NEVANLINNA THEORY FOR MEROMORPHIC MAPS FROM A CLOSED SUBMANIFOLD OF C l TO A COMPACT COMPLEX MANIFOLD

The purpose of this article is threefold. The first is to construct a Nevanlina theory for meromorphic mappings from a polydisc to a compact complex manifold. In particular, we give a simple proof of Lemma on logarithmic derivative for nonzero meromorphic functions on C l . The second is to improve the defi nition of the nonintegrated defect relation of H. Fujimoto 7 and to show two theorems on the new nonintegrated defect relation of meromorphic maps from a closed submanifold of C l to a compact complex manifold. The third is to give a unicity theorem for meromorphic mappings from a Stein manifold to a compact complex manifold

FROM A CLOSED SUBMANIFOLD OF Cl TO A

COMPACT COMPLEX MANIFOLD

DO DUC THAI AND VU DUC VIET

Abstract The purpose of this article is threefold The first is

to construct a Nevanlina theory for meromorphic mappings from

a polydisc to a compact complex manifold In particular, we give

a simple proof of Lemma on logarithmic derivative for nonzero

meromorphic functions on C l The second is to improve the

defi-nition of the non-integrated defect relation of H Fujimoto [7] and

to show two theorems on the new non-integrated defect relation

of meromorphic maps from a closed submanifold of C l to a

com-pact complex manifold The third is to give a unicity theorem

for meromorphic mappings from a Stein manifold to a compact

2.3 Pluri-subharmonic functions on complex manifolds 8

References 37

1 Introduction

To construct a Nevanlinna theory for meromorphic mappings tween complex manifolds of arbitrary dimensions is one of the mostimportant problems of the Value Distribution Theory Much attentionhas been given to this problem over the last few decades and severalimportant results have been obtained For instance, W Stoll [17] in-troduced to parabolic complex manifolds, i.e manifolds have exhaustedfunctions on the ones with the same role as the radius function in Cl

be-and constructed a Nevanlinna theory for meromorphic mappings from

a parabolic complex manifold into a complex projective space In thesame time, P Griffiths and J King [9] constructed a Nevanlinna theoryfor holomorphic mappings between algebraic varieties by establishingspecial exhausted functions on affine algebraic varieties There is being

a very interesting problem that is to construct explicitly a Nevanlinnatheory for meromorphic mappings from a Stein complex manifold or acomplete K¨ahler manifold to a compact complex manifold The firstmain aim of this paper is to deal with the above mentioned problem

in a special case when the Stein manifold is a polydisc In particular,

we give a simple proof of Lemma on logarithmic derivative for nonzeromeromorphic functions on Cl (cf Proposition 3.7 and Remark 3.8 be-low)

In 1985, H Fujimoto [7] introduced the notion of the non-integrateddefect for meromorphic maps of a complete K¨ahler manifold into thecomplex projective space intersecting hyperplanes in general positionand obtained some results analogous to the Nevanlinna-Cartan defectrelation We now recall this definition

Let M be a complete K¨ahler manifold of m dimension Let f be ameromorphic map from M into CPn, µ0 be a positive integer and D

be a hypersurface in CPn of degree d with f (M ) 6⊂ D We denote theintersection multiplicity of the image of f and D at f (p) by ν(f,D)(p)and the pull-back of the normalized Fubini-Study metric form on CPn

by Ωf The non-integrated defect of f with respect to D cut by µ0 isdefined by

¯

δ[µ0 ]

f (D) := 1 − inf{η ≥ 0 : η satisfies condition (∗)}

Here, the condition (*) means that there exists a bounded ative continuous function h on M with zeros of order not less thanmin{ν(f,D), µ0} such that dηΩf + ddclog h2 ≥ [min{ν(f,D), µ0}], where

Recently, M Ru and S Sogome [16] generalized the above result of

H Fujimoto for meromorphic maps of a complete K¨ahler manifold intothe complex projective space CPn intersecting hypersurfaces in generalposition After that, T.V Tan and V.V Truong [18] generalized suc-cessfully the above result of H Fujimoto for meromorphic maps of acomplete K¨ahler manifold into a complex projective variety V ⊂ CPn

intersecting global hypersurfaces in subgeneral position in V in theirsense Later, Q Yan [19] showed the non-integrated defect for mero-morphic maps of a complete K¨ahler manifold into CPn intersectinghypersurfaces in subgeneral position in the original sense in CPn Wewould like to emphasize that, in the results of the above mentionedauthors, there have been two strong restrictions

• The above mentioned authors always required a strong assumption(C) that functions h in the notion of the non-integrated defect arecontinuous By this request, their non-integrated defect is still small

• The above mentioned authors always asked a strong assumption asfollows: (H) The complete K¨ahler manifold M whose universal covering

is biholomorphic to the unit ball of Cl

Motivated by studying meromorphic mappings into compact plex manifolds in [3] and from the point of view of the Nevanlinnatheory on polydiscs, the second main aim of this paper is to improvethe above-mentioned definition of the non-integrated defect relation of

com-H Fujimoto by omiting the assumption (C) (cf Subsection 4.1 below)and to study the non-integrated defect for meromorphic mappings from

a Stein manifold without the assumption (H) into a compact complexmanifold sharing divisors in subgeneral position (cf Theorems 4.3 and4.7 below) As a direct consequence, we get the following Bloch-Cartantheorem for meromorphic mappings from Cl to a smooth algebraic va-riety V in CPm missing hypersurfaces in subgeneral position: a non-constant meromorphic mapping of Cl into an algebraic variety V of

CPm cannot omit (2N + 1) global hypersurfaces in N -subgeneral sition in V We would like to emphasize that, by using our argumentsand their techniques in [16], [18], [19] we can generalize exactly theirresults to meromorphic mappings from a Stein manifold without theassumption (H) into a smooth complex projective variety V ⊂ CPM(cf Remark 4.6 below)

po-In [8], the author gave a unicity theorem for meromorphic mappingsfrom a complete K¨ahler manifold satisfying the assumption (H) intothe complex projective space CPn The last aim of this paper is to give

an analogous unicity theorem for meromorphic mappings from a Steinmanifold without the assumption (H) to a compact complex manifold.

2 Some facts from pluri-potential theory

2.1 Derivative of a subharmonic function In this subsection, wegive an estimation of derivative of a subharmonic function Firstly, werecall some definitions

For R > 0, we consider the ball of radius R as follows:

BR = {x ∈ Rn: |x|< R},where |x| is the Euclidean norm in Rn

For R = (R1, · · · , Rn) with Rj > 0 for each 1 ≤ j ≤ n, we considerthe polydisc with a radius R as follows:

−|x|2−n/((n − 2)cn) if n > 2,where cn is the area of the unit sphere in Rn The classical Greenfunction of BR with pole at x ∈ BR is

For a proof of this theorem, we refer to [2, Proposition 4.22] The lowing has a crucial role in the proof of Proposition 3.7 on Logarithmicderivative lemma

fol-Proposition 2.2 Let u be a lower-bounded subharmonic function 6≡

−∞ in the ball BR Assume that u has the derivative a.e in BR Then

By a direct computation, we get

R/2|2) ak R/2

|a − y|n−1 + 1

Rn+3 + 1

R2n+3

,

|∂P (

a R/2, y)

|a − y|n−1 + 1

Rn+3 + 1

R2n+3

dµ(y)+

Z

∂B 1

u(Ry/2)dω(y)



In the Riezs representation formula of u, taking x = 0, we get

For convenience, in this proof, S(n) always stands for a constant pending only n Therefore,

Integrating the above inequality over ∆(4nR), we obtain

∆(4nR)

1

|a − y|n−1da+

Z

∆( R 4n )

(R−5+ R−n−5+ 1) sup

|x|≤R

|u(x)|da

≤Z

BR/2

dµ(y)Z

B(y,|y|+R/2)

1

|a − y|n−1da+ (R−5+ Rn) sup

Let Ω be a connected open subset of Cnand let a be a point in Ω If

u is a plurisubharmonic function in a neighborhood of a, we shall saythat u has a logarithmic pole at a if

u(z) − log|z − a|≤ O(1), as z → a,where |z − a| is the Euclidean norm in Cn The pluricomplex Greenfunction of Ω with pole at a is

Then the measure µris supported on S(r) and r → µris weakly uous on the left Denote by µΩ,a the weak-limit of µr as r → 0 We nowconsider Ω = ∆R = {(z1, z2, · · · , zn) ∈ Cn : |z1|< R1, · · · , |zn|< Rn} is

contin-a polydisc in Cn For brevity, we will denote the polydisc ∆R by ∆ inthe end of this subsection Then, we have

g∆,a = max1≤j≤nlog|Rj(zj− aj)

Theorem 2.3 Let V be a plurisubharmonic function on an open borhood of a polydisc ∆ of Cn Let g∆,a be a pluricomplex Green function

neigh-of ∆ with pole at a = (a1, · · · , an) ∈ ∆ Then

as r tends to 0 Now suppose that V is continuous Since the supports

of µr ⊂ ∆ and µr weakly converge to µΩ,a, we get µr(V ) → µΩ,a(V ) as

r tends to 0 In general, by taking a decreasing sequence of ous plurisubharmonic functions Vn converging to V, we get the desiredequality Notice that µΩ,a(V ) is finite by (1) and

continu-Z

∆

V (ddcg∆,a)n = (2π)nδ{a}(V ) = (2π)nV (a)

2.3 Pluri-subharmonic functions on complex manifolds Thissubsection is devoted to prove a version of [12, Theorem A] in thecase where M is Stein and u is a (not necessary continuous) plurisub-harmonic function Throughout this subsection M will denote an m-dimensional closed complex submanifold of Cn and the K¨ahler metric

of M is induced from the canonical one of Cn

Definition 2.4 Let N be a complex manifold and f be a locally grable real function in N We say that f is plurisubharmonic function(or psh function, for brevity) if ddcf ≥ 0 in the sense of currents.Lemma 2.5 (see [12, Lemma, p.552]) Let N1 be a K¨ahler manifoldand N2 be a complex manifold Let g be a holomorphic map of N1 to

inte-N2 Then for each C2-psh function f in N2, f ◦ g is subharmonic in

N1

Lemma 2.6 The volume of M is infinite

Proof Take a point a ∈ M Let BM(a, R) be the ball centered at a

of M and of radius R Put u = |z − a| Then u is a psh function on

Cn and hence, it is a subharmonic function on M Since the K¨ahlermetric on M is induced from the canonical one of Cn, it implies that

BM(a, R) ⊂ B(a, R), where B(a, R) is the usual ball centered at a and

of radius R in Cn Therefore u ≤ R in BM(a, R) By [12, Theorem A],

Proposition 2.7 Let u be a psh function on M and K be a compactsubset of M For each open subset U of M such that K ⊂ U b M,there exists a decreasing sequence of C∞-psh functions uk in U suchthat uk converge to u, a.e in U Moreover, if u is non-negative then uk

is non-negative

Proof By [10, Chapter VIII, Theorem 8], there exists a holomorphicretraction α of an open subset V of Cn containing M to M, i.e α isholomorphic and α|M = idM Then u ◦ α is a psh function on V Theconclusion now is deduced immediately from this fact 

As a direct consequence, we get the following

Corollary 2.8 Let ξ be an increasing convex function in R Let u be

a psh function on M Then ξ ◦ u is a psh function Specially, if u isnon-negative then up (p ≥ 1) is in the Sobolev space H0(M ) of degree

0 of M

Theorem 2.9 Let u be a non-negative psh function on M and p be apositive number greater than 1 Take a point a ∈ M Let BM(a, R) (orB(R) for brevity) be the ball centered at a of M and of radius R Thenone of the following two statements holds:

(ii) u is constant a.e in M

Proof Suppose that u is not constant a.e in M and

1

r2 j

Z

B(r j )

upjd vol ≤ 1

r2 j

Z

B M (a,r j )

upd vol + 1

For each j ≥ 1, let ϕj be a Lipschitz continuous function such that

ϕj(x) ≡ 1 on B(a, rj) and ϕj(x) ≡ 0 in M \B(a, rj+1) and gradϕj ≤ C

It is easy to see that for some j, IN

j > 0 for an infinite number ofvalues of N Since IN

j ≤ IN

k for j ≤ k ≤ N, it follows that there exist

an index j0 and a sequence Nk→ +∞ such that for each m ≥ j0, Nk ≥

j (from m to Nk), we obtain 1/INk

m ≥ C(Nk− m) for a constant C.Hence,

In the other words, ∆u = 0 in the sense of currents Hence, uq+1 ∈ C∞

by the regularity theorem (so that ”grad uq+1” makes sense) Put X =grad uq+1 Then,

by ∆R We now construct definitions in the case where Rj < ∞ for each

j The construction in the case where Rj = ∞ for some j is similar

3.1 First main theorem Let L−→ X be a holomorphic line bundleπover a compact complex manifold X and d be a positive integer Let E

be a C-vector subspace of dimension m + 1 of H0(X, Ld) Take a basis{ck}m+1k=1 a basis of E Put B(E) = ∩σ∈E{σ = 0} Then

∩1≤i≤m+1{ci = 0} = B(E)and

ω = ddclog(|c1|2+ · · · + |cm+1|2)1/d

is well-defined on X \ B(E)

Assume that R = (R1, · · · , Rn) and R0 = (R01, · · · , R0n), where Rj >

0 and R0j > 0 for each 1 ≤ j ≤ n Recall that R < R0 (R ≤ R0 resp.) if

0 < Rj < R0j (0 < Rj ≤ R0

j resp.) for each 1 ≤ j ≤ n

As usual, we say that the assertion P holds for a.e r ≤ R if theassertion P holds for each r ≤ R such that rj is excluded a Borelsubset Ej of the interval [0, Rj] with R

E jds < ∞ for each 1 ≤ j ≤ n.Let f be a meromorphic mapping of a polydisc ∆R of radius R =(R1, · · · , Rn) into X such that f (∆R) ∩ B(E) = ∅ We define thecharacteristic function of f with respect to E as follows

Cn Remark that the definition does not depend on choosing basic of

E Take a section σ of Ls for some s Let D be its zero divisor Put

∆ r

|g∆r,a| min{[f∗D], k} ∧ (ddcg∆r,a)n−1and the proximity function is

For brevity, we will omit the character [k] if k = ∞ Now, take morphic functions f0, f1, · · · , fm in ∆Rsuch that (f )0 = f∗(c1)0, fi+1=

holo-fici+1c (f )

i (f ) for i ≥ 0 We get a reduced representation (f0, f1, · · · , fm) of

f By the Lelong-Jensen formula, we get

Theorem 3.1 (First main theorem)

Tf(r, E) =

Z

∂ 0 ∆ r

log(|f0|2+ · · · + |fm|2)1/ddt1dt2· · · dtn+ O(1),where ∂0∆r is the distinguished boundary of ∆r Hence, Tf(r, E) is aconvex increasing function of log ri for each i

We also have an analogue for Nf(r, D)

Remark 3.3 Put log+s = max{log s, 0} for all s ≥ 0 Let g be ameromorphic function on ∆R Then g is considered as a mapping of

∆R into CP1 by sending z to [g1(z), g2(z)], where g = g1/g2 Let H1 bethe hyperplane bundle on CP1, D = [0, 1] (the divisor consists of onlyone component [0, 1] with coefficient 1) Put

mg(r) =

Z

∂ 0 ∆ r

log+|g(reit)| dt1· · · dtn.Then mg(r) = mg(r, [0, 1]) + O(1)

3.2 Second main theorem Let g be a meromorphic function on

a polydisc ∆R For an n-tuple α = (α1, α2, · · · , αn) of non-negativeintegers, we put

Lemma 3.4 Let f ∈ L1(∆R) Then f ∈ L1(∂0∆r) for e.a r ≤ R Put

f1(r) =

Z

∂ 0 ∆

f dt1· · · dtn (r ≤ R)

for a.e s ≤ r The proof now is deduced from the Fubini theorem and

Lemma 3.5 Let φ(r) ≥ 0 be a monotone increasing function for 0 <

r ≤ R Let δ be a positive real number Then

Proof We consider the case that |α|= 1 The general case follows easily

from this case Take r < r0 Write g = g1/g2 Applying Proposition 2.2

for log|g1|, log|g2| we get

where C(n) is a constant depending only on n Moreover, by Lemma

for all r ≤ R such that rj does not belong to a set Ej ⊂ [0, Rj] with

R

E j

1

R j −sds < ∞ The proof is completed 

Remark 3.8 We can apply the above argument to the characteristic

function in the classical sense of a nonzero meromorphic function g in

Cn to get a simple new proof of Lemma on logarithmic derivative

Now, put ∆(a) = {(z1, · · · , zn) : |zk|≤ a for all k} and ∆i(a) =

{(z1, · · · , zn) : |zk|≤ a for k 6= i, |zi|= a} The measure mi on ∆i(a) is

the product of the (n − 1) dimensional Lebesgue measure and the usual

measure on a circle in C

Proposition 3.9 Let g be a meromorphic function in ∆R Let p, p0

be positive real numbers such that p < p0 Assume R = (R0, · · · , R0)and p|α|≤ 1 Then, there exist a constant C depending only on n and

(R0− a)3Tf((a, · · · , a), E)

p0|α|n

Proof Let αk, k = 1, 2, · · · , |α| be a sequence of n-tuples satisfying:

α1 = 0, |αk|= |αk−1|+1, for all k ≥ 2 By the proof of Proposition 3.7and p < 1,

Proposition 3.10 Let F : ∆R → CPm be a linearly non-degeneratemapping Assume that F = (F0, · · · , Fm+1) is a reduced representation

of F Then there exist n-tuples α1, · · · , αm+1 such that

|α1|+ · · · + |αm+1|≤ m(m + 1)

2 and |αk|≤ m (1 ≤ k ≤ m + 1)

and the generalized Wronskian of F

of dimension m + 1 Let {ck}m+1k=1 be a basis of E and B(E) be the baselocus of E Define a mapping Φ : X \ B(E) → CPm by

Φ(x) := [c1(x) : · · · : cm+1(x)]

Denote by rankE the maximal rank of Jacobian of Φ on X \ B(E) It

is easy to see that this definition does not depend on choosing a basis

of E Take σj ∈ H0(X, L), Dj = {σj = 0} (1 ≤ j ≤ q) Assume that

Assume that {Dj} is located in N -subgeneral position with respect

to E Put u = rankE, b = dimB(E) + 1 if B(E) 6= ∅ and b = −1

if B(E) = ∅ Assume that u > b We set σi = P

H the hyperplane line bundle of CPm Put Hi := P

1≤j≤m+1aijzj−1,where [z0, z1, · · · , zm] is the homogeneous coordinate of CPm

For each K ⊂ Q, put c(K) = rank{Hi}i∈K We also set

n0({Dj}) = max{c(K) : K ⊂ Q with |K|≤ N + 1} − 1,

and

n({Dj}) = max{c(K) : K ⊂ Q} − 1

Then n({Dj}), n0({Dj}) are independent of the choice the C-vectorsubspace E of H0(X, L) containing σj(1 ≤ j ≤ q) We see that

u ≤ n0({Dj}) ≤ n({Dj}) ≤ m

Proposition 3.12 Let notations be as above Assume that D1, · · · , Dqare in N -subgeneral position with respect to E and q ≥ 2N − u + 2 + b.Put kN = 2N −u+2+b, sN = n0({Dj}) and tN = u − b

n({Dj}) − u + 2 + b.Then, there exist Nochka weights ω(j) for {Dj}, i.e there exist con-stants ω(j) (j ∈ Q) and Θ satisfying the following conditions:

By repeating the argument in [3], we get an analogous version of theSecond Main Theorem in [3] We state the following theorem withoutits proof

Theorem 3.13 Let X be a compact complex manifold Let L → X be

a holomorphic line bundle over X Fix a positive integer d Let E be

a C-vector subspace of dimension m + 1 of H0(X, Ld) Put u = rankEand b = dimB(E) + 1 if B(E) 6= ∅, otherwise b = −1 Take positivedivisors d1, d2, · · · , dq of d Let σj (1 ≤ j ≤ q) be in H0(X, Ld j) suchthat σ

d

d1

1 , · · · , σ

d dq

q ∈ E Set Dj = (σj)0 (1 ≤ j ≤ q) Assume that

D1, · · · , Dq are in N -subgeneral position with respect to E and u >

b Let f : ∆R → X be an analytically non-degenerate meromorphicmapping with respect to E, i.e f (∆R) 6⊂ supp((σ)) for any σ ∈ E \ {0}and f (∆R) ∩ B(E) = ∅ Then, for all r ≤ R such that rj does notbelong to a set Ej ⊂ [0, Rj] with RE

j

1

R j −sds < ∞, we have(q − (m + 1)K(E, N, {Dj}))Tf(r, E) ≤

where kN, sN, tN are defined as in Proposition 3.12 and

K(E, N, {Dj}) = kN(sN − u + 2 + b)

4 Non-integrated defect relation4.1 Definitions and basic properties Let all notations be as inSection 3 The defect of f with respect to D truncated by k in E isdefined by

δ[k]f,E(D) = lim inf

We now assume that f is a meromorphic mapping of a connectedcomplex manifold M into X such that f (M ) ∩ B(E) = ∅ Let D be

a divisor of H0(X, Ls) for some s > 0 For 0 ≤ k ≤ ∞, denote by

D[k]f,E the set of real numbers η ≥ 0 such that there exists a boundedmeasurable nonnegative function h on M such that

δf,E[k] (D) := 1 − inf{η : η ∈ D[k]f,E}

Note that this definition does not depend on choosing a base of E.Remark 4.1 In the original definition of H Fujimoto [7], when X =

CPk, L is the hyperplane bundle and s = 1 he required that functions

h, hϕare continuous, where ϕ is a holomorphic function in M such that(ϕ)0 = min{k, f∗D}

By [5, Theorem 1], there exists an open subset U of M such that U

is biholomorphic to a polydisc ∆R and M \ U has a zero measure, i.e

if (V, ϕ) is a local coordinate then ϕ((M \ U ) ∩ V ) is of zero Lebesguemeasure

Proposition 4.2 We have the following properties of the non-integrateddefect:

(i) 0 ≤ ¯δf,E[k] (D) ≤ 1

(ii) ¯δ[k]f,E(D) = 1 if f (M ) ∩ D = ∅

(iii) ¯δ[k]f,E(D) ≥ 1 − k

k if f∗D ≥ k0min{f∗D, 1}

(iv) Denote by fU the restriction of f to U and assume that

k0k

and η = k

k0

Since f∗D ≥ k0min{f∗D, 1}, we get (iii)

We now prove (iv) Take a holomorphic function ϕ in ∆R such that(ϕ) = min{fU∗D, k} For η ∈ D[k]f

4.2 Defect relation with a truncation Now we give the integrated defect with a truncation for meromorphic mappings from

non-a submnon-anifold of Cl to a compact complex manifold

Theorem 4.3 Let M be an n-dimensional closed complex fold of Cl and ω be its K¨ahler form that is induced from the canonicalK¨ahler form of Cl Let L → X be a holomorphic line bundle over

submani-a compsubmani-act msubmani-anifold X Fix submani-a positive integer d submani-and let d1, d2, · · · , dq

be positive divisors of d Let E be a C-vector subspace of dimension

m + 1 of H0(X, Ld) Put u = rankE and b = dimB(E) + 1 if B(E) 6=

∅, otherwise b = −1 Let σj (1 ≤ j ≤ q) be in H0(X, Ldj) suchthat σ

d

d1

1 , · · · , σ

d dq

q ∈ E Set Dj = (σj)0 (1 ≤ j ≤ q) Assume that