theorem. IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 17 Theorem 4.1. Let f , f , . . . , f k be k differentiably nondegenerate meromorphic map-pings from the complete K¨ahler manifold B m ( R ) (0 < R ≤ + ∞ ) into P n ( C ) , whichsatisfy the condition ( C ρ ) . Let H , . . . , H q be q hyperplanes of P n ( C ) in N − subgeneralposition, where q is a positive integer. Assume that there exists a non zero holomorphicfunction h ∈ B ( p, l ; f , . . . , f k ) such that ν h ≥ λ k X u =1 ν [1]( f u ,D ) , where D is the hypersurface H + · · · + H q , p, l are non-negative integers, λ is a positivenumber. Then we have q ≤ N − n + 1 + p (2 N − n + 1) λ ( n + 1) + ρ (cid:18) kn + l λ (cid:19) . (4.2) Moreover, if we assume further that ν h ≥ λ P ku =1 P qi =1 ν [1]( f u ,H i ) then we have q ≤ N − n + 1 + pn (2 N − n + 1) λ ( n + 1) N + ρ (cid:18) kn + l nλN (cid:19) . (4.3)Denote by ˜ ω, ω i (1 ≤ i ≤ q ) the Nochka’s weights of the family { H i } qi =1 . By Claim 3.5,we see that ν [1]( f u ,D ) ≥ q X i =1 ω i ν ( f u ,H i ) − ν W , where W is defined as in (3.3). Moreover, if we have ν h ≥ λ k X u =1 q X i =1 ν [1]( f u ,H i ) then ν h ≥ nN q X i =1 ˜ ων [1]( f u ,H i ) ≥ nN q X i =1 ω i ν ( f u ,H i ) − ν W ! (by Claim 3.11). Therefore, in order to prove Theorem 4.1 we only need to proof thefollowing lemma. Lemma 4.4. Let f , f , . . . , f k and H , . . . , H q be as in Theorem 4.1. Assume that thereexists a non zero holomorphic function h ∈ B ( p, l ; f , . . . , f k ) such that ν h ≥ λ q X i =1 ω i ν ( f u ,H i ) − ν W ! . Then we have q ≤ N − n + 1 + p (2 N − n + 1) λ ( n + 1) + ρ (cid:18) kn + l λ (cid:19) . , Proof. If R = + ∞ , by the second main theorem we have( q − N + n − k X u =1 T f u ( r ) ≤ k X u =1 N − n + 1 n + 1 N [1]( f u ,D ) ( r ) + o ( k X u =1 T f u ( r )) ≤ N − n + 1 λ ( n + 1) N h ( r ) + o ( k X u =1 T f u ( r )) ≤ p (2 N − n + 1) λ ( n + 1) k X u =1 T f u ( r ) + o ( k X u =1 T f u ( r )) , for all r ∈ [1; + ∞ ) outside a Lebesgue set of finite measure. Letting r → + ∞ , we obtain q ≤ N − n + 1 + p (2 N − n + 1) λ ( n + 1) . Now, we consider the case where R < + ∞ . Without loss of generality we assume that R = 1. Suppose contrarily that q > N − n + 1 + p (2 N − n +1) λ ( n +1) + ρl λ . Then, there is apositive constant ǫ such that q > N − n + 1 + p (2 N − n + 1) λ ( n + 1) + ρ (cid:18) kn + l + ǫλ (cid:19) . Put l ′ = l + ǫ > f u has a reduced representation f u = ( f u : · · · : f un ) for each 1 ≤ u ≤ k .Since f u is differentiably nondegenerate, there exists an admissible set α u = ( α u , . . . , α un ) ∈ ( N m ) n +1 with | α u | = 0 and | α ui | = 1 (1 ≤ i ≤ n ) such that the general Wronskian W α u ( f u ) := det (cid:0) D α ui ( f uj ); 0 ≤ i, j ≤ n (cid:1) . By Lemma 3.2, we have ν h ≥ λ k X u =1 ν [1]( f u ,D ) ≥ λ k X u =1 q X i =1 ω i ν ( f u ,H i ) − ν W αu ( f u ) ! . Put ζ u ( z ) := (cid:12)(cid:12)(cid:12)(cid:12) z α u + ··· + α un W α u ( f u ) Q qi =1 | ( f, H i ) | ω i (cid:12)(cid:12)(cid:12)(cid:12) (1 ≤ u ≤ k ). Since h ∈ B ( p, l ; f , . . . , f k ), thereexists a function g ∈ S ( l ; f , . . . , f k ) and β = ( β , . . . , β m ) ∈ Z m + with | β | ≤ l such that Z S ( r ) (cid:12)(cid:12) z β g (cid:12)(cid:12) t ′ σ m = O R m − R − r k X u =1 T f u ( r, r ) ! l , (4.5)for every 0 ≤ l t ′ < l < | h | ≤ k Y u =1 k f u k ! p | g | . (4.6)Put t = ρ ˜ ω ( q − N + n − − pλ > q − N + n − − pλ ˜ ω > q − N + n − − p (2 N − n +1) λ ( n +1) )and φ := | ζ | · · · | ζ k | · | z β h | /λ . Then a = t log φ is a plurisubharmonic function on B m (1) IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 19 and (cid:18) kn + l ′ λ (cid:19) t ≤ (cid:18) kn + l ′ λ (cid:19) ρ ˜ ω ( q − N + n − − pλ ≤ (cid:18) kn + l ′ λ (cid:19) ρ (2 N − n + 1)( q − N + n − n + 1) − p (2 N − n +1) λ < . Therefore, we may choose a positive number p ′ such that 0 ≤ ( kn + l ′ λ ) t < p ′ < . Since f u satisfies the condition ( C ρ ), then there exists a continuous plurisubharmonicfunction ϕ u on B m (1) such that e ϕ u dV ≤ k f u k ρ v m . We see that ϕ = ϕ + · · · + ϕ k + a is a plurisubharmonic function on B m (1). We have e ϕ dV = e ϕ + ··· + ϕ k + t log φ dV ≤ e t log φ k Y u =1 k f u k ρ v m = | φ | t k Y u =1 k f u k ρ v m ≤ | z β g | t/λ k Y u =1 ( | ζ u | t · k f u k ρ + pt/λ ) v m = | z β g | t/λ k Y u =1 ( | ζ u | t · k f u k ˜ ω ( q − N + n − t ) v m . Setting x = l ′ /λkn + l ′ /λ , y = nkn + l ′ /λ , then we have x + ky = 1. Therefore, by integratingboth sides of the above inequality over B m (1) and applying H¨older inequality, we have Z B m (1) e u dV ≤ Z B m (1) k Y u =1 ( | ζ u | t · k f u k ˜ ω ( q − N + n − t ) | z β g | t/λ v m ≤ (cid:18)Z B m (1) | z β g | t/ ( λx ) v m (cid:19) x × k Y u =1 (cid:18)Z B m (1) ( | ζ u | t/y · k f u k ˜ ω ( q − N + n − t/y ) v m (cid:19) y ≤ (cid:18) m Z r m − (cid:18)Z S ( r ) | z β g | t/ ( λx ) σ m (cid:19) dr (cid:19) x × k Y u =1 (cid:18) m Z r m − (cid:18)Z S ( r ) (cid:0) | ζ u | · k f u k ( P qi =1 ω i − n − (cid:1) t/y σ m (cid:19) dr (cid:19) y . (4.7)(a) We now deal with the case wherelim r → sup P ku =1 T f u ( r, r )log 1 / (1 − r ) < ∞ . We see that l tλx ≤ l ′ tλx = (cid:0) kn + l ′ λ (cid:1) t < p ′ and n ty = (cid:0) kn + l ′ λ (cid:1) t < p ′ . Similarly as (3.7)and (3.8), there exists a positive constant K such that, for every 0 < r < r < r ′ < , we , have Z S ( r ) (cid:0) | ζ u | · k f u k ( P qi =1 ω i − n − (cid:1) t/y σ m ≤ K (cid:18) r ′ m − r ′ − r T f u ( r ′ , r ) (cid:19) p ′ (1 ≤ u ≤ k )and Z S ( r ) | z β g | t/ ( λx ) σ m ≤ K r ′ m − r ′ − r k X u =1 T f u ( r ′ , r ) ! p ′ . Choosing r ′ = r + 1 − re max ≤ u ≤ k T f u ( r, r ) , we have T f u ( r ′ , r ) ≤ T f u ( r, r ), for all r outside a subset E of (0 , 1] with R E − r dr < + ∞ . Hence, the above inequality impliesthat Z S ( r ) (cid:0) | w u | · k f u k ( P qi =1 ω i − n − (cid:1) t/y σ m ≤ K ′ (1 − r ) p ′ (cid:18) log 11 − r (cid:19) p ′ (1 ≤ u ≤ k )and Z S ( r ) | z β g | t/ ( λx ) σ m ≤ K ′ (1 − r ) p ′ (cid:18) log 11 − r (cid:19) p ′ for all r outside E , and for some positive constant K ′ . Then the inequality (4.7) yieldsthat Z B m (1) e u dV ≤ m Z r m − K ′ − r (cid:18) log 11 − r (cid:19) p ′ dr < + ∞ . This contradicts the results of S.T. Yau [19] and L. Karp [6].(b) We now deal with the remaining case wherelim r → sup P ku =1 T f u ( r, r )log 1 / (1 − r ) = ∞ . As above, we have Z S ( r ) | z β g | t/ ( λx ) σ m ≤ K − r k X u =1 T f u ( r, r ) ! p ′ for every r < r < . By the concativity of the logarithmic function, we have Z S ( r ) log | z β | t/ ( λx ) σ m + Z S ( r ) log | g | t/ ( λx ) σ m ≤ K ′′ log + − r + log + k X u =1 T f u ( r, r ) ! . This implies that Z S ( r ) log | g | σ m = O log + − r + log + k X u =1 T f u ( r, r ) ! IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 21 By Theorem 4.6, we have k X u =1 pT f u ( r, r ) + Z S ( r ) log | g | σ m ≥ N h ( r, r ) + S ( r ) ≥ λ k X u =1 N [1]( f,D ) ( r, r ) + S ( r ) ≥ λ k X u =1 ( q − N + n − n + 1)2 N − n + 1 T f u ( r, r ) + S ( r ) , where S ( r ) = O (log + 11 − r + log + P ku =1 T f u ( r , r )) for every r excluding a set E with R E dr − r < + ∞ . Letting r → 1, we get pλ > ( q − N + n − n + 1)2 N − n + 1 , i.e., q < N − n + 1 + p (2 N − n + 1) λ ( n + 1) . This is a contradiction.Hence, the supposition is false. The proposition is proved. (cid:3) Now consider three mappings f , . . . , f k ∈ D ( f, { H i } qi =1 , S qi =1 { z : ( f, H i )( z ) = 0 } . For each γ ∈ Γ, we define V uγ to bethe set of all ( c , . . . , c n ) ∈ C n +1 such that γ ⊂ { z : c f u ( z ) + · · · + c un f n ( z ) = 0 } (1 ≤ u ≤ k ) . It easy to see that V uγ is a proper vector subspace of C n +1 . Then S ku =1 S γ ∈ Γ V uγ is theunion of finite proper vector spaces of C n +1 , and then is nowhere density C n +1 . We set C := C n +1 \ k [ u =1 [ γ ∈ Γ V uγ , (4.8)then C is a density open subset of C n +1 , and hence there exists c = ( c , . . . , c n ) ∈ C .By changing the coordinates if necessary, without loss of generality, from here we alwaysassume that c = (1 , , . . . , ∈ C . Then we havedim { z : f u ( z ) = 0 } ∩ { z : q Y i =1 ( f, H i )( z ) = 0 } ≤ m − ≤ u ≤ k ) . Proof of Theorem 1.2. (a) Suppose that f ∧ . . . ∧ f k 0. Then there three indices0 ≤ i < . . . < i k ≤ n such that P := det f i · · · f ki ... · · · ... f i k · · · f ki k . , We have P = f · · · f k · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · f i f i f i f i · · · f ki f i ... ... · · · ... f ik f i f ik f i · · · f kik f i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = f · · · f k · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f i f i − f i f i · · · f ki f i − f i f i ... · · · ... f ik f i − f ik f i · · · f kik f i − f ik f i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Hence, if a point z S ku =1 { f u = 0 } is a zero of ( f, D ) then it will be a zero of P withmultiplicity at least k − 1. Therefore, we have ν P ≥ ( k − ν [1]( f,D ) = k − k k X u =1 ν [1]( f u ,D ) . It also is easy to see that P ∈ B (1 , f , . . . , f k ). Then, by Proposition 4.1 we have q ≤ N − n + 1 + k (2 N − n + 1)( k − n + 1) + knρ. This is a contradiction.Then f ∧ · · · ∧ f k ≡ 0. The assertion (a) is proved.(b) Using the same notation and repeating the same argument as in the above part, wehave ν P ≥ ( k − ν [1]( f,D ) = k − k k X u =1 q X i =1 ν [1]( f u ,H i ) . Then, by Proposition 4.1 we have q ≤ N − n + 1 + kn (2 N − n + 1)( k − N ( n + 1) + kn ρN . This is a contradiction.Then f ∧ · · · ∧ f k ≡ 0. The theorem is proved. (cid:3) Proof of Theorem 1.4 Since the case where M = C m have already proved by the author in [9], without loss ofgenerality, in this proof we only consider the case where M = B m (1).We now define: • F ijk = ( f k , H i )( f k , H j ) (0 ≤ k ≤ , ≤ i, j ≤ n + 2) , • V i = (( f , H i ) , ( f , H i ) , ( f , H i )) ∈ M m , • ν i : the divisor whose support is the closure of the set { z ; ν ( f u ,H i ) ( z ) ≥ ν ( f v ,H i ) ( z ) = ν ( f t ,H i ) ( z ) for a permutation ( u, v, t ) of (1 , , } . We write V i ∼ = V j if V i ∧ V j ≡ 0, otherwise we write V i = V j . For V i = V j , we write V i ∼ V j if there exist 1 ≤ u < v ≤ F iju = F ijv , otherwise we write V i V j . IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 23 The following lemma is an extension of [5, Proposition 3.5] to the case of K¨ahler mani-folds. Lemma 5.1. With the assumption of Theorem 1.4, let f , f , f be three meromor-phic mappings in F ( f, { H i } qi =1 , . Assume that there exist i ∈ { , . . . , q } , c ∈ C and α ∈ N m with | α | = 1 such that Φ αic . Then there exists a holomophic function g i ∈ B (1 , f , f , f ) such that ν g i ≥ ν [1]( f,H i ) + 2 q X j =1 j = i ν [1]( f,H j ) Proof. We haveΦ αic = F ic · F ic · F ic · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ci F ci F ci D α ( F ci ) D α ( F ci ) D α ( F ci ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ic F ic F ic F ic D α ( F ci ) F ic D α ( F ci ) F ic D α ( F ci ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = F ic (cid:0) D α ( F ci ) F ci − D α ( F ci ) F ci (cid:1) + F ic (cid:0) D α ( F ci ) F ci − D α ( F ci ) F ci (cid:1) + F ic (cid:0) D α ( F ci ) F ci − D α ( F ci ) F ci (cid:1) . (5.2)This implies that ( Y u =1 ( f u , H c )) · Φ αic = g i , where g i =( f , H i ) · ( f , H c ) · ( f , H c ) · (cid:18) D α ( F ci ) F ci − D α ( F ci ) F ci (cid:19) + ( f , H c ) · ( f , H i ) · ( f , H c ) · (cid:18) D α ( F ci ) F ci − D α ( F ci ) F ci (cid:19) + ( f , H c ) · ( f , H c ) · ( f , H i ) · (cid:18) D α ( F ci ) F ci − D α ( F ci ) F ci (cid:19) . Hence, we easily see that | g i | ≤ C · k f k · k f k · k f k · X u =1 (cid:12)(cid:12)(cid:12)(cid:12) D α ( F ciu ) F ciu (cid:12)(cid:12)(cid:12)(cid:12) , where C is a positive constant, and then g i ∈ B (1; 1; f , f , f ). It is clear that ν g i = ν Φ αic + X u =1 ν ( f u ,H c ) . (5.3) , It is clear that g i is holomorphic on a neighborhood of each point of S u =1 ( f u , H c ) − { } which is not contained in S qi =1 ( f, H i ) − { } . Hence, we see that all zeros and poles of g i are points contained in some analytic sets ( f, H s ) − { } (1 ≤ s ≤ q ). We note that theintersection of any two of these set has codimension at least two. It is enough for us toprove that (5.3) holds for each regular point z of the analytic set S qi =1 ( f, H i ) − { } . Wedistinguish the following cases: Case 1: z ∈ Supp ν ( f,Hj ) ( j = i ). We write Φ αic in the formΦ αic = F ic · F ic · F ic × (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) F ci − F ci (cid:1) (cid:0) F ci − F ci (cid:1) D α (cid:0) F ci − F ci (cid:1) D α (cid:0) F ci − F ci (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) . Then by the assumption that f , f , f coincide on Supp ν ( f,Hj ) , we have F ci = F ci = F ci on Supp ν ( f,Hj ) . The property of the general Wronskian implies that ν Φ αic ( z ) ≥ ν [1]( f,H i ) ( z ) + 2 q X i =1 j = i ν [1]( f,H i ) ( z ) . Case 2: z ∈ Supp ν ( f,H i ) . Subcase 2.1: We may assume that 2 ≤ ν ( f ,H i ) ( z ) ≤ ν ( f ,H i ) ( z ) ≤ ν ( f ,H i ) ( z ). We writeΦ αic = F ic (cid:20) F ic ( F ci − F ci ) F ic D α ( F ci − F ci ) − F ic ( F ci − F ci ) F ic D α ( F ci − F ci ) (cid:21) It is easy to see that F ic ( F ci − F ci ), F ic ( F ci − F ci ) are holomorphic on a neighborhood of z , and ν ∞ F ic D α ( F ci − F ci ) ( z ) ≤ , and ν ∞ F ic D α ( F ci − F ci ) ( z ) ≤ . Therefore, it implies that ν Φ αic ( z ) ≥ ν [1]( f,H i ) ( z ) + 2 q X i =1 j = i ν [1]( f,H j ) ( z ) . Subcase 2.2: We may assume that ν ( f ,H i ) ( z ) = ν ( f ,H i ) ( z ) = ν ( f ,H i ) ( z ) = 1. We choosea neighborhood U of z and a holomorphic function h without multiple zero on U suchthat ν h = ν ( f u ,H i ) (1 ≤ u ≤ 3) on U . Hence F icu = hG icu for non-vanishing holomorphicfunctions G icu on U . By the properties of Gronskian, we have Φ αic = h Φ( G ic , G ic , G ic ) on U . This implies that ν Φ αic ( z ) = ν h ( z ) = ν [1]( f,H i ) ( z ) + 2 q X i =1 j = i ν [1]( f,H j ) ( z ) . From the above three cases, the inequality (5.3) holds. The lemma is proved. (cid:3) Proof of theorem 1.4. Denote by P the set of all i ∈ { , . . . , q } satisfying there exist c ∈ C , α ∈ N m with | α | = 1 such that Φ αij IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 25 If ♯P ≥ 3, for instance we suppose that 1 , , ∈ P , then there exist three correspondingholomorphic functions g , g , g as in Lemma 5.1. We have g g g ∈ B (3 , f , f , f ) and ν g g g ≥ X u =1 q X i =1 ν [1]( f u ,H i ) − X u =1 3 X i =1 ν [1]( f u ,H i ) ≥ X u =1 q X i =1 ν [1]( f u ,H i ) . Then, by Theorem 4.1 we have q ≤ (2 N − n + 1) (cid:18) n N ( n + 1) (cid:19) + ρ (cid:18) n + 9 n N (cid:19) . This is a contradiction.Hence ♯P ≤ 2. We suppose that i P ∀ i = 1 , . . . , q − . Therefore, for all i ∈{ , . . . , q − } and α ∈ N m with | α | = 1 we haveΦ αic ≡ ∀ c ∈ C . By the density of C in C n +1 , the above identification holds for all c ∈ C n +1 \ { } .In particular, Φ αij ≡ i ∈ { , . . . , q − } and α ∈ N m . Then for 1 ≤ i < j ≤ q − F ij = F ij or F ij = F ij or F ij = F ij .(ii) F ij F ij , F ij F ij and F ij F ij are all constant. Claim For any two indices i, j if there exists two mappings of { f , f , f } , forinstance they are f , f such that F ij = F ij then F ij = F ij = F ij . Indeed, suppose contrarily that F ij = F ij = F ij . Denote by M the field of all mero-morphic functions on B m (1). Then two vectors (cid:18) ( f , H i )( f , H j ) , ( f , H i )( f , H j ) , ( f , H i )( f , H j ) (cid:19) and (cid:18) ( f , H j )( f , H j ) , ( f , H j )( f , H j ) , ( f , H j )( f , H j ) (cid:19) are linear independent on M . Since f ∧ f ∧ f ≡ 0, the vector (cid:18) ( f , H s )( f , H j ) , ( f , H s )( f , H j ) , ( f , H s )( f , H j ) (cid:19) belongs to the vector space spanned by two above vectors on M for all s . Since ( f ,H i )( f ,H j ) = ( f ,H i )( f ,H j ) and ( f ,H j )( f ,H j ) = ( f ,H j )( f ,H j ) , it yields that ( f ,H s )( f ,H j ) = ( f ,H s )( f ,H j ) for all 1 ≤ s ≤ q . This impliesthat f = f , which contradicts to the supposition. Hence, we must have F ij = F ij = F ij .The claim is proved.From the above claim we see that for any two indices 1 ≤ i, j ≤ q − f u , f v we must have F iju = F ijv or there exists a constant α = 1 with F iju = αF ijv .Now we suppose that, there exists F ij = βF ij ( i < j ) with β = 1. Since F ij = F ij on S s = i,j ( f, H i ) − { } , it follows that S s = i,j ( f, H s ) − { } = ∅ . Take an index t ∈{ , . . . , q − } \ { i, j } , then we must have F it = F it or F jt = F jt . For instance, wesuppose that F it = F it . Similarly as above, we have S s = i,t ( f, H s ) − { } = ∅ . Therefore , S s = i ( f, H s ) − { } = ∅ . This implies that δ [1] f ( H s ) = 1 for all s ∈ { , . . . , q } \ { i } . ByTheorem 4.1, we have q − ≤ N − n + 1 + ρ (2 N − n + 1) nN + 1 . This is a contradiction.Therefore, F ij = F ij = F ij for all 1 ≤ i < j ≤ q − 2. This implies that f = f = f .The supposition is false.Hence, we must have f = f of f = f or f = f . The Theorem is proved. (cid:3) Acknowledgements This research is funded by Vietnam National Foundation for Science and TechnologyDevelopment (NAFOSTED) under grant number 101.04-2018.01. References [1] W. Chen and Q. Han, A non-integrated hypersurface defect relation for meromorphic maps overcomplete kahler manifolds into projective algebraic varieties , Kodai Math. J. (2018), no. 2, 284–300.[2] S. J. Drouilhet, A unicity theorem for meromorphic mappings between algebraic varieties , Trans.Amer. J. Math. 265, (1981), 349–358.[3] H. Fujimoto, Non-integrated defect relation for meromorphic mappings from complete K¨ahler mani-folds into P N ( C ) × · · · × P N k ( C ), Japan. J. Math. (1985) 233–264.[4] H. Fujimoto, A unicity theorem for meromorphic maps of a complete Khler manifold into P N ( C ),Tohoko Math. J. (1986), 327–341.[5] H. Fujimoto, Uniqueness problem with truncated multiplicities in value distribution theory , NagoyaMath. J. Vol. (1998), 131–152.[6] L. Karp, Subharmonic functions on real and complex manifolds , Math. Z. (1982) 535–554.[7] H. Fujimoto, On the Gauss mapping from a complete minimal surface in R m , J. Math. Soc. Japan (1983) 279–288.[8] J. Noguchi, A note on entire pseudo-holomorphic curves and the proof of Cartan-Nochka’s theorem ,Kodai Math. J. (2005) 336–346[9] S. D. Quang, Finiteness problem for meromorphic mappings sharing n + 3 hyperplanes of P n ( C ),Ann. Polon. Math. (2014), no. 2, 195–215.[10] S. D. Quang, N. T. Q. Phuong and N. T. Nhung, Non-integrated defect relation for meromorphicmaps from a Kahler manifold intersecting hypersurfaces in subgeneral position of P n ( C ), J. Math.Anal. Appl. 452 (2017), 1434–1452.[11] S. D. Quang, Algebraic relation of two meromorphic mappings on a Khler manifold having the sameinverse images of hyperplanes , J. Math. Anal. Appl. (2020), no. 1, 123888, 17 pp.[12] S. D. Quang, Meromorphic mappings of a complete connected K¨ahler manifold into a projective spacesharing hyperplanes , (2019), arXiv:1909.01849 [math.CV].[13] S. D. Quang, N. T. Nhung and L. N. Quynh, Non-integrated defect relation for meromorphic mapsfrom Kahler manifolds with hypersurfaces of a projective variety in subgeneral position , to appear inTohoku Journal of Mathematics (2020).[14] N. T. Nhung and P. D. Thoan, On Degeneracy of three meromorphic mappings from complete K¨ahlermanifolds into projective spaces , Comput. Methods Funct. Theory, (2019) 353–382..[15] M. Ru and M. Sogome, Non-integrated defect relation for meromorphic mappings from completeK¨ahler manifolds into P n ( C ) intersecting hypersurfaces , Trans. Amer. Math. Soc. (2012), 1145–1162. IFFERENTIABLY NONDEGENERATE MEROMORPHIC MAPPINGS ON K ¨AHLER MANIFOLDS 27 [16] T. V. Tan and V. V. Truong, A non-integrated defect relation for meromorphic mappings fromcomplete K¨ahler manifolds into a projective variety intersecting hypersurfaces , Bull. Sci. Math. (2012) 111–126.[17] D. D. Thai and S. D. Quang, Non-integrated defect of meromorphic maps on Khler manifolds , Math.Z. (2019), no. 1-2, 211–229.[18] Q. Yan, Non-integrated defect relation and uniqueness theorem for meromorphic mappings from acomplete K¨ahler manifold into P n ( C ), J. Math. Anal. Appl. (2013), 567–581.[19] S. T. Yau, Some function-theoretic properties of complete Riemannian manifolds and their applica-tions to geometry , Indiana U. Math. J. (1976), 659–670. Department of Mathematics, Hanoi National University of Education, 136-Xuan Thuy,Cau Giay, Hanoi, Vienam. Thang Long Institute of Mathematics and Applied Sciences, Nghiem Xuan Yem, HoangMai, Ha Noi. E-mail address ::