A finiteness theorem for meromorphic mappings sharing few hyperplanes

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  • D. Q. SIKODAI MATH. J.35 (2012), 463484

    A FINITENESS THEOREM FOR MEROMORPHIC MAPPINGS

    SHARING FEW HYPERPLANES

    Duc Quang Si

    Abstract

    In this article, we prove a finiteness theorem for meromorphic mappings of Cm into

    PnC sharing 2n 2 hyperplanes without counting multiplicity.

    1. Introduction

    In 1926, R. Nevanlinna showed that two distinct nonconstant meromorphicfunctions f and g on the complex plane C cannot have the same inverse imagesfor five distinct values, and that g is a special type of linear fractional trans-formation of f if they have the same inverse images counted with multiplicitiesfor four distinct values [9].

    In 1975, H. Fujimoto [5] generalized Nevanlinnas results to the case ofmeromorphic mappings of Cm into PnC. He considered two distinct meromor-phic maps f and g of Cm into PnC satisfying the condition that n f ;Hj ng;Hjfor q hyperplanes H1;H2; . . . ;Hq of P

    nC in general position, where n f ;Hjmeans the map of Cm into Z whose value n f ;Hja a A C

    m is the intersectionmultiplicity of the images of f and Hj at f a. He proved the following.

    Theorem A [5]. Let Hi, 1a ia 3n 2 be 3n 2 hyperplanes in PnClocated in general position, and let f and g be two nonconstant meromorphicmappings of Cm into PnC with f CmUHi and gCmUHi such that n f ;Hi ng;Hi for 1a ia 3n 2, where n f ;Hi and ng;Hi denote the pull-back of thedivisors Hi on PnC by f and g, respectively. Assume that either f or g islinearly non-degenerate over C, i.e., the image does not included in any hyperplanein PnC. Then f 1 g.

    Later on, the finiteness problem of meromorphic mappings sharing hyper-planes without counting multiplicities has been studied very intensively by manyauthors. Here we formulate some recent results on this problem.

    463

    2010 Mathematics Subject Classification. Primary 32H30, 32A22; Secondary 30D35.

    Key words and phrases. truncated multiplicity, meromorphic mapping, unicity, finiteness.

    Received September 30, 2011.

  • Take a meromorphic mapping f of Cm into PnC which is linearly non-degenerate over C, a positive integer d and q hyperplanes H1; . . . ;Hq of P

    nC ingeneral position with

    dim f 1Hi VHjam 2 1a i < ja qand consider the set F f ; fHigqi1; d of all linearly nondegenerate over Cmeromorphic maps g : Cm ! PnC satisfying the following two conditions:

    (a) minn f ;Hj; d minng;Hj; d 1a ja q;(b) f z gz on 6q

    j1 f1Hj.

    If d 1, we will say that f and g share q hyperplanes fHjgqj1 without countingmultiplicity.

    Denote byaS the cardinality of the set S. In 1999, H. Fujimoto proved afiniteness theorem for meromorphic mappings as follows.

    Theorem B. If q 3n 1 then aF f ; fHigqi1; 2a 2:

    Some recent years, Z. Chen - Q. Yang [2] and S. D. Quang [11] consideredthe case where q 2n 3 and they gave some uniqueness theorems for suchmeromorphic mappings. For the case where q 2n 2, in [11] S. D. Quangproved that

    Theorem C. Let f be a linearly nondegenerate meromorphic mapping ofCm into PnC and let H1; . . . ;H2n2 be 2n 2 hyperplanes of PnC in generalposition with

    dim f 1Hi VHjam 2 1a i < ja 2n 2:Let g be a linearly nondegenerate meromorphic mapping of Cm into PnCsatisfying:

    (i) minfn f ;Hj;an; 1g minfng;Hj;an; 1g and

    minfn f ;Hj;bn; 1g minfng;Hj;bn; 1g 1a ja 2n 2;

    (ii) f z gz on 62n2j1 f

    1Hj.If nb 2 then f 1 g.

    However, in Theorem C, the condition (i) means that the multiplicities ofthe zeros of the functions f ;Hj and g;Hj are considered to level n. Thereforethe following question arises naturally: Are there any finiteness theorems formeromorphic mappings sharing 2n 2 hyperplanes without counting multiplicity?It seems to us that the techniques used in the proofs of the above mentionedresults are not enough to apply to this case.

    Our main purpose in this paper is to give a positive answer for the abovequestion. To do so, we will introduce some new techniques.

    We would also like to note that in the definition of the familyF f ; fHigqi1; d, the meromorphic mapping g is assumed to be linearly non-

    464 duc quang si

  • degenerate. In this paper we will show that if qb 2n 2 then each map gsatisfying the conditions (a) and (b) is linearly nondegenerate.

    Now let f be a linearly nondegenerate meromorphic mapping of Cm intoPnC and let H1; . . . ;Hq be q hyperplanes of PnC in general position with

    dim f 1Hi VHjam 2 1a i < ja q:Let d be an integer. We consider the set G f ; fHigqi1; d of all meromorphicmaps g : Cm ! PnC satisfying the conditions:

    (a) minn f ;Hj; d minng;Hj; d 1a ja q,(b) f z gz on 6q

    j1 f1Hj.

    Our main result is stated as follows.

    Theorem 1.1. Let f be a linearly nondegenerate meromorphic mapping ofCm into PnC. Let H1; . . . ;H2n2 be 2n 2 hyperplanes of PnC in generalposition with

    dim f 1Hi VHjam 2 1a i < ja 2n 2:If nb 2 then aG f ; fHig2n2i1 ; 1a 2.

    In the section 4, we will consider meromorphic mappings with three familiesof hyperplanes in PnC and we will give a finiteness theorem for such maps.

    Throughout this paper, we always assume that nb 2.

    Acknowledgements. This work was done during a stay of the author atMathematisches Forschungsinstitut Oberwolfach. He wishes to express hisgratitude to this institute. The author would also like to thank ProfessorsJunjiro Noguchi and Do Duc Thai for their valuable suggestions concerning thismaterial. This work was supported in part by a NAFOSTED grant of Vietnam.

    2. Basic notions in Nevanlinna theory

    2.1. We set kzk jz1j2 jzmj21=2 for z z1; . . . ; zm A Cm and de-fine

    Br : fz A Cm : kzk < rg; Sr : fz A Cm : kzk rg 0 < r < y:Define

    sz : dd ckzk2m1 and hz : d c logkzk2 ^ dd c logkzk2m1 on Cmnf0g:

    2.2. Let F be a nonzero holomorphic function on a domain W in Cm. Fora set a a1; . . . ; am of nonnegative integers, we set jaj a1 am and

    DaF qjajF

    qa1z1 qamzm. We define the map nF : W ! Z by

    nF z : maxfl : DaFz 0 for all a with jaj < lg z A W:

    465meromorphic mappings sharing few hyperplanes

  • We mean by a divisor on a domain W in Cm a map n : W ! Z such that, foreach a A W, there are nonzero holomorphic functions F and G on a connectedneighborhood U HW of a such that nz nF z nGz for each z A U outsidean analytic set of dimensionam 2. Two divisors are regarded as the same ifthey are identical outside an analytic set of dimensionam 2. For a divisor non W we set jnj : fz : nz0 0g, which is a purely m 1-dimensional analyticsubset of W or empty set.

    Take a nonzero meromorphic function j on a domain W in Cm. For eacha A W, we choose nonzero holomorphic functions F and G on a neighborhood

    U HW such that j FG

    on U and dimF 10VG10am 2, and we define

    the divisors nj, nyj by nj : nF , nyj : nG, which are independent of choices of F

    and G and so globally well-defined on W.

    2.3. For a divisor n on Cm and for a positive integer M or M y, wedefine the counting function of n by

    nMz minfM; nzg;

    nt Bt nzs if mb 2;Pjzjat nz if m 1:

    (

    Nr; n r1

    ntt2m1

    dt 1 < r < y:

    For a meromorphic function j on Cm, we set Njr Nr; nj andN

    Mj r Nr; nMj . We will omit the character M if M y.

    2.4. Let f : Cm ! PnC be a meromorphic mapping. For arbitrarilyfixed homogeneous coordinates w0 : : wn on PnC, we take a reducedrepresentation f f0 : : fn, which means that each fi is a holomorphicfunction on Cm and f z f0z : : fnz outside the analytic set I f f f0 fn 0g of codimensionb 2. Set k f k j f0j2 j fnj21=2.

    The characteristic function of f is defined by

    Tf r Sr

    logk f khS1

    logk f kh:

    Let H be a hyperplane in PnC given by H fa0o0 anon 0g,where a : a0; . . . ; an0 0; . . . ; 0. We set f ;H

    Pni0 ai fi. It is easy to see

    that the divisor n f ;H does not depend on the choices of reduced representationof f and coecients a0; . . . ; an. Moreover, we define the proximity function of fwith respect to H by

    mf ;Hr Sr

    logk f k kHkj f ;Hj h

    S1

    logk f k kHkj f ;Hj h;

    where kHk Pn

    i0 jaij21=2.

    466 duc quang si

  • 2.5. Let j be a nonzero meromorphic function on Cm, which is occationallyregarded as a meromorphic map into P1C. The proximity function of j isdefined by

    mr; j :Sr

    logjjjh;

    where log t maxf0; log tg for t > 0. The Nevanlinna characteristic functionof j is defined by

    Tr; j N1=jr mr; j:There is a fact that

    Tjr Tr; j O1:The meromorphic function j is said to be small with respect to f i k Tr; j oTf r.

    Here as usual, by the notation k P we mean the assertion P holds for allr A 0;y excluding a Borel subset E of the interval 0;y with

    Edr < y.

    The following plays essential roles in Nevanlinna theory (see [10]).

    Theorem 2.6 (First main theorem). Let f : Cm ! PnC be a meromorphicmapping and let H be a hyperplane in PnC such that f CmQH. Then

    N f ;Hr mf ;Hr Tf r r > 1:

    Theorem 2.7 (Second main theorem). Let f : Cm ! PnC be a linearlynondegenerate meromorphic mapping and H1; . . . ;Hq be hyperplanes of P

    nC ingeneral position. Then

    k q n 1Tf raXqi1

    Nn f ;Hir oTf r:

    Lemma 2.8 (Lemma on logarithmic derivative). Let f be a nonzero mero-morphic function on Cm. Then m r;Da f f

    Olog Tf r a A Zm:

    3. Proof of Theorem 1.1

    Lemma 3.1. Let f be a linearly nondegenerate meromorphic mapping ofCm into PnC. Suppose that g A G f ; fHigqi1; 1 with qb 2n 2. Then g islinearly nondegenerate.

    Proof. Suppose that there exists a hyperplane H satisfying gCmHH.We assume that f has a reduce representation f f0 : : fn and H

    467meromorphic mappings sharing few hyperplanes

  • fo0 : : on jPn

    i0 aioi 0g. We consider function f ;H Pn

    i0 ai fi.Since f is linearly nondegenerate, f :H2 0. On the other hand f ;Hz g;Hz 0 for all z A 6q

    i1 f1Hi, hence

    N f ;HrbXqi1

    N1 f ;Hir:

    It yields that

    k Tf rbN f ;HrbXqi1

    N1 f ;Hir

    bq n 1

    nTf r oTf rb

    n 1n

    Tf r oTf r:

    Letting r ! y, we get Tf r 0. This is a contradiction. Hence gCm cannot be contained in any hyperplanes of PnC. Therefore g is linearly non-degenerate. 9

    Lemma 3.2. Suppose qb n 2. Thenk Tgr OTf r and k Tf r OTgr for each g A G f ; fHigqi1; 1:

    Proof. By the Second Main Theorem, we have

    k q n 1TgraXqi1

    Nng;Hir oTgr

    aXqi1

    nN1g;Hir oTgra qnTf r oTgr:

    Hence k Tgr OTf r. Similarly, we get k Tf r OTgr: 9

    Let f 1 and f 2 be two distinct meromorphic mappings in G f ; fHigqi1; 1.For t 1; . . . ; q, we set

    Stf 1; f 2;>n : fz A Cm;minfn f 1;Htz; n f 2;Htzg > ng;

    Stf 1; f 2;n UStf 1; f 2;

  • If there are two distinct maps f 1 and f 2 in G f ; fHig2n2i1 ; 1 then the followingassertions hold

    (i) k Nr;Stf 1; f 2;n

    oTf r, Et 1; . . . ; 2n 2,(ii) k q n 1Tf l r

    Pqt1 N

    n f l ;Htr oTf r, l 1; 2,

    (iii) Moreover, if we assume further that f 1;Hi f 2;Hi

    2 f 1;Hj f 2;Hj

    for all 1ai; ja q, then

    k Tf 1r Tf 2r Xvi; j

    N n f 1;Ht Nn f 2;Ht n 1N

    1 f 1;Ht

    Xqv1

    N1 f 1;Hv oTf r:

    Proof. By changing indices if necessary, we may assume that

    f 1;H1 f 2;H1

    1 f 1;H2 f 2;H2

    1 1 f1;Hk1

    f 2;Hk1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}group 1

    2 f 1;Hk11 f 2;Hk11

    1 1 f1;Hk2

    f 2;Hk2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}group 2

    2 f 1;Hk21 f 2;Hk21

    1 1 f1;Hk3

    f 2;Hk3|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}group 3

    2 2 f1;Hks11

    f 2;Hks111 1 f

    1;Hks f 2;Hks|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

    group s

    ;

    where ks 2n 2.For each 1a ta 2n 2, we set

    st t n if ta n 2;tn 2 if n 2 < ta 2n 2;

    and

    Pt f 1;Ht f 2;Hst f 2;Ht f 1;Hst:

    Since f 1 2 f 2, the number of elements of each group is at most n. Hence f 1;Ht f 2;Ht

    and f 1;Hst f 2;Hst

    belong to distinct groups for all ta q 2. This means

    that Pt 2 0 1a ta 2n 2. By changing indices again if necessary, we mayassume that i 1 and j s1.

    Fix an index t with 1a ta q. For z B I f 1U I f 2U6k0l f

    1Hk VHl,it is easy to see that:

    If z is a zero point of f 1;Ht then z is a zero point of Pt with multiplicityat least minfn f 1;Ht; n f 2;Htg. Similarly, if z is a zero point of f 1;Hst then zis a zero point of Pt with multiplicity at least minfn f 1;Hst; n f 2;Hstg.

    If z is a zero point of f 1;Hv with v B ft; stg then z is a zero point of Pt(because f 1z f 2z).

    469meromorphic mappings sharing few hyperplanes

  • Thus, we have

    nPtzbminfn f 1;Ht; n f 2;Htg minfn f 1;Hst; n f 2;Hstg3:4

    Xqv1

    v0t;st

    n1 f 1;Hvz;

    for all z outside the analytic set I f U IgU6k0l f

    1Hk VHl of dimensionam 2:

    For two integers a and b, set

    Sa;b;n 1 if maxfa; bg < n;0 if minfa; bga namaxfa; bg1 if minfa; bg > n:

    8