Guangsheng YU

Abstract In this paper,by using Seshadri constants for subschemes,the author establishes a second main theorem of Nevanlinna theory for holomorphic curves intersecting closed subschemes in (weak) subgeneral position. As an application of his second main theorem,he obtain a Brody hyperbolicity result for the complement of nef effective divisors. He also give the corresponding Schmidt’s subspace theorem and arithmetic hyperbolicity result in Diophantine approximation.

Keywords Second main theorem, In general position, Closed subscheme, Seshadri constant, Schmidt’s subspace theorem, Hyperbolicity

1 Introduction

In higher dimensional Nevanlinna theory, it mainly studies the second main theorem of holomorphic maps between complex manifolds intersecting subvarieties in the target manifold.Cartan[2]established the second main theorem for linearly non-degenerate holomorphic curves into complex projective space intersecting hyperplanes in general position, and Nochka [12]considered the case of subgeneral position. Ru [14–15] established second main theorems for algebraically non-degenerate holomorphic curves into complex projective varieties intersecting hypersurfaces in general position and there are many new developments (see [13, 16]).

Recently, there are many developments in extending the second main theorem to arbitrary subschemes case. Ru and Wang [17] obtained the following second main theorem for holomorphic curves intersecting closed subschemes.

Theorem 1.1(see[17]) Let X be a projective variety. Let Y1,··· ,Yqbe closed subschemes of X such that, for any x ∈X, there are at most m subschemes among Y1,··· ,Yqwhich contains x. Let A be a big Cartier divisor on X. Let f : C →X be a holomorphic curve with Zariski-dense image. Let

where πj:→X is the blowing-up of X along Yj, with associated exceptional divisor Ej.Then for every ε>0,

where “‖” means the estimate holds for all large r outside a set of finite Lebesgue measure.(Here we use some notations which will be explained later).

When the closed subschemes Yj=yjare distinct points, the Seshadri constants ?yj(A) and βA,yjhave the following relation (see [11])

where dim X =n. Then one may take m=1 in Theorem 1.1 and obtain the following inequality

More generally, by using the definition of Seshadri constants for general closed subscheme(see Section 2), Heier and Levin [9] obtained the following result.

Theorem 1.2(see [9, Theorem 1.3]) Let X be a projective variety of dimension n. Let Y1,··· ,Yqbe closed subschemes of X such that for every subset I ?{1,··· ,q} with#I ≤n+1,we have codim ∩Let A be an ample Cartier divisor on X. Let f :C →X be a holomorphic curve with Zariski-dense image. Then for every ε>0,

Let Y be a closed subscheme of X of codimension codim Y in X. Note that the elements of the list obtained by repeating Y up to codim Y times still satisfy the condition in Theorem 1.2, then from Theorem 1.2 we have the following corollary.

Corollary 1.1(see[9, Corollary 1.6]) Let X be a projective variety of dimension n. Let Y be a closed subscheme of X of codimension codim Y in X. Let A be an ample Cartier divisor on X. Let f :C →X be a holomorphic curve with Zariski-dense image. Then for every ε>0,

They also compared the constant βA,Yand the Seshadri constant ?Y(A) (see [9, Theorem 4.2]) and obtained that

under the assumption that X is smooth.

Recently, He and Ru [7] give more general results.

Theorem 1.3(see [7]) Let X be a projective variety of dimension n and let m ≥n be a positive integer. Let Y1,··· ,Yqbe closed subschemes of X such that for every subset I ?{1,··· ,q} with #I ≤m + 1 we have dim≤m ?#I, where we use the convention that dim ?= ?1. Let A be an ample Cartier divisor on X. Let f : C →X be a holomorphic curve with Zariski-dense image. Then for every ε>0,

We note that, in [7, 9, 17], the conditions on the subschemes Y1,··· ,Yqare called in“(sub)general position”, but actually they are different. In this paper, in order to distinguish them, we call the subschemes in “weak (sub)general position” and “(sub)general position”.

Let X be a projective variety of dimension n and let Y1,··· ,Yqbe closed subschemes of X.

Definition 1.1(Subgeneral position) Let m be a positive integer.

(a) The closed subschemes Y1,··· ,Yqare called in weak m-subgeneral position if, for any x ∈X, there are at most m subschemes among Y1,··· ,Yqwhich contains x. When m=n, the subschemes are called in weak general position.

(b) The closed subschemes Y1,··· ,Yqare called in m-subgeneral position if for any subset I ?{1,··· ,q} with #I ≤m+1, we have dim≤m ?#I. When m = n, the subschemes are called in general position.

Definition 1.1(a) is used in [17], Definition 1.1(b) is used in [7, 9].

We also note that the condition in (a) is weaker than that in (b), since when #I = m+1, dimimplies thatmust be empty. When Y1,··· ,Yqare hypersurfaces, (a) is equivalent to (b).

Motivated by the main theorem in [3], we give a second main theorem under“weak subgeneral position” condition.

Theorem 1.4Let X be a complex projective variety of dimension n and Y1,··· ,Yqbe closed subschemes of X. Let A be an ample Cartier divisor on X. Let f : C →X be a non-constant holomorphic curve such that f(C) /?Supp Yj, for j = 1,··· ,q. Assume that Y1,··· ,Yqare in weak m-subgeneral position. Then, for every ε>0,

Remark 1.1If Y1,··· ,Yqare general divisors, then Theorem 1.4 is a generalization of [3,Corollary 1.2].

Now we consider the case that Y1,··· ,Yqare divisors. As an application of Theorem 1.4,we can obtain a Brody hyperbolicity result for the complement of nef effective divisors, which is motivated by the work of Heier and Levin [8]. We first recall the definition of Brody hyperbolicity.

Definition 1.2(Brody hyperbolic) A complex variety is said to be quasi-Brody hyperbolic if the union of all images of nonconstant holomorphic maps from C is not Zariski dense in it.A complex variety is said to be Brody hyperbolic if it admits no nonconstant holomorphic maps from C.

Let X be a projective variety of dimension n. Let D1,··· ,Dqbe non-zero effective Cartier divisors in general position on X. For an ample divisor A on X, if there exist positive rational constants c1,··· ,cqsuch that for all j =1,··· ,q:

A ?cjDjis Q-nef.

Then (1.3) implies that

In[8],by using(1.8)and choosing appropriate A and cj,Heier and Levin obtained the following result.

If we use our Theorem 1.4 instead of Theorem 1.2,then we can obtain a hyperbolicity result under weak subgeneral poisition condition on the divisors.

2 Preliminaries

2.1 Seshadri constants

The Seshadari constants have the following non-decreasing property.

Proposition 2.1Let X be a projective variety and X′be a subvariety of X, denote by i : X′→X the inclusion map. Let A be a nef Cartier divisor on X and Y be a closed subscheme of X. Then we have

2.2 Weil functions

We briefly recall the basic definition of Weil functions, one can refer to[19]for more details.Let Y be a closed subscheme of a projective variety X. One can associate a Weil function λY: X Supp Y →R, well-defined up to O(1), which satisfies the following properties: If Y and Z are two closed subschemes of X, and φ:X′→X is a morphism of projective varieties,

(i) λY∩Z=min{λY,λZ};

(ii) λY+Z=λY+λZ;

(iii) λY≤λZ, if Y ?Z;

(iv) λY(φ(x))=λφ?Y(x).

In particular,let D be a Cartier divisor on a complex projective variety X. A Weil function with respect to D is a function λD: (X SuppD) →R such that for all x ∈X there is an open neighborhood U of x in X, a nonzero rational function f on X with D|U= (f), and a continuous function α:U →R such that

λD(x)=?log|f(x)|+α(x)

for all x ∈(U SuppD). Note that a continuous fiber metric ‖·‖ on the line sheaf OX(D)determines a Weil function for D given by λD(x)=?log‖s(x)‖,where s is the rational section of OX(D) such that D = (s). An example of Weil function for the hyperplane H = {a0x0+···+anxn=0} in Pn(C) is given by

where [x0,··· ,xn] are homogeneous coordinates for x. The Weil functions with respect to divisors satisfiy the following properties:

(a) Functoriality: If λ is a Weil function for a Cartier divisor D on X, and if φ : X′→X is a morphism such that φ(X′)SuppD, then xλ(φ(x)) is a Weil function for the Cartier divisor φ?D on X′.

(b) Additivity: If λ1and λ2are Weil functions for Cartier divisors D1and D2on X,respectively, then λ1+λ2is a Weil function for D1+D2.

(c) Uniqueness: If both λ1and λ2are Weil functions for a Cartier divisor on X, then λ1=λ2+O(1).

(d) Boundedness from below: If D is an effective divisor and λ is a Weil function for D,then λ is bounded from below.

Let X be a projective variety, and let Y ?X be a closed subscheme.

Lemma 2.1(see [19, Lemma 2.2]) There exist effective Cartier divisors D1,··· ,D?such that

By Lemma 2.1, we can assume that Y = D1∩··· ∩D?, where D1,··· ,D?are effective Cartier divisors. This means that IY= ID1+···+ID?, where IY,ID1,··· ,ID?are the defining ideal sheaves in OX. We set

Then we have λY:X Supp Y →R, which does not depend on the choice of Cartier divisors.

2.3 Nevanlinna functions

In this section, we briefly recall the definitions of characteristic function, proximity function and counting function in Nevanlinna theory.

2.3.1 Characteristic function

Let X be a complex projective variety and f : C →X be a holomorphic map. Let L →X be an ample line sheaf and ω be its Chern form. We define the characteristic function of f with respect to L by

Since any line sheaf L can be written as L = L1?with L1, L2being both ample, we define Tf,L(r)=Tf,L1(r)?Tf,L2(r). A divisor D on X defines a line bundle O(D), we denote by Tf,D(r)=Tf,O(D)(r). If X =Pn(C)and L=OPn(C)(1), then we simply write Tf,OPn(C)(1)(r)as Tf(r).

The characteristic function satisfies the following properties:

(a) Functoriality: If φ:X →X′is a morphism and if L is a line sheaf on X′, then

(b) Additivity: If L1and L2are line sheaves on X, then

(c) Positivity: If L is ample and f :C →X is non-constant, then

(d) Base locus: If the image of f is not contained in the base locus of |D|, then Tf,D(r) is bounded from below.

(e) Globally generated line sheaves: If L is a line sheaf on X, and is generated by its global sections, then Tf,L(r) is bounded from blow.

2.3.2 Counting and proximity functions

Let X be a projective variety and let Y ?X be a closed subscheme. For a holomorphic curve f :C →X with f(C)Supp Y, the proximity function of f with respect to Y is defined by

The proximity function satisfies the following properties:

(a) Functoriality: If φ : X →X′is a morphism and Y′is a closed subscheme on X′with φ ?f(C)/?Supp Y′, then

mf(r,φ?Y′)=mφ?f(r,Y′)+O(1).

(b) Additivity: If Y1and Y2are two closed subschemes on X, then

mf(r,Y1+Y2)=mf(r,Y1)+mf(r,Y2)+O(1).

(c) Boundedness from below: If D is an effective divisor, then mf(r,D) is bounded from below.

Next, we introduce the definition of counting function Nf(r,Y) given by Yamanoi in [23].Assume that Y can be written as Y = D1∩···∩D?with D1,··· ,D?being effective Cartier divisors, then we set

When f(C) ?Supp Di, we set ordzf?Di= +∞. The definition of ordzf?Y does not depend on the choice of the Cartier divisors D1,··· ,D?. We define the counting function by

For a closed subscheme Y as above, consider the blowing up π :→X of X along Y, let f : C →X be a holomorphic curve and: C →be its holomorphic lifting. Let A be an ample divisor on X. By using the functoriality property of characteristic function, proximity function and counting function, we have

2.3.3 First main theorem

Let X be a complex projective variety and f : C →X be a holomorphic map. Let D be a divisor on X. By using Poincar-Lelong formula, we have the first main theorem,

3 Proof of Theorem 1.4

Proof of Theorem 3.1Denote by ρj:=codim Yj, j =1,··· ,q.

Given z ∈C, we arrange {1,··· ,q} as {1(z),··· ,q(z)} so that

Since Y1,··· ,Yqare in weak m-subgeneral position, we have

The proof of the first inequality is similar to that of [21, Lemma 21.7] and is omitted here.

For each 1 ≤j ≤q, we observe that according to Definition 1.1(b), the elements of the list obtained by repeating Yjup to ρjtimes, i.e., the closed subschemes Yj,1,Yj,2,··· ,Yj,ρwith Yj,?= Yjfor ? = 1,··· ,ρj, are in general position (of Definition 1.1(b)). Note that the union of all closed subschemes Yj,?(1 ≤j ≤q,1 ≤? ≤ρj) is a finite set, which may be denoted by{,··· ,}. We rewrite (3.3) as

where the maximum is taken over all subsets K of {1,··· ,T} such that the closed subschemesu ∈K, are in general position.

Now, we need the following general form of second main theorem given in [9].

Theorem 3.2(see [9, Theorem 1.8]) Let X be a projective variety of dimension n. Let Y1,··· ,Yqbe closed subschemes of X. Let A be an ample Cartier divisor on X. Let f :C →X be a holomorphic curve with Zariski-dense image. Then for every ε>0,

where the maximum is taken over all subsets J of {1,··· ,q} such that the closed subschemes Yj,j ∈J, are in general position.

Then it follows from (3.4) and Theorem 3.2 that, for any given ε>0,

This completes the proof of Theorem 3.1.

Remark 3.1(1) If q =1, i.e., there is only one closed subscheme Y, we may take m =1,then (3.1) is exactly (1.4). If Y1,··· ,Yqare distinct points, then (1.2) can also be obtained from (3.1).

(2) Recently, Ru and Wang [18] proved that, if Y1,··· ,Yqare intersecting properly on X(which is, by [10, Theorem 17.4], equivalent to in general position when X is smooth), then

(Or see [22] for more general case.) This gives an improvement of Theorem 1.2 when X is smooth.

By using similar method, one can obtain that, if Y1,··· ,Yqare in weak m-subgeneral position on X, then

From Proposition 2.1, we have

Combining (3.9)–(3.12), we have

This completes our proof of Theorem 1.4.

Remark 3.2Combining the proof of Theorem 1.4 and He-Ru’s result(see[7,Main Theorem(Analytic Part)]), one can also obtain a second main theorem for “non-constant” holomorphic curves as follows.

Let X be a complex projective variety of dimension n and Y1,··· ,Yqbe closed subschemes of X. Let A be an ample Cartier divisor on X. Let f :C →X be a non-constant holomorphic curve. Assume that Y1,··· ,Yqare in m-subgeneral position. Then, for every ε>0,

Though the coefficient on the right-hand side of (3.13)is smaller than that in Theorem 3.1,but in our theorem, the subschemes only need to be in weak m-subgeneral position and m can be smaller than n.

4 Proof of Theorem 1.6

In this section, we give the proof of Theorem 1.6 by using our second main theorem and the method of Heier and Levin [8].

Proof of Theorem 1.6By assumption, for any proper subset T of the set of standard basis vectors {e1,··· ,er} ?Rr, at mostf the vectors P1,··· ,Pqare supported on T, then we may take

to be (discontinuous) functions of κ ∈(0,1] with the following properties.

(i) The function αj,i(κ) is identically equal to 0 if aj,i= 0. If, on the other hand, aj,i0,then αj,i(κ) takes on positive real values such that we have the limits

(ii) The R-divisors Bj(κ)=αj,1(κ)E1+···+αj,r(κ)Erare such that

unless both terms on the left are 0.

For Q=(b1,··· ,br)∈Rrwith positive coordinates, define

We need the following result in [8].

and

Then A′is Q-ample.

We define positive rational numbers

Note that

We first deal with the general case r ≥3.

Since

we have

Therefore,

Here, for R-divisors F1and F2, we write F1≥F2if the difference F1?F2is a nef R-divisor.

Finally, we find the inequalities

where the last inequality is due to (4.6). Therefore, by?2δ >0,

is Q-ample.

When r =1, since q ≥rm(n+1)+1, we have

Thus

Then we have

is Q-ample.

When r =2, since q ≥rm(n+1)+1, we have

Then the proof is the same as the case r ≥3 and we have

is Q-ample.

Now A′is an ample Q-divisor, there exists a positive integer N big enough such that NA′is an ample integral divisor. Set A:=NA′and cj:=for j =1,··· ,q.

To summarize, there exist an ample divisor A and positive rational constants c1,··· ,cq,δ such that for all j =1,··· ,q:

and

If f is not constant, we may apply Theorem 1.4 to conclude that,

5 Schmidt Subspace Theorem

In this section,we introduce the counterpart in number theory of our main results according to Vojta’s dictionary which gives an analogue between Nevanlinna theory and Diophantine approximation. The line of reasoning is by now well known and we omit the details here.

Let k be a number field. Denote by Mkthe set of places(i.e., equivalence classes of absolute values) of k and writefor the set of archimedean places of k.

Let X be a projective variety defined over k,let L be a line sheaf on X and let Y be a closed subscheme on X. For every place v ∈Mk, we can associate the local Weil functions λL,vand λY,vwith respect to v, which have similar properties as the Weil function introduced in Section 2. For more details, please refer to [20, Section 1.3].

Define

and

where S is a finite subset of Mkcontaining.

Now, we state the counterparts of Theorems 3.1 and 1.4.

Theorem 5.1Let X be a projective variety, defined over a number field k, of dimension n. Let Yjbe closed subschemes of codimension codim Yj(≥1) in X, j = 1,··· ,q. Let A be an ample Cartier divisor on X. Let S be a finite subset of Mkcontaining. Assume that Y1,··· ,Yqare in weak m-subgeneral position. Then, for every ε > 0, there exists a proper Zariski-closed subset Z ?X such that for all points x ∈X(k),

Theorem 5.2Let X be a projective variety, defined over a number field k, and Y1,··· ,Yqbe proper closed subschemes of X in weak m-subgeneral position. Let A be an ample Cartier divisor on X. Let S be a finite subset of Mkcontaining. Then, for every ε>0, the set of points x ∈X(k)Supp Yjwith

is a finite set.

Now, using the method shown in [8], we give an application of this theorem.

Definition 5.1(Arithmetically quasi-hyperbolic) Given a variety V =XD defined over a number field k.

(i) We say that V is arithmetically quasi-hyperbolic if there exists a proper closed subset Z ?X such that for every number field k′?k, every finite set of places S of k′containing the archimedean places, and every set R of (k′-rational) (D,S)-integral points on X, the set R is finite.

(ii) We say that XD is arithmetically hyperbolic if all sets of (D,S)-integral points on X are finite (i.e., one may take Z =?in the definition of quasi-hyperbolicity).

For the notion of (D,S)-integral sets of points, please refer to [20, Section 1.4].

In [8], Heier and Levin showed the following arithmetically quasi-hyperbolicity result as an application of [9, Corollary 1.4].

then X D is arithmetically quasi-hyperbolic.

As an application of Theorem 5.2,we have an arithmetically hyperbolicity result for divisors in weakly m-subgeneral position as follows.

then X D is arithmetically hyperbolic.