MINISTRY OF EDUCATION AND TRAINING

HANOI NATIONAL UNIVERSITY OF EDUCATION

Nguyen Thi Nhung

VALUE DISTRIBUTION OF MEROMORPHIC MAPPINGS

FROM KHLER MANIFOLDS INTO PROJECTIVE VARIETIES

AND ITS APPLICATION

Major: Geometry and Topology

Code: 9.46.01.05

SUMMARY OF MATHEMATICS DOCTOR THESIS

Ha Noi - 2019

The thesis was done at: Ha Noi National University of Education

The suppervisors: Asso. Prof. Dr. Si Duc Quang

Referee 1: Prof. Dr Pham Hoang Hiep, Institute of Mathematics - VAST.

Referee 2: Prof. Dr. Nguyen Quang Dieu, Ha Noi National University of Education

Referee 3: Asso. Prof. Dr. Nguyen Thac Dung, Hanoi University of Science - VNU

This Dissertation will be examined by Examination Board

At: Ha Noi National University of Education

On ......hour ............minute, day .............month ............2019

The thesis can be found at

- Library of Hanoi National University of Education

- National Library of Vietnam

WORKS RELATED TO THE THESIS

[1] S. D. Quang, N. T. Q. Phuong and N. T. Nhung (2017), Non-intergrated

defect relation for meromophic maps from a Kahler manifold intersecting hypersurfaces in subgeneral of P n (C), Journal of Mathematical Analysis and Application, 452 (2017), 4341452.

[2] N. T. Nhung and L. N. Quynh, Unicity of Meromorphic Mappings From

Complete K¨

ahler Manifolds into Projective Spaces, Houston Journal of Mathematics, 44(3) (2018), 769-785.

[3] N. T. Nhung and P. D. Thoan, On Degeneracy of Three Meromorphic Mappings From Complete K¨

ahler Manifolds into Projective Spaces, Comput. Methods Funct. Theory, 19(3) (2019), pp 353–382.

[4] S. D. Quang, L. N. Quynh and N. T. Nhung, Non-integrated defect relation for meromorphic mappings from a K¨ahler manifold with hypersurfaces of

a projective variety in subgeneral position, submitting.

INTRODUCTION

1. Rationale

Nevanlinna theory begins with the study on the value distribution of meromorphic functions. In 1926, R. Nevanlinna extended the classical little Picard’s

theorem by proving the two elegant theorems called the First and the Second

Main Theorem. The work of Nevanlinna has evoked a very strong interest of

research in his theory and a number of important papers have been published.

Recently, many mathematicians have generalized Nevanlinna theory to the

case of meromorphic mappings from K¨aher manifolds into projective varieties.

In 1985, H. Fujimoto studied value distribution theory to the case of a meromorphic map of a complete K¨aher manifold M whose universal covering is biholomorphic to a ball B(R0 ) in Cm . The difference is that there is no parabolic

exhaustion on gereral K¨aher manifolds. Therefore, notions of divisor counting

function, characteristic function as well as proximility function for meromorphic mappings can not be defined. In order to overcome this difficulty, using

property that distances on base spaces are less than or equal covering spaces,

H. Fujimoto transfered problems for meromorphic mapping f from B(R0 ) into

projective space Pn (C). He also introduced new notions as well as new methods

to solve the diferent cases when applying Nevanlinna theory on a ball B(R0 )

comparing with Cm . In more details, He introduced the notion of the nonintegrated defect and obtained some results analogous to the classical defect

relation. In 2012, T. V. Tan and V. V. Truong gave a non-integrated defect

relation for the family of hypersurfaces in subgeneral position. However, their

definition of ”subgeneral position” is quite special, which has an extra condition on the intersection of these q hypersurfaces. Independently, M. Ru and

S. Sogome generalized Fujimoto’s result to the case of meromorphic mappings

intersecting a family of hypersurfaces in general position. After that, some au-

1

thors such as Q.Yan, D. D. Thai and S. D. Quang extended the result of M. Ru

and S. Sogome by considering the case of hypersurfaces in subgeneral position.

However, the above results do not completely extend the results of H. Fujimoto

as well as M. Ru and S. Sogome. Thus, it raises a natural question ”Are there

any ways to establish a better non-integrated defect relation for the family of

hypersurfaces in subgeneral position?” In the thesis, we will give a new method

to answer this question.

Since R. Nevanlinna proved five points theorem, which is also called uniqueness theorem, many authors have extended this one to the case of meromorphic

mappings from Cm into Pn (C). The first results were obtained by H. Fujimoto

and L. Smiley. L. Smiley showed that if two meromorphic mappings f and g

have the same inverse images of 3n + 2 hyperplanes without counting multiplicities, the codimmension of the intersection of inverse images of any two

different hyperplanes is at least 2, f and g coincide on the inverse images of

these hyperplanes then f = g. This condition of Smiley helps us have better

estimation on counting function and many results that improve Smiley’s theorem have been given. Some of the best results in this direction were given by

Z. Chen and Q. Yan, H. H. Giang, L. N. Quynh and S. . Quang. In 1986, after

succeeding in establishing non-integrated defect relation, H. Fujimoto proved

uniqueness theorem for meromorphic mappings of M into Pn (C)) intersecting

family of hyperplanes. However, H.Fujimoto’s result is not the one in direction

of L.Smiley, consequently it doesn’t generalize the mentioned results when we

restrict it in case M = Cm . Therefore, the next our purpose is giving a theorem

that both extends Fujimoto’s result and generalizes ones on Cm .

Beside uniqueness problem, Algebraic dependences of meromorphic mappings have also been intensively studied by many authors. This direction started

with S. Ji’s work and there have been a number of results released. Some of

the best results belong to Z. Chen and Q. Yan, S. D. Quang, S. D. Quang and

L. N. Quynh. Thus, the following question arises naturally: ”Is it possible to

extend algebraic dependence theorem of meromorphic mapping f from Cm to

the case f from M into Pn (C)?” We note that although many authors have

generalized Fujimoto’s uniqueness theorem, generalizations of the dependency

theorems have not been obtained yet. In the last chapter of the thesis, we

introduced some new techniques to give a positive answer for this question.

2

From the above questions, we choose the topic ”Value distribution of

meromorphic mappings from K¨

ahler manifolds into projective varieties and its application” to investigate non-integrated defect relation

for meromorphic mappings intersecting hypersurfaces in subgeneral position,

uniqueness problems as well as algebraic dependence ones for meromorphic

mappings intersecting hyperplanes.

2. Objectives of research

The first aim of this thesis is establishing non-integrated defect relation for

meromorphic mappings from K¨ahler manifolds into projective varieties intersecting hypersurfaces in subgeneral position. The next one is studying uniqueness problems and the last one is examining algebraic dependence theorems of

meromorphic mappings from K¨ahler manifolds into projective spaces intersecting hyperplanes in general or subgeneral position.

3. Object and scope of research

Research objects: non-integrated defect relation, uniqueness problems and

algebraic dependence problems for meromorphic mappings from K¨ahler manifolds into projective varieties.

Research scope: Nevanlinna theory for meromorphic mappings from K¨ahler

manifolds.

4. Methodology

In order to solve problem given in the thesis, we use methods in value distribution and complex theory. Besides using traditional techniques, we also

introduce new techniques to achieve aims of the thesis.

5. Scientific and practical significances

The thesis gives more developed results on non-integrated defect relation for

meromorphic mappings from K¨ahler manifolds into projective varieties intersecting hypersurfaces in subgeneral position. It also proves more deepened theorems on uniqueness problem for meromorhic mappings from K¨ahler manifolds.

In addition, it presents some new results on algebraic dependence problems for

meromorphic mappings from K¨ahler manifolds.

3

This thesis acts as a helpful reference to bachelor, master as well as PhD

students majoring in Nevanlinna theory.

6. Structure

The thesis consists of four chapters (excluding the Introduction and Conclusion):

Chapter I. Overview.

Chapter II. Non-integrated defect relation for meromorphic mappings intersecting hypersurfaces in subgeneral position.

Chapter III. Unicity of meromorphic mappings sharing hyperplanes.

Chapter IV. Algebraic dependences of meromorphic mappings sharing few

hyperplanes.

4

Chapter 1

OVERVIEW

In this chapter, we deeply pay attention to analylizing history and results of

previous authors as well as our new results that we achieved regarding nonintegrated defect problems, uniqueness problems and algebraically dependent

problems for meromorphic mappings from K¨ahler M into projective space

Pn (C), where the universal covering of M is biholomorphic to a ball in Cm .

I. Non-integrated defect relation for meromorphic mappings intersecting hypersurfaces in subgeneral position

Let M be a complete K¨ahler manifold of dimension m. Let f : M −→ Pn (C)

be a meromorphic mapping and Ωf be the pull-back of the Fubini-Study form

Ω on Pn (C) by f .

Definition 1.0.1. For ρ ≥ 0 we say that f satisfies the condition (Cρ ) if

there exists a nonzero bounded continuous real-valued function h on M such

that ρΩf + ddc logh2 ≥ √

Ricω, where Ωf is the full-back of the Fubini-Study

−1

form Ω on Pn (C), ω =

ahler form on M , Ricω =

i,j hi¯j dzi ∧ dz j is K¨

2

√

−1

ddc log(det(hij )), d = ∂ + ∂ and dc =

(∂ − ∂).

4π

Definition 1.0.2. For a positive integer µ0 and a hypersurface D of degree d in

Pn (C) with f (M ) ⊂ D, we denote by νf (D)(p) the intersection multiplicity of

the image of f and D at f (p). The non-integrated defect of f with respect to D

[µ ]

truncated to level µ0 by δf 0 := 1 − inf{η ≥ 0 : η satisfies condition (∗)}. Here,

the condition (*) means that there exists a bounded non-negative continuous

function h on√M whose order of each zero is not less than min{νf (D), µ0 } such

−1 ¯

that dηΩf +

∂ ∂logh2 ≥ [min{νf (D), µ0 }].

2π

5

Definition 1.0.3. Let V be a subvariety of Pn (C) of dimension k > 0. Let N ≥

n and q ≥ N + 1. Let Q1 , ..., Qq be hypersurfaces in Pn (C). The hypersurfaces

Q1 , ..., Qq are said to be in N -subgeneral position with respect to V if

Qj1 ∩ · · · ∩ QjN +1 ∩ V = ∅ for every 1 ≤ j1 < · · · < jN +1 ≤ q.

If N = n then we say that Q1 , . . . , Qq are in general position w.r.t V .

In 1985, H. Fujimoto established non-integrated defect relation for meromorphic mappings intersecting hyperplanes in general position as follows.

Theorem A Let M be an m-dimensional complete K¨ahler manifold and ω be

a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic

to a ball in Cm . Let f : M → Pn (C) be a meromorphic map which is linearly

nondegenerate and satifies condition Cρ . If H1 , · · · , Hq be hyperplanes of Pn (C)

[n]

in general position then qi=1 δf (Hi ) ≤ n + 1 + ρn(n + 1).

In 2012, M. Ru-S. Sogome generalized Theorem A to the case of meromorphic

mappings intersecting a family of hypersurfaces in general position as follows

Theorem B Let M be an m-dimensional complete K¨ahler manifold and ω be

a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic

to a ball in Cm . Let f be an algebraically nondegenerate meromorphic map of M

into Pn (C) and satifies condition Cρ . Let Q1 , ..., Qq be hypersurfaces in Pn (C)

of degree PIj , in k-subgeneral position in Pn (C). Let d = l.c.m.{Q1 , ..., Qq }.

ρu(u − 1)

[u−1]

Then, for each > 0, we have qj=1 δf (Qj ) ≤ n + 1 + +

, where

d

2

u ≤ 2n +4n en d2n (nI(ε−1 ))n and here, for a real number x, we define I(x) :=

min{a ∈ Z ; a > x}.

After that, Q. Yan extended Theorem A to the case of the family of hypersurfaces in subgeneral position. He proved the following.

Theorem C Let M be an m-dimensional complete K¨ahler manifold and ω be

a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic

to a ball in Cm . Let f be an algebraically nondegenerate meromorphic map

of M into Pn (C) and satifies condition Cρ . Let Q1 , ..., Qq be hypersurfaces in

Pn (C) of degree PIj , in k-subgeneral position in Pn (C). Then, for each > 0,

ρu(u − 1)

K0 +n

[u−1]

q

we have

(Qj ) ≤ N (n + 1) + +

, where u =

≤

j=1 δf

n

d

(3eN dI( −1 ))n (n + 1)3n and K0 = 2N dn2 (n + 1)2 I( −1 ).

6

The above result of Q. Yan does not completely extend the results of H.

Fujimoto and M. Ru-S. Sogome. Indeed, when the family of hypersurfaces is in

general position, i.e., k = n, the first term in the right hand side of the defect

relation inequality is n(n + 1), which is bigger than (n + 1) as usual. In usual

principle, to deal with the case of family of hypersurfaces in subgeneral position,

we need to generalize the notion of Nochka weights. However, for the case of

hypersurfaces, there are no Nochka weights constructed. In order to overcome

this difficulty, we will use a technique ”replacing hypersurfaces” proposed by S.

D. Quang. Our main idea to avoid using the Nochka weights is that: each time

when we estimate the auxiliary functions, we will replace N+1 hypersurfaces by

n+1 other new hypersurfaces in general position so that this process does not

change the estimate. By using this technique, we prove the following theorem.

Theorem 2.2.4 Let M be an m-dimensional complete K¨ahler manifold and ω

be a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic to a ball in Cm . Let f be an algebraically nondegenerate meromorphic map

of M into Pn (C) and satifies condition Cρ . Let Q1 , ..., Qq be hypersurfaces in

Pn (C) of degree dj , in k-subgeneral position in Pn (C). Let d = l.c.m.{d1 , ..., dq }.

ρu(u − 1)

[u−1]

Then, for each > 0, we have qj=1 δf (Qj ) ≤ p(n + 1) + +

, where

d

L +n

p = N − n + 1, u = 0n ≤ en+2 (dp(n + 1)2 I( −1 ))n and L0 = (n + 1)d + p(n +

1)3 I( −1 )d.

Then we see that, if the family of hypersurfaces is in general position, i.e.,

k = n, then our result implies the results of H. Fujimoto and also of M. Ru-S.

Sogome.

In the above theorems, f is assumed to be algebraically nondegenerate. In

order to deal with cases that f may be algebraically degenerate, we need to

establish non-integrated defect relation for f from M into variety V of Pn (C)

intersecting hypersurfaces in subgeneral position. We continue to use technique

”replacing hypersurfaces”, we extend Therem ?? to the case of hypersurfaces

in subgeneral position with respect to a projective subvariety of Pn (C). Our

main theorem is stated as follows.

Theorem 2.2.10 Let M be an m-dimensional complete K¨ahler manifold and

ω be a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic to a ball in Cm . Let f be an algebraically nondegenerate meromorphic

map of M into a subvariety V of dimension k in Pn (C) and satifies condition

7

Cρ . Let Q1 , ..., Qq be hypersurfaces in Pn (C) of degree dj , in N -subgeneral position with respect to V . Let d = l.c.m.{d1 , ..., dq } (the least common multiple

[M −1]

of {d1 , ..., dq }). Then, for each ε > 0, we have qj=1 δf 0 (Qj ) ≤ p(k + 1) + ε +

ρεM0 (M0 − 1)

2

, where p = N −k+1, M0 = dk +k deg(V )k+1 ek pk (2k + 4)k lk ε−k + 1

d

and l = (k + 1)q!.

From the above theorem, we obtain the following corollary.

Corollary 1.0.4. Let f : M → Pn (C) be a meromorphic mapping and let

{Qi }qi=1 be hypersurfaces in Pn (C) of degree di , located in general position. Then,

2

ρM0 (M0 − 1)

n

[M −1]

+1 +1+

, for

for every ε > 0, we get qj=1 δf 0 (Qj ) ≤

2

d

some positive integer M0 .

II. Unicity of meromorphic mappings sharing hyperplanes

In 1975, H. Fujimoto proved that two meromorphic mappings of Cm into

Pn (C) which have the same inverse images of 3n + 2 hyperplane counted with

multiplicities must be concide. In 1983, by adding condition the codimmension

of the intersection of inverse images of any two different hyperplanes is at least 2,

L. Smiley proved the unicity of family of meromorphic mappings sharing 3n + 2

hyperplanes of Pn (C) without counting multiplicity. Later on, many authors

have improved the result of L. Smiley by reducing the number of hyperplanes.

The best one in this direction was given by Z. Chen and Q. Yan in 2009,

they showed uniqueness theorem for meromorphic mappings sharing 2n + 3

hyperplanes.

In all above results, authors fixed the condition that the codimmension of

the intersection of inverse images of any two different hyperplanes is at least

2. In 2012, H. H. Giang, L. N. Quynh and S. D. Quang generalized Z. Chen

and Q. Yan’s result under a general condition that the intersections of inverse

images of any k + 1 hyperplanes are of codimension at least two. Namely, they

proved the following theorem.

Theorem D. Let H1 , . . . , Hq be q hyperplanes of Pn (C) in general position

k+1

satifying dim f −1

≤ m − 2 (1 ≤ i1 < . . . < ik+1 ≤ q). Let f, g :

j=1 Hij

M → Pn (C) be linearly nondegenerate meromorphic mappings such that f = g

on ∪qj=1 f −1 (Hj ) ∪ g −1 (Hj ). If q = (n + 1)k + n + 2 then f ≡ g.

In 1986, Fujimoto firstly gave a new type of uniqueness theorem for meromorphic maps of M into Pn (C). His result is stated as follows.

8

Theorem E. Let M be a complete, connected K¨ahler manifold whose universal

covering is biholomorphic to B(R0 ) ⊂ Cm where 0 < R0 ≤ ∞, and let f and

g be linearly non-degenerate meromorphic maps of M into Pn (C). If f and g

satisfy the condition (Cρ ) for some non-negative constant ρ and there exist q

hyperplanes H1 , . . . , Hq in Pn (C) located in general position such that: f = g

on ∪qj=1 f −1 (Hj ) ∪ g −1 (Hj ). If q > n2 + 2n + 1 + ρn(n + 1) then f ≡ g.

As we presented in the Rationale, Theorem E is not in the direction of

dimention of L. Smiley. Therefore, it does not generalize the mensioned results

when we restrict it in the case M = Cm . Following H. H. Giang, L. N. Quynh

and S. D. Quang’s idea, by giving a general condition, we prove the following

theorem. In this result, we not only extend Theorem E but also generalize

Theorem of H. H. Giang, L. N. Quynh and S. D. Quang and the conclusion of

Z. Chen and Q. Yan as well.

Theorem 3.2.1 Let M be a complete, connected K¨ahler manifold whose universal covering is biholomorphic to B(R0 ) ⊂ Cm . Let f, g : M → Pn (C) be

linearly nondegenerate meromorphic mappings. Assume that f and g satisfy

the condition (Cρ ) for some ρ ≥ 0 and there are q hyperplanes H1 , ..., Hq of

Pn (C) in general position such that

(i) dim f −1

k+1

j=1 Hij

≤m−2

(1 ≤ i1 < ... < ik+1 ≤ q),

(ii) f = g on ∪qj=1 f −1 (Hj ) ∪ g −1 (Hj ).

2nkq

> n+1+

q + 2nk − 2k

ρn(n + 1) or q < 2(n + 1)k and q > (n + 1)(k + 1) + ρn(n + 1).

Then we have f ≡ g if either q ≥ 2(n + 1)k and q −

III. Algebraic dependences of meromorphic mappings sharing few

hyperplanes

Let f be a meromorphic mapping of Cm into Pn (C) which is linearly nondegenerate. Let d be a positive integer and let H1 , H2 , . . . , Hq be hyperplanes of

Pn (C) in general position with dim(f −1 (Hi ) ∩ f −1 (Hj )) m − 2 (1 i < j

q). Consider the set F(f, {Hi }qi=1 , d) of all linearly nondegenerate meromorphic

mappings g : Cm → Pn (C) satisfying the following two conditions:

(a) min(ν(f,Hj ) , d) = min(ν(g,Hj ) , d) (1

j

q), where ν(f,Hj ) (z) is the intersecting multiplicity of the mapping f with the hyperplane Hj at the point

z.

9

(b) f (z) = g(z) on

q

−1

j=1 f (Hj ).

In algebraic dependence problems of three meromorphic mappings, we find

conditions such that f 1 , f 2 , f 3 ∈ F(f, {Hi }qi=1 , d) is algebraically dependent

, namely we give conditions such that set {(f 1 (z), f 2 (z), f 3 (z)), z ∈ Cm } is

included in a proper algebraic subset of Pn (C) × Pn (C) × Pn (C).

Algebraic dependence of f 1 , f 2 , f 3 can be obtained by proving a stronger

result f 1 ∧f 2 ∧f 3 ≡ 0 or showing mapping f 1 ×f 2 ×f 3 is algebraically degenerate.

In 2015, S. D. Quang and L. N. Quynh proved the algebraic dependence theorem

of three meromorphic mappings sharing less than 2n + 2 hyperplanes in general

position as follows.

Theorem F. Let f 1 , f 2 , f 3 be three linearly nondegenerate meromorphic mappings of Cm into Pn (C). Let {Hi }qi=1 be a family of q hyperplanes of Pn (C)

in general position with dim f −1 (Hi ) ∩ f −1 (Hj )

m − 2 (1

i < j

q).

Assume that the following conditions are satisfied:

(a) min{ν(f 1 ,Hi ) , n} = min{ν(f 2 ,Hi ) , n} = min{ν(f 3 ,Hi ) , n} (1

i

q),

(b) f 1 = f 2 = f 3 on qi=1 (f 1 )−1 (Hi ).

√

2n + 5 + 28n2 + 20n + 1

If q >

then one of the following assertions holds:

4

(i) There exist 3q + 1 hyperplanes such that

(f u , Hi[ q ]+1 )

(f u , Hi1 ) (f u , Hi2 )

3

= v

= ··· = v

,

v

(f , Hi1 )

(f , Hi2 )

(f , Hi[ q ]+1 )

3

(ii) f 1 ∧ f 2 ∧ f 3 ≡ 0.

Theorem F showed algebraic dependence of three mappings with the number

of hyperplanes q less than 2n + 2 but truncation d was still n. In 2018, S. D.

Quang proved the following result of algebraic dependence of three mappings

in which the number of hyperplanes q was 2n + 1 and truncation was d n or

q was 2n + 2 and truncation was 1.

Theorem G. Let f be a linearly nondegenerate meromorphic mapping of Cm

into Pn (C). Let H1 , . . . , H2n+1 be 2n+1 hyperplanes of Pn (C) in general position

such that

dim f −1 (Hi ) ∩ f −1 (Hj )

m − 2 (1 i < j

2n + 2). Then the map f 1 ×

f 2 × f 3 of Cm into Pn (C) × Pn (C) × Pn (C) is linearly degenerate for every three

mappings f 1 , f 2 , f 3 ∈ F(f, {Hi }2n+1

i=1 , p).

10

Theorem H. If the three mappings f 1 , f 2 , f 3 belong to F(f, {Hi }2n+2

i=1 , 1), then

1

2

3

m

1

2

3

f ∧ f ∧ f ≡ 0 on C . It implies that f , f and f are algebraically dependent

over C.

By considering new auxiliary functions and rearranging hyperplanes into

groups, we will generalize Theorem F, G and H for meromorphic mappings f

from K¨ahler manifolds M into Pn (C). Moreover, we also extend the results from

hyperplanes in general position to subgeneral position. Concretely, we prove

the following theorems.

Theorem 4.2.3 Let M be a complete and connected K¨ahler manifold whose

universal covering is biholomorphic to B(R0 ) ⊂ Cm , where 0 < R0

∞. Let

1

2

3

n

f , f , f : M → P (C) be three linearly nondegenerate meromorphic mappings

which satisfy the condition (Cρ ) for some nonnegative constant ρ and there

are q hyperplanes H1 , H2 , . . . , Hq of Pn (C) in N -subgeneral position such that

dim f −1 (Hi ) ∩ f −1 (Hj )

m − 2 (1 i < j

q). Suppose that we have the

following conditions:

(a) min{ν(f 1 ,Hi ) , n} = min{ν(f 2 ,Hi ) , n} = min{ν(f 3 ,Hi ) , n} (1

(b) f 1 = f 2 = f 3 trn

i

q),

q

1 −1

i=1 (f ) (Hi ).

If q > 2N − n + 1 + ρn(n + 1) +

3nq

then one of the following assertions

2q + 3n − 3

holds:

(i) There exist

q

3

+ 1 hyperplanes such that

(f u , Hi[ q ]+1 )

(f u , Hi1 ) (f u , Hi2 )

3

= v

= ··· = v

,

v

(f , Hi1 )

(f , Hi2 )

(f , Hi[ q ]+1 )

3

(ii) f 1 ∧ f 2 ∧ f 3 ≡ 0 on M.

Theorem 4.2.6 Let M , f 1 , f 2 , f 3 and H1 , . . . , Hq be as in Theorem ??. Let

n 5 and p n be a positive integer. Assume that the following assertions are

satisfied:

(a) min{ν(f 1 ,Hi ) , p} = min{ν(f 2 ,Hi ) , p} = min{ν(f 3 ,Hi ) , p} (1

(b) f 1 = f 2 = f 3 on

i

q),

q

1 −1

i=1 (f ) (Hi ).

q(2n + p)

then the map f 1 × f 2 × f 3 of M

2q − 3 + 3p

n

n

n

into P (C) × P (C) × P (C) is linearly degenerate.

If q > 2N − n + 1 + ρn(n + 1) +

Theorem 4.2.4 Let M , f 1 , f 2 , f 3 and H1 , . . . , Hq be as in Theorem ?? Assume

11

that the following assertions are satisfied:

(a) min{ν(f 1 ,Hi ) , 1} = min{ν(f 2 ,Hi ) , 1} = min{ν(f 3 ,Hi ) , 1} (1

(b) f 1 = f 2 = f 3 on

i

q),

q

1 −1

i=1 (f ) (Hi ).

3nq

, then f 1 ∧ f 2 ∧ f 3 ≡ 0 on M . It

2q + 2n − 2

implies that three mappings f 1 , f 2 and f 3 are algebraically dependent on M .

If q > 2N − n + 1 + ρn(n + 1) +

12

Chapter 2

NON-INTEGRATED DEFECT RELATION FOR

MEROMORPHIC MAPPINGS INTERSECTING

HYPERSURFACES IN SUBGENERAL POSITION

As we presented in the Overview, the main goal of chapter two is establishing

defect relation for meromorphic mappings from K¨ahler manifolds into projective

varieties intersecting hypersurfaces in subgeneral position. By using technique

”replacing hypersurfaces” proposed by S. D. Quang, we succeeded in gereralizing the theorem of M. Ru and S. Sogome to the case of hypersurfaces in

subgeneral position and extended the results of previous authors as well.

Chapter two is written based on article [1] and [4] (in Works related to the

thesis).

2.1

Basic notions

In this section, first we present some basic notions and important results in

Nevanlinnas theory related to thesis such as: Nevanlinna’s basic functions,

Nevanlinna’s defect, Wronskian, First Main Theorem and Lemma on logarithmic derivative. Then, we recall definition as well as properties of non-integrate

defect. Last, we present Chow weight, Hilbert weight and some properties

which will be used later.

13

2.2

Non-integrate defect relation for meromorphic mappings

In this section, we prove two main theorems about non-integrate defect relation for meromorphic mappings from M into projective varieties and projective

spaces intersecting hypersurfaces in subgeneral position. We start with recalling some auxiliary lemmas. Lemma 2.2.1 and Lemma 2.2.2 give us important

statements which are: From a family of hypersurfaces in subgeneral position

(with respect to V ), we construct a family of hypersurfaces in general position

(with respect to V ) which each of these hypersurfaces can represent linearly

through given hypersurfaces. This is a basic idea of the technique ”replacing

hypersurfaces” mentioned in the Rationale as well as in the Overview. It is

also a key technique to find new results when we establish non-integrate defect

relation for hypersurfaces in subgeneral position.

Lemma 2.2.1. Let V be a smooth projective subvariety of Pn (C) of dimension

k. Let Q1 , ..., QN +1 be hypersurfaces in Pn (C) of the same degree d ≥ 1, such

N +1

that

i=1 Qi ∩ V = ∅. Then there exists k hypersurfaces P2 , ..., Pk+1 of the

−k+t

ctj Qj , ctj ∈ C, t = 2, ..., k +1, such that

forms Pt = N

j=2

where P1 = Q1 .

k+1

t=1 Pt

∩V = ∅,

When V = Pn (C), Lemma 2.2.1 is stated as follows.

Lemma 2.2.2. Let Q1 , ..., Qk+1 be hypersurfaces in Pn (C) of the same degree

k+1

d ≥ 1, such that

i=1 Qi = ∅. Then there exist n hypersurfaces P2 , ..., Pn+1 of

the forms Pt =

where P1 = Q1 .

k−n+t

ctj Qj ,

j=2

ctj ∈ C, t = 2, ..., n + 1, such that

n+1

t=1 Pt

= ∅,

Lemma 2.2.3. Let {Qi }i∈R be a family of hypersurfaces in Pn (C) of the common degree d and let f be a meromorphic mapping of Cm into Pn (C). Assume that i∈R Qi = ∅. Then, there exist positive constants α and β such that

α||f ||d ≤ maxi∈R |Qi (f )| ≤ β||f ||d .

Theorem 2.2.4. Let M be an m-dimensional complete K¨ahler manifold and

ω be a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic to a ball in Cm . Let f be an algebraically nondegenerate meromorphic

map of M into Pn (C) and satify condition Cρ for ρ ≥ 0. Let Q1 , ..., Qq be

hypersurfaces in Pn (C) of degree dj , in N -subgeneral position in Pn (C). Let

[u−1]

d = l.c.m.{d1 , ..., dq }. Then, for each > 0, we have qj=1 δf (Qj ) ≤ p(n +

14

ρu(u − 1)

, where p = N − n + 1, u =

d

and L0 = (n + 1)d + p(n + 1)3 I( −1 )d.

L0 +n

n

1) + +

In the above theorem, letting = 1 +

we obtain the following corollary.

with

≤ en+2 (dp(n + 1)2 I(

> 0 and then letting

−1

))n

−→ 0,

Corollary 2.2.5. With the assumption of Theorem 2.2.4, we have

ρu(u − 1)

[u−1]

q

δ

(Q

)

≤

p(n

+

1)

+

1

+

, where p = N − n + 1,

j

j=1 f

d

L +n

u = 0n ≤ en+2 (dp(n + 1)2 )n and L0 = (n + 1)d(1 + p(n + 1)2 ).

In order to prove Theorem 2.2.4 we need to prepare some following lemmas.

Now, for a positive integer L, we denote by VL the vector subspace of

C[x0 , . . . , xn ] which consists of all homogeneous polynomials of degree L and

zero polynomial. We see that L0 is divisible by d. Hence, for each (i) =

(i1 , . . . , in ) ∈ Nn0 with σ(i) = ns=1 is ≤ Ld0 , we set

j1

jn

PI1

· · · PIn

· VL0 −dσ(j) .

I

W(i)

=

(j)=(j1 ,...,jn )≥(i)

Lemma 2.2.6. Let (i) = (i1 , . . . , in ), (i) = (i1 , . . . , in ) ∈ Nn0 . Suppose that (i )

I

W(i)

I

follows (i) in the lexicographic ordering and defined m(i) = dim I . Then, we

W(i)

I

n

have m(i) = d , provided dσ(i) < L0 − nd.

I

I

I

We assume that VN = W(i)

⊃ W(i)

⊃ · · · ⊃ W(i)

, where (i)s = (i1s , ..., ins ),

1

2

K

I

I

W(i)

follows W(i)

in the ordering and (i)K = ( Nd , 0, . . . , 0). We see that K is

s+1

s

the number of n-tuples (i1 , . . . , in ) with ij ≥ 0 and i1 + · · · + in ≤ Ld0 . We define

WI

mIs = dim W I(i)s for all s = 1, . . . , K − 1 and set mIK = 1.

(i)s+1

Lemma 2.2.7. For L0 = (n + 1)d + p(n + 1)3 I( −1 )d as in the assumption, we

have

puL0

(a)

≤ (N − n + 1)(n + 1) + ,

db

n

(b) u ≤ en+2 dp(n + 1)2 I( −1 ) .

Proposition 2.2.8. b

q

j=1 νQj (f )

− pνW α (φs (f )) ≤ b

q

i=1 min{u

− 1, νQj (f ) }.

Theorem 2.2.9. With the assumption of Theorem 2.2.4 and suppose that M =

Bm (R0 ). Then, we have

q

(q − p(n + 1) − )Tf (r, r0 ) ≤

i=1

15

1 [u−1]

N

(r) + S(r),

d Qi (f˜)

where S(r) ≤ K(log+ R01−r + log+ Tf (r, r0 )) for all 0 < r0 < r < R0 outside a set

E ⊂ [0, R0 ] with E Rdt

< ∞.

0 −t

Theorem 2.2.10. Let M be an m-dimensional complete K¨ahler manifold and

ω be a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic to a ball in Cm . Let f be an algebraically nondegenerate meromorphic

map of M into a subvariety V of dimension k in Pn (C) satifying condition (Cρ )

for ρ ≥ 0. Let Q1 , ..., Qq be hypersurfaces in Pn (C) of degree dj , in N -subgeneral

position with respect to V . Let d = l.c.m.{d1 , ..., dq }.

ρεM0 (M0 − 1)

,

d

2

where p = N − k + 1, M0 = dk +k deg(V )k+1 ek pk (2k + 4)k lk ε−k + 1 and l =

(k + 1)q!.

Then, for each ε > 0, we have

[M0 −1]

q

(Qj )

j=1 δf

≤ p(k + 1) + ε +

In the case of the family of hypersurfaces is in general position, we get N = n.

Moreover, since (n−k+1)(k+1) ≤ ( n2 +1)2 for every 1 ≤ k ≤ n, letting ε = 1+ε

with ε > 0 and then letting ε −→ 0 from the above theorem, we obtain the

following corollary for the case f may be algebraically degenerate

Corollary 2.2.11. Let f : M → Pn (C) be a meromorphic mapping and let

{Qi }qi=1 be hypersurfaces in Pn (C) of degree di , located in general position. Then,

for every ε > 0, we get

q

[M0 −1]

δf

(Qj ) ≤

j=1

n

+1

2

2

+1+

ρM0 (M0 − 1)

,

d

for some positive integer M0 .

In order to prove Theorem 2.2.10, we need the following one.

Theorem 2.2.12. With the assumption of Theorem 2.2.10. Then, we have

q

(q − p(k + 1) − ε)Tf (r, r0 ) ≤

i=1

1 [M0 −1]

N

(r) + S(r),

d Qi (f˜)

where S(r) is evaluated as follows:

1

+ log+ Tf (r, r0 )), for all 0 <

R0 − r

dt

r0 < r < R0 outside a set E ⊂ [0, R0 ] with E

< ∞ and K is a positive

R0 − t

constant.

(i) In the case R0 < ∞, S(r) ≤ K(log+

(ii) In the case R0 = ∞, S(r) ≤ K(logr + log+ Tf (r, r0 )), for all 0 < r0 <

r < ∞ outside a set E ⊂ [0, ∞] with E dt < ∞ and K is a positive constant.

16

Chapter 3

UNICITY OF MEROMOPHIC MAPPINGS SHARING

FEW HYPERPLANES

Chapter three concentrates on extending the uniqueness theorem of H. Fujimoto

for meromorphic mappings from K¨ahler manifold M into Pn (C) and generalizing uniqueness theorem in the direction of L. Smiley for meromorphic mappings

from Cm into Pn (C). In H. Fujimoto’s result, he considered a family of hyperplanes {Hj }qj=1 in general position and L. Smiley added condition of dimention

for {Hj }qj=1 , dim f −1 (Hi ) ∩ f −1 (Hj ) ≤ m − 2, (1 ≤ i < j ≤ q). We replace

k+1

the above condition with the following dim f −1

≤ m − 2 (1 ≤

j=1 Hij

i1 < · · · < ik+1 ≤ q)(∗). Then, when the family of hyperplanes are in general

position then condition (∗) is always satified and when k = 1 then (∗) is exactly

Smiley’s condition. Thus, our result not only extend Fujimoto’s theorem but

also generalize results when we restrict to the case M = Cm .

Chapter three is written based on article [2] (in Works related to the thesis).

3.1

Second main theorem for meromorphic mapping from a ball and

hyperplanes in general position

In this section, we prove some auxiliary lemmas that will be used in the following

section.

Lemma 3.1.1. Let f be a linearly nondegenerate meromorphic mapping from

B(R0 ) into Pn (C) and H1 , . . . , Hq be q hyperplanes of Pn (C) in general position.

Set l0 = |α1 | + · · · + |αn+1 | and take t, p with 0 < tl0 < p < 1. Then, for

17

0 < r0 < R0 there exists a positive constant K such that for r0 < r < R < R0 ,

z

, Fn+1 ) t

σm ≤ K

L1 (F ) · · · Ln+1 (F )

α1 +···+αn+1 Wα1 ,...,αn+1 (F1 , · · ·

R2m−1

TF (R, r0 )

R−r

p

.

S(r)

Lemma 3.1.2. Let f be a linearly nondegenerate meromorphic mapping from

B(R0 ) into Pn (C) and H1 , ..., Hq be q hyperplanes of Pn (C) in general position.

Then we have

q

[n]

(q − n − 1)Tf (r, r0 ) ≤

NHj (f ) (r, r0 ) + Sf (r),

j=1

where Sf (r) is evaluated as follows:

1

+ log+ Tf (r, r0 )) for all 0 <

R0 − r

dt

r0 < r < R0 outside a set E ⊂ [0, R0 ) with E

< ∞, where K is a

R0 − t

positive constant.

(i) In the case R0 < ∞, Sf (r) ≤ K(log+

(ii) In the case R0 = ∞, Sf (r) ≤ K(log+ Tf (r, r0 )+logr) for all 0 < r0 < r < ∞

outside a set E ⊂ [0, ∞) with E dt < ∞, where K is a positive constant.

3.2

Uniqueness theorem for meromorphic mappings sharing few hyperplanes

Theorem 3.2.1. Let M be a complete, connected K¨ahler manifold whose universal covering is biholomorphic to B(R0 ) ⊂ Cm . Let f, g : M → Pn (C) be

linearly nondegenerate meromorphic mappings. Assume that f and g satisfy

the condition (Cρ ) for some ρ ≥ 0 and there are q hyperplanes H1 , ..., Hq of

Pn (C) in general position such that

(i) dim f −1

k+1

j=1 Hij

≤ m − 2 (1 ≤ i1 < ... < ik+1 ≤ q),

(ii) f = g on ∪qj=1 f −1 (Hj ) ∪ g −1 (Hj ).

2nkq

> n+1+

q + 2nk − 2k

ρn(n + 1) or q < 2(n + 1)k and q > (n + 1)(k + 1) + ρn(n + 1).

Then we have f ≡ g if either q ≥ 2(n + 1)k and q −

From the above theorem, we have the following remarks.

18

1) If k = 1 and q > 3n + 1 + ρn(n + 1) then

q−

2nkq

≥ q − 2n > n + 1 + ρn(n + 1).

q + 2nk − 2k

So the assumption of Theorem 3.2.1 is satisfied. Hence, our result is valid for

q > 3n + 1 + ρn(n + 1).

2) If k = n then the condition (i) in Theorem 3.2.1 is alway true since the

family of hyperplanes is in a general position. Let q > (n + 1)2 + ρn(n + 1). It

is easy to see that

2q

• If n = 1 then q > 4 + 2ρ implies q ≥ 4 and q −

> 2 + 2ρ.

q

n−1

• If n ≥ 2 and ρ <

then 2(n + 1)n > q > (n + 1)2 + ρn(n + 1).

n

Thus, the assumption of Theorem 3.2.1 is satisfied. Therefore, in the above

cases our theorem implies Theorem D of Fujimoto.

3) In the case M = Cm , by taking ρ = 0, we see that

• If k = 1 then q = (n + 1)k + n + 2 satifies the conditions q ≥ 2(n + 1)k

2nkq

and q −

> n + 1.

q + 2nk − 2k

• If k ≥ 2 then q = (n + 1)k + n + 2 satifies the conditions q < 2(n + 1)k

and q > (n + 1)(k + 1)

in Theorem 3.2.1. Hence, our result is also a generalization of the above conclusion of Giang-Quynh-Quang and the result of Z. Chen and Q. Yan as well.

19

Chapter 4

ALGEBRAIC DEPENDENCES OF MEROMORPHIC

MAPPINGS SHARING FEW HYPERPLANES

In Chapter four, we study algebraic dependences of three meromorphic mappings from K¨ahler M , namely we generalize results in which meromorphic mappings are from Cm into Pn (C). By giving new methods basing on rearrange

hyperplanes into groups and construct appropriate auxiliary lemmas, we succeeded in generalizing algebraically dependent theorems on K¨ahler M .

Chapter four is written based on article [3] (in Works related to the thesis).

4.1

Second main theorem for meromorphic mapping from a ball and

hyperplanes in subgeneral position

In this section, we prove the second main theorem for meromorphic mapping

from a ball into projective space intersecting hyperplanes in subgeneral position.

Proposition 4.1.1. Let H1 , . . . , Hq (q > 2N − n + 1) be hyperplanes in Pn (C)

located in N -subgeneral position. Then there exists a function ω : {1, . . . , q} →

(0, 1] called a Nochka weight and a real number ω

˜ ≥ 1 called a Nochka constant

satisfying the following conditions:

(i) If j ∈ {1, · · · , q}, then 0 < ω(j)˜

ω ≤ 1.

q

(ii) q − 2N + n − 1 = ω

˜ ( j=1 ω(j) − n − 1).

(iii) For R ⊂ {1, . . . , q} with |R| = N + 1, then i∈R ω(i) ≤ n + 1.

−n+1

(iv) Nn ≤ ω

˜ ≤ 2Nn+1

.

(v) Given real numbers λ1 , . . . , λq with λj ≥ 1 for 1 ≤ j ≤ q and given

any R ⊂ {1, . . . , q} and |R| = N + 1, there exists a subset R1 ⊂ R such that

20

|R1 | = rank {Hi }i∈R1 = n + 1 and

ω(j)

≤

λj

λi .

i∈R1

j∈R

We note that Lemma 4.1.2 and Lemma 4.1.3 generalize correspondingly

Lemma 3.1.1 and Lemma 3.1.2 in Chapter three for case hyperplanes in general

position.

Lemma 4.1.2. Let f be a linearly nondegenerate meromorphic mapping from

B(R0 ) into Pn (C) and H1 , . . . , Hq be q hyperplanes of Pn (C) in N − subgeneral

position. Set l0 = |α0 | + · · · + |αn | and take t, p with 0 < tl0 < p < 1. Then, for

0 < r0 < R0 there exists a positive constant K such that for r0 < r < R < R0 ,

z

α0 +···+αn

Wα0 ...αn (f )

(f, H1 )ω(1) · · · (f, Hq )ω(q)

t

q

j=1

f

ω(j)−n−1 t

σm

R2m−1

p

K

Tf (R, r0 ) ,

R−r

S(r)

where ω(j) are Nochka weights with respect to Hj , 1

j

q.

Lemma 4.1.3. Let f be a linearly nondegenerate meromorphic mapping from

B(R0 ) into Pn (C) and let H1 , H2 , . . . , Hq be q hyperplanes of Pn (C) in N subgeneral position. Then we have

q

[n]

(q − 2N + n − 1)Tf (r, r0 )

N(f,Hj ) (r, r0 ) + Sf (r),

j=1

where Sf (r) is evaluated as follows:

1

+ log+ Tf (r, r0 )) for all 0 <

R0 − r

r0 < r < R0 outside a set E ⊂ [0, R0 ) with E Rdt

< ∞, where K is a

0 −t

positive constant.

(i) In the case R0 < ∞, Sf (r)

K(log+

(ii) In the case R0 = ∞, Sf (r) K(log+ Tf (r, r0 )+logr) for all 0 < r0 < r < ∞

outside a set E ⊂ [0, ∞) with E dt < ∞, where K is a positive constant.

4.2

Algebraic dependences of three meromorphic function sharing

some hyperplanes

Lemma 4.2.1. Let f 1 , f 2 , f 3 be three meromorphic mappings from B(R0 ) into

Pn (C) and H1 , H2 , . . . , Hq are q hyperplanes in Pn (C) satifying

21

f 1 = f 2 = f 3 on ∪qj=1 (f 1 )−1 (Hj ) ∪ (f 2 )−1 (Hj ) ∪ (f 3 )−1 (Hj ). Suppose that

there exist s, t, l ∈ {1, . . . , q} such that

(f 1 , Hs ) (f 1 , Ht ) (f 1 , Hl )

P := Det (f 2 , Hs ) (f 2 , Ht ) (f 2 , Hl ) ≡ 0.

(f 3 , Hs ) (f 3 , Ht ) (f 3 , Hl )

Then we have

q

( min {ν(f u ,Hi ) (z)} −

νP (z)

i=s,t,l

1 u 3

[1]

ν(f 1 ,Hi ) (z))

[1]

ν(f 1 ,Hi ) (z), ∀z ∈

/ S.

+2

i=1

Lemma 4.2.2. Let M , f 1 , f 2 , f 3 and H1 , H2 , . . . , Hq be as in Theorem 4.2.3.

Let P be a holomorphic function on M and β be a positive real number such

that

3

q

[n]

ν(f u ,Hv ) (z)

βνP (z)

(4.1)

u=1 v=1

for all z outside an analytic subset of codimension two. If |P β |

with some positive constants C and α, then

q

C( f 1

f2

f 3 )α

2N − n + 1 + ρn(n + 1) + α.

Theorem 4.2.3. Let M be a complete and connected K¨ahler manifold whose

universal covering is biholomorphic to B(R0 ) ⊂ Cm , where 0 < R0

∞. Let

f 1 , f 2 , f 3 : M → Pn (C) be three linearly nondegenerate meromorphic mappings

which satisfy the condition (Cρ ) for some nonnegative constant ρ and there are

q hyperplanes H1 , H2 , . . . , Hq of Pn (C) in N -subgeneral position such that

dim f −1 (Hi ) ∩ f −1 (Hj )

m − 2 (1

i

q).

Suppose that we have the following conditions:

(a) min{ν(f 1 ,Hi ) , n} = min{ν(f 2 ,Hi ) , n} = min{ν(f 3 ,Hi ) , n} (1

(b) f 1 = f 2 = f 3 trn

3nq

then one of the following assertions

2q + 3n − 3

holds:

q

3

+ 1 hyperplanes Hi1 , . . . , Hi[ q ]+1 such that

3

(f u , Hi[ q ]+1 )

(f u , Hi1 ) (f u , Hi2 )

3

= v

= ··· = v

,

v

(f , Hi1 )

(f , Hi2 )

(f , Hi[ q ]+1 )

3

1

2

q),

q

1 −1

i=1 (f ) (Hi ).

If q > 2N − n + 1 + ρn(n + 1) +

(i) There exist

i

3

(ii) f ∧ f ∧ f ≡ 0 on M.

22

HANOI NATIONAL UNIVERSITY OF EDUCATION

Nguyen Thi Nhung

VALUE DISTRIBUTION OF MEROMORPHIC MAPPINGS

FROM KHLER MANIFOLDS INTO PROJECTIVE VARIETIES

AND ITS APPLICATION

Major: Geometry and Topology

Code: 9.46.01.05

SUMMARY OF MATHEMATICS DOCTOR THESIS

Ha Noi - 2019

The thesis was done at: Ha Noi National University of Education

The suppervisors: Asso. Prof. Dr. Si Duc Quang

Referee 1: Prof. Dr Pham Hoang Hiep, Institute of Mathematics - VAST.

Referee 2: Prof. Dr. Nguyen Quang Dieu, Ha Noi National University of Education

Referee 3: Asso. Prof. Dr. Nguyen Thac Dung, Hanoi University of Science - VNU

This Dissertation will be examined by Examination Board

At: Ha Noi National University of Education

On ......hour ............minute, day .............month ............2019

The thesis can be found at

- Library of Hanoi National University of Education

- National Library of Vietnam

WORKS RELATED TO THE THESIS

[1] S. D. Quang, N. T. Q. Phuong and N. T. Nhung (2017), Non-intergrated

defect relation for meromophic maps from a Kahler manifold intersecting hypersurfaces in subgeneral of P n (C), Journal of Mathematical Analysis and Application, 452 (2017), 4341452.

[2] N. T. Nhung and L. N. Quynh, Unicity of Meromorphic Mappings From

Complete K¨

ahler Manifolds into Projective Spaces, Houston Journal of Mathematics, 44(3) (2018), 769-785.

[3] N. T. Nhung and P. D. Thoan, On Degeneracy of Three Meromorphic Mappings From Complete K¨

ahler Manifolds into Projective Spaces, Comput. Methods Funct. Theory, 19(3) (2019), pp 353–382.

[4] S. D. Quang, L. N. Quynh and N. T. Nhung, Non-integrated defect relation for meromorphic mappings from a K¨ahler manifold with hypersurfaces of

a projective variety in subgeneral position, submitting.

INTRODUCTION

1. Rationale

Nevanlinna theory begins with the study on the value distribution of meromorphic functions. In 1926, R. Nevanlinna extended the classical little Picard’s

theorem by proving the two elegant theorems called the First and the Second

Main Theorem. The work of Nevanlinna has evoked a very strong interest of

research in his theory and a number of important papers have been published.

Recently, many mathematicians have generalized Nevanlinna theory to the

case of meromorphic mappings from K¨aher manifolds into projective varieties.

In 1985, H. Fujimoto studied value distribution theory to the case of a meromorphic map of a complete K¨aher manifold M whose universal covering is biholomorphic to a ball B(R0 ) in Cm . The difference is that there is no parabolic

exhaustion on gereral K¨aher manifolds. Therefore, notions of divisor counting

function, characteristic function as well as proximility function for meromorphic mappings can not be defined. In order to overcome this difficulty, using

property that distances on base spaces are less than or equal covering spaces,

H. Fujimoto transfered problems for meromorphic mapping f from B(R0 ) into

projective space Pn (C). He also introduced new notions as well as new methods

to solve the diferent cases when applying Nevanlinna theory on a ball B(R0 )

comparing with Cm . In more details, He introduced the notion of the nonintegrated defect and obtained some results analogous to the classical defect

relation. In 2012, T. V. Tan and V. V. Truong gave a non-integrated defect

relation for the family of hypersurfaces in subgeneral position. However, their

definition of ”subgeneral position” is quite special, which has an extra condition on the intersection of these q hypersurfaces. Independently, M. Ru and

S. Sogome generalized Fujimoto’s result to the case of meromorphic mappings

intersecting a family of hypersurfaces in general position. After that, some au-

1

thors such as Q.Yan, D. D. Thai and S. D. Quang extended the result of M. Ru

and S. Sogome by considering the case of hypersurfaces in subgeneral position.

However, the above results do not completely extend the results of H. Fujimoto

as well as M. Ru and S. Sogome. Thus, it raises a natural question ”Are there

any ways to establish a better non-integrated defect relation for the family of

hypersurfaces in subgeneral position?” In the thesis, we will give a new method

to answer this question.

Since R. Nevanlinna proved five points theorem, which is also called uniqueness theorem, many authors have extended this one to the case of meromorphic

mappings from Cm into Pn (C). The first results were obtained by H. Fujimoto

and L. Smiley. L. Smiley showed that if two meromorphic mappings f and g

have the same inverse images of 3n + 2 hyperplanes without counting multiplicities, the codimmension of the intersection of inverse images of any two

different hyperplanes is at least 2, f and g coincide on the inverse images of

these hyperplanes then f = g. This condition of Smiley helps us have better

estimation on counting function and many results that improve Smiley’s theorem have been given. Some of the best results in this direction were given by

Z. Chen and Q. Yan, H. H. Giang, L. N. Quynh and S. . Quang. In 1986, after

succeeding in establishing non-integrated defect relation, H. Fujimoto proved

uniqueness theorem for meromorphic mappings of M into Pn (C)) intersecting

family of hyperplanes. However, H.Fujimoto’s result is not the one in direction

of L.Smiley, consequently it doesn’t generalize the mentioned results when we

restrict it in case M = Cm . Therefore, the next our purpose is giving a theorem

that both extends Fujimoto’s result and generalizes ones on Cm .

Beside uniqueness problem, Algebraic dependences of meromorphic mappings have also been intensively studied by many authors. This direction started

with S. Ji’s work and there have been a number of results released. Some of

the best results belong to Z. Chen and Q. Yan, S. D. Quang, S. D. Quang and

L. N. Quynh. Thus, the following question arises naturally: ”Is it possible to

extend algebraic dependence theorem of meromorphic mapping f from Cm to

the case f from M into Pn (C)?” We note that although many authors have

generalized Fujimoto’s uniqueness theorem, generalizations of the dependency

theorems have not been obtained yet. In the last chapter of the thesis, we

introduced some new techniques to give a positive answer for this question.

2

From the above questions, we choose the topic ”Value distribution of

meromorphic mappings from K¨

ahler manifolds into projective varieties and its application” to investigate non-integrated defect relation

for meromorphic mappings intersecting hypersurfaces in subgeneral position,

uniqueness problems as well as algebraic dependence ones for meromorphic

mappings intersecting hyperplanes.

2. Objectives of research

The first aim of this thesis is establishing non-integrated defect relation for

meromorphic mappings from K¨ahler manifolds into projective varieties intersecting hypersurfaces in subgeneral position. The next one is studying uniqueness problems and the last one is examining algebraic dependence theorems of

meromorphic mappings from K¨ahler manifolds into projective spaces intersecting hyperplanes in general or subgeneral position.

3. Object and scope of research

Research objects: non-integrated defect relation, uniqueness problems and

algebraic dependence problems for meromorphic mappings from K¨ahler manifolds into projective varieties.

Research scope: Nevanlinna theory for meromorphic mappings from K¨ahler

manifolds.

4. Methodology

In order to solve problem given in the thesis, we use methods in value distribution and complex theory. Besides using traditional techniques, we also

introduce new techniques to achieve aims of the thesis.

5. Scientific and practical significances

The thesis gives more developed results on non-integrated defect relation for

meromorphic mappings from K¨ahler manifolds into projective varieties intersecting hypersurfaces in subgeneral position. It also proves more deepened theorems on uniqueness problem for meromorhic mappings from K¨ahler manifolds.

In addition, it presents some new results on algebraic dependence problems for

meromorphic mappings from K¨ahler manifolds.

3

This thesis acts as a helpful reference to bachelor, master as well as PhD

students majoring in Nevanlinna theory.

6. Structure

The thesis consists of four chapters (excluding the Introduction and Conclusion):

Chapter I. Overview.

Chapter II. Non-integrated defect relation for meromorphic mappings intersecting hypersurfaces in subgeneral position.

Chapter III. Unicity of meromorphic mappings sharing hyperplanes.

Chapter IV. Algebraic dependences of meromorphic mappings sharing few

hyperplanes.

4

Chapter 1

OVERVIEW

In this chapter, we deeply pay attention to analylizing history and results of

previous authors as well as our new results that we achieved regarding nonintegrated defect problems, uniqueness problems and algebraically dependent

problems for meromorphic mappings from K¨ahler M into projective space

Pn (C), where the universal covering of M is biholomorphic to a ball in Cm .

I. Non-integrated defect relation for meromorphic mappings intersecting hypersurfaces in subgeneral position

Let M be a complete K¨ahler manifold of dimension m. Let f : M −→ Pn (C)

be a meromorphic mapping and Ωf be the pull-back of the Fubini-Study form

Ω on Pn (C) by f .

Definition 1.0.1. For ρ ≥ 0 we say that f satisfies the condition (Cρ ) if

there exists a nonzero bounded continuous real-valued function h on M such

that ρΩf + ddc logh2 ≥ √

Ricω, where Ωf is the full-back of the Fubini-Study

−1

form Ω on Pn (C), ω =

ahler form on M , Ricω =

i,j hi¯j dzi ∧ dz j is K¨

2

√

−1

ddc log(det(hij )), d = ∂ + ∂ and dc =

(∂ − ∂).

4π

Definition 1.0.2. For a positive integer µ0 and a hypersurface D of degree d in

Pn (C) with f (M ) ⊂ D, we denote by νf (D)(p) the intersection multiplicity of

the image of f and D at f (p). The non-integrated defect of f with respect to D

[µ ]

truncated to level µ0 by δf 0 := 1 − inf{η ≥ 0 : η satisfies condition (∗)}. Here,

the condition (*) means that there exists a bounded non-negative continuous

function h on√M whose order of each zero is not less than min{νf (D), µ0 } such

−1 ¯

that dηΩf +

∂ ∂logh2 ≥ [min{νf (D), µ0 }].

2π

5

Definition 1.0.3. Let V be a subvariety of Pn (C) of dimension k > 0. Let N ≥

n and q ≥ N + 1. Let Q1 , ..., Qq be hypersurfaces in Pn (C). The hypersurfaces

Q1 , ..., Qq are said to be in N -subgeneral position with respect to V if

Qj1 ∩ · · · ∩ QjN +1 ∩ V = ∅ for every 1 ≤ j1 < · · · < jN +1 ≤ q.

If N = n then we say that Q1 , . . . , Qq are in general position w.r.t V .

In 1985, H. Fujimoto established non-integrated defect relation for meromorphic mappings intersecting hyperplanes in general position as follows.

Theorem A Let M be an m-dimensional complete K¨ahler manifold and ω be

a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic

to a ball in Cm . Let f : M → Pn (C) be a meromorphic map which is linearly

nondegenerate and satifies condition Cρ . If H1 , · · · , Hq be hyperplanes of Pn (C)

[n]

in general position then qi=1 δf (Hi ) ≤ n + 1 + ρn(n + 1).

In 2012, M. Ru-S. Sogome generalized Theorem A to the case of meromorphic

mappings intersecting a family of hypersurfaces in general position as follows

Theorem B Let M be an m-dimensional complete K¨ahler manifold and ω be

a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic

to a ball in Cm . Let f be an algebraically nondegenerate meromorphic map of M

into Pn (C) and satifies condition Cρ . Let Q1 , ..., Qq be hypersurfaces in Pn (C)

of degree PIj , in k-subgeneral position in Pn (C). Let d = l.c.m.{Q1 , ..., Qq }.

ρu(u − 1)

[u−1]

Then, for each > 0, we have qj=1 δf (Qj ) ≤ n + 1 + +

, where

d

2

u ≤ 2n +4n en d2n (nI(ε−1 ))n and here, for a real number x, we define I(x) :=

min{a ∈ Z ; a > x}.

After that, Q. Yan extended Theorem A to the case of the family of hypersurfaces in subgeneral position. He proved the following.

Theorem C Let M be an m-dimensional complete K¨ahler manifold and ω be

a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic

to a ball in Cm . Let f be an algebraically nondegenerate meromorphic map

of M into Pn (C) and satifies condition Cρ . Let Q1 , ..., Qq be hypersurfaces in

Pn (C) of degree PIj , in k-subgeneral position in Pn (C). Then, for each > 0,

ρu(u − 1)

K0 +n

[u−1]

q

we have

(Qj ) ≤ N (n + 1) + +

, where u =

≤

j=1 δf

n

d

(3eN dI( −1 ))n (n + 1)3n and K0 = 2N dn2 (n + 1)2 I( −1 ).

6

The above result of Q. Yan does not completely extend the results of H.

Fujimoto and M. Ru-S. Sogome. Indeed, when the family of hypersurfaces is in

general position, i.e., k = n, the first term in the right hand side of the defect

relation inequality is n(n + 1), which is bigger than (n + 1) as usual. In usual

principle, to deal with the case of family of hypersurfaces in subgeneral position,

we need to generalize the notion of Nochka weights. However, for the case of

hypersurfaces, there are no Nochka weights constructed. In order to overcome

this difficulty, we will use a technique ”replacing hypersurfaces” proposed by S.

D. Quang. Our main idea to avoid using the Nochka weights is that: each time

when we estimate the auxiliary functions, we will replace N+1 hypersurfaces by

n+1 other new hypersurfaces in general position so that this process does not

change the estimate. By using this technique, we prove the following theorem.

Theorem 2.2.4 Let M be an m-dimensional complete K¨ahler manifold and ω

be a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic to a ball in Cm . Let f be an algebraically nondegenerate meromorphic map

of M into Pn (C) and satifies condition Cρ . Let Q1 , ..., Qq be hypersurfaces in

Pn (C) of degree dj , in k-subgeneral position in Pn (C). Let d = l.c.m.{d1 , ..., dq }.

ρu(u − 1)

[u−1]

Then, for each > 0, we have qj=1 δf (Qj ) ≤ p(n + 1) + +

, where

d

L +n

p = N − n + 1, u = 0n ≤ en+2 (dp(n + 1)2 I( −1 ))n and L0 = (n + 1)d + p(n +

1)3 I( −1 )d.

Then we see that, if the family of hypersurfaces is in general position, i.e.,

k = n, then our result implies the results of H. Fujimoto and also of M. Ru-S.

Sogome.

In the above theorems, f is assumed to be algebraically nondegenerate. In

order to deal with cases that f may be algebraically degenerate, we need to

establish non-integrated defect relation for f from M into variety V of Pn (C)

intersecting hypersurfaces in subgeneral position. We continue to use technique

”replacing hypersurfaces”, we extend Therem ?? to the case of hypersurfaces

in subgeneral position with respect to a projective subvariety of Pn (C). Our

main theorem is stated as follows.

Theorem 2.2.10 Let M be an m-dimensional complete K¨ahler manifold and

ω be a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic to a ball in Cm . Let f be an algebraically nondegenerate meromorphic

map of M into a subvariety V of dimension k in Pn (C) and satifies condition

7

Cρ . Let Q1 , ..., Qq be hypersurfaces in Pn (C) of degree dj , in N -subgeneral position with respect to V . Let d = l.c.m.{d1 , ..., dq } (the least common multiple

[M −1]

of {d1 , ..., dq }). Then, for each ε > 0, we have qj=1 δf 0 (Qj ) ≤ p(k + 1) + ε +

ρεM0 (M0 − 1)

2

, where p = N −k+1, M0 = dk +k deg(V )k+1 ek pk (2k + 4)k lk ε−k + 1

d

and l = (k + 1)q!.

From the above theorem, we obtain the following corollary.

Corollary 1.0.4. Let f : M → Pn (C) be a meromorphic mapping and let

{Qi }qi=1 be hypersurfaces in Pn (C) of degree di , located in general position. Then,

2

ρM0 (M0 − 1)

n

[M −1]

+1 +1+

, for

for every ε > 0, we get qj=1 δf 0 (Qj ) ≤

2

d

some positive integer M0 .

II. Unicity of meromorphic mappings sharing hyperplanes

In 1975, H. Fujimoto proved that two meromorphic mappings of Cm into

Pn (C) which have the same inverse images of 3n + 2 hyperplane counted with

multiplicities must be concide. In 1983, by adding condition the codimmension

of the intersection of inverse images of any two different hyperplanes is at least 2,

L. Smiley proved the unicity of family of meromorphic mappings sharing 3n + 2

hyperplanes of Pn (C) without counting multiplicity. Later on, many authors

have improved the result of L. Smiley by reducing the number of hyperplanes.

The best one in this direction was given by Z. Chen and Q. Yan in 2009,

they showed uniqueness theorem for meromorphic mappings sharing 2n + 3

hyperplanes.

In all above results, authors fixed the condition that the codimmension of

the intersection of inverse images of any two different hyperplanes is at least

2. In 2012, H. H. Giang, L. N. Quynh and S. D. Quang generalized Z. Chen

and Q. Yan’s result under a general condition that the intersections of inverse

images of any k + 1 hyperplanes are of codimension at least two. Namely, they

proved the following theorem.

Theorem D. Let H1 , . . . , Hq be q hyperplanes of Pn (C) in general position

k+1

satifying dim f −1

≤ m − 2 (1 ≤ i1 < . . . < ik+1 ≤ q). Let f, g :

j=1 Hij

M → Pn (C) be linearly nondegenerate meromorphic mappings such that f = g

on ∪qj=1 f −1 (Hj ) ∪ g −1 (Hj ). If q = (n + 1)k + n + 2 then f ≡ g.

In 1986, Fujimoto firstly gave a new type of uniqueness theorem for meromorphic maps of M into Pn (C). His result is stated as follows.

8

Theorem E. Let M be a complete, connected K¨ahler manifold whose universal

covering is biholomorphic to B(R0 ) ⊂ Cm where 0 < R0 ≤ ∞, and let f and

g be linearly non-degenerate meromorphic maps of M into Pn (C). If f and g

satisfy the condition (Cρ ) for some non-negative constant ρ and there exist q

hyperplanes H1 , . . . , Hq in Pn (C) located in general position such that: f = g

on ∪qj=1 f −1 (Hj ) ∪ g −1 (Hj ). If q > n2 + 2n + 1 + ρn(n + 1) then f ≡ g.

As we presented in the Rationale, Theorem E is not in the direction of

dimention of L. Smiley. Therefore, it does not generalize the mensioned results

when we restrict it in the case M = Cm . Following H. H. Giang, L. N. Quynh

and S. D. Quang’s idea, by giving a general condition, we prove the following

theorem. In this result, we not only extend Theorem E but also generalize

Theorem of H. H. Giang, L. N. Quynh and S. D. Quang and the conclusion of

Z. Chen and Q. Yan as well.

Theorem 3.2.1 Let M be a complete, connected K¨ahler manifold whose universal covering is biholomorphic to B(R0 ) ⊂ Cm . Let f, g : M → Pn (C) be

linearly nondegenerate meromorphic mappings. Assume that f and g satisfy

the condition (Cρ ) for some ρ ≥ 0 and there are q hyperplanes H1 , ..., Hq of

Pn (C) in general position such that

(i) dim f −1

k+1

j=1 Hij

≤m−2

(1 ≤ i1 < ... < ik+1 ≤ q),

(ii) f = g on ∪qj=1 f −1 (Hj ) ∪ g −1 (Hj ).

2nkq

> n+1+

q + 2nk − 2k

ρn(n + 1) or q < 2(n + 1)k and q > (n + 1)(k + 1) + ρn(n + 1).

Then we have f ≡ g if either q ≥ 2(n + 1)k and q −

III. Algebraic dependences of meromorphic mappings sharing few

hyperplanes

Let f be a meromorphic mapping of Cm into Pn (C) which is linearly nondegenerate. Let d be a positive integer and let H1 , H2 , . . . , Hq be hyperplanes of

Pn (C) in general position with dim(f −1 (Hi ) ∩ f −1 (Hj )) m − 2 (1 i < j

q). Consider the set F(f, {Hi }qi=1 , d) of all linearly nondegenerate meromorphic

mappings g : Cm → Pn (C) satisfying the following two conditions:

(a) min(ν(f,Hj ) , d) = min(ν(g,Hj ) , d) (1

j

q), where ν(f,Hj ) (z) is the intersecting multiplicity of the mapping f with the hyperplane Hj at the point

z.

9

(b) f (z) = g(z) on

q

−1

j=1 f (Hj ).

In algebraic dependence problems of three meromorphic mappings, we find

conditions such that f 1 , f 2 , f 3 ∈ F(f, {Hi }qi=1 , d) is algebraically dependent

, namely we give conditions such that set {(f 1 (z), f 2 (z), f 3 (z)), z ∈ Cm } is

included in a proper algebraic subset of Pn (C) × Pn (C) × Pn (C).

Algebraic dependence of f 1 , f 2 , f 3 can be obtained by proving a stronger

result f 1 ∧f 2 ∧f 3 ≡ 0 or showing mapping f 1 ×f 2 ×f 3 is algebraically degenerate.

In 2015, S. D. Quang and L. N. Quynh proved the algebraic dependence theorem

of three meromorphic mappings sharing less than 2n + 2 hyperplanes in general

position as follows.

Theorem F. Let f 1 , f 2 , f 3 be three linearly nondegenerate meromorphic mappings of Cm into Pn (C). Let {Hi }qi=1 be a family of q hyperplanes of Pn (C)

in general position with dim f −1 (Hi ) ∩ f −1 (Hj )

m − 2 (1

i < j

q).

Assume that the following conditions are satisfied:

(a) min{ν(f 1 ,Hi ) , n} = min{ν(f 2 ,Hi ) , n} = min{ν(f 3 ,Hi ) , n} (1

i

q),

(b) f 1 = f 2 = f 3 on qi=1 (f 1 )−1 (Hi ).

√

2n + 5 + 28n2 + 20n + 1

If q >

then one of the following assertions holds:

4

(i) There exist 3q + 1 hyperplanes such that

(f u , Hi[ q ]+1 )

(f u , Hi1 ) (f u , Hi2 )

3

= v

= ··· = v

,

v

(f , Hi1 )

(f , Hi2 )

(f , Hi[ q ]+1 )

3

(ii) f 1 ∧ f 2 ∧ f 3 ≡ 0.

Theorem F showed algebraic dependence of three mappings with the number

of hyperplanes q less than 2n + 2 but truncation d was still n. In 2018, S. D.

Quang proved the following result of algebraic dependence of three mappings

in which the number of hyperplanes q was 2n + 1 and truncation was d n or

q was 2n + 2 and truncation was 1.

Theorem G. Let f be a linearly nondegenerate meromorphic mapping of Cm

into Pn (C). Let H1 , . . . , H2n+1 be 2n+1 hyperplanes of Pn (C) in general position

such that

dim f −1 (Hi ) ∩ f −1 (Hj )

m − 2 (1 i < j

2n + 2). Then the map f 1 ×

f 2 × f 3 of Cm into Pn (C) × Pn (C) × Pn (C) is linearly degenerate for every three

mappings f 1 , f 2 , f 3 ∈ F(f, {Hi }2n+1

i=1 , p).

10

Theorem H. If the three mappings f 1 , f 2 , f 3 belong to F(f, {Hi }2n+2

i=1 , 1), then

1

2

3

m

1

2

3

f ∧ f ∧ f ≡ 0 on C . It implies that f , f and f are algebraically dependent

over C.

By considering new auxiliary functions and rearranging hyperplanes into

groups, we will generalize Theorem F, G and H for meromorphic mappings f

from K¨ahler manifolds M into Pn (C). Moreover, we also extend the results from

hyperplanes in general position to subgeneral position. Concretely, we prove

the following theorems.

Theorem 4.2.3 Let M be a complete and connected K¨ahler manifold whose

universal covering is biholomorphic to B(R0 ) ⊂ Cm , where 0 < R0

∞. Let

1

2

3

n

f , f , f : M → P (C) be three linearly nondegenerate meromorphic mappings

which satisfy the condition (Cρ ) for some nonnegative constant ρ and there

are q hyperplanes H1 , H2 , . . . , Hq of Pn (C) in N -subgeneral position such that

dim f −1 (Hi ) ∩ f −1 (Hj )

m − 2 (1 i < j

q). Suppose that we have the

following conditions:

(a) min{ν(f 1 ,Hi ) , n} = min{ν(f 2 ,Hi ) , n} = min{ν(f 3 ,Hi ) , n} (1

(b) f 1 = f 2 = f 3 trn

i

q),

q

1 −1

i=1 (f ) (Hi ).

If q > 2N − n + 1 + ρn(n + 1) +

3nq

then one of the following assertions

2q + 3n − 3

holds:

(i) There exist

q

3

+ 1 hyperplanes such that

(f u , Hi[ q ]+1 )

(f u , Hi1 ) (f u , Hi2 )

3

= v

= ··· = v

,

v

(f , Hi1 )

(f , Hi2 )

(f , Hi[ q ]+1 )

3

(ii) f 1 ∧ f 2 ∧ f 3 ≡ 0 on M.

Theorem 4.2.6 Let M , f 1 , f 2 , f 3 and H1 , . . . , Hq be as in Theorem ??. Let

n 5 and p n be a positive integer. Assume that the following assertions are

satisfied:

(a) min{ν(f 1 ,Hi ) , p} = min{ν(f 2 ,Hi ) , p} = min{ν(f 3 ,Hi ) , p} (1

(b) f 1 = f 2 = f 3 on

i

q),

q

1 −1

i=1 (f ) (Hi ).

q(2n + p)

then the map f 1 × f 2 × f 3 of M

2q − 3 + 3p

n

n

n

into P (C) × P (C) × P (C) is linearly degenerate.

If q > 2N − n + 1 + ρn(n + 1) +

Theorem 4.2.4 Let M , f 1 , f 2 , f 3 and H1 , . . . , Hq be as in Theorem ?? Assume

11

that the following assertions are satisfied:

(a) min{ν(f 1 ,Hi ) , 1} = min{ν(f 2 ,Hi ) , 1} = min{ν(f 3 ,Hi ) , 1} (1

(b) f 1 = f 2 = f 3 on

i

q),

q

1 −1

i=1 (f ) (Hi ).

3nq

, then f 1 ∧ f 2 ∧ f 3 ≡ 0 on M . It

2q + 2n − 2

implies that three mappings f 1 , f 2 and f 3 are algebraically dependent on M .

If q > 2N − n + 1 + ρn(n + 1) +

12

Chapter 2

NON-INTEGRATED DEFECT RELATION FOR

MEROMORPHIC MAPPINGS INTERSECTING

HYPERSURFACES IN SUBGENERAL POSITION

As we presented in the Overview, the main goal of chapter two is establishing

defect relation for meromorphic mappings from K¨ahler manifolds into projective

varieties intersecting hypersurfaces in subgeneral position. By using technique

”replacing hypersurfaces” proposed by S. D. Quang, we succeeded in gereralizing the theorem of M. Ru and S. Sogome to the case of hypersurfaces in

subgeneral position and extended the results of previous authors as well.

Chapter two is written based on article [1] and [4] (in Works related to the

thesis).

2.1

Basic notions

In this section, first we present some basic notions and important results in

Nevanlinnas theory related to thesis such as: Nevanlinna’s basic functions,

Nevanlinna’s defect, Wronskian, First Main Theorem and Lemma on logarithmic derivative. Then, we recall definition as well as properties of non-integrate

defect. Last, we present Chow weight, Hilbert weight and some properties

which will be used later.

13

2.2

Non-integrate defect relation for meromorphic mappings

In this section, we prove two main theorems about non-integrate defect relation for meromorphic mappings from M into projective varieties and projective

spaces intersecting hypersurfaces in subgeneral position. We start with recalling some auxiliary lemmas. Lemma 2.2.1 and Lemma 2.2.2 give us important

statements which are: From a family of hypersurfaces in subgeneral position

(with respect to V ), we construct a family of hypersurfaces in general position

(with respect to V ) which each of these hypersurfaces can represent linearly

through given hypersurfaces. This is a basic idea of the technique ”replacing

hypersurfaces” mentioned in the Rationale as well as in the Overview. It is

also a key technique to find new results when we establish non-integrate defect

relation for hypersurfaces in subgeneral position.

Lemma 2.2.1. Let V be a smooth projective subvariety of Pn (C) of dimension

k. Let Q1 , ..., QN +1 be hypersurfaces in Pn (C) of the same degree d ≥ 1, such

N +1

that

i=1 Qi ∩ V = ∅. Then there exists k hypersurfaces P2 , ..., Pk+1 of the

−k+t

ctj Qj , ctj ∈ C, t = 2, ..., k +1, such that

forms Pt = N

j=2

where P1 = Q1 .

k+1

t=1 Pt

∩V = ∅,

When V = Pn (C), Lemma 2.2.1 is stated as follows.

Lemma 2.2.2. Let Q1 , ..., Qk+1 be hypersurfaces in Pn (C) of the same degree

k+1

d ≥ 1, such that

i=1 Qi = ∅. Then there exist n hypersurfaces P2 , ..., Pn+1 of

the forms Pt =

where P1 = Q1 .

k−n+t

ctj Qj ,

j=2

ctj ∈ C, t = 2, ..., n + 1, such that

n+1

t=1 Pt

= ∅,

Lemma 2.2.3. Let {Qi }i∈R be a family of hypersurfaces in Pn (C) of the common degree d and let f be a meromorphic mapping of Cm into Pn (C). Assume that i∈R Qi = ∅. Then, there exist positive constants α and β such that

α||f ||d ≤ maxi∈R |Qi (f )| ≤ β||f ||d .

Theorem 2.2.4. Let M be an m-dimensional complete K¨ahler manifold and

ω be a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic to a ball in Cm . Let f be an algebraically nondegenerate meromorphic

map of M into Pn (C) and satify condition Cρ for ρ ≥ 0. Let Q1 , ..., Qq be

hypersurfaces in Pn (C) of degree dj , in N -subgeneral position in Pn (C). Let

[u−1]

d = l.c.m.{d1 , ..., dq }. Then, for each > 0, we have qj=1 δf (Qj ) ≤ p(n +

14

ρu(u − 1)

, where p = N − n + 1, u =

d

and L0 = (n + 1)d + p(n + 1)3 I( −1 )d.

L0 +n

n

1) + +

In the above theorem, letting = 1 +

we obtain the following corollary.

with

≤ en+2 (dp(n + 1)2 I(

> 0 and then letting

−1

))n

−→ 0,

Corollary 2.2.5. With the assumption of Theorem 2.2.4, we have

ρu(u − 1)

[u−1]

q

δ

(Q

)

≤

p(n

+

1)

+

1

+

, where p = N − n + 1,

j

j=1 f

d

L +n

u = 0n ≤ en+2 (dp(n + 1)2 )n and L0 = (n + 1)d(1 + p(n + 1)2 ).

In order to prove Theorem 2.2.4 we need to prepare some following lemmas.

Now, for a positive integer L, we denote by VL the vector subspace of

C[x0 , . . . , xn ] which consists of all homogeneous polynomials of degree L and

zero polynomial. We see that L0 is divisible by d. Hence, for each (i) =

(i1 , . . . , in ) ∈ Nn0 with σ(i) = ns=1 is ≤ Ld0 , we set

j1

jn

PI1

· · · PIn

· VL0 −dσ(j) .

I

W(i)

=

(j)=(j1 ,...,jn )≥(i)

Lemma 2.2.6. Let (i) = (i1 , . . . , in ), (i) = (i1 , . . . , in ) ∈ Nn0 . Suppose that (i )

I

W(i)

I

follows (i) in the lexicographic ordering and defined m(i) = dim I . Then, we

W(i)

I

n

have m(i) = d , provided dσ(i) < L0 − nd.

I

I

I

We assume that VN = W(i)

⊃ W(i)

⊃ · · · ⊃ W(i)

, where (i)s = (i1s , ..., ins ),

1

2

K

I

I

W(i)

follows W(i)

in the ordering and (i)K = ( Nd , 0, . . . , 0). We see that K is

s+1

s

the number of n-tuples (i1 , . . . , in ) with ij ≥ 0 and i1 + · · · + in ≤ Ld0 . We define

WI

mIs = dim W I(i)s for all s = 1, . . . , K − 1 and set mIK = 1.

(i)s+1

Lemma 2.2.7. For L0 = (n + 1)d + p(n + 1)3 I( −1 )d as in the assumption, we

have

puL0

(a)

≤ (N − n + 1)(n + 1) + ,

db

n

(b) u ≤ en+2 dp(n + 1)2 I( −1 ) .

Proposition 2.2.8. b

q

j=1 νQj (f )

− pνW α (φs (f )) ≤ b

q

i=1 min{u

− 1, νQj (f ) }.

Theorem 2.2.9. With the assumption of Theorem 2.2.4 and suppose that M =

Bm (R0 ). Then, we have

q

(q − p(n + 1) − )Tf (r, r0 ) ≤

i=1

15

1 [u−1]

N

(r) + S(r),

d Qi (f˜)

where S(r) ≤ K(log+ R01−r + log+ Tf (r, r0 )) for all 0 < r0 < r < R0 outside a set

E ⊂ [0, R0 ] with E Rdt

< ∞.

0 −t

Theorem 2.2.10. Let M be an m-dimensional complete K¨ahler manifold and

ω be a K¨

ahler form of M. Assume that the universal covering of M is biholomorphic to a ball in Cm . Let f be an algebraically nondegenerate meromorphic

map of M into a subvariety V of dimension k in Pn (C) satifying condition (Cρ )

for ρ ≥ 0. Let Q1 , ..., Qq be hypersurfaces in Pn (C) of degree dj , in N -subgeneral

position with respect to V . Let d = l.c.m.{d1 , ..., dq }.

ρεM0 (M0 − 1)

,

d

2

where p = N − k + 1, M0 = dk +k deg(V )k+1 ek pk (2k + 4)k lk ε−k + 1 and l =

(k + 1)q!.

Then, for each ε > 0, we have

[M0 −1]

q

(Qj )

j=1 δf

≤ p(k + 1) + ε +

In the case of the family of hypersurfaces is in general position, we get N = n.

Moreover, since (n−k+1)(k+1) ≤ ( n2 +1)2 for every 1 ≤ k ≤ n, letting ε = 1+ε

with ε > 0 and then letting ε −→ 0 from the above theorem, we obtain the

following corollary for the case f may be algebraically degenerate

Corollary 2.2.11. Let f : M → Pn (C) be a meromorphic mapping and let

{Qi }qi=1 be hypersurfaces in Pn (C) of degree di , located in general position. Then,

for every ε > 0, we get

q

[M0 −1]

δf

(Qj ) ≤

j=1

n

+1

2

2

+1+

ρM0 (M0 − 1)

,

d

for some positive integer M0 .

In order to prove Theorem 2.2.10, we need the following one.

Theorem 2.2.12. With the assumption of Theorem 2.2.10. Then, we have

q

(q − p(k + 1) − ε)Tf (r, r0 ) ≤

i=1

1 [M0 −1]

N

(r) + S(r),

d Qi (f˜)

where S(r) is evaluated as follows:

1

+ log+ Tf (r, r0 )), for all 0 <

R0 − r

dt

r0 < r < R0 outside a set E ⊂ [0, R0 ] with E

< ∞ and K is a positive

R0 − t

constant.

(i) In the case R0 < ∞, S(r) ≤ K(log+

(ii) In the case R0 = ∞, S(r) ≤ K(logr + log+ Tf (r, r0 )), for all 0 < r0 <

r < ∞ outside a set E ⊂ [0, ∞] with E dt < ∞ and K is a positive constant.

16

Chapter 3

UNICITY OF MEROMOPHIC MAPPINGS SHARING

FEW HYPERPLANES

Chapter three concentrates on extending the uniqueness theorem of H. Fujimoto

for meromorphic mappings from K¨ahler manifold M into Pn (C) and generalizing uniqueness theorem in the direction of L. Smiley for meromorphic mappings

from Cm into Pn (C). In H. Fujimoto’s result, he considered a family of hyperplanes {Hj }qj=1 in general position and L. Smiley added condition of dimention

for {Hj }qj=1 , dim f −1 (Hi ) ∩ f −1 (Hj ) ≤ m − 2, (1 ≤ i < j ≤ q). We replace

k+1

the above condition with the following dim f −1

≤ m − 2 (1 ≤

j=1 Hij

i1 < · · · < ik+1 ≤ q)(∗). Then, when the family of hyperplanes are in general

position then condition (∗) is always satified and when k = 1 then (∗) is exactly

Smiley’s condition. Thus, our result not only extend Fujimoto’s theorem but

also generalize results when we restrict to the case M = Cm .

Chapter three is written based on article [2] (in Works related to the thesis).

3.1

Second main theorem for meromorphic mapping from a ball and

hyperplanes in general position

In this section, we prove some auxiliary lemmas that will be used in the following

section.

Lemma 3.1.1. Let f be a linearly nondegenerate meromorphic mapping from

B(R0 ) into Pn (C) and H1 , . . . , Hq be q hyperplanes of Pn (C) in general position.

Set l0 = |α1 | + · · · + |αn+1 | and take t, p with 0 < tl0 < p < 1. Then, for

17

0 < r0 < R0 there exists a positive constant K such that for r0 < r < R < R0 ,

z

, Fn+1 ) t

σm ≤ K

L1 (F ) · · · Ln+1 (F )

α1 +···+αn+1 Wα1 ,...,αn+1 (F1 , · · ·

R2m−1

TF (R, r0 )

R−r

p

.

S(r)

Lemma 3.1.2. Let f be a linearly nondegenerate meromorphic mapping from

B(R0 ) into Pn (C) and H1 , ..., Hq be q hyperplanes of Pn (C) in general position.

Then we have

q

[n]

(q − n − 1)Tf (r, r0 ) ≤

NHj (f ) (r, r0 ) + Sf (r),

j=1

where Sf (r) is evaluated as follows:

1

+ log+ Tf (r, r0 )) for all 0 <

R0 − r

dt

r0 < r < R0 outside a set E ⊂ [0, R0 ) with E

< ∞, where K is a

R0 − t

positive constant.

(i) In the case R0 < ∞, Sf (r) ≤ K(log+

(ii) In the case R0 = ∞, Sf (r) ≤ K(log+ Tf (r, r0 )+logr) for all 0 < r0 < r < ∞

outside a set E ⊂ [0, ∞) with E dt < ∞, where K is a positive constant.

3.2

Uniqueness theorem for meromorphic mappings sharing few hyperplanes

Theorem 3.2.1. Let M be a complete, connected K¨ahler manifold whose universal covering is biholomorphic to B(R0 ) ⊂ Cm . Let f, g : M → Pn (C) be

linearly nondegenerate meromorphic mappings. Assume that f and g satisfy

the condition (Cρ ) for some ρ ≥ 0 and there are q hyperplanes H1 , ..., Hq of

Pn (C) in general position such that

(i) dim f −1

k+1

j=1 Hij

≤ m − 2 (1 ≤ i1 < ... < ik+1 ≤ q),

(ii) f = g on ∪qj=1 f −1 (Hj ) ∪ g −1 (Hj ).

2nkq

> n+1+

q + 2nk − 2k

ρn(n + 1) or q < 2(n + 1)k and q > (n + 1)(k + 1) + ρn(n + 1).

Then we have f ≡ g if either q ≥ 2(n + 1)k and q −

From the above theorem, we have the following remarks.

18

1) If k = 1 and q > 3n + 1 + ρn(n + 1) then

q−

2nkq

≥ q − 2n > n + 1 + ρn(n + 1).

q + 2nk − 2k

So the assumption of Theorem 3.2.1 is satisfied. Hence, our result is valid for

q > 3n + 1 + ρn(n + 1).

2) If k = n then the condition (i) in Theorem 3.2.1 is alway true since the

family of hyperplanes is in a general position. Let q > (n + 1)2 + ρn(n + 1). It

is easy to see that

2q

• If n = 1 then q > 4 + 2ρ implies q ≥ 4 and q −

> 2 + 2ρ.

q

n−1

• If n ≥ 2 and ρ <

then 2(n + 1)n > q > (n + 1)2 + ρn(n + 1).

n

Thus, the assumption of Theorem 3.2.1 is satisfied. Therefore, in the above

cases our theorem implies Theorem D of Fujimoto.

3) In the case M = Cm , by taking ρ = 0, we see that

• If k = 1 then q = (n + 1)k + n + 2 satifies the conditions q ≥ 2(n + 1)k

2nkq

and q −

> n + 1.

q + 2nk − 2k

• If k ≥ 2 then q = (n + 1)k + n + 2 satifies the conditions q < 2(n + 1)k

and q > (n + 1)(k + 1)

in Theorem 3.2.1. Hence, our result is also a generalization of the above conclusion of Giang-Quynh-Quang and the result of Z. Chen and Q. Yan as well.

19

Chapter 4

ALGEBRAIC DEPENDENCES OF MEROMORPHIC

MAPPINGS SHARING FEW HYPERPLANES

In Chapter four, we study algebraic dependences of three meromorphic mappings from K¨ahler M , namely we generalize results in which meromorphic mappings are from Cm into Pn (C). By giving new methods basing on rearrange

hyperplanes into groups and construct appropriate auxiliary lemmas, we succeeded in generalizing algebraically dependent theorems on K¨ahler M .

Chapter four is written based on article [3] (in Works related to the thesis).

4.1

Second main theorem for meromorphic mapping from a ball and

hyperplanes in subgeneral position

In this section, we prove the second main theorem for meromorphic mapping

from a ball into projective space intersecting hyperplanes in subgeneral position.

Proposition 4.1.1. Let H1 , . . . , Hq (q > 2N − n + 1) be hyperplanes in Pn (C)

located in N -subgeneral position. Then there exists a function ω : {1, . . . , q} →

(0, 1] called a Nochka weight and a real number ω

˜ ≥ 1 called a Nochka constant

satisfying the following conditions:

(i) If j ∈ {1, · · · , q}, then 0 < ω(j)˜

ω ≤ 1.

q

(ii) q − 2N + n − 1 = ω

˜ ( j=1 ω(j) − n − 1).

(iii) For R ⊂ {1, . . . , q} with |R| = N + 1, then i∈R ω(i) ≤ n + 1.

−n+1

(iv) Nn ≤ ω

˜ ≤ 2Nn+1

.

(v) Given real numbers λ1 , . . . , λq with λj ≥ 1 for 1 ≤ j ≤ q and given

any R ⊂ {1, . . . , q} and |R| = N + 1, there exists a subset R1 ⊂ R such that

20

|R1 | = rank {Hi }i∈R1 = n + 1 and

ω(j)

≤

λj

λi .

i∈R1

j∈R

We note that Lemma 4.1.2 and Lemma 4.1.3 generalize correspondingly

Lemma 3.1.1 and Lemma 3.1.2 in Chapter three for case hyperplanes in general

position.

Lemma 4.1.2. Let f be a linearly nondegenerate meromorphic mapping from

B(R0 ) into Pn (C) and H1 , . . . , Hq be q hyperplanes of Pn (C) in N − subgeneral

position. Set l0 = |α0 | + · · · + |αn | and take t, p with 0 < tl0 < p < 1. Then, for

0 < r0 < R0 there exists a positive constant K such that for r0 < r < R < R0 ,

z

α0 +···+αn

Wα0 ...αn (f )

(f, H1 )ω(1) · · · (f, Hq )ω(q)

t

q

j=1

f

ω(j)−n−1 t

σm

R2m−1

p

K

Tf (R, r0 ) ,

R−r

S(r)

where ω(j) are Nochka weights with respect to Hj , 1

j

q.

Lemma 4.1.3. Let f be a linearly nondegenerate meromorphic mapping from

B(R0 ) into Pn (C) and let H1 , H2 , . . . , Hq be q hyperplanes of Pn (C) in N subgeneral position. Then we have

q

[n]

(q − 2N + n − 1)Tf (r, r0 )

N(f,Hj ) (r, r0 ) + Sf (r),

j=1

where Sf (r) is evaluated as follows:

1

+ log+ Tf (r, r0 )) for all 0 <

R0 − r

r0 < r < R0 outside a set E ⊂ [0, R0 ) with E Rdt

< ∞, where K is a

0 −t

positive constant.

(i) In the case R0 < ∞, Sf (r)

K(log+

(ii) In the case R0 = ∞, Sf (r) K(log+ Tf (r, r0 )+logr) for all 0 < r0 < r < ∞

outside a set E ⊂ [0, ∞) with E dt < ∞, where K is a positive constant.

4.2

Algebraic dependences of three meromorphic function sharing

some hyperplanes

Lemma 4.2.1. Let f 1 , f 2 , f 3 be three meromorphic mappings from B(R0 ) into

Pn (C) and H1 , H2 , . . . , Hq are q hyperplanes in Pn (C) satifying

21

f 1 = f 2 = f 3 on ∪qj=1 (f 1 )−1 (Hj ) ∪ (f 2 )−1 (Hj ) ∪ (f 3 )−1 (Hj ). Suppose that

there exist s, t, l ∈ {1, . . . , q} such that

(f 1 , Hs ) (f 1 , Ht ) (f 1 , Hl )

P := Det (f 2 , Hs ) (f 2 , Ht ) (f 2 , Hl ) ≡ 0.

(f 3 , Hs ) (f 3 , Ht ) (f 3 , Hl )

Then we have

q

( min {ν(f u ,Hi ) (z)} −

νP (z)

i=s,t,l

1 u 3

[1]

ν(f 1 ,Hi ) (z))

[1]

ν(f 1 ,Hi ) (z), ∀z ∈

/ S.

+2

i=1

Lemma 4.2.2. Let M , f 1 , f 2 , f 3 and H1 , H2 , . . . , Hq be as in Theorem 4.2.3.

Let P be a holomorphic function on M and β be a positive real number such

that

3

q

[n]

ν(f u ,Hv ) (z)

βνP (z)

(4.1)

u=1 v=1

for all z outside an analytic subset of codimension two. If |P β |

with some positive constants C and α, then

q

C( f 1

f2

f 3 )α

2N − n + 1 + ρn(n + 1) + α.

Theorem 4.2.3. Let M be a complete and connected K¨ahler manifold whose

universal covering is biholomorphic to B(R0 ) ⊂ Cm , where 0 < R0

∞. Let

f 1 , f 2 , f 3 : M → Pn (C) be three linearly nondegenerate meromorphic mappings

which satisfy the condition (Cρ ) for some nonnegative constant ρ and there are

q hyperplanes H1 , H2 , . . . , Hq of Pn (C) in N -subgeneral position such that

dim f −1 (Hi ) ∩ f −1 (Hj )

m − 2 (1

i

q).

Suppose that we have the following conditions:

(a) min{ν(f 1 ,Hi ) , n} = min{ν(f 2 ,Hi ) , n} = min{ν(f 3 ,Hi ) , n} (1

(b) f 1 = f 2 = f 3 trn

3nq

then one of the following assertions

2q + 3n − 3

holds:

q

3

+ 1 hyperplanes Hi1 , . . . , Hi[ q ]+1 such that

3

(f u , Hi[ q ]+1 )

(f u , Hi1 ) (f u , Hi2 )

3

= v

= ··· = v

,

v

(f , Hi1 )

(f , Hi2 )

(f , Hi[ q ]+1 )

3

1

2

q),

q

1 −1

i=1 (f ) (Hi ).

If q > 2N − n + 1 + ρn(n + 1) +

(i) There exist

i

3

(ii) f ∧ f ∧ f ≡ 0 on M.

22

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