Second main theorems for meromorphic mappings and moving hyperplanes with truncated counting functions
aa r X i v : . [ m a t h . C V ] S e p SECOND MAIN THEOREMS FOR MEROMORPHIC MAPPINGS ANDMOVING HYPERPLANES WITH TRUNCATED COUNTINGFUNCTIONS
SI DUC QUANG
Abstract.
In this article, we establish some new second main theorems for meromor-phic mappings of C m into P n ( C ) and moving hyperplanes with truncated counting func-tions. Our results are improvements of the previous second main theorems for movinghyperplanes with truncated (to level n ) counting functions. Introduction
The second main theorem for meromorphic mappings into projective spaces with movinghyperplanes was first given by W. Stoll, M. Ru [7] and M. Shirosaki in 1990’s [9, 10],where the counting functions are not truncated. In 2000, M. Ru [6] proved a secondmain theorem with trucated counting functions for nondegenerate mappings of C into P n ( C ) and moving hyperplanes. After that, this result was reproved for the case ofseveral complex variables by Thai-Quang [12]. For the case of degenerate meromorphicmappings, in [8], Ru and Wang gave a second main theorem for moving hyperplanes withcounting function truncated to level n . And then, the result of Ru-Wang was improvedby Thai-Quang [13] and Quang-An [5]. In 2016, the author have improved and extendedall those results to a better second main theorem. To state their results, we recall thefollowing.Let a , . . . , a q ( q ≥ n +1) be q meromorphic mappings of C m into the dual space P n ( C ) ∗ with reduced representations a i = ( a i : · · · : a in ) (1 ≤ i ≤ q ) . We say that a , . . . , a q arelocated in general position if det( a i k l ) ≤ i < i < · · · < i n ≤ q. Let M m be the field of all meromorphic functions on C m . Denote by R { a i } qi =1 ⊂ M m the smallestsubfield which contains C and all a ik a il with a il . Thoughout this paper, if without anynotification, the notation R is always stands for R { a i } qi =1 .In 2004, M. Ru and J. Wang proved the following. Theorem A [8, Theorem 1.3]
Let f : C → P n ( C ) be a holomorphic map. Let { a j } qj =1 bemoving hyperplanes of P n ( C ) in general position such that ( f, a j ) ≤ j ≤ q ) . If q ≥ n + 1 then (cid:12)(cid:12)(cid:12)(cid:12) qn (2 n + 1) T f ( r ) ≤ q X i =1 N [ n ]( f i ,a ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) . Date : Here, by the notation “ || P ” we mean the assertion P holds for all r ∈ [0 , ∞ ) outside aBorel subset E of the interval [0 , ∞ ) with R E dr < ∞ .In 2008, D. D. Thai and S. D. Quang improved the above result by increasing thecoefficent qn (2 n +1) in front of the characteristic function to q n +1 . In 2016, S D. Quang [3]improved these result to the following. Theorem B [3, Theorem 1.1]
Let f : C m → P n ( C ) be a meromorphic mapping. Let { a j } qj =1 ( q ≥ n − k + 2) be meromorphic mappings of C m into P n ( C ) ∗ in general positionsuch that ( f, a j ) ≤ j ≤ q ) , where rank R { a j } ( f ) = k + 1 . Then the followingassertion holds:(a) (cid:12)(cid:12)(cid:12)(cid:12) q n − k + 2 T f ( r ) ≤ q X i =1 N [ k ]( f i ,a ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) , (b) (cid:12)(cid:12)(cid:12)(cid:12) q − ( n + 2 k − n + k + 1 T f ( r ) ≤ q X i =1 N [ k ]( f i ,a ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) . The main purpose of the present paper is to establish a stronger second main theoremfor meromorphic mappings of C m into P n ( C ) and moving hyperplanes. Namely, we willprove the following theorem. Theorem 1.1.
Let f : C m → P n ( C ) be a meromorphic mapping. Let { a i } qi =1 ( q ≥ n − k + 2) be meromorphic mappings of C m into P n ( C ) ∗ in general position such that ( f, a i ) ≤ i ≤ q ) , where k + 1 = rank R{ a i } ( f ) . Then the following assertions hold:(a) || q − ( n − k ) n + 2 T f ( r ) ≤ q X i =1 N [ k ]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) , (b) || q − n − k ) k ( k + 2) T f ( r ) ≤ q X i =1 N [1]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) . Here by rank R{ a i } ( f ) we denote the rank of the set { f , f , . . . , f n } over the field R{ a i } ,where ( f : f : · · · : f n ) is a representation of the mapping f . Remark:
1) The assertion (a) is an improvement of Theorem B.2) It is easy to see that q − n − k ) k ( k +2) ≥ qn ( n +2) . Therefore, the assertion (b) immediatelyimplies the following corollary. Corollary 1.2.
With the assumptions of Theorem A, we have || qn ( n + 2) T f ( r ) ≤ q X i =1 N [1]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) . In order to prove the above result, beside developing the method used in [3, 8, 12],we also propose some new techniques. Firstly, we will rearrange the family hyperplanesin the increasing order of the values of the counting functions (of their inverse images).After that, we find the smallest number of the first hyperplanes in this order such that thesum of their counting functions exceed the characteristic functions. And then, we have to
ECOND MAIN THEOREMS FOR MEROMORPHIC MAPPINGS 3 compare the characteristic functions with this sum of counting functions with explicitlyestimating the truncation level. From that, we deduce the second main theorem.For the case where the number of moving hyperplanes is large enough, we will prove abetter second main theorem as follows.
Theorem 1.3.
With the assumptions of Theorem 1.1, we assume further more that q ≥ ( n − k )( k + 1) + n + 2 . Then we have || qk + 2 T f ( r ) ≤ q X i =1 N [ k ]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) . In this case, we may see that the coefficient in front of the characteristic functions areexactly the same as the case where the mappings are assumed to be non-degenerate.
Acknowledgements.
This work was done during a stay of the author at the VietnamInstitute for Advanced Study in Mathematics. He would like to thank the institute fortheir support. This research is funded by Vietnam National Foundation for Science andTechnology Development (NAFOSTED) under grant number 101.04-2018.01.2.
Basic notions and auxiliary results from Nevanlinna theory
Throughout this paper, we use the standart notation on Nevanlina theory from [3, 4] and[13]. For a meromorphic mapping f : C m → P n ( C ), we denote by T f ( r ) its characteristicfuntion. Let ϕ be a meromorphic funtion on C m . We denote by ν ϕ its divisor, N [ k ] ϕ ( r ) thecounting function with the trucation level k of its zeros divisor and m ( r, ϕ ) its proximityfunction. The lemma on logarithmic derivative in Nevanlinna theory is stated as follows. Lemma 2.1 ([11, Lemma 3.11]) . Let f be a nonzero meromorphic function on C m . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:18) r, D α ( f ) f (cid:19) = O (log + T f ( r )) ( α ∈ Z m + ) . The first main theorem states that T ϕ ( r ) = m ( r, ϕ ) + N ϕ ( r ) . We assume that thoughout this paper, the homogeneous coordinates of P n ( C ) is chosenso that for each given meromorphic mapping a = ( a : · · · : a n ) of C m into P n ( C ) ∗ then a
0. We set ˜ a i = a i a and ˜ a = (˜ a : ˜ a : · · · : ˜ a n ) . Supposing that f has a reduced representation f = ( f : · · · : f n ) , we put ( f, a ) := P ni =0 f i a i and ( f, ˜ a ) := P ni =0 f i ˜ a i . Let { a i } qi =1 be q meromorphic mappings of C m into P n ( C ) ∗ with reduced representations a i = ( a i : · · · : a in ) (1 ≤ i ≤ q ) . Definition 2.2.
The family { a i } qi =1 is said to be in general position if det( a i j l ; 0 ≤ j ≤ n, ≤ l ≤ n ) for any ≤ i ≤ · · · ≤ i n ≤ q . SI DUC QUANG
Definition 2.3.
A subset L of M (or M n +1 ) is said to be minimal over the field R if itis linearly dependent over R and each proper subset of L is linearly independent over R . Repeating the argument in [1, Proposition 4.5], we have the following proposition.
Proposition 2.4 (see [1, Proposition 4.5]) . Let Φ , . . . , Φ k be meromorphic functionson C m such that { Φ , . . . , Φ k } are linearly independent over C . Then there exists anadmissible set { α i = ( α i , . . . , α im ) } ki =0 ⊂ Z m + with | α i | = P mj =1 | α ij | ≤ k (0 ≤ i ≤ k ) satisfying the following two properties:(i) {D α i Φ , . . . , D α i Φ k } ki =0 is linearly independent over M , i.e., det ( D α i Φ j ) , (ii) det (cid:0) D α i ( h Φ j ) (cid:1) = h k +1 det (cid:0) D α i Φ j (cid:1) for any nonzero meromorphic function h on C m . Proof of Theorem 1.1 and Theorem 1.3
In order to prove Theorem 1.1 we need the following lemma, which is an improvementof Lemma 3.1 in [3].
Lemma 3.1.
Let f : C m → P n ( C ) be a meromorphic mapping. Let { a i } pi =1 be p mero-morphic mappings of C m into P n ( C ) ∗ in general position with rank { ( f, ˜ a i ); 1 ≤ i ≤ q } = rank R ( f ) , where R = R{ a i } pi =1 . Assume that there exists a partition { , . . . , q } = I ∪ I · · · ∪ I l satisfying: (i) { ( f, ˜ a i ) } i ∈ I is minimal over R , { ( f, ˜ a i ) } i ∈ I t is linearly independent over R (2 ≤ t ≤ l ) , (ii) For any ≤ t ≤ l, i ∈ I t , there exist meromorphic functions c i ∈ R \ { } such that X i ∈ I t c i ( f, ˜ a i ) ∈ (cid:18) t − [ j =1 [ i ∈ I j ( f, ˜ a i ) (cid:19) R . Then we have T f ( r ) ≤ t X i =1 q X j ∈ I i N [ n i ]( f,a j ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ p T a i ( r )) , where n = ♯I − and n t = ♯I t − for t = 2 , ..., l . Proof.
Let f = ( f : · · · : f n ) be a reduced representation of f . By changing thehomogeneous coordinate system of P n ( C ) if necessary, we may assume that f . Without loss of generality, we may assume that I = { , . . . ., k } and I t = { k t − + 1 , . . . , k t } (2 ≤ t ≤ l ) , where 1 = k < · · · < k l = q. Since { ( f, ˜ a i ) } i ∈ I is minimal over R , there exist c i ∈ R \ { } such that k X i =1 c i · ( f, ˜ a i ) = 0 . Define c i = 0 for all i > k . Then k l X i =1 c i · ( f, ˜ a i ) = 0 . ECOND MAIN THEOREMS FOR MEROMORPHIC MAPPINGS 5
Because { c i ( f, ˜ a i ) } k i = k +1 is linearly independent over R , Proposition 2.4 yields that thereexists an admissible set { α k +1) , . . . , α k } ⊂ Z m + ( | α i | ≤ k − k − n ) such thatthe matrix A = ( D α i ( c j ( f, ˜ a j )); k + 1 ≤ i, j ≤ k )has nonzero determinant.Now consider t ≥ . By the construction of the set I t , there exist meromorphic mappings c ti k t − + 1 ≤ i ≤ k t ) such that k t X i = k t − +1 c ti · ( f, ˜ a i ) ∈ (cid:18) t − [ j =1 [ i ∈ I t ( f, ˜ a i ) (cid:19) R . Therefore, there exist meromorphic mappings c ti ∈ R (1 ≤ i ≤ k t − ) such that k t X i =1 c ti · ( f, ˜ a i ) = 0 . Define c ti = 0 for all i > k t . Then k l X i =1 c ti · ( f, ˜ a i ) = 0 . Since { c ti ( f, ˜ a i ) } k t i = k t − +1 is R -linearly independent, by again Proposition 2.4 there existsan admissible set { α t ( k t − +1) , . . . , α tk t } ⊂ Z m + ( | α ti | ≤ k t − k t − − n t ) such that thematrix A t = ( D α ti ( c j ( f, ˜ a j )); k t − + 1 ≤ i, j ≤ k t )has nonzero determinant.Consider the following ( k l − × k l matrix T = (cid:20)(cid:18) D α ti ( c tj ( f, ˜ a j )); 1 ≤ t ≤ l, k t − + 1 ≤ i ≤ k t (cid:19) ; 1 ≤ j ≤ k l (cid:21) Denote by D i the subsquare matrix obtained by deleting the i -th column of the minormatrix T . Since the sum of each row of T is zero, we havedet D i = ( − i − det D = ( − i − l Y j =1 det A j . Since { a i } qi =1 is in general position, we havedet(˜ a ij , ≤ i ≤ n + 1 , ≤ j ≤ n ) . By solving the linear equation system ( f, ˜ a i ) = ˜ a i · f + . . . + ˜ a in · f n (1 ≤ i ≤ n + 1) , weobtain f v = n +1 X i =1 A vi ( f, ˜ a i ) ( A vi ∈ R ) for each 0 ≤ v ≤ n. (3.2)Put Ψ( z ) = P n +1 i =1 P nv =0 | A vi ( z ) | ( z ∈ C m ) . Then || f ( z ) || ≤ Ψ( z ) · max ≤ i ≤ n +1 (cid:0) | ( f, ˜ a i )( z ) | (cid:1) ≤ Ψ( z ) · max ≤ i ≤ q (cid:0) | ( f, ˜ a i )( z ) | (cid:1) ( z ∈ C m ) , SI DUC QUANG and Z S ( r ) log + Ψ( z ) η ≤ n +1 X i =1 n X v =0 T ( r, A vi ) + O (1) = O ( max ≤ i ≤ q T a i ( r )) + O (1) . Fix z ∈ C m \ S qj =1 (cid:18) Supp ( ν f, ˜ a j ) ) ∪ Supp ( ν ∞ ( f, ˜ a j ) ) (cid:19) . Take i (1 ≤ i ≤ q ) such that | ( f, ˜ a i )( z ) | = max ≤ j ≤ q | ( f, ˜ a j )( z ) | . Then log | det D ( z ) | . || f ( z ) || Q qj =1 | ( f, ˜ a j )( z ) | ≤ log + (cid:18) Ψ( z ) · (cid:18) | det D i ( z ) | Q qj =1 ,j = i | ( f, ˜ a j )( z ) | (cid:19)(cid:19) ≤ log + (cid:18) | det D i ( z ) | Q qj =1 ,j = i | ( f, ˜ a j )( z ) | (cid:19) + log + Ψ( z ) . Thus, for each z ∈ C m \ S qj =1 (cid:18) Supp ( ν f, ˜ a j ) ) ∪ Supp ( ν ∞ ( f, ˜ a j ) ) (cid:19) , we havelog | det D ( z ) | . || f ( z ) || Q qi =1 | ( f, ˜ a i )( z ) | ≤ q X i =1 log + (cid:18) | det D i ( z ) | Q qj =1 ,j = i | ( f, ˜ a j )( z ) | (cid:19) + log + Ψ( z ) . Hence log || f ( z ) || ≤ log Q qi =1 | ( f, ˜ a i )( z ) || det D ( z ) | + q X i =1 log + (cid:18) | det D i ( z ) | Q qj =1 ,j = i | ( f, ˜ a j )( z ) | (cid:19) + log + Ψ( z ) . (3.3)Integrating both sides of the above inequality and using Jensen’s formula and the lemmaon logarithmic derivative, we have || T f ( r ) ≤ N Q qi =1 ( f, ˜ a i ) ( r ) − N ( r, ν det D ) + q X i =1 m (cid:18) r, det D i Q qj =1 ,j = i ( f, ˜ a j ) (cid:19) + O ( max ≤ i ≤ q T a i ( r ))= N Q qi =1 ( f, ˜ a i ) ( r ) − N ( r, ν det D ) + O (log + T f ( r )) + O ( max ≤ i ≤ q − T a i ( r )) . (3.4) Claim || N Q qi =1 ( f, ˜ a i ) ( r ) − N ( r, ν det D ) ≤ P ls =1 P qi ∈ I s N [ n s ]( f,a i ) ( r ) + O (max ≤ i ≤ q T a i ( r )) . Indeed, fix z ∈ C m \ I ( f ), where I ( f ) = { f = · · · f n = 0 } . We call i the indexsatisfying ν f, ˜ a i ) ( z ) = min ≤ i ≤ n +1 ν f, ˜ a i ) ( z ) . For each i = i , i ∈ I s , we easily have ν D αsks − j ( c si ( f, ˜ a i )) ( z ) ≥ max { , ν f ˜ a i ) ( z ) − n s } − C (cid:0) ν ∞ c si ( z ) + ν a i ( z ) (cid:1) , where C is a fixed constant. ECOND MAIN THEOREMS FOR MEROMORPHIC MAPPINGS 7
Since each element of the matrix D i is of the form D α sks − j ( c si ( f, ˜ a i )) ( i = i ), oneestimates ν D ( z ) = ν D i ( z ) ≥ l X s =1 X i ∈ Is i = i (cid:0) max { , ν f ˜ a i ) ( z ) − n s } − ( k + 1) (cid:0) ν ∞ c si ( z ) + ν a i ( z ) (cid:1)(cid:1) . (3.6)We see that there exists v ∈ { , . . . , n } with f v ( z ) = 0. Then by (3.2), there exists i ∈ { , . . . , n + 1 } such that A v i ( z ) · ( f, ˜ a i )( z ) = 0. Thus ν f, ˜ a i ) ( z ) ≤ ν f, ˜ a i ) ( z ) ≤ ν ∞ A v i ( z ) ≤ X A vi ν ∞ A vi ( z ) . (3.7)Combining the inequalities (3.6) and (3.7), we have ν Q qi =1 ( f, ˜ a i ) ( z ) − ν det D ( z ) ≤ l X s =1 X i ∈ Is i = i (cid:0) min { ν f, ˜ a i ) ( z ) , n s } + ( k + 1) (cid:0) ν ∞ c si ( z ) + ν a i ( z ) (cid:1)(cid:1) + X A vi ν ∞ A vi ( z ) ≤ l X s =1 X i ∈ I s (cid:0) min { ν f, ˜ a i ) ( z ) , n s } + ( k + 1) (cid:0) ν ∞ c si ( z ) + ν a i ( z ) (cid:1)(cid:1) + X A vi ν ∞ A vi ( z ) . Integrating both sides of this inequality, we easily obtain || N Q qi =1 ( f, ˜ a i ) ( r ) − N ( r, ν det D ) ≤ l X s =1 X i ∈ I s N [ n s ]( f,a i ) ( r ) + O ( max ≤ i ≤ q T a i ( r )) . (3.8)The claim is proved.From the inequalities (3.4) and the claim, we get || T f ( r ) ≤ l X s =1 X i ∈ I s N [ n s ]( f,a i ) ( r ) + O (log + T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) . The lemma is proved. (cid:3)
Proof of Theorem 1.1.
We denote by I the set of all permutations of q -tuple (1 , . . . , q ). For each element I = ( i , . . . , i q ) ∈ I , we set N I = { r ∈ R + ; N [ k ]( f,a i ) ( r ) ≤ · · · ≤ N [ k ]( f,a iq ) ( r ) } ,M I = { r ∈ R + ; N [1]( f,a i ) ( r ) ≤ · · · ≤ N [1]( f,a iq ) ( r ) } . We now consider an element I = ( i , . . . , i q ) of I . We will construct subsets I t of the set A = { , . . . , n − k + 2 } as follows.We choose a subset I of A which is the minimal subset of A satisfying that { ( f, ˜ a i j ) } j ∈ I is minimal over R . If rank R { ( f, ˜ a i j ) } j ∈ I = k + 1 then we stop the process.Otherwise, set I ′ = { i ; ( f, ˜ a i ) ∈ (cid:0) { ( f, ˜ a i j ) } j ∈ I (cid:1) } , A = A \ ( I ∪ I ′ ) and see that ♯I ∪ I ′ ≤ n + 1. We consider the following two cases: SI DUC QUANG • Case 1. Suppose that ♯A ≥ n + 1. Since { ˜ a i j } j ∈ A is in general position, we have (cid:0) ( f, ˜ a i j ); j ∈ A (cid:1) R = ( f , . . . , f n ) R ⊃ (cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R . • Case 2. Suppose that ♯A < n + 1. Then we have the following:dim R (cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R ≥ k + 1 − ( n + 1 − ♯I ∪ I ′ ) = k − n + ♯I ∪ I ′ , dim R (cid:0) ( f, ˜ a i j ); j ∈ A (cid:1) R ≥ k + 1 − ( n + 1 − ♯A ) = k − n + ♯A . We note that ♯I ∪ I ′ + ♯A = 2 n − k + 2. Hence the above inequalities imply thatdim R (cid:18)(cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R ∩ (cid:0) ( f, ˜ a i j ); j ∈ A (cid:1) R (cid:19) ≥ dim R (cid:0) ( f, ˜ a i j ); j ∈ I ∪ I ′ (cid:1) R + dim R (cid:0) ( f, ˜ a i j ); j ∈ A (cid:1) R − ( k + 1)= k − n + ♯I ∪ I ′ + k − n + ♯A − ( k + 1) = 1 . Therefore, from the above two cases, we see that (cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R ∩ (cid:0) ( f, ˜ a i j ); j ∈ A (cid:1) R = { } . Therefore, we may chose a subset I ⊂ A which is the minimal subset of A satisfyingthat there exist nonzero meromorphic functions c i ∈ R ( i ∈ I ), X i ∈ I c i ( f, ˜ a i ) ∈ (cid:18) [ i ∈ I ( f, ˜ a i ) (cid:19) R . We see that ♯I ≥
2. By the minimality of the set I , the family { ( f, ˜ a i j ) } j ∈ I is linearlyindependent over R , and hence ♯I ≤ k + 1 and ♯ ( I ∪ I ) ≤ min { n − k + 2 , n + k + 1 } . Moreover, we will show thatdim (cid:18)(cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R ∩ (cid:0) ( f, ˜ a i j ); j ∈ A (cid:1) R (cid:19) = 1 . Indeed, suppose contrarily there exist two linearly independent vectors x, y ∈ (cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R ∩ (cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R , with x = X i ∈ I x i ( f, ˜ a i ) ∈ (cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R ,y = X i ∈ I y i ( f, ˜ a i ) ∈ (cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R , where x i , y i ∈ R . By the minimality of the se I , all functions x i , y i are not zero. Therefore,fixing i ∈ I , we have X i ∈ I i = i ( y x i − x y i )( f, ˜ a i ) ∈ (cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R . Since x, y are linearly independent, the left hand side is not zero. This contradics theminimality of the set I . Hencedim (cid:18)(cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R ∩ (cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R (cid:19) = 1 . ECOND MAIN THEOREMS FOR MEROMORPHIC MAPPINGS 9
On the other hand, we will see that ♯I ∪ I ≤ n + 2. If rank R { ( f, ˜ a i j ) } j ∈ I ∪ I = k + 1then we stop the process.Otherwise, by repeating the above argument, we have a subset I ′ = { i ; ( f, ˜ a i ) ∈ (cid:0) { ( f, ˜ a i j ) } j ∈ I ∪ I (cid:1) } , a subset I of A = A \ ( I ∪ I ∪ I ′ ), which satisfy the following: • there exist nonzero meromorphic functions c i ∈ R ( i ∈ I ) so that X i ∈ I c i ( f, ˜ a i ) ∈ (cid:18) [ i ∈ I ∪ I ( f, ˜ a i ) (cid:19) R , • { ( f, ˜ a i j ) } j ∈ I is linearly independent over R , • ≤ ♯I ≤ k + 1 and ♯ ( I ∪ · · · ∪ I ) ≤ min { n − k + 2 , n + k + 1 } , • dim (cid:18)(cid:0) ( f, ˜ a i j ); j ∈ I ∪ I (cid:1) R ∩ (cid:0) ( f, ˜ a i j ); j ∈ I (cid:1) R (cid:19) = 1 . Continuing this process, we get a sequence of subsets I , . . . , I l , which satisfy:(1) { ( f, ˜ a i j ) } j ∈ I is minimal over R , ♯I t ≥ { ( f, ˜ a i j ) } j ∈ I t is linearly independentover R (2 ≤ t ≤ l ) , (2) for any 2 ≤ t ≤ l, j ∈ I t , there exist meromorphic functions c j ∈ R \ { } suchthat X j ∈ I t c j ( f, ˜ a i j ) ∈ (cid:18) t − [ s =1 [ j ∈ I s ( f, ˜ a i j ) (cid:19) R , and dim (cid:18)(cid:0) ( f, ˜ a i j ); j ∈ I ∪ · · · ∪ I t − (cid:1) R ∩ (cid:0) ( f, ˜ a i j ); j ∈ I t (cid:1) R (cid:19) = 1 , (3) rank R { ( f, ˜ a i j ) } j ∈ I ∪···∪ I l = k + 1.If ♯I = 2 we will remove one element from I and combine the remaining element with I to become a new set I . Therefore, we will get a sequence I , ..., I l which satisfy the abovethree properties and ♯I ≥ , ♯I t ≥ ≤ t ≤ l ). We set n = ♯I − , n s = ♯I s − ≤ s ≤ l ) , n = max ≤ s ≤ l n s , J = I ∪ · · · ∪ I l and d + 2 = ♯J . One estimates( n + 2) + ( n + 1) + · · · + ( n l + 1) = d + 2 , ( n + 1) + n + · · · + n l = k + 1 . Since the rank of the set of any n + 1 functions ( f, ˜ a i ) ′ s is equal to k + 1, we have( n + 1) − ♯ ( I ∪ · · · ∪ I l − ) ≥ ( k + 1) − rank { ( f, ˜ a i ); i ∈ I ∪ · · · ∪ I l − } , i . e ., ( n + 1) − ( n + 2) − ( n + 1) − · · · − ( n l − + 1) ≥ ( k + 1) − ( n + 1) − n − · · · − n l − . This implies that d + 2 ≤ n + 2 . On the other hand, we see that k + 1 + l = d + 2, and hence n s = k − l X i =1 i = s n i ≤ k − ( l − ≤ k ( k + 2) k + l + 1 = k ( k + 2) d + 2 . Thus n ≤ k ( k +2) d +2 . Now the family of subsets I , . . . , I l satisfies the assumptions of the Lemma 3.1. There-fore, we have || T f ( r ) ≤ l X s =1 X j ∈ I s N [ n s ]( f,a j ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) . (3.9)(a) For all r ∈ N I (may be outside a finite Borel measure subset of R + ), from (3.9) wehave || T f ( r ) ≤ X j ∈ J N [ k ]( f,a j ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) ≤ ♯Jq − (2 n − k + 2) + ♯J (cid:18)X j ∈ J N [ k ]( f,a ij ) ( r ) + q X j =2 n − k +3 N [ k ]( f,a ij ) ( r ) (cid:19) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) . Since ♯J = d + 2 ≤ n + 2, the above inequality implies that || T f ( r ) ≤ n + 2 q − ( n − k ) q X i =1 N [ k ]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) , r ∈ N I . (3.10)We see that S I ∈I N I = R + and the inequality (3.10) holds for every r ∈ N I , I ∈ I .This yields that T f ( r ) ≤ n + 2 q − ( n − k ) q X i =1 N [ k ]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r ))for all r outside a finite Borel measure subset of R + . Thus || q − ( n − k ) n + 2 T f ( r ) ≤ q X i =1 N [ k ]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) . The assertion (a) is proved.
ECOND MAIN THEOREMS FOR MEROMORPHIC MAPPINGS 11 (b) We repeat the same argument as in the proof of the assertion (a). For all r ∈ M I (may be outside a finite Borel measure subset of R + ) we have || T f ( r ) ≤ l X s =1 X j ∈ I s N [ n s ]( f,a j ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) ≤ X j ∈ J n N [1]( f,a j ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) ≤ n · d + 2 q − (2 n − k + 2) + d + 2 (cid:18)X j ∈ J N [1]( f,a ij ) ( r ) + q X j =2 n − k +3 N [1]( f,a ij ) ( r ) (cid:19) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) ≤ k ( k + 2) q − (2 n − k + 2) + d + 2 q X i =1 N [1]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) ≤ k ( k + 2) q − n − k ) q X i =1 N [1]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) . Repeating again the argument in the proof of assertion (a), we see that the above in-equality holds for all r ∈ R + outside a finite Borel measure set. Then the assertion (b) isproved. (cid:3) Proof of Theorem 1.3.
We denote by I the set of all permutations of q -tuple (1 , . . . , q ).For each element I = ( i , . . . , i q ) ∈ I , we set N I = { r ∈ R + ; N [ k ]( f,a i ) ( r ) ≤ · · · ≤ N [ k ]( f,a iq ) ( r ) } . We now consider an element I of I , for instance it is I = (1 , ..., q ). Then there is amaximal linearly independent subfamily of the set { ( f, ˜ a i ); 1 ≤ i ≤ n + 1 } which is ofexactly k + 1 elements and contains ( f, ˜ a ). We assume that they are { ( f, ˜ a i j ); 1 = i < · · · < i k +1 ≤ n + 1 } . For each 1 ≤ j ≤ k + 1, we set J = { i , ..., i k +1 } V j = (cid:26) i ∈ { , . . . , q } ; ( f, ˜ a j ) ∈ (cid:18) ( f ˜ a i s ); 1 ≤ s ≤ k + 1 , s = j (cid:19) R (cid:27) . Since the space (cid:18) ( f ˜ a i s ); 1 ≤ s ≤ k + 1 , s = j (cid:19) R is of dimension k , the set V j has at most n elements. Hence ♯ k +1 [ j =1 V j = ♯ k +1 [ j =1 ( V j \ J ) + ( k + 1) ≤ ( n − k )( k + 1) + ( k + 1) = ( n − k + 1)( k + 1) . Therefore, there exists an index i ≤ ( n − k + 1)( k + 1) + 1 such that i S k +1 j =1 V j . Thisyields that the set { ( f, ˜ a i j ); 0 ≤ j ≤ k + 1 } is minimal over R . Then by Lemma 3.1, for all r ∈ N I we have || T ( r, f ) ≤ k +1 X j =0 N [ k ]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) ≤ N [ k ]( f,a ) ( r ) + n +1 X i = n − k +2 N [ k ]( f,a i ) ( r ) + N [ k ]( f,a ( n − k +1)( k +1)+1 ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) ≤ n − k + 1 n − k +1 X i =1 N [ k ]( f,a i ) ( r ) + ( n − k +1)( k +1) X i = n − k +2 N [ k ]( f,a i ) ( r ) + ( n − k +1)( k +2) X i =( n − k +1)( k +1)+1 N [ k ]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r ))= 1 n − k + 1 ( n − k +1)( k +2) X i =1 N [ k ]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) ≤ n − k + 1 · ( n − k + 1)( k + 2) q q X i =1 N [ k ]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r ))= k + 2 q q X i =1 N [ k ]( f,a i ) ( r ) + o ( T f ( r )) + O ( max ≤ i ≤ q T a i ( r )) . Repeating again the argument in the proof of Theorem 1.1, we see that the above inequal-ity holds for all r ∈ R + outside a finite Borel measure set. Hence, the theorem is proved. (cid:3) References [1] H. Fujimoto,
Non-integrated defect relation for meromorphic maps of complete K¨ahler manifolds into P N ( C ) × . . . × P N k ( C ) , Japanese J. Math. (1985), 233-264.[2] J. Noguchi and T. Ochiai, Introduction to Geometric Function Theory in Several Complex Variables ,Trans. Math. Monogr. 80, Amer. Math. Soc., Providence, Rhode Island, 1990.[3] S. D. Quang,
Second main theorems for meromorphic mappings intersecting moving hyperplanes withtruncated counting functions and unicity problem , Abh. Math. Semin. Univ. Hambg. (2016) 1–18.[4] S. D. Quang, Second main theorems with weighted counting functions and algebraic dependence ofmeromorphic mappings , Proc. Amer. Soc. Math. (2016), 4329–4340.[5] S. D. Quang and D. P. An,
Unicity of meromorphic mappings sharing few moving hyperplanes ,Vietnam Math. J. (2013), 383-398.[6] M. Ru, A uniqueness theorem with moving targets without counting multiplicity , Proc. Amer. Math.Soc. (2001), 2701-2707.[7] M. Ru and W. Stoll,
The second main theorem for moving targets , Journal of Geom. Anal. , No. 2(1991) 99–138.[8] M. Ru and J. T-Y. Wang, Truncated second main theorem with moving targets , Trans. Amer. Math.Soc. (2004), 557-571.[9] M. Shirosaki,
Another proof of the defect relation for moving target , Tohoku Math. J., (1991),355–360.[10] M. Shirosaki, On defect relations of moving hyperplanes , Nogoya Math. J. (1990), 103–112.[11] B. Shiffman,
Introduction to the Carlson - Griffiths equidistribution theory , Lecture Notes in Math. (1983), 44-89.
ECOND MAIN THEOREMS FOR MEROMORPHIC MAPPINGS 13 [12] D. D. Thai and S. D. Quang,
Uniqueness problem with truncated multiplicities of meromorphicmappings in several complex variables for moving targets , Internat. J. Math., 16 (2005), 903-939.[13] D. D. Thai and S. D. Quang,
Second main theorem with truncated counting function in severalcomplex variables for moving targets,
Forum Mathematicum (2008), 145-179.[14] K. Yamanoi, The second main theorem for small functions and related problems , Acta Math. (2004), 225-294.
Si Duc Quang
Department of Mathematics, Hanoi National University of Education,136-Xuan Thuy, Cau Giay, Hanoi, Vietnam.