The first is to show the Second Main Theorem for degenerate holomorphic curves intoPn C with hypersurface targets located in n-subgeneral position.. The second is to show the Second Main
Trang 1Kyushu J Math 65 (2011), 219–236
doi:10.2206/kyushujm.65.219
THE SECOND MAIN THEOREM FOR HYPERSURFACES
DO DUC THAI and NINH VAN THU
(Received 7 November 2009)
Abstract The purpose of this article is twofold The first is to show the Second Main
Theorem for degenerate holomorphic curves intoPn ( C) with hypersurface targets located
in n-subgeneral position The second is to show the Second Main Theorem with truncated
counting functions for nondegenerate holomorphic curves into Pn ( C) with hypersurface
targets in general position Finally, by applying the above result, a unicity theorem for algebraically nondegenerate curves intoP2( C) with hypersurface targets in general position
is also given
1 Introduction and main results
Let f : C → Pn ( C) be a holomorphic map Let ˜ f = (f0, , f n )be a reduced representation
of f , where f0, , f n are entire functions on C and have no common zeros The
Nevanlinna–Cartan characteristic function T f (r)is defined by
T f (r)= 1
2π
2π
0
log ˜f (re iθ ) dθ,
where
˜f (z) = max{|f0(z) |, , |f n (z) |}.
The above definition is independent, up to an additive constant, of the choice of a reduced
representation of f Let D be a hypersurface inPn ( C) of degree d Let Q be the homogeneous polynomial (form) of degree d defining D The proximity function m f (r, D)is defined as
m f (r, D)=
2π
0
log ˜f (re iθ )d Q
|Q( ˜ f )(re iθ )|
dθ 2π ,
whereQ is the sum of the absolute values of the coefficients of Q The above definition
is independent, up to an additive constant, of the choice of a reduced representation of f To define the counting function, let n f (r, D) be the number of zeros of Q( ˜ f )in the disk|z| < r,
counting multiplicity The counting function is then defined by
N f (r, D)=
2π
0
n f (t, D) − n f ( 0, D)
t dt + n f ( 0, D) log r.
2000 Mathematics Subject Classification: Primary 32H30; Secondary 32H04, 32H25, 14J70 Keywords: holomorphic curves; algebraic degeneracy; defect relation; Nochka weight.
c
2011 Faculty of Mathematics, Kyushu University
Trang 2220 Do Duc Thai and Ninh Van Thu
Note that
N f (r, D)=
2π
0
log|Q( ˜ f )(re iθ )|dθ
2π + O(1).
The Poisson–Jensen formula implies the following
FIRST MAIN THEOREM Let f : C → Pn ( C) be a holomorphic map, and let D be a hypersurface in Pn ( C) of degree d If f (C) ⊂ D, then for every real number r with 0 <
r < +∞,
m f (r, D) + N f (r, D) = dT f (r) + O(1), where O(1) is a constant independent of r.
Recall that hypersurfaces D1, , D qinPn ( C)(q > n) are said to be in general position
if∩n+1
k=1Supp(D j k ) = ∅ for any distinct j1, , j n+1∈ {1, , q}.
In [8], Ru showed the Second Main Theorem (SMT, for short) for algebraically
nondegenerate holomorphic curves into Pn ( C) and fixed hypersurface targets in general
position in Pn ( C) As a corollary of this theorem, he proved the Shiffman conjecture for
algebraically nondegenerate holomorphic curves into Pn ( C) and fixed hypersurfaces in
general position in Pn ( C) Later, Dethloff and Tan [4] showed a SMT for algebraically
nondegenerate meromorphic maps ofCm into Pn ( C) and q slowly moving hypersurfaces
targets inPn ( C) (q ≥ n + 2) in (weakly) general position (cf the detailed definitions in [4]).
The following question arose naturally at this moment: is there the SMT for degenerate holomorphic curves into Pn ( C) and hypersurface targets in P n ( C), i.e., a theorem of
Cartan–Nochka type? The first aim of this article is to show the SMT for degenerate holomorphic curves intoPn ( C) with hypersurface targets located in n-subgeneral position
(cf Definition 2.5) Namely, we show the following
THEOREM1.1 (SMT for algebraically degenerate holomorphic curves) Let f : C → Pn ( C)
be a holomorphic map whose image is contained in some k-dimensional subspace but not in any subvariety of dimension lower than k Let {D j}q
j=1 be hypersurfaces inPn ( C) of degree
d j , located in n-subgeneral position in that subspace Then for every > 0,
q
j=1
d−1
j m f (r, D j ) ≤ (2n − k + 1 + )T f (r).
As usual, by notation ‘P ’ we mean that the assertion P holds for all r ∈ [0, ∞)
excluding a Borel subset E of the interval [0, ∞) withE dr <∞
We remark that in the above-mentioned original SMT of Ru [8], there is no truncated
counting function that can be used in the study of uniqueness problems Later, An and
Phuong [1] and Dethloff and Tan [4] gave truncated counting functions in this theorem.
Namely, they showed the following
Let f : C → Pn ( C) be an algebraically nondegenerate holomorphic map, and let {D j}q
j=1
be fixed hypersurfaces inPn ( C) of degree d j in general position Let d be the least common multiple of the d j Then, for every 0 < < 1,
(q − n − 1 − )T f (r)≤
q
j=1
1
d j N (M j ) (r, div(f, D j )),
where:
Trang 3The Second Main Theorem 221
(i) M1= · · · = M q ≥ 2d[2 n (n + 1)n(d + 1)−1]n (see An and Phuong [1]);
(ii) M j ≤d j·
n +N
n
− d j
N = d · [2(n + 1)(2 n − 1)(nd + 1)−1+ n + 1] (see Dethloff and Tan [4]).
Here and in what follows [x] = min{n ∈ Z : n ≥ x} (the ceiling function) and N (α) (r, ν)
is the counting function with the truncation level to α of divisor ν.
However, their truncation level is very big The second aim of this article is to give a
better truncation in their SMT To state our second result, we need some notation Let N be any positive integer For each i = (i1, , i n )∈ Nn with σ (i) = i1+ · · · + i n ≤ N/d, denote
by (i) the number of n-tuples (j1, , j n )∈ Nn such that j1+ · · · + j n ≤ N − dσ(i) and
0≤ j1, , j n ≤ d − 1 Let = (1/n)σ (i) ≤N/d σ (i) (i) and M=n +N
n
Here is our result
THEOREM1.2 Let f : C → Pn ( C) be an algebraically nondegenerate holomorphic curve, and let {D j}q
j=1be hypersurfaces inPn ( C) of degree d j in general position Let d be the least common multiple of the d j Then
dq−MN
T f (r) ≤ N ( [M2/ ]) (r, div(f, D)) + O(ln T f (r)), where D = D1∪ · · · ∪ D q and div(f, D)=q
j=1(d/d j ) div(f, D j ).
In the above result, we have the following explicit estimate
COROLLARY1.3 Let f : C → Pn ( C) be an algebraically nondegenerate holomorphic curve, and let {D j}q
j=1 be hypersurfaces inPn ( C) of degree d j in general position Let d
be the least common multiple of the d j Then, for every 0 < ≤ (n + 1)/d,
d(q − n − 1 − )T f (r) ≤ N (L) (r, div(f, D)) + O(ln T f (r)),
where
L=
2de n d(n + 1)2
n−1
, D = D1∪ · · · ∪ D q and
div(f, D)=
q
j=1
d
d j
div(f, D j ).
In the last part of this article, by applying the SMT with explicit truncated counting functions for nondegenerate holomorphic curves intoP2( C) and hypersurface targets located
in general position, a unicity theorem for algebraically nondegenerate curves intoP2( C) with
hypersurface targets in general position is also given
THEOREM1.4 Let f, g: C → P2( C) be algebraically nondegenerate curves, and let
Q jq
j=1 be homogeneous polynomials of degree d j in general position inP2( C) Let d be the least common multiple of the d j Assume that:
(i) f = g onq
j=1(f−1(Q j ) ∪ g−1(Q j ));
(ii) f−1(Q i ) ∩ f−1(Q j ) = ∅ for i = j.
Trang 4222 Do Duc Thai and Ninh Van Thu
Then f ≡ g for each
q≥4(d + 1)2( 2d + 1)2
d(d2+ 3d + 4) +
4(d + 1)(2d + 1)
d2+ 3d + 4 +
2
d .
2 Second Main Theorem for the degenerate case
First of all, we need the following general form of the SMT for holomorphic curves
intersecting hyperplanes They are stated and proved in [6] and [7].
THEOREM2.1 (See [6, Theorem 2.1]) Let F = [f0: · · · : f n] : C → Pn ( C) be a holomor-phic map whose image is not contained in any proper linear subspace Let H1, , H q be arbitrary hyperplanes inPn ( C) Let L j ,1≤ j ≤ q, be the linear forms defining H1, , H q Then, for every > 0,
02π max
β ∈Klog
n
t=0
˜F (re iθ ) L β(t )
|L β(t ) ( ˜ F (re iθ ))| − (n + 1) log ˜ F (re
iθ )
dθ 2π < T (r, F ), where the maximum is taken over all subsets K of {1, , q} such that the linear forms
L j , j ∈ K, are linearly independent, and L j is the maximum of the absolute values of the coefficients in L j
THEOREM2.2 (See [7, Theorem A3.1.3]) Let f = [f0: · · · : f n] : C → Pn ( C) be a holo-morphic map whose image is not contained in any proper linear subspace Let H1, , H q
be arbitrary hyperplanes in Pn ( C) Let L j ,1≤ j ≤ q, be the linear forms defining
H1, , H q Denote by W (f0, , f n ) the Wronskian of f0, , f n Then for every > 0,
02πmax
β ∈Klog
n
t=0
˜f (re iθ ) L β(t )
|L β(t ) ( ˜ F (re iθ ))|
dθ 2π + N W (r, 0)
< (n + 1)T f (r)+n(n + 1)
2 ( ln T f (r) + (1 + ) ln+( ln T
f (r))) + O(1), (1)
where the maximum is taken over all subsets K of {1, , q} such that the linear forms
L j , j ∈ K, are linearly independent, and L j is the maximum of the absolute values of the coefficients in L j
Hyperplanes b1, , b q∈ Pk ( C)∗ are said to be in n-subgeneral position if, for every
1≤ j1< · · · < j n+1≤ q, the linear span of b j1, , b j n+1 is Ck+1 Recall the following
lemma about Nochka weights; for details, see [10].
LEMMA2.3 (Nochka) Let B = {b j∈ Pk ( C)∗,1≤ j ≤ q} be the set of hyperplanes in n-subgeneral position with 2n − k + 1 ≤ q For each ∅ = P ⊂ {1, , q}, let L(P ) be the linear space generated by { ˜b j | j ∈ P }, where ˜b j is a reduced representation of b j Then there exists numbers ω1, , ω q ∈ (0, 1], called the Nochka weights, and a real number ≥ 1, called the Nochka constant, satisfying the properties:
(i) if j ∈ {1, 2, , q}, then 0 < ω j ≤ 1;
(ii) q − 2n + k − 1 ≤ (q
j=1ω j − k − 1);
(iii) if ∅ = P ⊂ {1, , q} with #P ≤ n + 1, thenj ∈P ω j ≤ dim L(P );
(iv) 1≤ (n + 1)/(k + 1) ≤ ≤ (2n − k + 1)/(k + 1);
Trang 5The Second Main Theorem 223
(v) given E1, , E q , real numbers ≥ 1, and given any Y ⊂ {1, , q} with 0 < #Y ≤
n + 1, there exists a subset M of Y such that { ˜b j}j ∈M is a basis for L(Y ) and
j ∈Y
E j ω j ≤
j ∈M
E j
LEMMA2.4 Let Q1, , Q n be n homogeneous polynomials of the same degree d > 0 in C[x1, , x n ] If the Jacobian ∂(Q1, , Q n )/∂(x1, , x n ) ≡ 0 on C n , then Q1, , Q n have a common zero (x1, , x n ) = (0, , 0).
Proof Let S be a variety defined by
S = {x = (x1, , x n )∈ Cn | Q3(x) = · · · = Q n (x) = 0}.
If dim S > 2, then dim{x = (x1, , x n )∈ Cn | Q j (x) = 0, 1 ≤ j ≤ n} > 0 Therefore,
Q1, , Q n have a common zero (x1, , x n ) = (0, , 0) So, we may assume that dim S = 2 Denote by ˜S the set of all smooth points in S Let us consider an arbitrary smooth point x0∈ ˜S Without loss of generality, we may assume that there exist a neighborhood
U of x0 in S and a homeomorphism h: C2→ U ∩ S such that (U, h) is a chart on
S and ∂(Q3, , Q n )/∂(x3, , x n ) = 0 on U ∩ S It means that Q j (h(u, v))= 0 for
every (u, v)∈ C2 and for every 3≤ j ≤ n Let ˜ Q1(u, v) := Q1(h(u, v)) and ˜Q2(u, v):=
Q2(h(u, v)) for every (u, v)∈ C2 Note that
∂ ˜ Q l
∂u =
n
k=1
∂Q l
∂x k
∂h k
∂u ,
∂ ˜ Q l
∂v =
n
k=1
∂Q l
∂x k
∂h k
∂v , l = 1, 2,
n
k=1
∂Q j
∂x k
∂h k
∂u = 0,
n
k=1
∂Q j
∂x k
∂h k
∂v = 0, j = 3, , n.
By using Euler’s formula and ∂(Q1, , Q n )/∂(x1, , x n )≡ 0 on Cn, it follows that
˜
∂ ˜ Q1
∂u (u, v) ∂ ˜ Q2
∂Q1(h(u,v))
∂x3
∂Q2(h(u,v))
∂x3
∂Q3(h(u,v))
∂x3 · · · ∂Q n (h(u,v))
∂x3
∂Q1(h(u,v))
∂x n
∂Q2(h(u,v))
∂x n
∂Q3(h(u,v))
∂x n · · · ∂Q n (h(u,v))
∂x n
≡ 0 on C2
(2)
Since ∂(Q3, , Q n )/∂(x3, , x n ) = 0 on U ∩ S,
˜
Q1 Q˜2
∂ ˜ Q1
∂u
∂Q2
∂u
≡ 0 on C2.
Similarly, we also have
˜
Q1 Q˜2
∂ ˜ Q1 ∂Q2
≡ 0 on C2.
Trang 6224 Do Duc Thai and Ninh Van Thu
Therefore, there exists a constant C such that ˜ Q1(u, v) = C ˜ Q2(u, v) for every (u, v)∈ C2;
that is, Q1(x) = CQ2(x) for all x ∈ U ∩ S Since x0 is an arbitrary smooth point in S,
Q1(x) = CQ2(x) for all x ∈ ˜S Hence, Q1= CQ2on S since ˜ S is dense in S Consequently, there is a common zero (x1, , x n ) = (0, , 0) of Q1, , Q n This completes the
Let D1, , D q , q > n be hypersurfaces in Pk ( C) Denote by Q j the homogeneous
polynomial (form) of degree d j (smallest) defining D j, 1≤ j ≤ q For each 1 ≤ j ≤ q and for each p∈ Pk ( C), the hyperplane H p
j is defined by
k
l=0
∂Q j
∂z l (p)z l = 0.
Let T be the set of all injective maps α : {0, , n} → {1, , q} For each α ∈ T , denote
by A [k]
α (p)the matrix
∂Q α(i)
∂z j (p)
0≤i≤n
0≤j≤k
.
Denote by M α the set of all p∈ Pk ( C) such that rank A [k] α (p) < k + 1 and let M =
∪α ∈T M α Let K be the set of all injective maps β : {0, , k} → {1, , q} such that
∩k
l=0Supp(D β(l) ) = ∅ For each β ∈ K, let
˜
M β:=
p∈ Pk ( C)
∂(Q β( 0) , , Q β(k) )
∂(z0, , z k ) (p)= 0
.
Let ˜M= Pk ( C) if K = ∅ and let ˜ M= ∩β ∈K M˜βif otherwise
Definition 2.5 Hypersurfaces D1, , D q , q > n, inPk ( C) are said to be in n-subgeneral
position if ˜M \ M = ∅.
Definition 2.6 Let D1, , D q , q > n, be hypersurfaces in Pn ( C) Let V be a linear
subspace ofPn ( C) {D1, , D q } are called in n-subgeneral position in V if ˆ D i1, , ˆ D i t are in n-subgeneral position, where ˆ D i s := D i s ∩ V if D i s ∩ V = V
Example 2.7 Let D1, , D q , q > n , be hypersurfaces of the same degree d≥ 1 in Pk ( C).
Assume that:
(i) D j1∩ · · · ∩ D j n+1= ∅ for any distinct j1, , j n+1∈ {1, 2, , q},
(ii) for any distinct i0, , i k ∈ {1, , q} such that Q i0, , Q i k are linearly
independent, D i0∩ · · · ∩ D i k= ∅
Then D1, , D q are in n-subgeneral position inPk ( C).
Proof By Lemma 2.4, M is a proper analytic set inPk ( C) On the other hand, ˜ M= Pk ( C).
By Definition 2.5, we have the following
LEMMA2.8 Assume that D1, , D q , q > n, are hypersurfaces inPk ( C), located in n-subgeneral position Then for each p∈ ˜M \ M and for any distinct i0, , i k ∈ {1, , q},
D i0 ∩ · · · ∩ D i k = ∅ iff H p
i , , H i p are linearly independent.
Trang 7The Second Main Theorem 225
Proof of Theorem 1.1 Without loss of generality, we may assume that f : C → Pk ( C) ⊂
Pn ( C) is a holomorphic map So f : C → P k ( C) is a nondegenerate holomorphic map We also assume that q ≥ 2n − k + 1 Let ˜ f = (f0, , f k ) be a reduced representation of f , where f0, , f k are entire functions onC and have no common zeros For simplicity, we just write ˜f as f Let D1, , D q be hypersurfaces inPn ( C), located in general position Let Q j ,1≤ j ≤ q, be the homogeneous polynomials in C[x0, , x n ] of degree d jdefining
D j Replacing Q j by Q d/d j j if necessary, where d is the least common multiple of the d j,
we may assume that Q1, , Q q have the same degree of d Let ˆ D j = D j∩ Pk ( C) and let
ˆ
Q(z) = Q(z) for z ∈ P k ( C) Without loss of generality, we may assume that ˆ D1, , ˆ D q
are hypersurfaces inPk ( C) Then ˆ D1, , ˆ D q are in n-subgeneral position in Pk ( C) By Lemma 2.8, there exist hyperplanes H1, , H q , located in n-subgeneral position inPk ( C) such that, for any distinct i0, , i k ∈ {1, , q}, if D i0 ∩ · · · ∩ D i k = ∅ then H i0, , H i k
are linearly independent Let ω1, , ω q be the Nochka weights and let be the Nochka constant associated to the hyperplanes H1, , H qby Lemma 2.3
Given z ∈ C, there exists a renumbering {i1, , i q } of the indices {1, , q} such that
| ˆQ i1◦ f (z)| ≤ | ˆ Q i2 ◦ f (z)| ≤ · · · ≤ | ˆ Q i q ◦ f (z)|. (3) Since ˆQ1, , ˆ Q q are in n-subgeneral position and by Hilbert’s Nullstellensatz, for any integer l (0 ≤ l ≤ n), there is an integer m l ≥ d such that
x m l
l =
n+1
j=1
b j l (x0, , x k ) ˆ Q i j (x0, , x k ),
where b j l, 1≤ j ≤ n + 1, 0 ≤ l ≤ k, are the homogeneous forms with coefficients in C of degree m l − d So
|f l (z)|m l = c1f (z) m l −d max{| ˆQ i1(f )(z) |, , | ˆ Q i n+1(f )(z) |},
where c1 is a positive constant and depends only on the coefficients of b j l, 1≤ i ≤ n + 1,
0≤ l ≤ k, thus depends only on the coefficients of Q i j, 1≤ j ≤ n + 1 Therefore,
f (z) d ≤ c1max{| ˆQ i1(f )(z) |, , | ˆ Q i n+1(f )(z) |}. (4) Since ˆH1, , ˆ H q are in n-subgeneral position inPk ( C), there exists an injective map α : {0, , k} → {i1, , i n+1} such that ˆH α( 0) , , ˆ H α(k) are linearly independent Hence,
∩k
l=0D α(l)∩ Pk ( C) = ∅ By (3), (4), and Lemma 2.3, we have
q
j=1
f (z) d ˆQ j
| ˆQ j (f )(z)|
ω j
≤ c
n+1
j=1
d ˆQ i j
| ˆQ i j (f )(z)|
ω ij
≤ c
k+1
l=0
f (z) d ˆQ α(l)
| ˆQ α(l) (f )(z)| , (5)
Trang 8226 Do Duc Thai and Ninh Van Thu
where c > 0 is constant, depending only on the coefficients of ˆ Q i j ,1≤ j ≤ n + 1 Therefore,
q
j=1
ω jlog f (z) d ˆQ j
| ˆQ j (f )(z)|
≤
k
l=0
log f (z) d ˆQ α(l)
| ˆQ α(l) (f )(z)|
+ O(1)
≤ max
α ∈T
k
l=0
log f (z) d ˆQ α(l)
| ˆQ α(l) (f )(z)|
where T is the set of all injective maps α : {0, , k} → {1, , q} such that
∩k
l=0Supp(D α(l) )∩ Pk ( C) = ∅ Thus,
q
j=1
logf (re iθ )d ˆQ j
| ˆQ j (f )(re iθ )| =
q
j=1
(1− ω j )logf (re iθ )d ˆQ j
| ˆQ j (f )(re iθ )| +
q
j=1
ω jlogf (re iθ )d ˆQ j
| ˆQ j (f )(re iθ )|
≤ d[2n − k + 1 − (k + 1)] log f (re iθ )
−
q
j=1
[1 − ω j] log | ˆQ j (f )(re iθ )|
+ max
α ∈T
k
l=0
log f (re iθ )d ˆQ α(l)
| ˆQ α(l) (f )(re iθ )| + O(1). (7)
We now use Ru’s argument (see the proof of Main Theorem in [9]) to estimate
max
α ∈T
k
l=0
logf (re iθ )d ˆQ α(l)
| ˆQ α(l) (f )(re iθ )| .
Define a map φ : x ∈ P k ( C) → [ ˆ Q1(x): · · · : ˆQ q (x)] ∈ Pq−1( C) and let Y = φ(P k ( C)) By the ‘in n-subgeneral position’ assumption, φ is a finite morphism on Pk ( C) and Y is
a complex projective subvariety of Pq−1( C) We also have that dim Y = n For every
a = (a1, , a q ) ∈ Z q
≥0, denote by y a = y a1
1 · · · y a q
q Let m be a positive integer Put
n m := H Y (m) − 1, q m:= q + m − 1
m
− 1, where H Y is the Hilbert function of Y Consider the Veronese embedding
φ m: Pq−1( C) → P q m ( C) : y → [y a0 : · · · : y a qm ], where y a0, , y a qm are the monomials of degree m in y1, , y q, in some order Denote
by Y m the smallest linear subvariety of Pq m ( C) containing φ m (Y ) It is known that Y m
is an n m-dimensional linear subspace of Pq m ( C), where n m = H Y (m) − 1 Since Y m is
an n m-dimensional linear subspace of Pq m ( C), there are linear forms L0, , L q m ∈
C[w0, , w n m] such that the map
ψ m : w ∈ P q m ( C) → [L0: · · · : L q ] ∈ Y m
Trang 9The Second Main Theorem 227
is a linear isomorphism from Pq m ( C) to Y m Thus ψ−1
m ◦ φ m is an injective map from Y
into Pn m ( C) Let f : C → P k ( C) be the given holomorphic map and let F = ψ−1
m ◦ φ m◦
φ ◦ f : C → P n m ( C) Then F is a holomorphic map Furthermore, since f is algebraically nondegenerate, F is linearly nondegenerate.
For > 0 given in Theorem 1.1, we have the following estimate (see inequality (3.19)
of [9]):
max
α ∈T log
k
l=0
f (re iθ )d ˆQ α(l)
| ˆQ α(l) (f (re iθ ))|
3mNd
max
β ∈Klog
n m
t=0
˜F(re iθ ) L β(t )
|L β(t ) ( ˜ F (re iθ ))| − (n m + 1) log ˜F (re
iθ )
+ d(k + 1) +
3N
where N := (2n − k + 1)(k + 1) and ˜F is a reduced representation of F Therefore, we get
q
j=1
logf (re iθ )d ˆQ j
| ˆQ j (f )(re iθ )|
≤ d(2n − k + 1) log f (re iθ ) −
q
j=1
[1 − ω j] log | ˆQ j (f )(re iθ )|
3mNd
max
β ∈Klog
n m
t=0
˜F (re iθ ) L β(t )
|L β(t ) ( ˜ F (re iθ ))| − (n m + 1) log ˜F(re
iθ )
+
Applying Theorem 2.1 with = 1 to a holomorphic map F and linear forms L0, , L q m,
we obtain that
02π max
β ∈Klog
n m
t=0
˜F (re iθ ) L β(t )
|L β(t ) ( ˜ F (re iθ ))| − (n m + 1) log ˜F(re
iθ )
dθ 2π
≤ T F (r)
Since 0≤ 1 − ω j < 1, (n + 1)/(k + 1) ≤ ≤ (2n − k + 1)(k + 1),
2π
0
log| ˆQ j (f )(re iθ )|dθ
2π > O( 1)
and by (10), after taking integration on both sides of (9) we get
q
j=1
m f (r, D j ) ≤ (d(2n − k + 1) + 2/3)T f (r) + C,
where C is a constant, independent of r Take r big enough so we can make C ≤ /3T f (r)
Trang 10228 Do Duc Thai and Ninh Van Thu
Hence, we have
q
j=1
m f (r, D j ) ≤ (d(2n − k + 1) + )T f (r).
3 Second Main Theorem with truncated counting functions for the
nondegenerate case
In this section we recall Corvaja and Zannier’s filtration as made more explicit in [3] Details
of proofs can be found in [3], and also [2] For a fixed positive integer N , denote by V N the
space of homogeneous polynomials of degree N in C[x0, , x n]
LEMMA3.1 (See [2, Lemma 5]) Let γ1, , γ n be homogeneous polynomials in C[x0, , x n ] and assume that they define a subvariety of P n ( C) of dimension 0 Then for all N≥n
j=1deg γ j ,
(γ1, , γ n ) ∩ V N = deg γ1· · · deg γ n Throughout the rest of this paper, we shall use the lexicographic ordering on n-tuples (i1, , i n )∈ Nn of natural numbers Namely, (j1, , j n ) > (i1, , i n ) if and only if
for some b ∈ {1, , n} we have j l = i l for l < b and j b > i b Given an n-tuple (i)=
(i1, , i n ) of non-negative integers, we denote σ (i):=j i j
Let γ1, , γ n ∈ C[x0, , x n ] be the homogeneous polynomials of degree d that
define a zero-dimensional subvariety ofPn ( C) We now recall Corvaja and Zannier’s filtration
of V N Arrange, by the lexicographic order, the n-tuples (i) = (i1, , i n )of non-negative
integers such that σ (i) ≤ N/d Define the spaces W (i) = W N,(i)by
W (i)=
(e) ≥(i)
γ e1
1 · · · γ e n
n V N −dσ (e) Clearly, W ( 0, ,0) = V N and W (i) ⊂ W (i) if (i) > (i) , so the W
(i) is a filtration of V N Next, we recall some results about the quotients of consecutive spaces in the filtration
LEMMA3.2 (See [4, Proposition 3.3]) For any non-negative integer k, the dimension of the
vector space V k /(γ1, , γ n ) ∩ V k is equal to the number of n-tuples (i1, , i n )∈ Nn such that i1+ · · · + i n ≤ k and 0 ≤ i1, , i n ≤ d − 1 In particular, for all k ≥ n(d − 1), we have
(γ1, , γ n ) ∩ V k = d n
LEMMA3.3 (See [2, Lemma 6]) There is an isomorphism
W (i) /W (i)∼ V N −dσ (i)
(γ1, , γ n ) ∩ V N −dσ (i) .
Furthermore, we may choose a basis of W (i) /W (i) from the set containing all equivalence classes of the form γ i1
1 · · · γ i n
n q modulo W (i) with q being a monomial in x0, , x n with total degree N − dσ(i).
...2 Second Main Theorem for the degenerate case
First of all, we need the following general form of the SMT for holomorphic curves
intersecting hyperplanes They are stated... 7
The Second Main Theorem 225
Proof of Theorem 1.1 Without loss of generality, we may assume that f : C → Pk (... (r).
3 Second Main Theorem with truncated counting functions for the< /b>
nondegenerate case
In this section we recall Corvaja and Zannier’s filtration as made more