Holomorphic mappings into compact complex manifolds

0.34 MB·English

by  Do Duc Thai

HOLOMORPHIC MAPPINGS INTO COMPACT
COMPLEX MANIFOLDS
3
1 DO DUC THAI AND VU DUC VIET
0
2

n
a
Abstract. The purpose of this article is to show a second main
J
theorem with the explicit truncation level for holomorphic map-
9
2 pingsofC(orofacompactRiemannsurface)intoacompactcom-
plex manifold sharing divisors in subgeneral position.

]
V
C
Contents
.
h
t 1. Introduction and main results 1
a
m 2. Hypersurfaces in N-subgeneral position 8
3. Basic notions and auxiliary results from Nevanlinna theory 16
[
4. Some lemmas 19
1
5. The proof of Theorem A 21
v
4 6. The proof of Theorems B 26
9
7. Applications 28
9
6 7.1. Unicity theorem 28
1. 7.2. Five-Point Theorem of Lappan in high dimension 31
0 References 36
3
1
:
v
i
X 1. Introduction and main results
r
a Let {H }q be hyperplanes of CPn. Denote by Q the index set
j j=1
{1,2,··· ,q}. Let N ≥ n and q ≥ N + 1. We say that the family
{H }q are in N-subgeneral position if for every subset R ⊂ Q with
j j=1
the cardinality |R| = N +1
H = ∅.
j
j∈R
\
If they are in n-subgeneral position, we simply say that they are in
general position.
The research of the authors is supported by an NAFOSTED grant of Vietnam
(Grant No. 101.01-2011.29).
1
2 DO DUCTHAI AND VU DUCVIET
Let f : Cm → CPn be a linearly nondegenerate meromorphic map-
ping and {H }q be hyperplanes in N-subgeneral position in CPn.
j j=1
Then the Cartan-Nochka’s second main theorem (see [10], [13]) stated
that
q
|| (q −2N +n−1)T(r,f) ≤ N[n](r,div(f,H ))+o(T(r,f)).
i
i=1
X
TheaboveCartan-Nochka’ssecondmaintheoremplaysanextremely
important role in Nevanlinna theory, with many applications to Alge-
braic or Analytic geometry. Over the last few decades, there have
been several results generalizing this theorem to abstract objects. The
theory of the second main theorems for algebraically nondegenerate
holomorphic curves into an arbitrary nonsingular complex projective
variety V sharing curvilinear divisors in general position in V began
about 40 years ago and has grown into a huge theory. Many con-
tributed. We refer readers to the articles [1], [7], [11], [12], [14], [16],
[17], [15], [18], [19], [21], [22] and references therein for the development
of related subjects. We recall some recent results and which are the
best results available at present.
In 2004, Min Ru [18] established a defect relation for algebraically
nondegenerate holomorphic curves f : C → CPn intersecting curvilin-
ear hypersurfaces in general position in CPn, which settled a long-
standing conjecture of B. Shiffman (see [20]). Recently, in [19] he
further extended the above mentioned result to holomorphic curves
f : C → V intersecting hypersurfaces in general position in V, where
V is an arbitrary nonsingular complex projective variety in CPk. We
now state his celebrated theorem.
Let V ⊂ CPk be a smooth complex projective variety of dimension
n ≥ 1. Let D ,··· ,D be hypersurfaces in CPk, where q > n. Also,
1 q
D ,··· ,D are said to be in general position in V if for every subset
1 q
{i ,··· ,i } ⊂ {1,··· ,q},
0 n
V ∩suppD ∩···∩suppD = ∅,
i0 in
where suppD means the support of the divisor D. A map f : C → V is
saidtobealgebraicallynondegenerate iftheimageoff isnotcontained
in any proper subvarieties of V.
Theorem of Ru (see [19]) Let V ⊂ CPk be a smooth complex
projective variety of dimension n ≥ 1. Let D ,··· ,D be hypersurfaces
1 q
in CPk of degree d , located in general position in V. Let f : C → V
j
be an algebraically nondegenerate holomorphic map. Then, for every
HOLOMORPHIC MAPPINGS INTO COMPACT COMPLEX MANIFOLDS 3
ǫ > 0,
q
d−1m (r;D ) ≤ (n+1+ǫ)T(r,f),
j f j
j=1
X
where the inequality holds for all r ∈ (0,∞) except for a possible set E
with finite Lebesgue measure.
As the first steps towards establishing the second main theorems for
curvilinear divisors in subgeneral position in a (nonsingular) complex
projective variety, recently, D. T. Do and V. T. Ninh in [4] and G.
Dethloff, V. T. Tran and D. T. Do in [3] gave the Cartan-Nochka’s
second main theorem with the truncation for holomorphic curves f :
C → V intersecting hypersurfaces located in N-subgeneral position in
an arbitrary smooth complex projective variety V. We now state their
theorem in [3].
Let N ≥ n and q ≥ N +1. Hypersurfaces D ,··· ,D in CPM with
1 q
V 6⊆ D for all j = 1,...,q are said to be in N-subgeneral position in V
j
if the two following conditions are satisfied:
(i) For every 1 ≤ j < ··· < j ≤ q, V ∩D ∩···∩D = ∅.
0 N j0 jN
(ii) For any subset J ⊂ {1,··· ,q} such that 0 < |J| ≤ n and
{D , j ∈ J} are in general position in V and V ∩(∩ D ) 6= ∅, there
j j∈J j
exists an irreducible component σ of V ∩ (∩ D ) with dimσ =
J j∈J j J
dim V ∩ (∩ D ) such that for any i ∈ {1,··· ,q} \ J, if dim V ∩
j∈J j
(∩ D ) = dim V ∩D ∩(∩ D ) , then D contains σ .
j∈(cid:0)J j (cid:1) i j∈J j i J (cid:0)
Theor(cid:1)em of (cid:0)Dethloff-Tran-Do(cid:1) (see [3]) Let V ⊂ CPM be a
smooth complex projective variety of dimension n ≥ 1. Let f be an alge-
braically nondegenerate holomorphic mapping of C into V. Let D , ··· ,
1
D (V 6⊆ D ) be hypersurfaces in CPM of degree d , in N-subgeneral
q j j
position in V, where N ≥ n and q ≥ 2N − n + 1. Then, for ev-
ery ǫ > 0, there exist positive integers L (j = 1,...,q) depending on
j
n,degV,N,d (j = 1,...,q),q,ǫ in an explicit way such that
j
q
1
(q −2N +n−1−ǫ)T (r) ≤ N[Lj](r,D ).
f d f j
j
(cid:13) Xj=1
(cid:13)
(cid:13)
We would like to emphasize the following.
(i) The condition (ii) in the above definition on N-subgeneral position
of Dethloff-Tan-Thai is hard. Thus, their results may not be very use-
ful and applicable due to this reason.
(ii) In the above-mentioned papers and in other papers (see [2], [5]
for instance), either there is no the truncation levels or the truncation
4 DO DUCTHAI AND VU DUCVIET
levels obtained depend on the given ǫ. When ǫ goes to zero, the trun-
cation level goes to infinite (so the truncation is totally lost). The most
serious and difficult problem (which is supposed to be extremely hard)
is to get the truncation which is independent of ǫ.
(iii)ThefamilyD ,··· ,D arehypersurfacesinCPk,i.etheyareglobal
1 q
hypersurfaces. The more difficult problem is to consider the case where
the family D ,··· ,D are hypersurfaces in V, i.e they are local hyper-
1 q
surfaces.
Motivated by studing holomorphic mappings into compact complex
manifolds, the following arised naturally.
Problem 1. To show a second main theorem with the explicit trun-
cation level for holomorphic mappings of C into a compact complex
manifold sharing divisors in subgeneral position.
Unfortunately, this problem is extremely difficult and while a sub-
stantialamountofinformationhasbeenamassedconcerningthesecond
main theorem for holomorphic curves into complex projective varieties
through the years, the present knowledge of this problem for arbitrary
compact complex manifolds has remained extremely meagre. So far
there has been no literature of such results.
The purpose of this paper is to solve Problem 1 in the case where
divisors aredefined by global sections ofa holomorphicline bundle over
a given compact complex manifold. To state the results, we recall some
definitions of Nevanlinna theory.
π
Let L −→ X be a holomorphic line bundle over a compact com-
plex manifold X. Let R ,··· ,R be divisors of global sections of
1 q
H0(X,Ldi) respectively, where d ,··· ,l are positive numbers. Take a
1 q
positive integer d such that d is divided by lcm(d ,d ,··· ,d ). Let
1 2 q
E be a C-vector subspace of dimension m + 1 of H0(X,Ld) such
d d
that σd1,··· ,σdq ∈ E. Take a basis {c }m+1 a basis of E. Then
1 q k k=1
d
∩ {c = 0} = B(E) and σdi is a linear combination of {c }m+1.
1≤i≤m+1 i i k k=1
Take trivializations p = (p1,p2) : π−1(U ) → U × C of L and {U }
i i i i i i
covers X. Denote by λ the transition function system of this local
ij
trivialization covering. Set h = ( |p2 ◦c |2)1/d on each U . It
i 1≤i≤m+1 i i i
is easy to check that h = |λ |2h on U ∩ U . Put h = {h }. Then
i ij j i j i
P
ddclogh = ddclogh for U ∩U 6= ∅ and hence, ddclogh := ddclogh
j i i j i
is well-defined in X −B(E).
HOLOMORPHIC MAPPINGS INTO COMPACT COMPLEX MANIFOLDS 5
Letf beaholomorphicmappingofCintoX suchthatf(C)∩B(E) =
∅. We define the characteristic function of f with respect to E to be
r 1
T (r,L) = f∗ddclogh,
f
t
Z1 Z|z|≤t
where r > 1. Remark that the definition does not depend on choosing
basic of E. Take a section σ of L. Assume that D = ν = s a is its
σ i i i
zero divisor, where a ∈ C and s is a positive integer. For 1 ≤ k ≤ ∞,
i i
P
we put ν[k](D) = min{s ,k}a and n[k](t,D) = min{s ,k}.
i i i |ai|<t i
The truncated counting function of f to level k with respect to D is
P P
r n[k](t,D)
[k]
N (r,D) = dt.
f t
Z1
σ(·)
Forbrevitywewillomitthecharacter[k] ifk = ∞. Put||σ(·)||:=
h(·)
By the Jensen formula and by the Poincare-Lelong theorem, we have
p
Theorem 1.1. (First main theorem) Let the notations be as above.
Then,
2π 1
T (r,L) = log dφ+N (r,D)+O(1).
f ||σ(f(reiφ))|| f
Z0
Let L → X be a holomorphic line bundle over a compact complex
manifold X and E be a C−vector subspace of H0(X,L). Let {c }m+1
k k=1
be a basis of E and B(E) be the base locus of E. Define a mapping
Φ : X −B(E) → CPm by Φ(x) := [c (x) : ··· : c (x)]. Denote by
1 m+1
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.
We now state our results.
Theorem A. Let X be a compactcomplex manifold. Let L → X be a
holomorphic line bundle over X. Fix a positive integer d. Let E be a C-
vector subspace of dimension m+1 of H0(X,Ld). Put u = rankE and
b = dimB(E)+1 if B(E) 6= ∅, otherwise b = −1. Takepositivedivisors
d ,d ,··· ,d of d. Let σ (j = 1,2,··· ,q) be in H0(X,Ldj) such that
1 2 q j
d d
σd1,··· ,σdq ∈ E. Set D = {σ = 0} and denote by R the zero
1 q j j j
divisors of σ . Assume that D ,··· ,D are in N-subgeneral position
j 1 q
with respect to E and u > b. Let f : C → X be an analytically non-
degenerateholomorphicmappingwith respectto E, i.e f(C) 6⊂ supp(ν )
σ
for any σ ∈ E \{0} and f(C)∩B(E) = ∅. Then,
q
1
[m]
(q −(m+1)K(E,N,{D }))T (r,L) ≤ N (r,R ),
j f d f i
i
(cid:13) Xi=1
(cid:13)
(cid:13)
6 DO DUCTHAI AND VU DUCVIET
where k ,s ,t are defined as in Proposition 2.12 and
N N N
k (s −u+2+b)
N N
K(E,N,{D }) = .
j
t
N
We would like to point out that, in general, the above constants m,u
do not depend on the dimension of X (cf. Example 7.2 below).
Since u ≤ n ({D }) ≤ n({D }) ≤ m (cf. Subsection 2.1 below), we
0 j j
have a nice corollary in the case m = u,B(E) = ∅.
Corollary 1. Let X be a compact complex manifold. Let L → X be
a holomorphic line bundle over X. Fix a positive integer d. Let E be a
C-vector subspace of dimension m+1 of H0(X,Ld). Put u = rankE.
Take positive divisors d ,d ,··· ,d of d. Let σ (j = 1,2,··· ,q) be
1 2 q j
d d
in H0(X,Ldj) such that σd1,··· ,σdq ∈ E. Set D = {σ = 0} and
1 q j j
denote by R the zero divisors of σ . Assume that B(E) = ∅, m = u
j j
and D ,··· ,D are in N-subgeneral position with respect to E. Let
1 q
f : C → X be an analytically non-degenerate holomorphic mapping
with respect to E, i.e f(C) 6⊂ supp(ν ) for any σ ∈ E \{0}. Then,
σ
q
1
[m]
(q −2N +u−1)T (r,L) ≤ N (r,R ).
f d f i
i
(cid:13) Xi=1
(cid:13)
In the spe(cid:13)cial case where X = CPn,d = d = ··· = d = 1,L is the
1 q
hyperplane line bundle over CPn and E = H0(X,L), then m = u = n.
Thus, the original Cartan-Nochka’s second main theorem is deduced
immediately from Corollary 1.
As we know well, the Cartan’s original second main theorem has
been extended to holomorphic mappings f from a compact Riemann
surface into CPn sharing hyperplanes located in general position in
CPn. For instance, J. Noguchi [15] has established the Nevanlinna
theory for holomorphic mappings of compact Riemann surfaces into
complex projective spaces, and obtained the second main theorem for
hyperplanes with truncation level. Recently, Yan Xu and Min Ru in
[25] also have obtained a similar second main theorem. Motivated by
the above Problem 1 and observations, we now study the following
problem.
Problem 2.To show the second main theorem with the explicit trun-
cation level which is independent of ǫ for holomorphic mappings of a
compact Riemann surface into a compact complex manifold sharing di-
visors in subgeneral position.
Thenextpartofthisarticleistoshowthesecondmaintheoremswith
an explicit truncation level which is independent of ǫ for holomorphic
HOLOMORPHIC MAPPINGS INTO COMPACT COMPLEX MANIFOLDS 7
curves of a compact Riemann surface into a compact complex manifold
sharingdivisorsinN-subgeneralposition. Tostatetheresults, werecall
some definitions of Nevanlinna theory.
Let f be a holomorphic mapping of a compact Riemann surface S
into a compact complex manifold X of dimension n. Let L → X be
a holomorphic line bundle and ω be a curvature form of a hermitian
metric in L. Let E and h be as above. We can see that f∗logh is a
singular metric in the pulled-back line bundle f∗L of L (see Section 3
for definitions). We put
T(f,L) = f∗ω
ZS
Note that this definition does not depend on choosing ω and by (iii)
of Lemma 3.1 we get T(f,L) = f∗ddclogh. Let R be a divisor
S
in H0(X,L). Put f∗R = a V (this is a finite sum). Then, the
i i
R
truncated counting function to level T of f with respect to R is defined
P
by
N[T](f,R) = min{T,a }.
i
i
X
We now state our results.
Theorem B. Let S be a compact Riemann surface with genus g and
X be a compact complex manifold of dimension n. Let L → X be a
holomorphic line bundle over X. Fix a positive integer d. Let E be a C-
vector subspace of dimension m+1 of H0(X,Ld). Put u = rankE, and
b = dimB(E)+1 if B(E) 6= ∅, b = −1 otherwise. Takepositive divisors
d ,d ,··· ,d of d. Let σ (j = 1,2,··· ,q) be in H0(X,Ldj) such that
1 2 q j
d d
σd1,··· ,σdq ∈ E. Set D = {σ = 0} and denote by R the zero
1 q j j j
divisors of σ . Assume that R ,··· ,R are in N-subgeneral position in
j 1 q
X and u > b. Let f : S → X be a holomorphic mapping such that f is
analytically nondegenerate with respect to E, i.e f(S) 6⊂ supp(ν ) for
σ
any σ ∈ E \{0} and f(S)∩B(E) = ∅. Then,
q
1
[m]
(q −(m+1)K(E,N,{D }))T (r,L) ≤ N (r,R )+A(d,L),
j f d f i
i
i=1
X
where k ,s ,t are defined as in Proposition 2.12 and
N N N
k (s −u+2+b)
N N
K(E,N,{D }) = ,
j
t
N
m(m+1)k (g −1)
N
if g ≥ 1
A(d,L) = t
 N
0 if g = 0.


8 DO DUCTHAI AND VU DUCVIET
We would like to emphasize the following at this moment. By Re-
mark 5.1 below, in Theorem A and in Theorem B, we have
T (r,L)
f
liminf > 0.
r→∞ logr
Finally, we will give some applications of the above main theorems.
Namely, weshowaunicitytheoremforholomorphiccurvesofacompact
Riemann surface into a compact complex manifold sharing divisors in
N-subgeneral position. Moreover, we also generalize the Five-Point
Theorem of Lappan to a normal family from an arbitrary hyperbolic
complex manifold to a compact complex manifold.
The paper is organized as follows. In Section 2, we give a definition
of hypersurfaces located in N-subgeneral position and construct Nocka
weights fordivisors defined bysections ofaholomorphic linebundle. In
Section3, weintroducetoNevanlinnatheoryforholomorphicmappings
from a compact Riemann surface into a compact complex manifold. In
Section 4, we show some lemmas which needed later. In Section 5 and
Section 6, we end the proof of our main theorems. In Section 7, some
applications of the above main theorems are given.
2. Hypersurfaces in N-subgeneral position
LetX beacompactcomplexmanifoldofdimensionn. LetL → X be
a holomorphic line bundle over X. Take σ ∈ H0(X,L),D = {σ = 0}
j j j
and R is the zero divisor of σ (j = 1,2,··· ,q). Let E be a C-vector
j j
subspace of dimension m+1 of H0(X,L) containing σ ,...,σ .
1 q
Definition 2.1. The hypersurfaces D ,D ,··· ,D is said to be located
1 2 q
in N-subgeneral position with respect to E if for any 1 ≤ i < ··· <
0
i ≤ q, we have ∩N D = B(E).
N j=0 ij
Assume that {D } is located in N−subgeneral position with respect
j
to E. Let {c }m+1 be a basis of E. Put u = rankE, b = dimB(E)+1
k k=1
if B(E) 6= ∅ and b = −1 if B(E) = ∅. Assume that u > b.
We set σ = a c , where a ∈ C. Define a mapping
i 1≤j≤m+1 ij j ij
Φ : X → CPm by Φ(x) := [c (x) : ··· : c (x)]. The above mapping
1 m+1
P
is a meromorphic mapping. Let G(Φ) be the graph of Φ. Define
p : G(Φ) → X,p : G(Φ) → CPm
1 2
by p (x,z) = x,p (x,z) = z. Since X is compact, p ,p is proper
1 2 1 2
and hence, Y = Φ(X) = p (p−1(X)) is an algebraic variety of CPm.
2 1
Moreover, by definitionof rankE, Y isofdimension rankE = u.Denote
HOLOMORPHIC MAPPINGS INTO COMPACT COMPLEX MANIFOLDS 9
by H the hyperplane line bundle of CPm. Put H := a z ,
i 1≤j≤m+1 ij j−1
where [z ,z ,··· ,z ] is the homogeneous coordinate of CPm.
0 1 m
P
For each K ⊂ Q, put c(K) = rank{H } . We also set
i i∈K
n ({D }) = max{c(K) : K ⊂ Q with |K|≤ N +1}−1,
0 j
and
n({D }) = max{c(K) : K ⊂ Q}−1.
j
Then n({D }),n ({D }) are independent of the choice the C-vector
j 0 j
subspace E of H0(X,L) containing σ (1 ≤ j ≤ q). By Lemma 2.4
j
below, we see that
u ≤ n ({D }) ≤ n({D }) ≤ m.
0 j j
Theorem 2.2. Let notations be as above. Assume that R ,··· ,R are
1 q
in N-subgeneral position with respect to E and q ≥ 2N−u+2+b. Then,
there exist Nocka weights ω(j) for {D }, i.e they satisfy properties in
j
Proposition 2.12.
In order to prove Theorem 2.2 we need the following lemmas.
Lemma 2.3. Let notations be as above. Then dimp (p−1(B(E))) ≤
2 1
dimB(E)+1. Moreover, dimp (p−1(B(E))) = 0 if B(E) = ∅.
2 1
Proof. The second assertion is trivial. Let us consider the case of
B(E) 6= ∅. We have ∩N D = B(E) 6= ∅. Denote by j the biggest in-
j=0 j 0
dexamong{0,1,··· ,N}suchthat∩j0 D 6= B(E).Usingthefactthat
j=0 j
if the intersection of a hypersurface and an analytic set of dimension α
is not empty, its dimension is at least α−1, we see that dim∩j0 D =
j=0 j
dimB(E) + 1. Put V = ∩j0 D . Then dimV = dimB(E) + 1, V is
j=0 j
compact and B(E) ⊂ V. Consider the restriction Φ of Φ into V. Then
1
Φ is a meromorpic mapping between V and CPm. Therefore, we get
1
dimp (p−1(B(E))) ≤ dimΦ (V) ≤ dimV = dimB(E)+1. (cid:3)
2 1 1
Lemma 2.4. Let Y, H be as above. Then, rank{H ,0 ≤ j ≤ N} ≥
j j
u−b.
Proof. Put k = rank{H ,j ∈ R}. Then, there are 0 ≤ j < ··· < j ≤
j 1 k
N such that rank{H ,··· ,H } = k and H (j ∈ {0,1,··· ,N}) is a
j1 jk j
linear combination of H ,··· ,H . Therefore, we have H ∩H ···∩
j1 jk 0 1
H ∩Y = H ∩···∩H ∩Y. We have
N j1 jk
p (p−1(H ∩H ···∩H ∩Y)) = p (p−1(H )∩···∩p−1(H )∩G(Φ))
1 2 0 1 N 1 2 0 2 N
⊂ D ∩···∩D = B(E)
0 N
Hence, p−1(H ∩H ···∩H ∩Y) ⊂ p−1(B(E)). That means
2 0 1 N 1
H ∩H ···∩H ∩Y ⊂ p (p−1(B(E))).
0 1 N 2 1
10 DO DUCTHAI AND VU DUCVIET
Therefore, by Lemma 2.3, we get dimH ∩H ···∩H ∩Y ≤ b. On the
0 1 N
other hand, since H ∩···∩H is an algebraic variety of dimension
j1 jk
m−k, it implies that b ≥ m−k+u−m. This yields that k ≥ u−b. (cid:3)
′
Lemma 2.5. For each K ⊂ Q, c(K) ≤ |K|. And for K ⊂ K ⊂ Q
′ ′
with c(K ) = |K |, we have c(K) = |K|.
(cid:3)
Proof. The proof is trivial.
Lemma 2.6. Let K,R ⊂ Q such that K ⊂ R and c(K) = |K|. Then,
there exists a set K′ such that K ⊂ K′ ⊂ R and c(K′) = |K′|= c(R).
(cid:3)
Proof. The proof is trivial.
Lemma 2.7. i) Let R ,R ⊂ Q. Then,
1 2
c(R ∪R )+c(R ∩R ) ≤ c(R )+c(R )
1 2 1 2 1 2
ii) Let S ⊂ S ⊂ Q. Then, |S |−c(S ) ≤ |S |−c(S ). Furthermore, if
1 2 1 1 2 2
|S |≤ N +1, then |S |−c(S ) ≤ N −u+b+1.
2 2 2
Proof. i) By Lemma 2.5, there exist subsets K,K ,K with K ⊂ R ∩
1 2 1
R , K ⊂ K ⊂ R , K ⊂ K ⊂ R ∪R such that
2 1 1 1 3 1 2
|K|= c(K) = c(R ∩R ),|K |= c(K ) = c(R ),
1 2 1 1 1
and
|K |= c(K ) = c(R ∪R ).
3 3 1 2
Set K = K − K . We show that K ⊂ R . Indeed, otherwise there
2 3 1 2 2
exists i ∈ K − R . Then i ∈ R − K and hence, K ∪ {i} ⊂ K
2 2 1 1 1 3
and K ∪ {i} ⊂ R . This implies that if |K |= c(K ) = c(R ), then
1 1 1 1 1
c(R ) ≥ c(K ∪ {i}) = |K ∪ {i}|= c(K ) + 1 = c(R ) + 1. This is a
1 1 1 1 1
contradiction. Thus, K ⊂ R and hence, K ∪K ⊂ R . On the other
2 2 2 2
hand, K ∪K ⊂ K and K ∩K ⊂ K ∩K = (K −K )∩K = ∅.
2 3 2 2 1 3 1 1
By Lemma 2.5, we get c(R ) ≥ c(K ∪ K) = |K ∪ K|= |K |+|K|=
2 2 2 2
(|K |−|K |)+|K|≥ c(R ∪R )−c(R )+c(R ∩R ). Hence, theassertion
3 1 1 2 1 1 2
(i) holds.
′ ′ ′ ′
ii) By Lemma 2.5, there exist S (v = 1,2) such that S ⊂ S , S ⊂ S
v v v 1 2
′ ′ ′ ′
and |S |= c(S ) = c(S ). We show that (S −S )∩ S = ∅. Indeed,
v v v 2 1 1
′ ′ ′ ′
otherwise there exists i 6∈ S such that S ∪{i} ⊂ S and S ∪{i} ⊂ S .
1 1 2 1 1
If |S′|= c(S′) = c(S ), then |S′|= c(S ) ≥ c(S′ ∪{i}) = c(S′)+1. This
1 1 1 1 1 1 1
′ ′ ′ ′
is a contradiction. Thus, (S − S ) ∩ S = ∅ and hence, S − S ⊂
2 1 1 2 1
′ ′ ′ ′
S −S . Therefore, c(S )−c(S ) ≤ |S |−|S |+1 = |S −S |≤ |S −S |=
2 1 2 1 2 1 2 1 2 1
|S |−|S |.
2 1
If |S |≤ N + 1, then we choose S such that S ⊂ S ⊂ Q and
2 3 2 3
|S |= N +1. By Lemma 2.4 , we have c(S ) ≥ u−b. Hence, c(S )−
3 3 2
|S |≤ N −u+b+1. (cid:3)
2