In this paper, we establish a second main theorem with a good defect relation for entire curves in a projective variety whose derivatives vanish on inverse image of hypersurface targets. Our method is a combination of the techniques in [7-9].
Trang 1Natural Science, 2020, Volume 65, Issue 6, pp 31-40
This paper is available online at http://stdb.hnue.edu.vn
A SECOND MAIN THEOREM FOR ENTIRE CURVES IN A PROJECTIVE VARIETY WHOSE DERIVATIVES VANISH ON INVERSE IMAGE
OF HYPERSURFACE TARGETS
Nguyen Thi Thu Hang1, Nguyen Thanh Son2 and Vu Van Truong3
1Department of Mathematics, Hai Phong University, Hai Phong
2Lam Son High School for the Gifted, Thanh Hoa
3Department of Mathematics, Hoa Lu University, Ninh Binh
Abstract We establish a second main theorem for algebraically nondegenerate
entire curves f in a projective variety V ⊂ Pn(C) and a hypersurface target
{D1, D2, , Dq} satisfying f∗,z = 0 for all z ∈ ∪qj=1f−1(Dj)
Keywords: second main theorem, Nevanlinna theory.
1 Introduction
During the last century, several Second Main Theorems have been established for linearly nondegenerate holomorphic curves in complex projective spaces intersecting (fixed or moving) hyperplanes, and we now have satisfactory knowledge about it Motivated by a paper of Corvaja-Zannier [1] in Diophantine approximation, in 2004
Ru [2] proved a Second Main Theorem for algebraically nondegenerate holomorphic curves in the complex projective space CPnintersecting (fixed) hypersurface targets One
of the most important developments in 15 years pass in Nevanlinna theory is the work on the Second Main Theorem for hypersurface targets The interested reader is referred to [2-9] for many interesting results on this topic
In this paper, we establish a second main theorem with a good defect relation for entire curves in a projective variety whose derivatives vanish on inverse image of hypersurface targets Our method is a combination of the techniques in [7-9]
Received June 10, 2020 Revised June 18, 2020 Accepted June 25 2020
Contact Nguyen Thanh Son, e-mail address: k16toannguyenthanhson@gmail.com
Trang 22 Notations
Let ν be a nonnegative divisor on C For each positive integer (or+∞) p, we define
the counting function of ν (where multiplicities are truncated by p) by
N[p](r, ν) :=
Z r 1
n[p]ν
t dt (1 < r < ∞)
where n[p]ν (t) = P
|z|≤tmin{ν(z), p} For brevity we will omit the character [p] in the
counting function if p= +∞
For a meromorphic function ϕ on C, we denote by(ϕ)0 the divisor of zeros of ϕ
We have the following Jensen’s formula for the counting function
N(r, (ϕ)0) − N(r,
1 ϕ
0
) = 1 2π
Z 2π 0
log (ϕ(reiθ) dθ + O(1)
Let f be a holomorphic mapping of C into Pn(C) with a reduced representation bf = (f0, , fn) The characteristic function Tf(r) of f is defined by
Tf(r) := 1
2π
Z 2π 0
log kf (reiθ)kdθ − 1
2π
Z 2π 0
log kf (eiθ)kdθ, r > 1,
wherekf k = max
i=0, ,n|fi|
Denote by f∗,z the tangent mapping at z ∈ C of f
Let D be a hypersurface in Pn(C) defined by a homogeneous polynomial Q ∈
C[x0, , xn], deg Q = deg D Asumme that f (C) 6⊂ D, then the counting function of f
with respect to D is defined by Nf[p](r, D) := N[p](r, (Q(f0, , fn))0)
Let V ⊂ Pn(C) be a projective variety of dimension k Denote by I(V ) the
prime ideal in C[x0, , xn] defining V Denote by C[x0, , xn]m the vector space of homogeneous polynomials in C[x0, , xn] of degree m (including 0) Put I(V )m :=
C[x0, , xn]m∩ I(V )
Assume that f(C) ⊂ V , then we say that f is algebraically nondegenerate in V if
there is no hypersurface D⊂ Pn(C), V 6⊂ D such that f (C) ⊂ D
The Hilbert function HV of V is defined by HV(m) := dimC[x0 , ,x n ] m
I(V ) m
Consider two integer numbers q, N satisfying q ≥ N + 1, N ≥ k Hypersurfaces
D1, , Dq in Pn(C) are said to be in N-subgeneral position with respect to V if V ∩ (∩N
i=0Dj i) = ∅, for all 1 ≤ j0 <· · · < jN ≤ q
Theorem 3.1 Let V ⊂ Pn(C) be a complex projective variety of dimension k (1 ≤ k ≤
n) Let Q1, , Qq be hypersurfaces in Pn(C) in N-subgeneral position with respect to
Trang 3V ,deg Qj = dj, where N ≥ k and q > (N − k + 1)(k + 1) Denote by d the common
multiple of dj’s Let f be an algebraically entire curve in V satisfying f∗,z = 0 for all
z ∈ ∪qj=1f−1(Qj) Then, for each ǫ > 0,
f(r) ≤ M
2+ M − 1
M2+ M
q
X
j=1
1
dj
Nf(r, Qj) + o(Tf(r)),
where M = k + dkdeg V [(2k + 1)(N − k + 1)2(k + 1)2dk−1deg V ǫ−1] + 1k
Here,
we denote [x] := max{t ∈ Z : t ≤ x} for each real number x, and as usual, by the
notation
E of (1, +∞) withR
Edr <+∞
We would like to remark that Chen-Ru-Yan [10], Giang [11], Quang [7] established degeneracy second main theorems with truncated counting functions With notations as
in Theorem 3.1, Quang [7] gave the following inequality:
f(r) ≤
q
X
j=1
1
dj
N[M0 ]
f (r, Qj) + o(Tf(r))
Proof Firstly, we prove the theorem for the case where all hypersurfaces Qj’s have the same degree d Denote byI the set of all permutations of the set {1, , q} We have
n0 := #I = q! We write I = {I1, , In 0} and Ii = (Ii(1), , Ii(q)) where I1 < I2 <
· · · < In 0 in the lexicographic order Since Q1, , Qqare in N -subgeneral position with respect to V , we have QIi(1)∩ · · · ∩ QI i (N +1)∩ V = ∅ for all i ∈ {1, , n0} Therefore,
by Lemma 3.1 in [7], for each Ii ∈ I, there are linearly combinations QI i (1), , QIi(N +1)
in the following forms:
Pi,1 := QI i (1), Pi,s :=
NX−k+s j=2
bsjQI i (j) (2 ≤ s ≤ k + 1, bsj ∈ C) (3.1)
such that Pi,1∩ · · · ∩ Pi,k+1∩ V = ∅
We define a mapΦ : V → Pℓ−1(C) (ℓ := n0(k + 1)) by
Φ(x) = (P1,1(x) : · · · : P1,k+1(x) : · · · : Pn 0 ,1(x) : · · · : Pn 0 ,k+1(x))
ThenΦ is a finite morphism on V We have that Y := ImΦ is a complex projective variety
of Pℓ−1(C) and dim Y = k and
△ := deg Y ≥ dkdeg V
Let bf = (f0, , fn) be a reduced presentation of f For each positive integer u,
we take v1, , vH Y (u) in C[y1,1, , y1,k+1, , yn 0 ,1, , yn 0 ,k+1]u such that they form
Trang 4a basis of C[y1,1, ,y1,k+1, ,yI n0,1, ,yn0,k+1]u
Y (u) We consider an entire curve F in PH Y (u)−1(C)
with a reduced representation
b
F(z) = (v1(Φ( bf(z))), , vH Y (u)(Φ( bf(z))))
Since f is algebraically nondegenerate, we have that F is linearly nondegenerate
By (3.12) in [7], for every ǫ′ >0 (which will be chosen later) we have
(q − (N − k + 1)(k + 1)) Tf(r)
≤
q
X
j=1
1
dNf(r, Qj) −(N − k + 1)(k + 1)
duHY(u)
N(r, (W ( bF))0) − ǫ′duTf(r)
+ (N − k + 1)(2k + 1)(k + 1)△
ud
X
1≤i≤n 0 ,1≤j≤k+1
mf(r, Pi,j) (3.2)
For each i∈ {1, , HY(u)}, we have
vi(Φ( bf(z))′
=
n
X
s=0
∂(viΦ)
∂xs
( bf(z)) · fs′(z) (3.3)
On the other hand, since f∗,z = 0 for all z ∈ ∪qj=1f−1(Qj), we have
(f0(z) : · · · : fn(z)) = (f0′(z) : · · · : fn′(z))
for all z ∈ ∪qj=1f−1(Qj)
Hence, by (3.3) and by Euler formula (for homogenous polynomials vi(Φ(x)) ∈
C[x0, , xn]), for all z ∈ ∪qj=1f−1(Qj)
v1(Φ( bf(z)))′
: · · · :
vHY(u)(Φ( bf(z)))′
=
n
X
s=0
∂(v1Φ)
∂xs
( bf(z)) · fs′(z) : · · · :
n
X
s=0
∂(vH Y (u)Φ)
∂xs
( bf(z)) · fs′(z)
!
=
n
X
s=0
∂(v1Φ)
∂xs
( bf(z)) · fs(z) : · · · :
n
X
s=0
∂(vH Y (u)Φ)
∂xs
( bf(z)) · fs(z)
!
=
v1(Φ( bf(z))) : · · · : vH Y (u)(Φ( bf(z)))
We consider an arbitrary a ∈ ∪qj=1f−1(Qj) (if this set is nonempty) Then there exists
Ip ∈ I such that
(QI p (1)( bf))0(a) ≥ (QI p (2)( bf))0(a) ≥ · · · ≥ (QI p (q)( bf))0(a) (3.5)
Trang 5Since Q1, , Qq are in N -subgeneral position with respect to V , we have
(QI p (j)( bf))0(a) = 0 for all j ∈ {N + 1, , q} (3.6) Set ct,s := (Pt,s( bf))0(a) and
c:= (c1,1, , c1,k+1, , cn 0 ,1, , cn 0 ,k+1)
Then there are ai = (ai1,1, , ai1,k+1, , ain0,1, , ain0,k+1), i = 1, 2, , HY(u), such
that ya 1, , yaHY (u)form a basis of C[y1,1, ,y1,k+1 , ,yn0,1, ,yn0,k+1] u
SY(u, c) =
HXY (u) i=1
ai· c,
where y= (y1,1, , y1,k+1, , yn 0 ,1, , yn 0 ,k+1)
Hence, there are linearly independent (over C) forms L1, , LH Y (u) such that ya i =
Li(v1, , vHY(u)) in C[y1,1 , ,y1,k+1, ,yn0,1, ,yn0,k+1] u
I Y (u) Then we have
Li( bF) = Li(v1(Φ( bf)), · · · , vH Y (u)(Φ( bf)))
= P1,1ai1,1( bf) · · · P1,k+1ai1,k+1( bf) · · · Pna0in0,1,1 ( bf) · · · Pna0in0,k+1,k+1 ( bf), (3.7)
for all i∈ {1, 2, HY(u)}
Hence, for al i∈ {1, 2, HY(u)}
(Li( bF))0(a) = X
1≤u≤n 0 ,1≤v≤k+1
ait,s(Pi t,s( bf))0(a) = ai · c
Hence,
HXY (u) i=1
(Li( bF))0(a) =
HXY (u) i=1
ai· c = SY(u, c) (3.8)
By (3.4), we have
(L1( bF(a)) : · · · : LHY(u)( bF(a))) = ((L1( bF))′(a) : · · · : (LHY(u)( bF))′(a)) (3.9)
Trang 6By Laplace expansion Theorem, we have
W(L1( bF)) : · · · : LH Y (u)( bF))
=
L1( bF) L2( bF) LH Y (u)( bF) (L1( bF))′ (L2( bF))′ (LHY(u)( bF))′
(L1( bF))(H Y (u)−1) (L2( bF))(H Y (u)−1) (LH Y (u)( bF))(H Y (u)−1)
1≤s<t≤H Y (u)
(−1)1+s+t
Ls( bF) Lt( bF)
Ls( bF)′ Lt( bF)′
det Ast (3.10)
where Ast is the matrix which is defined from the matrix
Li( bF)(v)
1≤i,v+1≤H Y (u) by omitting two first rows and sth, tth columns
For each1 ≤ s < t ≤ HY(u), it is clear that
(det Ast)0 ≥
H Y (u)
X
v∈{1, ,H Y (u)}\{s,t}
max{(Lv(f ))0− HY(u) + 1, 0} (3.11)
We now prove that
Ls( bF) · Lt( bF)′− Lt( bF) · Ls( bF)′
0(a) ≥ max{(Ls( bF))0(a) − HY(u) + 1, 0} + max{(Lt( bF))0(a) − HY(u) + 1, 0} + 1 (3.12)
We distinguish three cases
Case 1.(Ls( bF))0(a) ≤ HY(u) − 1 and (Hi t( bF))0(a) ≤ HY(u) − 1
Then, the right side of (3.12) is equal to 1, but by (3.9), the left side of (3.12) is not less than 1
Case 2.(Ls( bF))0(a) > HY(u) − 1 and (Lt( bF))0(a) > HY(u) − 1
We have
Ls( bF) · (Lt( bF))′− Lt( bF) · (Ls( bF))′
0(a) ≥
Ls( bF)
0(a) +
Lt( bF)
0(a) − 1
≥
(Ls( bF))0(a) − HY(u) + 1
+ (Lt( bF))0(a) − HY(u) + 1
+ 1
= max{(Ls( bF))0(a) − HY(u) + 1, 0} + max{(Lt( bF))0(a) − HY(u) + 1, 0} + 1
Case 3. (Ls( bF))0(a) > HY(u) − 1 and (Lt( bF))0(a) < HY(u) − 1 (and similarly for the
case where(Ls( bF))0(a) < HY(u) − 1 and (Lt( bF))0(a) > HY(u) − 1)
Trang 7We have
Ls( bF) · (Lt( bF))′− Lt( bF) · (Ls( bF))′
0(a) ≥ (Ls( bF))0(a) − 1
≥
(Ls( bF))0(a) − HY(u) + 1
+ 1
= max{(Ls( bF))0(a) − HY(u) + 1, 0} + max{(Lt( bF))0(a) − HY + 1, 0} + 1
We have completed the proof of (3.12)
By (3.10), (3.11) and (3.12), we have
(W ( bF))0(a) =
W(L1( bF), , LH Y (u)( bF))
0(a)
≥
H Y (u)
X
i=1
max{(Li( bF))0(a) − HY(u) + 1, 0} + 1
=
H Y (u)
X
i=1
max{(Li( bF))0(a) − HY(u) + 1, 0} + 1
HY(u)
HY(u)(HY(u) − 1)
H Y (u)
X
i=1
(Li( bF))0(a)
(note thatmax{x − y, 0} + 1z ≥ yz1 x for all x≥ 0, y, z > 1)
Combining with (3.8), we get
(W ( bF))0(a) ≥ 1
HY(u)(HY(u) − 1)SY(u, c). (3.13)
By the definition of Pi,j, Pp,1 ∩ · · · ∩ Pp,k+1 ∩ V = ∅, hence, by Lemma 3.2 in [5] (or
Theorem 2.1 and Lemma 3.2 in [3]), we have
1
uHY(u)SY(u, c) ≥
1 (k + 1)(cp,1+ · · · + cp,k+1) −
(2k + 1)△
1≤t≤n 0 ,1≤s≤k+1ct,s
(k + 1)
k+1
X
s=1
(Pp,s( bf))0(a) − (2k + 1)△
u
X
1≤t≤n 0 ,1≤s≤k+1
(Pt,s( bf))0(a) (3.14)
By (3.13) and (3.5), we have(Pp,1( bf))0(a) = (QI p (1)( bf))0(a) and
(Pp,s( bf))0(a) ≥ (QI p (N −k+s)( bf))0(a)
for all s∈ {1, , k + 1}
Trang 8Hence, by (3.5), (3.6), we have
q
X
j=1
(Qj( bf))0(a) =
N+1X
t=1
(QI p (t)( bf))0(a)
≤ (N − k + 1)(QI p (1)( bf))0(a) +
N+1X
t=N −k+2
(QI p (t)( bf))0(a)
≤ (N − k + 1)(Pp,1( bf))0(a) +
k+1
X
s=2
(Pp,s( bf))0(a)
≤ (N − k + 1)
k+1
X
s=1
(Pp,s( bf))0(a)
Hence, by (3.14), we have
1
uHY(u)SY(u, c) ≥
1 (k + 1)(N − k + 1)
q
X
j=1
(Qj( bf))0(a)
− (2k + 1)△
u
X
1≤t≤n 0 ,1≤s≤k+1
(Pt,s( bf))0(a)
Combining with (3.13) we get
(N − k + 1)(k + 1)
duHY(u) (W ( bF))0(a) ≥
(N − k + 1)(k + 1) duH2
Y(u)(HY(u) − 1)SY(u, c)
dHY(u)(HY(u) − 1)
q
X
j=1
(Qj( bf))0(a)
−(2k + 1)(N − k + 1)(k + 1)△
duHY(u)(HY(u) − 1)
X
1≤t≤n 0 ,1≤s≤k+1
(Pt,s( bf))0(a)
dHY(u)(HY(u) − 1)
q
X
j=1
(Qj( bf))0(a)
−(2k + 1)(N − k + 1)(k + 1)△
du
X
1≤t≤n 0 ,1≤s≤k+1
(Pt,s( bf))0(a),
for all a∈ ∪qj=1f−1(Qj)
Hence,
(N − k + 1)(k + 1)
duHY(u) N(r, W ( bF))0) ≥
1
dHY(u)(HY(u) − 1)
q
X
j=1
Nf(r, Qj)
− (2k + 1)(N − k + 1)(k + 1)△
du
X
1≤t≤n 0 ,1≤s≤k+1
N(r, Pt,s( bf))0)
Trang 9Combining with (3.2) we have
f(r) ≤
q
X
j=1
1
dNf(r, Qj)
dHY(u)(HY(u) − 1)
q
X
j=1
Nf(r, Qj) + (N − k + 1)(k + 1)ǫ
′
HY(u) Tf(r) +(2k + 1)(N − k + 1)(k + 1)△
du
X
1≤i≤n 0 ,1≤j≤k+1
(N(r, Pt,s( bf))0) + mf(r, Pi,j))
≤ HY(u)(HY(u) − 1) − 1
HY(u)(HY(u) − 1)
q
X
j=1
1
dNf(r, Qj) +
(N − k + 1)(k + 1)ǫ′
HY(u) +
(2k + 1)(N − k + 1)(k + 1)△
du
Tf(r) (3.15)
For each ǫ > 0, we choose u = u0 := [(2k+1)(N −k+1)dǫ 2(k+1)2△] + 1, and ǫ′ :=
ǫ
(N −k+1)(k+1) −(2k+1)(N −k+1)(k+1)△du
Then, we have
HY(u0) ≤ k + deg Y uk0
≤ k + dkdeg V [(2k + 1)(N − k + 1)2(k + 1)2dk−1deg V ǫ−1] + 1k
= M,
(note thatdeg Y = △ ≤ dkdeg V )
Hence, by (3.15) we have
f(r) ≤ M(M − 1) − 1
M(M − 1)
q
X
j=1
1
dNf(r, Qj)
REFERENCES
[1] P Corvaja and U Zannier, 2004 On a general Thue’s equation Amer J Math 126,
pp 1033-1055
[2] M Ru, 2004 A defect relation for holomorphic curves intersecting hypersurfaces
Amer J Math., 126, pp 215-226.
[3] M Ru, 2009 Holomorphic curves into algebraic varieties Ann of Math., 169, pp.
255-267
Trang 10[4] G Dethloff and T V Tan, 2011 A second main theorem for moving hypersurface
targets Houston J Math., 37, pp 79-111.
[5] G Dethloff, T V Tan and D D Thai, 2011 An extension of the Cartan-Nochka
second main theorem for hypersurfaces Int J Math., 22, pp 863-885.
[6] G Dethloff and T V Tan, 2020 Holomorphic curves into algebraic varieties
intersecting moving hypersurface targets Acta Math Vietnam, 45, pp 291-308.
[7] S D Quang, 2019 Degeneracy second main theorems for meromorphic mappings
into projective varieties with hypersurfaces Trans Amer Math Soc., 371, pp.
2431-2453
[8] T V Tan, Higher dimensional generalizations of some theorems on normality of
meromorphic functions, to appear in the Michigan Math J.
[9] N T Son and T V Tan, A property of the spherical derivative of an entire curve in
complex projective space, to appear in the J Math Anal Appl.
[10] Z Chen, M Ru and Q Yan, 2012 The degenerated second main theorem and
Schmidts subspace theorem Sci China Math., 55, pp 1367-1380.
[11] L Giang, 2016 An explicit estimate on multiplicity truncation in the degenerated
second main theorem Houston J Math 42, pp 447-462.
... Dethloff and T V Tan, 2011 A second main theorem for moving hypersurface< /p>targets Houston J Math., 37, pp 79-111.
[5] G Dethloff, T V Tan and D D Thai, 2011 An extension...
intersecting moving hypersurface targets Acta Math Vietnam, 45, pp 291-308.
[7] S D Quang, 2019 Degeneracy second main theorems for meromorphic mappings
into projective. .. extension of the Cartan-Nochka
second main theorem for hypersurfaces Int J Math., 22, pp 863-885.
[6] G Dethloff and T V Tan, 2020 Holomorphic curves into algebraic varieties