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

NEVANLINNA THEORY FOR MEROMORPHIC MAPS 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 Cl . The second is to improve the definition 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 Cl 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. Contents 1. Introduction 2. Some facts from pluri-potential theory 2.1. Derivative of a subharmonic function 2.2. Pluri-complex Green function 2.3. Pluri-subharmonic functions on complex manifolds 3. Nevanlinna theory in polydiscs 3.1. First main theorem 3.2. Second main theorem 4. Non-integrated defect relation 4.1. Definitions and basic properties 4.2. Defect relation with a truncation 4.3. Defect relation with no truncation 5. A unicity theorem 2 4 4 6 8 10 11 12 18 18 19 26 31 2000 Mathematics Subject Classification. Primary 32H30; Secondary 32H04, 32H25, 14J70. Key words and phrases. N -subgeneral position, Non-integrated defect relation, Second main theorem. The research of the authors is supported by an NAFOSTED grant of Vietnam (Grant No. 101.01-2011.29). 1 2 DO DUC THAI AND VU DUC VIET References 37 1. Introduction To construct a Nevanlinna theory for meromorphic mappings between complex manifolds of arbitrary dimensions is one of the most important problems of the Value Distribution Theory. Much attention has been given to this problem over the last few decades and several important results have been obtained. For instance, W. Stoll [17] introduced to parabolic complex manifolds, i.e manifolds have exhausted functions on the ones with the same role as the radius function in Cl and constructed a Nevanlinna theory for meromorphic mappings from a parabolic complex manifold into a complex projective space. In the same time, P. Griffiths and J. King [9] constructed a Nevanlinna theory for holomorphic mappings between algebraic varieties by establishing special exhausted functions on affine algebraic varieties. There is being a very interesting problem that is to construct explicitly a Nevanlinna theory for meromorphic mappings from a Stein complex manifold or a complete K¨ahler manifold to a compact complex manifold. The first main 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 nonzero meromorphic functions on Cl (cf. Proposition 3.7 and Remark 3.8 below). In 1985, H. Fujimoto [7] introduced the notion of the non-integrated defect for meromorphic maps of a complete K¨ahler manifold into the complex projective space intersecting hyperplanes in general position and obtained some results analogous to the Nevanlinna-Cartan defect relation. We now recall this definition. Let M be a complete K¨ahler manifold of m dimension. Let f be a meromorphic map from M into CP n , µ0 be a positive integer and D be a hypersurface in CP n of degree d with f (M ) ⊂ D. We denote the intersection 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 CP n by Ωf . The non-integrated defect of f with respect to D cut by µ0 is defined by [µ ] δ¯f 0 (D) := 1 − inf{η ≥ 0 : η satisfies condition (∗)}. Here, the condition (*) means that there exists a bounded nonnegative continuous function h on M with zeros of order not less than min{ν(f,D) , µ0 } such that dηΩf + ddc log h2 ≥ [min{ν(f,D) , µ0 }], where NEVANLINNA THEORY √ 3 −1 ¯ dc = 4π (∂ − ∂) and we mean by [ν] the (1, 1)-current associated with the divisor ν. Recently, M. Ru and S. Sogome [16] generalized the above result of H. Fujimoto for meromorphic maps of a complete K¨ahler manifold into the complex projective space CP n intersecting hypersurfaces in general position. After that, T.V. Tan and V.V. Truong [18] generalized successfully the above result of H. Fujimoto for meromorphic maps of a complete K¨ahler manifold into a complex projective variety V ⊂ CP n intersecting global hypersurfaces in subgeneral position in V in their sense. Later, Q. Yan [19] showed the non-integrated defect for meromorphic maps of a complete K¨ahler manifold into CP n intersecting hypersurfaces in subgeneral position in the original sense in CP n . We would like to emphasize that, in the results of the above mentioned authors, 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 are continuous. By this request, their non-integrated defect is still small. • The above mentioned authors always asked a strong assumption as follows: (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 complex manifolds in [3] and from the point of view of the Nevanlinna theory on polydiscs, the second main aim of this paper is to improve the above-mentioned definition of the non-integrated defect relation of 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 complex manifold sharing divisors in subgeneral position (cf. Theorems 4.3 and 4.7 below). As a direct consequence, we get the following Bloch-Cartan theorem for meromorphic mappings from Cl to a smooth algebraic variety V in CP m missing hypersurfaces in subgeneral position: a nonconstant meromorphic mapping of Cl into an algebraic variety V of CP m cannot omit (2N + 1) global hypersurfaces in N -subgeneral position in V . We would like to emphasize that, by using our arguments and their techniques in [16], [18], [19] we can generalize exactly their results to meromorphic mappings from a Stein manifold without the assumption (H) into a smooth complex projective variety V ⊂ CP M (cf. Remark 4.6 below). In [8], the author gave a unicity theorem for meromorphic mappings from a complete K¨ahler manifold satisfying the assumption (H) into the complex projective space CP n . The last aim of this paper is to give 4 DO DUC THAI AND VU DUC VIET an analogous unicity theorem for meromorphic mappings from a Stein manifold 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, we give an estimation of derivative of a subharmonic function. Firstly, we recall 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 consider the polydisc with a radius R as follows: ∆R = {x = (x1 , · · · , xn ) ∈ Rn : |xj |< Rj for each 1 ≤ j ≤ n}. If R1 = · · · = Rn := R > 0, then the polydisc ∆R is denoted by ∆R := ∆(R). For x ∈ Rn − {0}, put E(x) = (2π)−1 log|x| if n = 2, −|x|2−n /((n − 2)cn ) if n > 2, where cn is the area of the unit sphere in Rn . The classical Green function of BR with pole at x ∈ BR is  2  |x|  E(x − y) − E R ( R|x|2x − y) if x = 0, x = y GR (x, y) =  1 1 1  − Rn−2 if x = 0.  (n−2)c |y|n−2 n Note that GR (x, y) = GR (y, x). The Poisson kernel of B1 is given by 1 P (x, y) = (1 − |x|2 )|y − x|−n , |y|= 1, |x|< 1. cn Theorem 2.1. (Riezs representation formula) Let u be a subharmonic function ≡ −∞ in the ball BR = {x ∈ Rn : |x|< R}. Take 0 < R < R. Then x u(x) = GR (x, y)dµ(y) + u(R y)P , y dω(y), x ∈ BR , R BR ∂B1 where dµ = ∆u as distributions. For a proof of this theorem, we refer to [2, Proposition 4.22]. The following has a crucial role in the proof of Proposition 3.7 on Logarithmic derivative lemma. NEVANLINNA THEORY 5 Proposition 2.2. Let u be a lower-bounded subharmonic function ≡ −∞ in the ball BR . Assume that u has the derivative a.e in BR . Then ∂u | (a)| da ≤ S(n)(R−5 + Rn ) sup |u(x)| R |x|≤R ∆( 4n ) ∂xk for each 1 ≤ k ≤ n, where S(n) is a constant depending only on n. Proof. By Theorem 2.1 and the hypothesis, we have ∂u (a) = ∂xk BR/2 ∂GR/2 (a, y) dµ(y)+ ∂xk u(Ry/2) a ∂P ( R/2 , y) ∂xk ∂B1 dω(y) By a direct computation, we get ∂GR/2 (a, y) ak − yk cn =− + ∂xk |a − y|n R 2 n−2 |( R2 )2 a − y|a|2 |2 ak |( R2 )2 a − y|a|2 |n+2 |a|2 (( R2 )2 ak − yk |a|2 ) − |( R2 )2 a − y|a|2 |n+2 and a ∂P ( R/2 , y) ∂xk = ak a a 2 ak −2 R/2 | R/2 − y|2 −n(1 − | R/2 | ) R/2 a cn | R/2 − y|n+2 . Take a such that |a|≤ R/4. Then, there exists S(n) depending only on n such that ∂GR/2 (a, y) 1 1 1 | |≤ S(n) + n+3 + 2n+3 , n−1 ∂xk |a − y| R R a ∂P ( R/2 , y) | |≤ S(n). ∂xk Therefore, for |a|< R/4, | ∂u (a)|≤ S(n) ∂xk BR/2 1 1 1 + n+3 + 2n+3 dµ(y)+ n−1 |a − y| R R u(Ry/2)dω(y) . ∂B1 In the Riezs representation formula of u, taking x = 0, we get u(0) = G3R/4 (0, y)dµ(y) + B3R/4 u(3Ry/4)P (0, y)dω(y). ∂B1 Hence dµ(y) ≤ S(n)Rn−2 sup |u(x)|. BR/2 |x|≤R 6 DO DUC THAI AND VU DUC VIET For convenience, in this proof, S(n) always stands for a constant depending only n. Therefore, | ∂u (a)|≤ S(n) ∂xk BR/2 1 dµ(y)+(R−5 +R−n−5 +1) sup |u(x)| . |a − y|n−1 |x|≤R R ), we obtain Integrating the above inequality over ∆( 4n 1 S(n) | R ) ∆( 4n ∂u (a)| da ≤ ∂xk dµ(y) R ) ∆( 4n BR/2 1 da |a − y|n−1 (R−5 + R−n−5 + 1) sup |u(x)|da + R ∆( 4n ) ≤ |x|≤R dµ(y) BR/2 + (R −5 B(y,|y|+R/2) 1 da |a − y|n−1 n + R ) sup |u(x)| |x|≤R ≤ (R −5 n + R ) sup |u(x)|. |x|≤R 2.2. Pluri-complex Green function. For detailed explosion of this subsection, one may consult [13, Chapter 6, Section 6.5] and [2, Chapter 1 ¯ as (∂ − ∂) III, Section 6]. Firstly, we denote d = ∂ + ∂¯ and dc = 4iπ usual. Let Ω be a connected open subset of Cn and let a be a point in Ω. If u is a plurisubharmonic function in a neighborhood of a, we shall say that 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 Green function of Ω with pole at a is gΩ,a (z) = sup{u(z) : u ∈ PSH(Ω, [−∞, 0)) and u has a logarithmic pole at a}. (It is assumed here that sup ∅ = ∞). We would like to notice that if V is a plurisubharmonic function, then the ddc V ∧ (ddc gΩ,a )n−1 is well-defined (see [2, Proposition 4.1]). For r ∈ (−∞, 0], put −1 gr (z) = max{gΩ,a (z), r}, S(r) = gΩ,a (r). Define µr = (ddc gr (z))n − 1{gΩ,a ≥r} (ddc gΩ,a )n , r ∈ (−∞, 0). NEVANLINNA THEORY 7 Then the measure µr is supported on S(r) and r → µr is weakly continuous on the left. Denote by µΩ,a the weak-limit of µr as r → 0. We now consider Ω = ∆R = {(z1 , z2 , · · · , zn ) ∈ Cn : |z1 |< R1 , · · · , |zn |< Rn } is a polydisc in Cn . For brevity, we will denote the polydisc ∆R by ∆ in the end of this subsection. Then, we have g∆,a = max1≤j≤n log| Rj (zj − aj ) |. Rj2 − zj a¯j Moreover, µ∆,a is concentrated on the distinguished boundary ∂ ∆ of ∆ and n (1) dµ∆,a = Rj2 − |aj |2 dt1 · · · dtn . it |2 |a − R e j j j=1 Theorem 2.3. Let V be a plurisubharmonic function on an open neighborhood of a polydisc ∆ of Cn . Let g∆,a be a pluricomplex Green function of ∆ with pole at a = (a1 , · · · , an ) ∈ ∆. Then n V ∂∆ Rj2 − |aj |2 dt1 · · · dtn − (2π)n V (a) = it |2 |a − R e j j j=1 0 ddc V ∧ (ddc g∆,a )n−1 dt −∞ {g∆,a k→+∞ = 0. 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, ||X||2 ϕd vol = =− < grad uq+1 , ϕX > d vol uq+1 div(ϕX)d vol = − lim k→+∞ uq+1 Nk div(ϕX)d vol = (q + 1) lim k→+∞ uq+1 Nk < grad uNk , ϕX > d vol = 0. Therefore, X = 0. That means u is constant, a contradiction. Corollary 2.10. Let u be a psh function on M. Then eu d vol = ∞. M 3. Nevanlinna theory in polydiscs In Cn , consider a polydisc ∆(a, R) = {(z1 , z2 , · · · , zn ) ∈ Cn : |z1 − a1 |< R1 , · · · , |zn − an |< Rn }, where 0 < R1 , · · · , Rn ≤ ∞. In case of a = 0, we simply denote ∆(a, R) 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. NEVANLINNA THEORY 11 π 3.1. First main theorem. Let L → − X be a holomorphic line bundle over a compact complex manifold X and d be a positive integer. Let E be a C-vector subspace of dimension m + 1 of H 0 (X, Ld ). Take a basis {ck }m+1 k=1 a basis of E. Put B(E) = ∩σ∈E {σ = 0}. Then ∩1≤i≤m+1 {ci = 0} = B(E) and ω = ddc log(|c1 |2 + · · · + |cm+1 |2 )1/d is well-defined on X \ B(E). Assume that R = (R1 , · · · , Rn ) and R = (R1 , · · · , Rn ), where Rj > 0 and Rj > 0 for each 1 ≤ j ≤ n. Recall that R < R (R ≤ R resp.) if 0 < Rj < Rj (0 < Rj ≤ Rj resp.) for each 1 ≤ j ≤ n. As usual, we say that the assertion P holds for a.e r ≤ R if the assertion P holds for each r ≤ R such that rj is excluded a Borel subset Ej of the interval [0, Rj ] with Ej ds < ∞ 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 the characteristic function of f with respect to E as follows Tf (r, E) = = 1 m(∆r0 ) 1 m(∆r0 ) 0 f ∗ ω ∧ (ddc g∆r ,a )n−1 ds dm(a) −∞ ∆r0 g∆r ,a b. Let f : ∆R → X be an analytically non-degenerate meromorphic mapping with respect to E, i.e f (∆R ) ⊂ supp((σ)) for any σ ∈ E \ {0} and f (∆R ) ∩ B(E) = ∅. Then, for all r ≤ R such that rj does not belong to a set Ej ⊂ [0, Rj ] with Ej Rj1−s ds < ∞, we have q (q − (m + 1)K(E, N, {Dj }))Tf (r, E) ≤ i=1 1 [mdi /d] N (r, Di ) + Sf (r), di f 18 DO DUC THAI AND VU DUC VIET where kN , sN , tN are defined as in Proposition 3.12 and K(E, N, {Dj }) = kN (sN − u + 2 + b) . tN 4. Non-integrated defect relation 4.1. Definitions and basic properties. Let all notations be as in Section 3. The defect of f with respect to D truncated by k in E is defined by [k] Nf (r, D) [k] . δf,E (D) = lim inf 1 − r→R sTf (r, E) We now assume that f is a meromorphic mapping of a connected complex manifold M into X such that f (M ) ∩ B(E) = ∅. Let D be a divisor of H 0 (X, Ls ) for some s > 0. For 0 ≤ k ≤ ∞, denote by [k] Df,E the set of real numbers η ≥ 0 such that there exists a bounded measurable nonnegative function h on M such that 1 ηf ∗ ddc log(|c1 |2 + · · · + |cm+1 |2 )1/d + ddc log h2 ≥ min{k, f ∗ D} s in the sense of currents. The non-integrated defect of f with respect to D in E truncated by k is defined by [k] [k] δ¯f,E (D) := 1 − inf{η : η ∈ Df,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 = CP k , L is the hyperplane bundle and s = 1 he required that functions h h, ϕ are 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 Lebesgue measure. Proposition 4.2. We have the following properties of the non-integrated defect: [k] (i) 0 ≤ δ¯ (D) ≤ 1. f,E [k] (ii) δ¯f,E (D) = 1 if f (M ) ∩ D = ∅. [k] (iii) δ¯ (D) ≥ 1 − k if f ∗ D ≥ k0 min{f ∗ D, 1}. f,E k0 NEVANLINNA THEORY 19 (iv) Denote by fU the restriction of f to U and assume that lim TfU (r, E) = ∞. r→R [k] [k] Then 0 ≤ δ¯f,E (D) ≤ δfU ,E (D) ≤ 1. Proof. The properties (i) and (ii) are evident. To prove (iii), put h= k k0 σ(f ) s 2 (|c1 (f )| + · · · + |cm+1 (f )|2 ) d and η = k . k0 Since f ∗ D ≥ k0 min{f ∗ D, 1}, we get (iii). We now prove (iv). Take a holomorphic function ϕ in ∆R such that [k] (ϕ) = min{fU∗ D, k}. For η ∈ DfU ,E , put 1 v = η log(|c1 (fU )|2 + · · · + |cm+1 (fU )|2 ) d + log h − 1 log ϕ. s Then ddc η ≥ 0 and hence, by Corollary 2.3, we get 0≤ ∆r 0 dm(a) m(∆r0 ) vdµ∆r ,a − ∂ ∆r ∆r0 dm(a) m(∆r0 ) v(ddc g∆r ,a )n ∆r s log(|c1 (fU )|2 + · · · + |cm+1 (fU )|2 ) d dµ∆r ,0 + =η − ∂ ∆r log h dµ∆r ,0 ∂ ∆r ∂ ∆r 1 log ϕ dµ∆r ,0 − s 1 [k] v dm(a) ≤ ηTfU (r, E) − NfU (r, D) + K, s ∆r0 where K is a constant, because h is bounded from above. This implies that [k] NfU (r, D) K 1− ≥1−η+ . sTfU (r, E) TfU (r, E) [k] [k] Letting r → R, we obtain δ¯ (D) ≤ δ (D). f,E fU ,E 4.2. Defect relation with a truncation. Now we give the nonintegrated defect with a truncation for meromorphic mappings from a submanifold of Cl to a compact complex manifold. Theorem 4.3. Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler form of Cl . Let L → X be a holomorphic line bundle over a compact manifold X. Fix a positive integer d and let d1 , d2 , · · · , dq be positive divisors of d. Let E be a C-vector subspace of dimension m + 1 of H 0 (X, Ld ). Put u = rankE and b = dimB(E) + 1 if B(E) = ∅, otherwise b = −1. Let σj (1 ≤ j ≤ q) be in H 0 (X, Ldj ) such d d d that σ1d1 , · · · , σq q ∈ E. Set Dj = (σj )0 (1 ≤ j ≤ q). Assume that 20 DO DUC THAI AND VU DUC VIET D1 , · · · , Dq are in N -subgeneral position with respect to E and u > b. Let f : M → X be an analytically non-degenerate meromorphic mapping with respect to E, i.e f (M ) ⊂ supp((σ)) for any σ ∈ E \ {0} and f (M ) ∩ B(E) = ∅. Assume that, for some ρ ≥ 0 and for some basis {ck }m+1 k=1 of E, there exists a bounded measurable function h ≥ 0 on M such that ρf ∗ ddc log(|c1 |2 + · · · + |cm+1 |2 )1/d + ddc log h2 ≥ Ric ω. Then, q [md /d] δ¯f,E i (Di ) ≤ kN + K (E, N, {Dj }), i=1 where kN , sN , tN are defined as in Proposition 3.12 and K (E, N, {Dj }) is the constant given in the end of the proof. d di Proof. Put σi = 1≤j≤m+1 aij cj , where aij ∈ C. We define a meromorphic mapping Φ : X → CP m by Φ(x) := [c1 (x) : · · · : cm+1 (x)]. Also since X compact, Y = Φ(X) is an algebraic variety of CP m . Moreover, by definition of rankE, Y is of dimension rankE = u. Put F = Φ ◦ f . Since f (M ) ∩ B(E) = ∅ and f is non-degenerate with respect to E, F is meromorphic and linearly non-degenerate. Denote by Hm the hyperplane bundle of CP m . Put Hi := 1≤j≤m+1 aij zj−1 , where [z0 , z1 , · · · , zm ] is the homogeneous coordinate of CP m . 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. For convenience, we still denote by f, F their restrictions to U. It is easy to see that 1 Tf (r, L) = TF (r, Hm ) and Nf (r, Di ) = NF (r, Hi ). (8) d Moreover, we get the following. [kd /d] [k] [kd /d] • NFk (r, Hi ) = Nf i (r, Di ), δ¯F,Hm (Hi ) = δ¯f,Ei (r, Di ). • Hj1 ∩ · · · ∩ Hjt ∩ Y ⊂ Φ(B(E)) if Dj1 ∩ · · · ∩ Djt = B(E) . Put K1 = {R ⊂ {1, 2, · · · , q + m − u + b + 1} : |R |= rank(R ) = m + 1}. Without loss of generality we may assume Rj = R∗ (1 ≤ j ≤ n). From now on, we just consider n-tuples r = (r1 , · · · , rn ) such that rj = r∗ (1 ≤ j ≤ n). Suppose that lim sup r→R Tf (r, E) = ∞. − log(R∗ − r∗ ) NEVANLINNA THEORY 21 Then by Theorem 3.13, we have q [md /d] δf,E i (Dj ) ≤ (m+1)K(E, N, {Dj }) ≤ j=1 kN (m+1)(sN +m−2(u−b−1)). tN By Proposition 4.2, we get q kN [md /d] (m+1)(sN +m−2(u−b−1)). δ¯f,E i (Dj ) ≤ (m+1)K(E, N, {Dj }) ≤ t N j=1 Hence, we can assume lim sup r→R Tf (r, E) < ∞. − log(R∗ − r∗ ) Let α1 , · · · , αm+1 be as in Proposition 3.10. Set l0 = |α1 |+ · · · + |αm+1 | and take t, p with 0 < l0 t ≤ p < 1. Put ω(j) − (m + 1)(sN − u + 2 + b) + m − u + 1 + b. lN = j∈Q By Proposition 3.9 and the proof of Theorem A in [3], we get the following. Claim 1. Let p be a real positive number such that m+1 |αi |≤ p p i=1 m(m + 1) < 1. 2 Then there exists a positive constant K such that n |W (F )(z)|sN −u+2+b i i=1 ∆ (r∗ ) q+m−u+b+1 |Hj (F (z))|ω(j) ) j=1 p ||F (z)||plN dmi (z) (R∗ )3n ≤C TF (r∗ , L) (R∗ − r∗ )3n for each 0 < r∗ < R∗ and r∗ outside a set E satisfying Claim 2. p m(m+1) 2 1 dt E R∗ −t < ∞. [m] ω(j)(νHj (F ) − νHj (F ) ) ≤ (sN − u + 2 + b)νW (F ) . 1≤j≤q+m−u+b+1 By definition of the non-integrated defect, there exist ηi ≥ 0 (1 ≤ i ≤ q + m − u + b + 1) and a nonnegative functions hi such that ηi F ∗ ddc log(|z1 |2 + · · · + |zm+1 |2 ) + ddc log h2i ≥ min{m, F ∗ Hi } 22 DO DUC THAI AND VU DUC VIET [m] and 1 − ηi ≤ δ¯F,Hm (Hi ) ≤ 1. Take a holomorphic function ϕi in M such that (ϕi ) = min{m, F ∗ Hi }. Put ui = Θ log h2i + log(|F1 |2 + · · · |Fm+1 |2 )ηi /2 , Ki where Ki is a constant which is greater than h2i and Θ is the constant in Proposition 3.12. By the above inequality, we see that ui − Θ log ϕi is a plurisubharmonic function on M and eui ≤ ||F ||ηi Θ . Put q+m−u+b+1 (1 − ηi ) s= i=1 and v := log |W (F )(z)|sN −u+2+b q+m−u+b+1 |Hj (F (z))|ω(j) j=1 q+m−u+b+1 + ui . i=1 Since ui − Θ log ϕi (Θ ≥ ω(j)) is a psh function and by virtue of Claim 2, it implies that v is a psh function. On the other hand, by the hypothesis, there exist ρ > 0 and a nonnegative bounded function h such that ρ ΩF + ddc log h2 ≥ Ric ω, d where ω is the K¨ahler form on M. √ ||F ||ρ/d h2 Put w = log K0 det(hij ) , where ω = 2−1 i,j hij dzi ∧ d¯ zj in U , K0 is a 2 constant which is greater than h . Then we have ew ω n ≤ ||F ||ρ/d dx1 ∧ · · · ∧ dxn and ω is a psh function. Hence, ew d vol ≤ ||F ||ρ/d dx1 ∧ · · · ∧ dxn in U, where d vol stands for the volume form of M with respect to the given K¨ahler metric. Put ρ/d α= , lN + Θ(s − q) χ= |W (F )(z)|sN −u+2+b q+m−u+b+1 |Hj (F (z))|ω(j) j=1 NEVANLINNA THEORY 23 and w1 = w + αv. Then w1 is plurisubharmonic. Hence, ew1 is also plurisubharmonic. We have |χ|α ||F ||ρ/d+Θα(q+m−u+b+1−s) dm(z) ew1 d vol ≤ ∆r ∆r |χ|α ||F ||αlN dm(z) = ∆r Suppose α = 3nm(m + 1)α < 1. Then by Claim 1, for each r∗ outside a set E with E R∗1−s ds < ∞, we have n α αlN |χ| ||F || i ∆ (r∗ ) i=1 (R∗ )3n dmi (z) ≤ C TF (r∗ , L) (R∗ − r∗ )3n Conbining with the fact that lim supr→R n i |χ|α ||F ||αlN dmi (z) ≤ K1 ∆ (r∗ ) i=1 Tf (r,E) − log(R∗ −r∗ ) 1 ∗ (R − r∗ )α α m(m+1) 2 . < ∞, we get log 1 ∗ R − r∗ α /4 for some K1 and r∗ ∈ [0, R∗ ) − \E. Varying K1 slightly, we may assume the above inequality holds for all r∗ ∈ [0, R∗ ) by [6, Proposition 5.5]. From this, we conclude that R∗ w1 e d vol ≤ K1 U 0 1 (R∗ − t)α log 1 R∗ − t α /4 dt + O(1) < ∞ Combining with the fact that M \ U has zero measure, we get ew1 d vol < ∞. M By Proposition 2.10, we get a contradiction. Hence, 3nm(m + 1)α ≥ 1. This means lN 3ρnm(m + 1) +q+m−u+b+1− ≥ s. Θd Θ Put tN − (m + 1)(sN − u + 2 + b) + m − u + b + 1 + Θ 3ρnm(m + 1) . Θd By a direct computation and note that K (E, N, {Dj }) = q+m−u+b+1 [md /d] δ¯f,E i (Di )|< q |s − i=1 24 DO DUC THAI AND VU DUC VIET for > 0 small enough and Θ ≥ tN /kN , and lN ≥ Θ(q − kN ) + tN − (m + 1)(sN − u + 2 + b) + m − u + b + 1. we obtain the desired inequality. In the case where X is the complex projective space, L is the hyperplane bundle of X and Dj are hyperplanes in N -subgeneral position, we get the following. Corollary 4.4. Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler form of Cl . Let f : M → CP m be a linear non-degenerate meromorphic map. Let {Hj } be a family of hyperplanes in N -subgeneral position in CP m . Denote by Ωf the pull-back of the Fubini-Study form of CP m by f. Assume that, for some ρ ≥ 0, there exists a bounded measurable function h ≥ 0 on M such that ρΩf + ddc log h2 ≥ Ric ω. Then, q [m] δ¯f,E (Di ) ≤ (2N − m + 1) + 2ρnm(2N − m + 1). i=1 Corollary 4.5. Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler form of Cl . Let f : M → CP m be a meromorphic mapping. Denote by Ωf the pull-back of the Fubini-Study form of CP m by f. Assume that f satisfies the following two conditions: (i) Assume that, for some ρ ≥ 0, there exists a bounded measurable function h ≥ 0 on M such that ρΩf + ddc log h2 ≥ Ric ω, (ii) f omits ((2N − m + 2) + 2ρnm(2N − m + 1)) hyperplanes in N -subgeneral position in CP m . Then f is linearly degenerate. Remark 4.6. By using our arguments and their techniques in [16], [18], [19], we can generalize exactly their results to meromorphic mappings from a Stein manifold without the assumption (H) into a smooth complex projective variety V ⊂ CP M . Let D1 , · · · , Dq be hypersurfaces in CP n , where q > n. Also, the hupersurfaces D1 , · · · , Dq are said to be in general position in CP n if for every subset {i0 , · · · , in } ⊂ {1, · · · , q}, Di0 ∩ · · · ∩ Din = ∅. NEVANLINNA THEORY 25 We now can prove the following improvement of [16, Theorem 1.1]). Theorem 4.3’. Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler form of Cl . Let f : M → CP n be a meromorphic map which is algebraically nondegenerate (i.e. its image is not contained in any proper subvariety of CP n ). Denote by Ωf the pull-back of the FubiniStudy form of CP n by f. Let D1 , · · · , Dq be hypersurfaces of degree dj in CP n , located in general position. Let d = l.c.m.{d1 , · · · , dq } (the least common multiple of {d1 , · · · , dq }). Assume that, for some ρ ≥ 0, there exists a bounded continuous function h ≥ 0 on M such that ρΩf + ddc log h2 ≥ Ric ω. Then for every > 0, q i=1 2 ρl(l − 1) [l−1] δ¯f,Hn (Di ) ≤ (n + 1) + + , d where l ≤ 2n +4n en d2n (nI( a positive real number x. −1 ))n and I(x) := min{k ∈ N : k > x} for We now recall the definition of the subgeneral position in the sense of [18, Theorem 1.2]. Let V ⊂ CP N be a smooth complex projective variety of dimension n ≥ 1. Let n1 ≥ n and q ≥ 2n1 − n + 1. Hypersurfaces D1 , · · · , Dq in CP N with V ⊆ Dj for all j = 1, ..., q are said to be in n1 -subgeneral position in V if the two following conditions are satisfied: (i) For every 1 ≤ j0 < · · · < jn1 ≤ q, V ∩ Dj0 ∩ · · · ∩ Djn1 = ∅. (ii) For any subset J ⊂ {1, · · · , q} such that 0 < |J| ≤ n and {Dj , j ∈ J} are in general position in V and V ∩ (∩j∈J Dj ) = ∅, there exists an irreducible component σJ of V ∩ (∩j∈J Dj ) with dimσJ = dim V ∩ (∩j∈J Dj ) such that for any i ∈ {1, · · · , q} \ J, if dim V ∩ (∩j∈J Dj ) = dim V ∩ Di ∩ (∩j∈J Dj ) , then Di contains σJ . We now can prove the following improvement of [18, Theorem 1.2]). Theorem 4.3”. Let V ⊂ CP N be a smooth complex projective variety of dimension n ≥ 1. Let n1 ≥ n and q ≥ 2n1 − n + 1. Let D1 , · · · , Dq be hypersurfaces in CP N of degree dj , in n1 -subgeneral position in V. Let d = l.c.m.{d1 , · · · , dq } (the least common multiple of {d1 , · · · , dq }). Let be an arbitrary constant with 0 < < 1. Set m := 4dn (2n + 1)(2n1 − n + 1) deg V · 1 +1, 26 DO DUC THAI AND VU DUC VIET where [x] := max{k ∈ Z : k ≤ x} for a real number x. Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler form of Cl . Let f be an algebraically nondegenerate meromorphic map of M into V . Denote by Ωf the pull-back of the Fubini-Study form of CP N by f. For some ρ ≥ 0, if there exists a bounded continuous function h ≥ 0 on M such that ρΩf + ddc log h2 ≥ Ric ω, then we have q [l] δ¯f,HN (Dj ) ≤ 2n1 − n + 1 + q + ρT j=1 for some positive integers l, T satisfying N +md l≤ N +md md (2n1 − n + 1) · md and T ≤ . d(m − (n + 1)(2n + 1)dn deg V ) With the same definition of hypersurfaces in subgeneral position as in Definition 3.11, we can also prove the following improvement of [19, Theorem 1.1]). Theorem 4.3”’. Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler form of Cl . Let f be an algebraically nondegenerate meromorphic map of M into CP n . Let D1 , · · · , Dq be hypersurfaces in CP n of degree dj , in k-subgeneral position in CP n . Let d = l.c.m.{d1 , · · · , dq } (the least common multiple of {d1 , · · · , dq }). Denote by Ωf the pull-back of the Fubini-Study form of CP n by f. Assume that for some ρ ≥ 0, there exists a bounded continuous function h ≥ 0 on M such that ρΩf + ddc log h2 ≥ Ric ω. Then, for each > 0, we have q ρl(l − 1) [l−1] δ¯f,Hn (Dj ) ≤ k(n + 1) + + , d j=1 where l = N +n n ≤ (3ekdI( −1 ))n (n+1)3n for N := 2kdn2 (n+1)2 I( −1 ). 4.3. Defect relation with no truncation. In this case, we get the following sharp defect relation. Theorem 4.7. Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler form of Cl . Let L → X be a holomorphic line bundle over NEVANLINNA THEORY 27 a compact manifold X. Fix a positive integer d and let d1 , d2 , · · · , dq be positive divisors of d. Let E be a C-vector subspace of dimension m + 1 of H 0 (X, Ld ). Let σj (1 ≤ j ≤ q) be in H 0 (X, Ldj ) such that d d d σ1d1 , · · · , σq q ∈ E. Set Dj = (σ)0 (1 ≤ j ≤ q). Assume that D1 , · · · , Dq are in N -subgeneral position with respect to E Let f : M → X be a meromorphic mapping satisfying f (M ) ⊂ Dj for 1 ≤ j ≤ q and f (M ) ∩ B(E) = ∅. Assume that, there exists a holomorphic section −1 ν of KM such that for some basis {c1 , c2 , · · · , cm+1 } of E and l large enough, ddc log(|c1 (f )|2 + · · · + |cm+1 (f )|2 )l/d ≥ ddc (ννω n ). Then, q δ¯f,E (Di ) ≤ 2N. i=1 Before proving the above theorem, we will give a modification of [4, Theorem 2’]. Proposition 4.8. Let M, δ1 , δ2 > 0 and q, n ∈ N, q > 2n. Then, there is a number α = α(δ1 , δ2 , M, q, n) > 0 with the following property: If u, u1 , · · · , uq are subharmonic functions in an open neighborhood of ∆1 ⊂ C with Riesz charges ν, ν1 , · · · , νq , respectively such that q νi )(∆1 ) ≤ M, (ν + i=1 then | sup ∆1 1≤k≤n+1 uik − u|dxdy ≤ α, for all 1 ≤ i1 < · · · < in+1 ≤ q. Moreover, there exists r ∈ [1 − δ1 , 1] such that q νi − (q − 2n)ν) (∆r ) > −δ2 . i=1 Proof. Suppose the theorem is false. Then there are a number δ > 0 and a sequence (uj , uj1 , · · · , ujq ), j ∈ N with Riesz charge ν j , νij respectively such that q νij )(∆1 ) ≤ M, j (ν + i=1 and for all r ∈ [1 − δ1 , 1] one has q νi − (q − 2n)ν) (∆r ) ≤ −δ2 . i=1 28 DO DUC THAI AND VU DUC VIET Hence, by passing to a subsequence if necessary, we can assume νij → νi , ν j → ν, Let G ∗ λ be the Green potential of the charge λ in the disk ∆1 . By the Riesz representation formula, we have uji − uj = hji + G ∗ (νij − ν j ), where hji is harmonic in ∆1 . By the proof of [4, Theorem 2’], we get the followings: (i) G ∗ νij → G ∗ νi , G ∗ ν j → G ∗ ν in L1 (∆1 ). (ii) hji → hi uniformly on compact subsets, some of hi may be identical −∞. We can suppose hi = −∞ for 1 ≤ i ≤ q and hi = −∞ for i > q . Note that q − q ≤ n. (iii) Put ui = hi + G ∗ νi (1 ≤ i ≤ q ), u = G ∗ ν, n = n − (q − q ). Then u = sup1≤k≤n+1 uik for all 1 ≤ i1 < · · · < in+1 ≤ q . Hence, by [4, Theorem 2], we get q νi − (q − 2n )ν ≥ 0. i=1 Consequently, q νi − (q − 2n)ν κ := i=1 q q νi − (q − 2n)ν + = i=1 νi + {(q − 2n ) − (q − 2n)}ν. i=q +1 Obviously, the expression in the braces is nonnegative. Therefore, κ ≥ 0. Finally, for a Radon measure λ in a neighborhood of ∆1 , we have λ(∂∆r ) = 0 for all r outside a countable subset of [0, 1]. Thus, we can choose a sequence rn increasingly tending to 1 such that ν j (∂∆rn ) = 0. Hence, q νij (∆rn ) − (q − 2n)ν j (∆rn ) → κ(∆rn ) i=1 as j → ∞. From these, we get a contradiction. Corollary 4.9. Let M, δ > 0 and q, n ∈ N, q > 2n. Let u, u1 , · · · , uq be subharmonic functions in an open neighborhood of ∆R ⊂ C with Riesz charges ν, ν1 , · · · , νq , respectively such that the following two statements satisfied (i) ν(∆r ) → +∞ as r tends to R, NEVANLINNA THEORY 29 (ii) For all 1 ≤ i1 < · · · < in+1 ≤ q, we have 1 r2 | sup ∆r 1≤k≤n+1 uik − u|dxdy = O(1) as r tends to R. Then for each δ > 0, q q νi − (q − 2n)ν (∆r ) ≥ −δ ν(∆r ) + νj (∆r ) i=1 i=1 for r close enough to R. Proof. Put w(z) = u(rz) ui (rz) , wi (z) = q ν(∆r ) + i=1 νj (∆r ) ν(∆r ) + qi=1 νj (∆r ) for z in a neighborhood of ∆1 and r < R. By the condition (i) and Proposition 4.8, we obtain the assertion. By the Jensen formula and Corollary 4.9, we have the EremenkoSodin second main theorem. Corollary 4.10. Let the notations and the hypothesis be as in Corollary 4.9. Then for each δ > 0, 2π q 2π ui (reit ) − (q − 2n)u(reit ) dt > −δ 0 u(reit )dt + O(1) 0 i=1 for all r close enough to R. Here the term O(1) is a constant as r → R, but depends on ui , u. In high dimension, we have Corollary 4.11. Let M, δ > 0 and q, n ∈ N, q > 2n. Let u, u1 , · · · , uq be psh functions in an open neighborhood of ∆R ⊂ Cl such that the following two statements are satisfied (i) ∂ ∆r u dt1 · · · dtl → +∞ as r tends to R, (ii) For all 1 ≤ i1 < · · · < in+1 ≤ q, | sup 1≤k≤n+1 as |z| tends to R. uik (z) − u(z)|= O(1) 30 DO DUC THAI AND VU DUC VIET Then for each δ > 0, q ui − (q − 2n)u dt1 · · · dtl > −δ ∂ ∆r udt1 · · · dtl + O(1) ∂ ∆r i=1 for r close enough to R. Proof. Put 2π 2π u(·, r2 eit2 , · · · , rl eitl )dt2 · · · dtl . ··· w(·) = 0 0 And we define wi (1 ≤ i ≤ q) in the similar manner. The radius r = (r1 , · · · , rn ) is chosen close enough to R. It is easy to see that w, wi satisfy conditions in Corollary 4.9. Proof of Theorem 4.7. We still use notations as in the first paragraph of the proof of Theorem 4.3. We now suppose on the contrary. By definition of the non-integrated defect, there exist ηi ≥ 0 (1 ≤ i ≤ q) and nonnegative functions hi such that ηi F ∗ ddc log(|z1 |2 + · · · + |zm+1 |2 ) + ddc log h2i ≥ F ∗ Hi and 1 − ηi ≤ δ¯F,Hm (Hi ) ≤ 1. Put η = qi=1 (1 − ηi ). Therefore, q F ∗ Hi , (q − η)ddc log||F ||2 +ddc log h 2 ≥ i=1 where h is measurable and bounded. Subtracting (q − 2n)ddc log||F ||2 from the two sides of the above inequality, we get q c F ∗ Hi − (q − 2n)ddc log||F ||2 +(η − 2n)ddc log||F ||2 . 2 dd log h ≥ i=1 Note that by f (M ) ∩ B(E) = ∅, we have (9) |log||F ||2 −max1≤j≤N +1 log|Hij (F )|2 |= O(1) for all 1 ≤ i1 < · · · iN +1 ≤ q. Claim 1. ννω n = +∞ ∆R 2 Indeed, denote by |ν| the trivialization of νν on ∆R . If |ν|2 is not integrable over M then we are done. Otherwise, it is integrable hence since ν holomorphic |ν|2 is plurisubharmonic function on M. Taking into account of Theorem 2.9 ones get Claim 1. Claim 2. lim log||F (z)||= +∞. r→R ∂ ∆r NEVANLINNA THEORY 31 Applying Theorem 2.3 for dl log||F || and ννω n , and integrating in a over ∆r0 (for some r0 fixed) and the inequality in the hypothesis of Theorem 4.7 we get d log||F (z)||≥ ννω n + C, l ∂ ∆R ∂ ∆R where R < R and C is a constant that does not depend on R . On the other hand, Rn R1 ννω n = +∞ = ∆R r1 dr1 · · · 0 ννω n rn drn 0 ∂ ∆r n Hence, ∂ ∆ ννω tends to +∞ as r closes to R. That yields the claim R 2. Now, by applying Corollary 4.11 for u = log||F (z)||, ui = Hi (F ) and Jensen’s formula, we get the desired conclusion. We recall the following version of the Bloch-Cartan theorem which plays an essential role in Geometric Function Theory. Theorem 4.12. (see [14, Corollary 3.10.8, p.137]) If a holomorphic map f : C → CP m misses 2m + 1 or more hyperplanes in general position, then it is a constant map. The Bloch-Cartan theorem is generalized to hypersurfaces in general position in CP m by Babets and Eremenko-Sodin Theorem 4.13. (see [1] and [4]) If a holomorphic map f : C → CP m misses 2m + 1 or more hypersurfaces in general position, then it is a constant map. From Theorem 4.7, we have the following Bloch-Cartan theorem for meromorphic mappings from Cl to a smooth algebraic variety V in CP m missing 2N + 1 or more hypersurfaces in N -general position. Corollary 4.14. Let f be a meromorphic mapping of Cl to a smooth algebraic variety V in CP m . Let D1 , · · · , D2N +1 be hypersurfaces of CP m such that V ⊂ Dj and Dj ∩ V are in N -subgeneral position in V. Assume that f omits Dj (1 ≤ j ≤ 2N + 1). Then f is constant. 5. A unicity theorem In [8], the author gave a unicity theorem for meromorphic mappings from a complete K¨ahler manifold satisfying the assumption (H) into the complex projective space CP n . The last aim of this paper is to give an analogous unicity theorem for meromorphic mappings from a Stein manifold without the assumption (H) to a compact complex manifold. 32 DO DUC THAI AND VU DUC VIET Denote by Aρ (M, X) the set of holomorphic mappings f : M → X satisfying the following condition: There exist ρ > 0 and a bounded measurable function h ≥ 0 on M such that ρf ∗ ddc log(|c1 |2 + · · · + |cm+1 |2 )1/d + ddc log h2 ≥ Ric ω, where {ck }m+1 k=1 are a basis of E. In this section, we assume the hypothesis as in the statement of Theorem 4.3 and also keep the notations as in the first part of the proof of Theorem 4.3. Let f, g be in Aρ (M, X). Set F = Φ ◦ f, G = Φ ◦ g. Theorem 5.1. Assume that the following are satisfied. i) f = g on ∪qi=1 (f −1 (Di ) ∪ g −1 (Di )), kN ii) q > (m+1)K(E, N, {Dj })+ (3ρ(γF +γG )+mF +mG +m−u+b+1) tN (for the definition of mF , mG , see below). Then f ≡ g. Firstly, some notations and auxiliary lemmas in [7], [8] are re-used. Let ξ be a holomorphic mapping of M to CP m . Take a point p ∈ M and a reduced representation of ξ as ξ = (ξ0 , · · · , ξm ) in a neighborhood of p. Denote by Mp the field of germs of meromorphic functions in an generated open subset containing p. Let Fpk be the submodule of Mm+1 p |α| α by ∂ ξ/∂z with |α|≤ k, where z = (z1 , z2 , · · · , zn ) is a holomorphic local coordinate around p. Clearly, this definition does not depend on the coordinate z and the reduced representation of f. The k-th rank of f is defined by rξ (k) = rankMp Fpk − rankMp Fpk−1 which is independent of the choice of p ∈ M, if M is connected. Set krξ (k), γξ = k l−1 + (k − l) mξ = min{Aln−1 , (rξ (k) k,l where An−1 l Aλn−1 )}, − λ=1 denotes the number of solutions of the equation t1 +t2 +· · · tn−1 = l, where ti (1 ≤ i ≤ n−1) is a non-negative integer. Lemma 5.2. (see [8, Def. 3.1, Example 3.3 and Paragraph (3.5)] ) (i) We have 0 ≤ mξ ≤ γξ ≤ m(m+1) . 2 (ii) Suppose that n = m and rankξ = n. Then mξ = 1, γξ = n. NEVANLINNA THEORY 33 (iii) Let αi (1 ≤ i ≤ m+1) be n-tuples satisfying the properties given in Proposition 3.10 with respect to ξ. Put αi = (αi1 , αi2 , · · · , αin ). j Then m+1 j=1 αi ≤ mξ for each 1 ≤ i ≤ n. For convenience, we denote by W (ξ) one of the generalized Wronskians Wα1 ···αm+1 (ξ) for some {α1 , · · · , αm+1 } being as in the statement iii) of Lemma 5.2. Remark 5.3. Let notations and hypothesis be as in the statement of Theorem 3.13 and its proof. Then, for all r ≤ R such that rj does not belong to a set Ej ⊂ [0, Rj ] with Ej Rj1−s ds < ∞, we have (q − (m + 1)K(E, N, {Dj }))Tf (r, E) ≤ q+m−u+b+1 ω(j)NHj (F ) (r, 0) − (sN − u + 2 + b)NW (F ) (r, 0) + Sf (r), j=1 where K(E, N, {Dj }) = kN (sN − u + 2 + b)/tN . Remark 5.4. By [7, Proposition 4.5 and Proposition 4.10], the righthanded side of the last inequality in Proposition 3.10 can be improved to be m+1 min{(Fi )0 , mF }. i=1 and the right-handed side of its first inequality can be chosen to be γF . Lemma 5.5. The coefficients of the divisor q+m−u+b+1 νHj (F ) − (sN − u + 2 + b)νW (F ) j=1 are smaller than mF . Proof. Denote by K the set of all subsets K of {1, · · · , q} such that |K|= sN + 1 and ∩j∈K Dj = B(E). Then K is the set of all subsets K ⊂ {1, 2 · · · , q} such that |K|= sN + 1 and ∩j∈K Hj ∩ Y = Φ(B(E)). By [3, Lemma 4.1 and 4.3], there are (m − u) hyperplanes Hq+1 , · · · , Hq+m−u+b+1 in CP m such that {Hj , Hq+i : j ∈ R, 1 ≤ i ≤ m − u + b + 1} are in (sN + m − u + b + 1)-subgeneral position in the usual sense, where R ∈ K. Put K1 = {R ⊂ {1, 2, · · · , q + m − u + b + 1} : |R|= rank(R) = m + 1}. 34 DO DUC THAI AND VU DUC VIET By Proposition 3.12, for any z ∈ M and for any J ⊂ {1, 2, · · · , q} with |J|= N + 1, there exists a subset K (J, z) ∈ K such that νHj (F ) (z) ≤ maxK∈K ω(j)νHj (F ) (z) ≤ j∈J νHj (F ) (z). j∈K j∈K (J,z) Hence, (10) ω(j)νHj (F ) (z) ≤ maxK∈K max|J|=N +1 j∈J νHj (F ) (z). j∈K On the other hand, we have q (11) ω(j)νHj (F ) (z) = max|J|=N +1 j=1 ω(j)νHj (F ) (z). j∈J ω(j)νHj (F ) (z). Combining (11) and (10) and Put LH = q+m−u+b+1 j=1 by Lemma 4.2 [3], we have q+m−u+b+1 (12) LH ≤ maxK∈K νHj (F ) (z) + j∈K νHj (F ) (z) j=q+1 ≤ maxR∈K1 (sN − u + 2 + b) νHj (F ) (z) . j∈R On the other hand, for R ∈ K1 we deduce from [8, Lemma 3.4] that νHj (F ) (z) − νW (F ) (z) ≤ mF . j∈R It follows that νW (F ) (z) + mF ≥ maxR∈K1 (13) νHj (F ) (z). j∈R Therefore we get the conclusion. Proof of Theorem 5.1. Suppose that f ≡ g. We consider two cases: Case 1. lim sup r→R Tf (r, E) Tg (r, E) = ∞ or lim sup = ∞. − log(R − r) r→R − log(R − r) NEVANLINNA THEORY 35 By the remark above and by noting that NHj (F )Hj (G) (r, 0) ≤ Tf (r, E) + Tg (r, E), one can see that q − (m + 1)K(E, N, {Dj }) − (m − u + b + 1) (Tf (r, E) + Tg (r, E)) ≤ q ω(j)NHj (F )Hj (G) (r, 0)−(sN −u+2+b)NW (F )W (G) (r, 0)+(Sf (r)+Sg (r)), j=1 Choose two indices i0 , j0 such that χ := Fi0 Gj0 − Fj0 Gi0 ≡ 0. By the assumption, we have supp ∪qj=1 Hj (F ) ∪ Hj (G) ⊂ suppνχ . By Lemma 5.5, it implies that q ω(j)NHj (F )Hj (G) (r, 0) − (sN − u + 2 + b)NW (F )W (G) (r, 0) j=1 ≤ (mF + mG )Nχ (r, 0) + (Sf (r) + Sg (r)). For |χ|≤ ||F ||||G||, one deduces q−(m+1)K(E, N, {Dj })−(m−u+b+1)−mF −mG (Tf (r, E)+Tg (r, E)) ≤ Sf (r) + Sg (r). This is a contradiction by the condition posed in this case. Case 2. Tf (r, E) Tg (r, E) lim sup < ∞ and lim sup < ∞. r→R − log(R − r) r→R − log(R − r) By asumption, we can take psh functions u1 , u2 such that eu1 det(hij )1/2 ≤ ||F ||ρ/d , eu2 det(hij )1/2 ≤ ||G||ρ/d . Put ω(j) − (m + 1)(sN − u + 2 + b) + m − u + 1 + b, lN = j∈Q t= Also set φ(F ) = ρ . lN − mF − mG |W (F )(z)|sN −u+2+b q+m−u+b+1 |Hj (F (z))|ω(j) j=1 , and a similar notation for G. By Lemma 5.5, the function v := t log φ(F )φ(G) + t(mF + mG ) log|χ| 36 DO DUC THAI AND VU DUC VIET is plurisubharmonic. Since |χ|≤ 2||F ||||G||, we have ||F ||tlN φ(F )t ||G||tlN φ(G)t dx1 ∧ · · · ∧ dxn in U, ev+u1 +u2 d vol ≤ where d vol stands for the volume form of M with respect to the given K¨ahler metric. Put p1 = (γF + γG )/γF , p2 = (γF + γG )/γG . Applying Holder’s inequality with the powers (p1 , p2 ), it implies that 1/p1 v+u1 +u2 e p1 tlN d vol ≤ ∆r ||F || φ(F ) tp1 × ∆r 1/p2 p2 tlN ||G|| tp2 φ(G) ∆r By Proposition 3.12, we get Θ ≥ tN /kN and ω(j) ≥ Θ(q − kN ) + tN j∈Q tN + tN kN ≥ 3ρ(γF + γG ) + mF + mG . ≥ (q − kN ) F +γG ) This yields that 3p1 tγF = 3 lNρ(γ < 1, and similarly 3p2 tγG < 1. −mF −mG Now, proceeding as the last part of the proof of Theorem 4.3 we get a contradiction. Hence, f ≡ g. We have a nice corollary in case of equi-dimension. Corollary 5.6. Let M be a Stein manifold of dimension m. Let f, g be two holomorphic mappings of M to CP m such that they have rank m and belong to Aρ (M, CP m ). Let {Dj }qj=1 be a family of hyperplanes in general position. Suppose that i) f = g on ∪qi=1 f −1 Di ∪ g −1 Di . ii) q > m + 3 + 6mρ. Then f ≡ g. Remark 5.7. In the case where X = CP m and {Dj } is a family of hyperplanes in general position, the difference between our result and the unicity theorem of Fujimoto in [8] only is the coefficient 3 corresponding ρ(γF +γG ). That is caused by the power in the left-handed side of the inequality in Proposition 3.9. Acknowledgements. This work was completed during a stay of the first author at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the staff there, in particular the partially support of VIASM. NEVANLINNA THEORY 37 References [1] V. A. Babets, Picard-type theorems for holomorphic mappings, Siberian Math. J, 25(1984), 195-200. [2] J. Demailly, Complex Analytic and Differential Geometry, Textbook, 2009. [3] Do Duc Thai and Vu Duc Viet, Holomorphic mappings into compact complex manifolds, Preprint, 2011, arXiv:1301.6994. [4] A. E. Eremenko and M. L. 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[19] Q. Yan, Non-integrated defect relation and uniqueness theorem for meromorphic maps of a complete K¨ ahler manifold into P n (C), To appear in J. Math. Anal. and Appl., DOI: 10.1016/j.jmaa.2012.09.008. 38 DO DUC THAI AND VU DUC VIET Department of Mathematics Hanoi National University of Education 136 XuanThuy str., Hanoi, Vietnam E-mail address: doducthai@hnue.edu.vn E-mail address: vuducviet@hnue.edu.vn [...]... + 1) i=1 Corollary 4.5 Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler form of Cl Let f : M → CP m be a meromorphic mapping Denote by Ωf the pull-back of the Fubini-Study form of CP m by f Assume that f satisfies the following two conditions: (i) Assume that, for some ρ ≥ 0, there exists a bounded measurable function h... hypersurfaces in general position, then it is a constant map From Theorem 4.7, we have the following Bloch-Cartan theorem for meromorphic mappings from Cl to a smooth algebraic variety V in CP m missing 2N + 1 or more hypersurfaces in N -general position Corollary 4.14 Let f be a meromorphic mapping of Cl to a smooth algebraic variety V in CP m Let D1 , · · · , D2N +1 be hypersurfaces of CP m such that... because h is bounded from above This implies that [k] NfU (r, D) K 1− ≥1−η+ sTfU (r, E) TfU (r, E) [k] [k] Letting r → R, we obtain δ¯ (D) ≤ δ (D) f,E fU ,E 4.2 Defect relation with a truncation Now we give the nonintegrated defect with a truncation for meromorphic mappings from a submanifold of Cl to a compact complex manifold Theorem 4.3 Let M be an n-dimensional closed complex submanifold of Cl and... from the canonical K¨ahler form of Cl Let f : M → CP n be a meromorphic map which is algebraically nondegenerate (i.e its image is not contained in any proper subvariety of CP n ) Denote by Ωf the pull-back of the FubiniStudy form of CP n by f Let D1 , · · · , Dq be hypersurfaces of degree dj in CP n , located in general position Let d = l .c. m.{d1 , · · · , dq } (the least common multiple of {d1 , · ·... small enough and Θ ≥ tN /kN , and lN ≥ Θ(q − kN ) + tN − (m + 1)(sN − u + 2 + b) + m − u + b + 1 we obtain the desired inequality In the case where X is the complex projective space, L is the hyperplane bundle of X and Dj are hyperplanes in N -subgeneral position, we get the following Corollary 4.4 Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from. .. submanifold of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler form of Cl Let f be an algebraically nondegenerate meromorphic map of M into V Denote by Ωf the pull-back of the Fubini-Study form of CP N by f For some ρ ≥ 0, if there exists a bounded continuous function h ≥ 0 on M such that ρΩf + ddc log h2 ≥ Ric ω, then we have q [l] δ¯f,HN (Dj ) ≤ 2n1 − n + 1 + q + ρT j=1 for some... over a compact complex manifold X of dimension n and E be a C vector subspace of H 0 (X, L) of dimension m + 1 Let {ck }m+1 k=1 be a basis of E and B(E) be the base locus of E Define a mapping Φ : X \ B(E) → CP m 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... and Dj ∩ V are in N -subgeneral position in V Assume that f omits Dj (1 ≤ j ≤ 2N + 1) Then f is constant 5 A unicity theorem In [8], the author gave a unicity theorem for meromorphic mappings from a complete K¨ahler manifold satisfying the assumption (H) into the complex projective space CP n The last aim of this paper is to give an analogous unicity theorem for meromorphic mappings from a Stein manifold. . .NEVANLINNA THEORY 11 π 3.1 First main theorem Let L → − X be a holomorphic line bundle over a compact complex manifold X and d be a positive integer Let E be a C- vector subspace of dimension m + 1 of H 0 (X, Ld ) Take a basis {ck }m+1 k=1 a basis of E Put B(E) = ∩σ∈E {σ = 0} Then ∩1≤i≤m+1 {ci = 0} = B(E) and ω = ddc log( |c1 |2 + · · · + |cm+1 |2 )1/d is well-defined on X \ B(E) Assume that R =... integers l, T satisfying N +md l N +md md (2n1 − n + 1) · md and T ≤ d(m − (n + 1)(2n + 1)dn deg V ) With the same definition of hypersurfaces in subgeneral position as in Definition 3.11, we can also prove the following improvement of [19, Theorem 1.1]) Theorem 4.3”’ Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler form of Cl ... nonintegrated defect with a truncation for meromorphic mappings from a submanifold of Cl to a compact complex manifold Theorem 4.3 Let M be an n-dimensional closed complex submanifold of Cl and ω... Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler form of Cl Let f be an algebraically nondegenerate meromorphic. .. i=1 Corollary 4.5 Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler form of Cl Let f : M → CP m be a meromorphic mapping