Degeneracy of entire curves into higher dimensional complex manifolds

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Degeneracy of entire curves into higher dimensional complex manifolds

Ya Deng

To cite this version:

Ya Deng. Degeneracy of entire curves into higher dimensional complex manifolds. 2016. �hal-01283719�

Degeneracy of entire curves into higher dimensional complex manifolds

Ya Deng

Abstract

Pursuing McQuillan’s philosophy in proving the Green-Griffiths conjecture for certain surfaces of general type, we deal with the algebraic degeneracy of entire curves tangent to holomorphic foliations by curves. Inspired by the recent work [PS14], we study the intersec- tion of Ahlfors current T [f ] with tangent bundle T

F

of F , and derive some consequences. In particular, we introduce the definition of weakly reduced singularities for foliations by curves, which requires less work than the exact classification for foliations. Finally we discuss the strategy to prove the Green-Griffiths conjecture for complex surfaces.

Contents

1 Introduction 1

2 Proof of the Main Theorems 3

2.1 Notions and definitions . . . . 3

2.2 Basic results about Ahlfors currents . . . . 5

2.3 Intersection with the tangent bundle . . . . 9

2.4 Siu’s refined tautological inequality . . . . 14

2.5 Intersection with the normal bundle . . . . 17

2.6 A generalization of McQuillan’s theorem . . . . 20

3 Towards the Green-Griffiths conjecture 26

1 Introduction

In [McQ98], McQuillan proved the following striking theorem:

Theorem 1.1. Let X be a surface of general type and F a holomorphic foliation on X. Then any entire curve F : C → X tangent to F can not be Zariski dense.

The original proof of Theorem 1.1 is rather involved. Later on, several simplified proofs appeared, cf. [Bru99] and [PS14]. The idea in proving Theorem (1.1) is to argue by contradiction.

One assumes that there exists a Zariski dense entire curve f : C → X which is tangent to F .

Then, one studies the intersection of the Ahlfors current T [f ], which can be treated as a (1, 1)-

cohomology class in X, with the tangent bundle and the normal bundle of the foliation F . The

above works proved that both of the intersections numbers are positive. However, since K

X

is big, then T [f ] · K

_X

> 0, and by the equality K

_X⁻¹

= T

_F

+ N

_F

, a contradiction is obtained.

In our paper, we give a generalization of McQuillan’s result dealing with entire curves on higher dimensional complex manifolds, by pursuing the same philosophy. Firstly we obtain a formula for T [f ] · T

_F

, which is an improvement of that of [PS14]. Our first result is the following theorem:

Theorem 1.2. Let (X, F ) be a K¨ ahler 1-foliated pair. If f : C → X is a transcendental entire curve (see Definition 2.2 below) tangent to F whose image is not contained in Sing( F ), then

h T [f ], c

₁

(T

_F

) i + T (f, J

F

) = h T[f

₁

], O

X1

( − 1) i ≥ 0,

where J

F

is a coherent ideal sheaf determined by the singularity of F , and T (f, J

F

) is a non- negative real number representing the “intersection” of T [f ] with J

F

; this number will be defined later.

If X is a complex surface, as is proved by McQuillan, Theorem 1.2 can be improved to the extent that

T [f ] · T

_F

≥ 0. (1.1)

On the one hand, by pursuing his philosophy of “diaphantine approximation”, one can generalize (1.1) to higher dimensional manifolds, under some assumptions on the foliation:

Theorem 1.3. Let F be a foliation by curves on the n-dimensional complex manifold X, such that the singular set Sing( F ) of the foliation F is a set of absolutely isolated singularities (this will be defined later). If f : C → X is an Zariski dense entire curve which is tangent to F , then one can blow-up X a finite number of times to get a new birational model ( X, e F e ) such that

T [ f e ] · T

_Fe

= 0.

Moreover, if F is a foliation by curves with absolutely isolated singularities that are simple (see Definition 2.6), and whose canonical bundle K

_F

is big, then every entire curve f : C → X that is tangent to F is algebraically degenerate.

On the other hand, Theorem 1.2 leads us to the fact that the error term T (f, J

F

) is control- lable, if the singularities of F are not too “bad” (they are called weakly reduced singularities in Section 2.5). The theorem is as follows:

Theorem 1.4. Let X be a projective manifold of dimension n endowed with a 1-dimensional foliation F with weakly reduced singularities. If f is a Zariski dense entire curve tangent to F , satisfying h T [f ], K

_X

i > 0 (e.g. K

_X

is big), then we have

h T [ ˆ f ], c

₁

(N

_Fˆ

) i < 0 for some birational pair ( ˆ X, F ˆ ).

Remark 1.1. Our definition of “weakly reduced singularities” is actually weaker than the usual

concept of reduced singularities, which always requires a lot of checking (e.g. a classification of

singularities). We only need to focus on the multiplier ideal sheaf of J

F

, instead of trying to

understand the exact behavior of singularities.

It is notable that the following strong result due to M. Brunella implies a conclusive contra- diction in combination with Theorem 1.4, in the case of dimension 2.

Theorem 1.5. Let X be a complex surface endowed with a foliation F (no assumption is made for singularities of F here). If f : C → X is a Zariski dense entire curve tangent to F , then we have

h T [f ], N

_F

i ≥ 0.

Therefore, we get another proof of Theorem 1.1 without using the refined formula (1.1) im- mediately. This leads us to observe that if one can resolve any singularities of the 1-dimensional foliation F into weakly reduced ones, and generalize the previous Brunella Theorem to higher dimensional manifolds, one could infer the Green-Griffiths conjecture for surfaces of general type.

Theorem 1.6. Assume that Theorem 1.5 holds for a directed variety (X, F ) where X is a base of arbitrary dimension and F has rank 1, and that one can resolve the singularities of F into weakly reduced ones. Then every entire curve drawn in a projective surface of general type must be algebraically degenerate.

2 Proof of the Main Theorems

2.1 Notions and definitions

For any coherent ideal sheaf J ⊂ O

X

, one can construct a global quasi-plurisubarmonic function ϕ

_J

on X such that

ϕ

_J

= log( X

i

| g

_i

|

²

) + O (1)

where (g

i

) are local holomorphic functions that generate the ideal J . We call ϕ

J

the charac- teristic function associated to the coherent sheaf. For any entire curve f which is not contained in the subscheme Z( J ), we can write

ϕ

J

◦ f (t) |

B(r)

= X

|tj|<r

ν

j

log | t − t

j

|

²

+ O (1), and here we call ν

_j

the multiplicity of f along J .

In a related way, we define the proximity function of f with respect to J by m

_f,J

(r) := − 1

2π Z

2π

0

ϕ

J

◦ f (re

^iθ

)dθ, and the counting function of f with respect to J by

N

_f,J

(r) := X

|tj|<r

ν

_j

log r

| t

_j

| .

Let us take a log resolution of F with a birational morphism p : ˆ X → X such that p

⁻¹

( J ) = O

Xˆ

( − D), let ˆ f denote the lift of f to ˆ X (so that p ◦ f ˆ = f ), and let Θ

_D

be the curvature form of D.

Now we recall the following formula, which will be very useful in what follows.

Theorem 2.1. (Jensen formula) For r ≥ 1 we have Z

_r

1

dt t

Z

B(t)

dd

^c

ϕ = 1 2π

Z

_2π

0

ϕ(re

^iθ

)dθ − 1 2π

Z

_2π

0

ϕ(e

^iθ

)dθ, (2.1)

in particular if ϕ is a quasi-plurisubharmonic function, then for r large enough we have Z

_r

1

dt t

Z

B(t)

dd

^c

ϕ = 1 2π

Z

_2π

0

ϕ(re

^iθ

)dθ + O (1).

Then the following First Main Theorem due to Nevanlinna is an immediate consequence.

Theorem 2.2. As r → ∞ , one has T

_{f ,Θˆ}

D

(r) = N

_f,J

(r) + m

_f,J

(r) + O (1).

Let F be a 1-dimension foliation. Then we can take an open covering { U

_α

}

α∈I

such that on each U

_α

there exists v

_α

∈ H

⁰

(U

_α

, T

_X

|

Uα

) which generates F , and such that the v

_α

coincide up to multiplication by nowhere vanishing holomorphic functions { g

_αβ

} :

v

_α

= g

_αβ

v

_β

if U

_α

∩ U

_β

6 = ∅ . The functions { g

_αβ

} define a cohomology class H

¹

(X, O

_X^∗

) that corresponds to the cotangent bundle of F , denoted here by T

_F^∗

. Let ω be a hermitian metric on X. Then ω induces a natural singular metric h

_s

on T

_F

. Indeed, on each U

_α

the local weight ϕ

_α

is given by

ϕ

_α

= − log | v

_α

|

²ω

= − log X

i,j

a

ⁱ_α

a

^j_α

ω

_ij

, (2.2)

where v

α

= P

n

i=1

a

ⁱ_α∂z^∂i

with respect to the coordinate system (z

1

, . . . , z

n

) on U

α

.

We are going to define a coherent sheaf J

F

reflecting the behavior of the singularies of F : on each U

_α

the generators of J

F

are precisely the coefficients (a

ⁱ_α

) of the vector v

_α

defining F , i.e.

v

_α

= X

n i=1

a

ⁱ_α

∂

∂z

ⁱ

.

If we fix a smooth metric h on T

_F

, then there exists a globally defined function ϕ

_s

such that h = h

_s

e

^−ϕs

We know that

ϕ

_s

= log | v

_α

|

²ω

(2.3)

modulo a bounded function, and ϕ

_s

is the characteristic function associated to the coherent sheaf J

F

.

All the constructions explained above can be generalized to log pairs. We first begin with

the following definition.

Definition 2.1. Let X be a smooth K¨ ahler manifold, D a simple normal crossing divisor and F a foliation by curves defined on X. We say that F is defined on the log pair (X, D) if each component of D is invariant by F . For brevity we say that (X, F , D) a K¨ ahler 1-foliated triple.

The logarithmic tangent bundle T

_X

h− log D i with respect to the pair (X, D) is the subsheaf of T

_X

whose local sections are given by

v = X

k j=1

z

_j

v

_j

∂

∂z

j

+ X

n

i=k+1

v

_i

∂

∂z

i

, where z

₁

z

₂

. . . z

_k

= 0 is the local equation of D.

We can take a hermitian metric ω

_X,D

on T

_X

h− log D i defined as follows ω

_X,D

= √

− 1 X

k i,j=1

ω

_i¯j

dz

_i

∧ d¯ z

_j

z

_i

z ¯

_j

+ 2Re √

− 1 X

i>k≥j

ω

_i¯j

dz

_i

∧ d¯ z

_j

¯

z

_j

+ √

− 1 X

i,j≥k+1

ω

_i¯j

dz

_i

∧ d¯ z

_j

, where (ω

_i¯j

) is smooth positive definite hermitian matrix. In other words, the local model of ω

_X,D

is given by

ω

X,D

≡ √

− 1 X

k

i=1

dz

_i

∧ d¯ z

_i

| z

_i

|

²

+ + √

− 1 X

i≥k+1

dz

i

∧ d¯ z

i

.

Assume that the foliation F is defined on (X, D). Locally we have v

_α

=

X

k j=1

z

_j

a

^j_α

∂

∂z

j

+ X

n

i=k+1

a

ⁱ_α

∂

∂z

i

as the generator of F . Then ω

_X,D

induces a singular hermitian metric h

_s,D

on T

_F

whose local weight is given by

ϕ

α,D

= − log | v

α

|

²ωX,D

= − log X

i,j

a

ⁱ_α

a

^jα

ω

_ij

.

We denote by J

F,D

the coherent sheaf defined by the functions (a

^j_α

). In general we have J

F

⊂ J

F,D

,

and the inclusion may be strict. If we find a smooth metric h = h

_s,D

e

^−ϕs,D

on T

_F

, then it is easy to check that ϕ

s,D

is the characteristic function associated with J

F,D

.

We set ¯ X

₁

:= P (T

_X

h− log D i ), then ω

_X,D

induces a natural smooth metric h

₁

on O

X^¯1

( − 1).

2.2 Basic results about Ahlfors currents

Let X be a compact complex manifold, and let f be an entire curve. Then we can associate a closed positive current of (n − 1, n − 1) type as follows: for any smooth 2-form η, we let

h T

_rk

[f ], η i := T

_f,η

(r

_k

)

T

_f,ω

(r

_k

) ,

where T

_f,ω

(r) := R

r 1

dt t

R

B(t)

f

^∗

ω. Here B(t) is the ball of radius t in C and S(t) is its boundary.

It is known that one can find a suitable sequence of (r

_k

) that tends to infinity, such that the weak limit of T

_rk

[f ] is a closed positive current (ref. [Bru99]). It is denoted by T [f ] and called the Ahlfors current of f . Indeed, to ensure that the weak limit of T

_rk

[f] is closed, we only need r

_k

to satisfy the following condition

rk

lim

→∞

length(f (S(r

_k

)))

area(f (B(r

_k

))) = 0. (*)

Remark 2.1. In the definition of the Ahlfors current, it is not indispensable to assume ω to be a K¨ ahler form. In fact, it suffices to assume that ω is a semi-positive form satisfying

rk

lim

→∞

T

_f,ω

(r

_k

)

T

_f,eω

(r

_k

) > C > 0 with respect to some K¨ ahler form e ω.

Theorem 2.3. Let L be a big line bundle on a K¨ ahler manifold X. If f : C → X is an entire curve on X such that its image is not contained in the augmented base locus B

₊

(L) of L (ref. [Laz04]), then h T[f ], c

₁

(L) i > 0.

Proof. Since the image of f is not contained in B

₊

(L), by the definition of the augmented base locus one can find an effective divisor E whose support does not contain the image of f , such that

L ≡ A + E,

where A is an Q ample divisor, and “ ≡ ” means numerically equivalent. Then the counting function of f with respect to E is non-negative and one can find a smooth hermitian metric h

_E

on E such that the proximity function of f with respect to E is also non-negative. Therefore

h T [f ], Θ

_hE

(E) i ≥ 0.

By the ampleness of A, we have

h T [f], c

₁

(A) i > 0, and thus

h T [f ], c

₁

(L) i = h T[f ], c

₁

(A) + c

₁

(E) i > 0.

Definition 2.2. An entire curve f : C → X is said to be an algebraic curve iff f admits a factorization in the form f = g ◦ R, where R : C → P

¹

is a rational function and g : P

¹

→ X is a rational curve; otherwise f will be called transcendental.

We have the following criterion for an entire curve to be algebraic (ref. [Dem97]):

Theorem 2.4. Any entire curve f : C → X is an algebraic curve if and only if T

_f,ω

(r) = O (log r). In particular, if f is Zariski dense, then

r→∞

lim

T

_f,ω

(r)

log r = + ∞ .

From now on, for brevity we write T

_f

(r) in place of T

_f,ω

(r), where ω can be some semi- positive (1, 1)-form satisfying the condition in Remark 2.1.

The following logarithmic derivative lemma will be very useful in our arguments.

Theorem 2.5. (Logarithmic derivative lemma) Let f be a meromorphic function on C. Then 1

2π Z

2π

0

log

⁺

| f

^′

(re

^iθ

)

f (re

^iθ

) | dθ ≤ O (log T

_f

(r) + log r). (2.4) Let (X, V ) be a smooth directed variety. It is easy to see that there is a canonical lift of f to X

1

:= P (V ), defined by f

1

(t) := (f (t), [f

^′

(t)]) and satisfying π(f

1

) = f , where π : X

1

→ X is the natural projection map. Fix a hermitian metric ω on X. It induces a smooth metric h on the line bundle O

X1

(1). Then 0 < δ ≪ 1 , ω

₁

:= π

^∗

ω + δΘ

_h

( O

X1

(1)) is a hermitian metric on X

1

, and we have the following lemma:

Lemma 2.1. Assume that f is a transcendental entire curve on X. Let π : X

₁

→ X be the projection map. Then we have

r→∞

lim

T

_f1,π^∗_ω

(r) T

_f1,ω1

(r) ≥ 1.

In particular, we can define the Ahlfors current T [f

1

] with respect to π

^∗

ω in such a way that π

_∗

T [f

₁

] = T [f ].

Proof. Since f

^′

(τ ) can be seen as a section of the bundle f

₁^∗

( O

X1

( − 1)), by the Lelong-Poincar´e formula we have

dd

^c

log | f

^′

(τ ) |

²ω

= X

|tj|<r

µ

_j

δ

_tj

− f

₁^∗

Θ

_h^∗

( O

X1

( − 1)) (2.5) on B(r), where µ

_j

is the vanishing order of f

^′

(t) at t

_j

. Thus we get

Z

r 1

dt t

Z

B(t)

f

₁^∗

Θ

_h^∗

( O

X1

( − 1)) = X

|tj|<r

µ

_j

log r

| t

j

| − Z

r

1

dt t

Z

B(t)

dd

^c

log | f

^′

(τ ) |

²ω

= X

|tj|<r

µ

j

log r

| t

_j

| − 1 2π

Z

2π 0

log | f

^′

(re

^iθ

) |

²ω

dθ + 1 2π

Z

2π 0

log | f

^′

(e

^iθ

) |

²ω

dθ, (2.6) where the last equality is a consequence of the Jensen formula (2.1). Let (ϕ

_α

)

_α∈J

be a partition of unity subcoordinate to the covering (U

_α

)

_α∈J

of X. We can take a finite family of logarithms of global meromorphic fuctions (log u

_αj

)

_{α∈J,1≤j≤n}

as local coordinates for U

_α

, and by using the logarithmic derivative lemma (2.4) we have

1 2π

Z

_2π

0

log

⁺

| f

^′

(re

^iθ

) |

²ω

dθ = X

α∈J

1 2π

Z

_2π

0

ϕ

_α

log

⁺

| f

^′

(re

^iθ

) |

²ω

dθ

≤ X

α∈J

X

n j=1

C Z

_2π

0

log

⁺

| u

^′_αj

(re

^iθ

) u

_αj

(re

^iθ

) |

²

dθ

≤ O (log

⁺

T

_f

(r) + log r),

where C is some constant. Since f is non-algebraic, from Theorem 2.4 we know that

r→∞

lim T

_f

(r)

log r = + ∞ , thus we have

r→∞

lim − 1 T

_f

(r)

Z

_2π

0

log | f

^′

(re

^iθ

) |

²ω

dθ ≥ 0.

By (2.6) we have

r→+∞

lim 1 T

_f

(r)

Z

_r

1

dt t

Z

B(t)

f

₁^∗

Θ

_h^∗

( O

X1

( − 1)) ≥ lim

r→∞

− 1

T

_f

(r) Z

_2π

0

log | f

^′

(re

^iθ

) |

²ω

dθ ≥ 0. (2.7) From the definition of ω

₁

, we have

T

_f1,ω1

(r) = Z

_r

1

dt t

Z

B(t)

f

₁^∗

ω

₁

= T

_f1,π^∗_ω

(r) + δ Z

_r

1

dt t

Z

B(t)

f

₁^∗

Θ

_h

( O

X1

(1)).

By T

_f,ω

(r) = T

_f1,π^∗_ω

(r) we get

r→∞

lim

T

_f1,π^∗ω

(r) T

_f1,ω1

(r) ≥ 1.

By Remark 2.1 we can replace ω

₁

by π

^∗

ω in the definition of Ahlfors current T[f

₁

], and the equality T

_f1,π^∗_η

(r) = T

_f,η

(r) for any (1, 1) form η then yields

π

_∗

T [f

₁

] = T [f ].

Similarly we have the following lemma:

Lemma 2.2. Let p : Y → X be a bimeromorphic morphism between K¨ ahler manifolds X and Y , obtained by a sequence of blow-ups with smooth centers. If f : C → X is an entire curve whose image is not contained in the critical value of p, then we can lift f as f e : C → Y and define the Ahlfors current T [ f] e with respect to p

^∗

ω in such a way that p

_∗

T [ f e ] = T [f ].

Proof. We first fix a K¨ ahler metric on X. Since Y is obtained by a finite sequence of blow-ups, we can find a K¨ ahler metric ω e on Y defined by

e

ω = p

^∗

ω − X

i

ǫ

_i

Θ

_hi

(E

_i

),

where all E

_i

are irreducible divisors supported in the exceptional locus of p, h

_i

is some smooth hermitian metric on E

_i

, and each ǫ

_i

is some positive real number. Since the image of f is not contained in the critical value of p, the image of f e is not contained in the exceptional locus of p, and a fortiori in any of the E

_i

. We claim that for all E

_i

we have

h T [f], Θ

_hi

(E

i

) i ≥ 0. (2.8)

Indeed, since the image of f e is not contained in E

_i

, the counting function of f with respect to

E

_i

is non-negative, and if we normalize the hermitian metric h

_i

, the proximity function is also

non-negative. By Nevanlinna’s First Main Theorem, we infer (2.8).

Therefore

r→∞

lim

T

_{f ,pe} ∗ω

(r) T

_{f ,eeω}

(r) ≥ 1,

and by Remark 2.1 we can define the Ahlfors current T [ f e ] with respect to p

^∗

ω in such a way that p

_∗

T [ f e ] = T [f ].

Remark 2.2. For any coherent ideal sheaf J whose zero scheme does not contain the image of f : C → X, one can take a log resolution p : ˆ X → X of J with p

^∗

J = O

Xˆ

( − D), and by Lemma 2.2 one can find a suitable sequence (r

_k

) such that

h T [ ˆ f ], Θ(D) i = lim

rk→∞

T

_{f ,Θ(D)ˆ}

(r

_k

)

T

_{f ,pˆ} ∗ω

(r

_k

) = lim

rk→∞

T

_{f ,Θ(D)ˆ}

(r

_k

) T

_f,ω

(r

_k

) ,

where ˆ f is the lift of f to ˆ X and Θ(D) is a curvature form of O

Xˆ

(D) with respect to some smooth metric. By Theorem 2.2, we know that h T [ ˆ f], Θ(D) i does not depend on the log resolution of J . We will denote this intersection number by T (f, J ).

2.3 Intersection with the tangent bundle

Theorem 2.6. (Tautological inequality) Let f : (C, T

_C

) → (X, V ) be a transcendental entire curve in X, where (X, V ) is a smooth directed variety. We denote by f

1

: C → P(V ) the lift of f . Let ω be a hermitian metric on X, and consider the smooth metric induced by h on the line bundle O

P(V)

(1). Then we have

h T [f

₁

], O

P(V)

( − 1) i = h T [f

₁

], Θ

_h^∗

( O

P(V)

( − 1)) i ≥ 0.

Proof. By (2.7) above we have

r→+∞

lim 1 T

_f,ω

(r)

Z

r 1

dt t

Z

B(t)

f

₁^∗

Θ

_h^∗

( O

P(V)

( − 1)) ≥ 0.

By Lemma 2.1 we can take π

^∗

ω as the semi-positive form used in the definition of the Ahlfors current of T [f

₁

], where π : P(V ) → X is the natural projection. The equality T

_f,ω

(r) = T

_f1,π^∗_ω

(r) then implies

h T [f

₁

], O

P(V)

( − 1) i = lim

r→+∞

1 T

_f1,π^∗_ω

(r)

Z

_r

1

dt t

Z

B(t)

f

₁^∗

Θ

_h^∗

( O

P(V)

( − 1)) ≥ 0.

Remark 2.3. Recall the following well known formula: if C ⊂ X is a smooth algebraic curve

and C e ⊂ P (T

_X

) is its lift to P (T

_X

), then c

₁

( O

P(TX)

( − 1)) · [ C] = e χ(C). Thus the above

tautological inequality intuitively means that the “Euler characteristic” of the transcendental

curve f : C → X is non-negative.

Theorem 2.7. Let X be a K¨ ahler manifold endowed with a 1-dimensional foliation F , and f : C → X be a transcendental entire curve tangent to F such that its image is not contained in Sing( F ). Then we have

h T [f ], c

₁

(T

_F

) i + T (f, J

F

) = h T [f

₁

], Θ

_g

( O

X1

( − 1)) i ≥ 0, (2.9) where g is the smooth metric on O

X1

( − 1) induced by ω.

Proof. Let (U

_α

) be a partition of unit of X, and let us denote Ω

_α

= f

⁻¹

(U

_α

). Then we have

f

^′

(τ ) = λ

_α

(τ )v

_α

|

f(τ)

(2.10)

for some holomorphic function λ

α

(τ ) on Ω

α

\ f

⁻¹

(Sing( F )) (notice that f

⁻¹

(Sing( F )) ∩ Ω

α

is a set of isolated points since its image is not contained in Sing( F )). We denote by η

_j

the multiplicities of λ

_α

(τ ) at t

_j

(they may be negative if f(t

_j

) ∈ Sing( F ), however we have η

_j

+ ν

_j

≥ 0), where ν

_j

is the multiplicity of f along J

F

, i.e.

dd

^c

log | v

_α

|

²ω

◦ f (τ ) |

B(r)

= X

|tj|<r

ν

_j

log | τ − t

_j

|

²

+ O (1).

Since v

_α

= g

_αβ

v

_β

, then if t

_j

∈ Ω

_α

∩ Ω

_β

, λ

_α

and λ

_β

have the same multiplicity at t

_j

, and thus η

_j

does not depend on the partition on unity. By the Lelong-Poincar´e formula and (2.10) we have

dd

^c

log | f

^′

(τ ) |

²ω

= X

|tj|<r

η

_j

δ

_tj

+ dd

^c

log | v

_α

|

²ω

◦ f (τ )

= X

|tj|<r

η

_j

δ

_tj

− f

^∗

Θ

_hs

, (2.11)

where h

_s

is the singular metric on T

_F

whose local weight is ϕ

_α

= − log | v

_α

|

²ω

(see (2.2) above).

If we fix a smooth metric h on T

_F

, then there exists a globally defined function ϕ

_s

such that h = h

_s

e

^−ϕs

,

and ϕ

_s

is the characteristic function associated to the coherent sheaf J

F

. By (2.11) we have dd

^c

log | f

^′

(τ ) |

²ω

− f

^∗

dd

^c

ϕ

s

= X

|tj|<r

η

j

δ

tj

− f

^∗

Θ

_h

(T

F

).

on B(r). From (2.5) we know that X

|tj|<r

µ

_j

δ

_tj

− f

₁^∗

Θ

_g

( O

X1

( − 1)) − f

^∗

dd

^c

ϕ

_s

= X

|tj|<r

η

_j

δ

_tj

− f

^∗

Θ

_h

(T

_F

),

where g is the smooth metric on O

X1

( − 1) induced by ω, and µ

_j

is the multiplicity of f

^′

(τ ) at t

_j

. Therefore, as µ

_j

− η

_j

= ν

_j

, we have

f

^∗

Θ

_h

(T

_F

) = − X

|tj|<r

ν

_j

δ

_tj

+ f

₁^∗

Θ

_g

( O

X1

( − 1)) + f

^∗

dd

^c

ϕ

_s

on each B(r). Then we have h T

_r

[f ], Θ

_h

(T

_F

) i := 1

T

_f

(r) Z

_r

1

dt t

Z

B(t)

f

^∗

Θ

_h

(T

_F

)

= 1

T

_f

(r) Z

r

1

dt t

Z

B(t)

f

₁^∗

Θ

_g

( O

X1

( − 1)) − 1 T

_f

(r)

X

|tj|<r

ν

_j

log r

| t

_j

|

+ 1

T

_f

(r) 1 2π

Z

_2π

0

ϕ

_s

◦ f (re

^iθ

)dθ − 1 T

_f

(r)

1 2π

Z

_2π

0

ϕ

_s

◦ f (e

^iθ

)dθ

= 1

T

_f

(r) Z

_r

1

dt t

Z

B(t)

f

₁^∗

Θ

g

( O

X1

( − 1))

− 1

T

_f

(r) N

_f,JF

(r) − 1

T

_f

(r) m

_f,JF

(r), (2.12)

where N

_f,JF

(r) and m

_f,JF

(r) are the counting and proximity function of f with respect to J

F

, and the second equality above is a consequence of the Jensen formula. Since T [f ] is the weak limit of the positive current T

_rk

[f ] for some sequence r

_k

→ ∞ , we have

h T [f ], Θ

_h

(T

_F

) i = lim

rk→+∞

h T

_rk

[f ], Θ

_h

(T

_F

) i . From Theorem 2.2, Remark 2.2 and Lemma 2.1 we conclude that

h T [f], Θ

_h

(T

_F

) i + T(f, J

F

) = h T [f

₁

], Θ

_g

( O

X1

( − 1)) i ≥ 0.

In fact, K

_F

⊗ J

F

is the canonical sheaf K

F

of F defined in [Dem14], by using admissible metric. We recall the following definition in [Dem14].

Definition 2.3. We say that the canonical sheaf K

F

is “ big” if there exists some birational morphism ν

_α

: (X

_α

, F

α

) → (X, F ) of (X, F ) such that the invertible sheaf µ

^∗_α

K

Fα

is big in the usual sense for some log resolution µ

_α

: Y

_α

→ X

_α

of J

Fα

. The base locus Bs( F ) of F is defined to be

Bs( F ) := \

α

ν

_α

◦ µ

_α

B

₊

(µ

^∗_α

K

Fα

),

where (X

α

, F

α

) varies among all the birational models with µ

^∗_α

K

Fα

big.

The above Theorem 2.7 then gives another proof of the following Generalized Green-Griffiths conjecture for rank 1 foliations formulated in [Dem12]. Moreover we can specify more precisely the subvariety containing the images of all transcendental curves tangent to the foliation. The theorem is as follows:

Corollary 2.1. Let (X, F ) be a projective 1-foliated manifold and assume that K

F

is big. If

f : C → X is a transcendental entire curve tangent to F , its image must be contained in

Sing( F ) ∪ Bs( F ). In particular, any entire curve tangent to F must be algebraically degenerate.

Proof. Assume that the image of f is not contained in Sing( F ) ∪ Bs( F ). We proceed by con- tradiction. By Definition 2.3, there exists a birational morphism ν

_α

: (X

_α

, F

α

) → (X, F ) such that the invertible sheaf µ

^∗_α

K

Fα

is big in the usual sense, for some log resolution µ

_α

: Y

_α

→ X

_α

of J

Fα

, and such that the image of f

α

is not contained in B

₊

(µ

^∗_α

K

Fα

), where f

α

is the lift of f to Y

_α

. We denote by f e

_α

the lift of f to X

_α

. By Theorem 2.3 we have

h T [f

_α

], c

₁

(µ

^∗_α

K

Fα

) i > 0.

By Remark 2.2 and the fact that ν

_α∗

T[f

_α

] = T [ f e

_α

] i , we get

h T [ f e

_α

], K

_Fα

i − T( f e

_α

, J

Fα

) = h T [f

_α

], µ

^∗_α

K

Fα

i > 0.

However, since f is transcendental, by Theorem 2.7 we infer h T [ f e

_α

], T

_Fα

i + T ( f e

_α

, J

Fα

) ≥ 0,

and the contradiction is obtained by observing that c

₁

(K

_Fα

) = − c

₁

(T

_Fα

).

Remark 2.4. In [Den15] we have generalized the above theorem to any singular directed variety (X, V ) (without assuming V to be involutive), by applying the Ahlfors-Schwarz Lemma. In the proof, the canonical sheaf plays a crucial role (and it arises in a natural way).

By a result due to Takayama (ref. [Tak08]) we know that on a projective manifold X of general type, every irreducible component of B

₊

(K

_X

) is uniruled. It is natural to ask the following analogous question:

Problem 2.1. Let (X, F ) be a projective 1-foliated pair with K

F

big. Is every irreducible component of Bs( F ) uniruled?

From (2.9), we see that the positivity of T [f ] · T

_F

is controlled by the error term T (f, J

F

).

This gives us the advantage that we only need to compute the coherent ideal sheaf J

F

instead of knowing the exact form of F at the singularities.

Remark 2.5. If X is a complex surface endowed with a foliation F , and C is an algebraic curve, then by Proposition 2.3 in [Bru04] we have the formula:

C · T

_F

+ Z ( F , C) = χ(C), (2.13)

where Z( F , C) is the multiplicity of the singularities of F along the curve C. By Remark 2.3, we know that h T [f

₁

], O

X1

( − 1) i can be seen as the “Euler characteristic” of f . Thus (2.9) is a transcendental version of the above formula, and we have the following metamathematical correspondences between any algebraic leaf C and any transcendental leaf f : C → X:

Z( F , C) ∼ T (f, J

F

), C · T

_F

∼ h T [f ], c

₁

(T

_F

) i , χ(C) ∼ h T [f

₁

], O

X1

( − 1) i .

We also need the following logarithmic version of Theorem 2.7:

Theorem 2.8. Let (X, F , D) be a K¨ ahler 1-foliated triple, and let f : C → X be a transcendental entire curve tangent to F such that its image is not contained in Sing( F ) ∪ | D | , where | D | is the support of D. Then we have

h T [f ], c

1

(T

F

) i + T (f, J

F,D

) = h T[ ¯ f

1

], O

X^¯1

( − 1) i ≥ − lim inf

r→∞

N

_f,D⁽¹⁾

(r)

T

_f

(r) =: − N

⁽¹⁾

(f, D), where f ¯

₁

is the lift of f on X ¯

₁

:= P(T

_X

h− log D i ), and N

_f,D⁽¹⁾

(r) is the truncated counting function of f with respect to D defined by

N

_f,D⁽¹⁾

(r) := X

|tj|<r,f(tj)∈D

log r

| t

_j

| .

Proof. Since the image of f is not contained in | D | , the condition that f (t

_j

) ∈ D implies f (t

_j

) ∈ Sing( F ).

We use the notation and concepts introduced in Section 2.1. Let (U

_α

) be a partition of unity on X. On each U

_α

we have

v

_α

= X

k j=1

z

_j

a

^j_α

∂

∂z

_j

+ X

n i=k+1

a

ⁱ_α

∂

∂z

_i

as the generator of F , where z

1

· · · z

_k

= 0 is the local equation of D in U

α

. The hermitian metric ω

_X,D

induces a singular metric h

_s,D

on T

_F

with local weight

ϕ

α,D

= − log | v

α

|

²ωX,D

= − log X

i,j

a

ⁱ_α

a

^jα

ω

_ij

.

If h = h

_s

e

^−ϕs,D

is a smooth metric on T

_F

, then ϕ

_s,D

is the characteristic function associated with J

F,D

.

Since the image of f is not contained in Sing( F ) ∪ | D | , on U

_α

we have f ¯

^′

(τ ) := ( f

₁^′

f

₁

, . . . , f

_k^′

f

_k

, f

_k+1^′

, . . . , f

_n^′

) = λ

_α

(τ )(a

¹_α

(f ), . . . , a

ⁿ_α

(f)), (2.14) where λ

_α

(τ ) is the meromorphic functions with poles contained in f

⁻¹

(Sing( F ) ∪| D | ). By (2.14) we know that f(t

j

) ∈ D implies f (t

j

) ∈ Sing( F ). Indeed, if f (t

j

) ∈ D, then λ

α

has a pole of order at least 1 at t

_j

, and such a pole can only occur at a point of Sing( F ).

Observe that ¯ f

^′

(τ ) can be seen as a meromorphic section of ¯ f

₁^∗

O

X^¯1

( − 1), where ¯ f

₁

(τ ) is the lift of f to ¯ X

1

, and denote Ω

α

= f

⁻¹

(U

α

). Then on Ω

α

∩ B (r) we have

dd

^c

log | f

^′

(τ ) |

²ωX,D

= X

|tj|<r,tj∈Ωα

η

_j

δ

_tj

+ f

^∗

dd

^c

log | v

_α

|

²ωX,D

,

where η

j

is the vanishing order of λ

α

(τ ) on t

j

. Since v

α

= g

_αβ

v

_β

, we see that η

j

does not depend on the partition of unity, and thus on B(r) we have

dd

^c

log | f

^′

(τ ) |

²ωX,D

= X

|tj|<r

η

_j

δ

_tj

− f

^∗

Θ

_h

(T

_F

) + f

^∗

dd

^c

ϕ

_s,D

. (2.15)

On the other hand, since ω

X,D

induces a natural smooth metric on T

X

h− log D i , it also induces a smooth hermitian metric ¯ h

₁

on O

X^¯1

( − 1), thus

dd

^c

log | f

^′

(τ ) |

²ωX,D

= dd

^c

log | f ¯

^′

(τ ) |

²¯h1

= X

0<|tj|<r

µ

_j

δ

_tj

− f ¯

₁^∗

Θ

h¯1

( O

X^¯1

( − 1)) (2.16)

on B(r), where µ

_j

is the vanishing order of ¯ f

^′

(t). By (2.14) we know that µ

_j

= − 1 if and only if f (t

j

) ∈ | D | , and otherwise µ

j

≥ 0. Then by using the logarithmic derivative lemma again as in Lemma 2.1, we find

h T [ ¯ f

₁

], O

X^¯1

( − 1) i ≥ − lim inf

r→∞

N

_f,D⁽¹⁾

(r)

T

_f

(r) . (2.17)

We can combine (2.15) and (2.16) together to obtain f

^∗

Θ

_h

(T

_F

) = − X

0<|tj|<r

µ

_j

δ

_tj

+ X

|tj|<r

η

_j

δ

_tj

+ ¯ f

₁^∗

Θ

¯h1

( O

X^¯1

( − 1)) + f

^∗

dd

^c

ϕ

_s,D

on B(r). By arguments very similar to those in the proof of Theorem 2.7, we get

h T [f ], Θ

_h

(T

_F

) i = h T [ ¯ f

₁

], O

X^¯1

( − 1) i − T (f, J

F,D

), and the theorem follows from (2.17).

2.4 Siu’s refined tautological inequality

In [Siu02] Y-T. Siu proved McQuillan’s “refined tautological inequality” by applying the tradi- tional function-theoretical formulation. We will give here an improvement of this result. First we begin with the following lemma due to Siu.

Lemma 2.3. Let U be an open neighborhood of 0 in C

ⁿ

and π : U e → U be the blow-up at 0.

Then π

^∗

( O

U

(Ω

¹_U

)) ⊂ I

E

⊗ (Ω

¹_e

U

h− log E i ), where I

E

is the ideal sheaf of the exceptional divisor E.

Theorem 2.9. Let H be an ample line bundle on a projective manifold X of dimension n. Let Z be a finite subset of X and f : C → X be an entire curve. Let σ ∈ H

⁰

(X, S

^l

Ω

_X

⊗ (klH)) be such that f

^∗

σ is not identically zero on C. Let W be the zero divisor of σ in X

1

:= P(T

X

), and π : Y → X be the blow-up of Z with E := π

⁻¹

(Z). Then we have

1

l N

_f1,W

(r) + T

_{f ,Θˆ}

E

(r) − N

_f,m⁽¹⁾

Z

(r) ≤ kT

_f,ΘH

(r) + O (log T

_f,ΘH

(r) + log r), (2.18) where N

_f,m⁽¹⁾

Z

(r) is the truncated counting function with respect to the ideal m

_Z

, and Θ

_H

(resp.

Θ

E

) is the curvature of H (resp. Θ

H

) with respect to some smooth metric h

H

(resp. h

E

).

Remark 2.6. In [Siu02] and [McQ98], a slightly weaker inequality is obtained comparatively to (2.18), with m

_f,mZ

(r) in place of T

_{f ,Θˆ}

E

(r) − N

_f,m⁽¹⁾

Z

(r).

Proof of Theorem 2.9. Let τ = π

^∗

σ. By Lemma 2.3, τ is a holomorphic section of S

^l

(Ω

Y

(log E)) ⊗ π

^∗

(klH ) over Y and τ vanishes to order at least l on E. Let s

_E

be the canonical section of E. If we divide τ by s

^⊗l_E

, then e τ :=

^τ

s^⊗l_E

is a holomorphic section of S

^l

(Ω

_Y

(log E)) ⊗ π

^∗

(klH) ⊗ ( − lE) over Y . We now prove that

k τ ( f ¯ ˆ

^′

(t)) k

_π^∗_h^⊗kl

H

= k σ(f

^′

(t)) k

_h^⊗kl

H

, (2.19)

where f ¯ ˆ

^′

(t) is the derivative of ˆ f in T

_Xˆ

h− log E i (see Definition 2.14). To make things simple we assume l = 1. Let p be a point in Z and let U be a small open set containing p such that locally we have

σ = X

n i=1

a

_i

dz

_i

⊗ e

^⊗k

,

where e is the local section of H and p is the origin. The blow-up at p is the complex submanifold of U × P

ⁿ⁻¹

defined by w

_j

z

_k

= w

_k

z

_j

for 1 ≤ j 6 = k ≤ n, where [w

₁

: · · · : w

_n

] are the homogeneous coordinates of P

ⁿ

. In the affine coordinate chart w

₁

6 = 0 we have the relation

(z

₁

, z

₁

w

₂

, . . . , z

₁

w

_n

) = (z

₁

, z

₂

, . . . , z

_n

), thus

τ = z

₁

(a

₁

+ X

n

i=2

a

_i

w

_i

)d log z

₁

+ X

n

i=2

a

_i

dw

_i

!

⊗ (π

^∗

e)

^⊗k

, and ˆ f (t) = (f

1

,

^f_f²₁

, . . . ,

^f_fⁿ₁

) in the local coordinate (z

1

, w

2

, . . . , w

n

). Thus

¯ ˆ f

^′

(t) :=

f

₁^′

f

₁

, ( f

₂

f

₁

)

^′

, . . . , ( f

_n

f

₁

)

^′

with respect to the local section (z

_1∂z^∂

1

,

_∂w^∂

2

, . . . ,

_∂w^∂

n

) of T

_Xˆ

h− log E i . It is easy to check the equality (2.19).

By the logarithmic derivative lemma again we know that

_2π¹

R

2π

0

log

⁺

k τ ( f ¯ ˆ

^′

(re

^iθ

)) k

_π^∗_h^⊗kl

H

and

1 2π

R

_2π

0

log

⁺

ke τ ( f ¯ ˆ

^′

(re

^iθ

)) k

_π^∗_h^⊗kl

H ⊗h^∗⊗l_E

are both of the order O (log T

_f,ΘH

(r) + log r). Using log x =

log

⁺

x − log

^{+ 1}_x

for any x > 0, we obtain 2lm

_{f ,Eˆ}

(r) = 1

2π Z

_2π

0

log

⁺

1

k s

^⊗l_E

◦ f(re ˆ

^iθ

) k

²_hE

≤ 1 2π

Z

_2π

0

log

⁺

1

k τ ( f ¯ ˆ

^′

(re

^iθ

)) k

²_π∗h^⊗kl_H

+ 1 2π

Z

_2π

0

log

⁺

ke τ ( f ¯ ˆ

^′

(re

^iθ

)) k

²_π^∗_h^⊗kl

H ⊗h^∗⊗l_E

+ O (1)

= − 1 2π

Z

_2π

0

log k τ ( f ¯ ˆ

^′

(re

^iθ

)) k

²_π∗h^⊗kl_H

+ 1 2π

Z

_2π

0

log

⁺

k τ ( f ¯ ˆ

^′

(re

^iθ

)) k

²_π∗h^⊗kl_H

+ 1

2π Z

_2π

0

log

⁺

ke τ ( f ¯ ˆ

^′

(re

^iθ

)) k

²_π∗h^⊗kl_H ⊗h^∗⊗l_E

+ O (1)

= − 1 2π

Z

2π 0

log k τ ( f ¯ ˆ

^′

(re

^iθ

)) k

²_π∗h^⊗kl_H

+ O (log T

_f,ΘH

(r) + log r)

= − 1 2π

Z

2π 0

log k σ(f

^′

(re

^iθ

)) k

²_h^⊗kl

H

+ O (log T

_f,ΘH

(r) + log r), (2.20) where the last equality is due to equality (2.19). Observe that there is a natural isomorphism betweem H

⁰

(X, S

^l

(Ω

_X

) ⊗ (klH)) and H

⁰

(X

₁

, O

X1

(l) ⊗ p

^∗

(klH)), where X

₁

:= P (T

_X

) and p is its projection to X. We denote by P

_σ

the corresponding section of σ in H

⁰

(X

₁

, O

X1

(l) ⊗ p

^∗

(klH)), whose zero divisor is W . Then we have

k P

_σ

(f

₁

(t)) · (f

^′

(t))

^l

k

_p^∗_h^⊗kl

H

= k σ(f

^′

(t)) k

_h^⊗kl

H

, and thus on B(r) we have

N (σ(f

^′

(t)), r) = N

_f1,W

(r) + l X

|tj|<r

µ

_j

r

| t

_j

| ,

where N (σ(f

^′

(t)), r) is the counting function of σ(f

^′

(t)) and µ

j

is the vanishing order of f

^′

(t) at t

_j

. Therefore, by applying the Jensen formula to the last term in (2.20) we obtain

2lm

_{f ,Eˆ}

(r) + 2N (σ(f

^′

(t)), r) ≤ 2klT

_f,ΘH

(r) + O (log T

_f,ΘH

(r) + log r)). (2.21) Since we have

N

_f,m⁽¹⁾

Z

(r) + X

|tj|<r,f(tj)∈Z

µ

_j

r

| t

_j

| = N

_f,mZ

(r) = N

_{f ,Eˆ}

(r), then by applying Nevanlinna’s First Main Theorem to ( ˆ f , E) we get

T

_{f ,Θˆ}

E

(r) = N

_{f ,Eˆ}

(r) + m

_{f ,Eˆ}

(r) + O (1), and we can combine this with (2.21) to obtain

T

_{f ,Θˆ}

E

(r) − N

_f,m⁽¹⁾

Z

(r) + 1

l N

_f1,W

(r) ≤ kT

_f,ΘH

(r) + O (log T

_f,ΘH

(r) + log r)).

Now we have the following refined tautological equality:

Theorem 2.10. Let X be a K¨ ahler manifold of dimension n and f : C → X be a transcendental entire curve. Then for any finite set Z we have

T [f

₁

] · O

X1

( − 1) ≥ T (f, m

_Z

) − N

_f,m⁽¹⁾

Z

≥ m(f, m

_Z

) := lim

r→∞

m

_f,mZ

(r) T

_f,ΘH

(r) .

Proof. First we choose k large enough, in such a way that O

X1

(1) ⊗ p

^∗

(kH) is ample over X

₁

. When we choose l sufficient large, O

X1

(l) ⊗ p

^∗

(lkH) will be very ample over X

₁

. Hence there exists a section σ ∈ H

⁰

(X

₁

, O

X1

(l) ⊗ p

^∗

(lkH)) whose defect is zero, i.e.

N (f

₁

, W ) := lim

rk→∞

N

_f1,W

(r

_k

)

T

_f,ΘH

(r

_k

) = h T [f

₁

], O

X1

(l) ⊗ p

^∗

(lkH) i = h T [f

₁

], O

X1

(l) i + kl,

where W is the zero divisor of σ, and where the last equality comes from h T [f ], Θ

H

i = 1. Since f is transcendental, by Theorem 2.4 we have

r→∞

lim

T

_f,ΘH

(r)

log r = + ∞ .

We can thus divide both sides in (2.18) by T

_f,ΘH

(r) and take r → ∞ to obtain 1

l N (f

₁

, W ) + T (f, m

_Z

) − N

_f,m⁽¹⁾

Z

≤ k, and we obtain the formula in the theorem.

2.5 Intersection with the normal bundle

As an application for Theorem 2.7, we will study the intersection of T [f ] with the normal bundle;

i.e. with c

₁

(N

_F

). Before anything else, we begin with the following definition.

Definition 2.4. Let X be a K¨ ahler manifold endowed with a foliation F by curves. We say that F has weakly reduced singularities if

1. For some log resolution π : ˆ X → X of J

F

, we have T

_Fˆ

= π

^∗

T

_F

, where F ˆ is the induced foliation of π

^∗

F ;

2. the L

²

multiplier ideal sheaf I ( J

F

) of J

F

is equal to O

X

, i.e., at each p ∈ X, assume that v is the local generator of F around p, then for any f ∈ O

X,p

, we have

| f |

²

| v |

²ω

∈ L

¹loc

, where ω is any smooth hermitian metric on X.

With the previous definition, we have the following theorem.

Theorem 2.11. Let X be a projective manifold of dimension n endowed with a foliation F by curves with weakly reduced singularities. If f : C → X is a transcendental entire curve tangent to F , whose image is not contained in Sing( F ) and satisfies h T [f ], K

_X

i > 0 (e.g. K

_X

is big), then we have

h T[ ˆ f ], c

₁

(N

_Fˆ

) i < 0 for some birational modification ( ˆ X, F ˆ ) of (X, F ).

Proof. From the standard short exact sequence

0 −→ T

F

−→ T

X

−→ N

F

−→ 0 that holds outside of a codimension 2 subvariety, we find

c

₁

(K

_X

) + c

₁

(T

_F

) = − c

₁

(N

_F

).

By the definition of multiplier ideal sheaves (ref. [Laz04]) we have I ( J

F

) = π

_∗

(K

_X/Xˆ

− D),

where π : ˆ X → X is a log resolution of J

F

such that π

^∗

J

F

= O

Xˆ

( − D). We know that I ( J

F

) = O

X

if and only if K

_X/Xˆ

− D is effective. By the assumption that the image of f is not contained in Sing( F ), i.e. in the zero scheme of J

F

, we know that the image of ˆ f is not contained in the support of the exceptional divisor, and thus

h T [ ˆ f ], K

_X/Xˆ

− D i ≥ 0.

Therefore we have

h T [ ˆ f ], K

_Xˆ

+ T

_Fˆ

i = h T [ ˆ f ], π

^∗

K

X

+ π

^∗

T

F

+ K

_X/Xˆ

i

≥ h T [ ˆ f ], π

^∗

K

_X

+ π

^∗

T

_F

+ D i

= h T [f ], K

_X

+ T

_F

i + T (f, J

F

),

where the last equality follows from the fact that π

∗

T [ ˆ f ] = T [f ] and T (f, J

F

) = h T [ ˆ f ], D i (see Remark 2.2). By Theorem 2.7 we have

h T [f], T

F

i + T (f, J

F

) ≥ 0, thus

−h T [ ˆ f ], c

₁

(N

_Fˆ

) i = h T [ ˆ f ], K

_Xˆ

+ T

_Fˆ

i ≥ h T[f ], K

_X

i > 0.

The theorem is proved.

By the reduction theorem of singularities for surface foliations due to Seidenberg, for any

pair (X

₀

, F

0

) there exists a finite sequence of blow-ups such that the induced pair (X, F ) has

singularities of one of the following two types:

• a non-degenerate singular point x

0

, in the sense that log | a

₁

|

²

+ | a

₂

|

²

| z

1

|

²

+ | z

2

|

²

= O (1).

• a degenerate singular point x

₁

“of type k”, such that log | a

₁

|

²

+ | a

₂

|

²

| z

₁

|

²

+ | z

₂

|

^2k

= O (1), where k ≥ 2 (we also call this a “type k” singularity).

Moreover

• Any singular point of F is either non-degenerate or degenerate of type k.

• For any blow-up π : ˆ X → X of a point on X, we have π

^∗

T

_F

= T

_Fˆ

for the induced foliation F ˆ .

• For any blow-up of the non-degenerate point x

₀

, ˆ F has two singularities on the exceptional divisor and both of them are non-degenerate.

• For the blow-up of a degenerate point x

₁

of type k, on the exceptional divisor ˆ F has a non-degenerate singular point and a degenerate one with the same type as x

₁

.

By the property above it is easy to show that I ( J

F

) = O

X

if F is reduced, thus F is weakly reduced in the sense of Definition 2.4. Then we can get another proof of McQuillan’s theorem without using his “Diophantine approximation analysis” (still, we will make use of “Diophantine approximation” in the next section):

Theorem 2.12. Let X be a complex surface endowed with a foliation F . If h T [f ], K

_X

i > 0 (e.g. if K

_X

is big), then any entire curve tangent to F is algebraically degenerate.

Proof. Assume that we have a Zariski dense entire curve f : C → X tangent to F . We proceed by contradiction.

By Seidenberg’s theorem there is a sequence of blow-ups π : X e → X such that the singular- ities of F e = π

⁻¹

F are reduced, and the lift f e of f to X e is still Zariski dense. Thus J e is weakly reduced and by Theorem 2.11 we have

h T[ ˆ f ], c

₁

(N

_Fˆ

) i < 0

for some birational pair ( ˆ X, F ˆ ) which is obtained by resolving the ideal J

Fe

. However, Theorem 1.5 tells us that

h T[ ˆ f ], c

₁

(N

_Fˆ

) i ≥ 0

and we get a contradiction.

2.6 A generalization of McQuillan’s theorem

Thanks to the above theorems, we can obtain certain generalizations of McQuillan’s theorem to higher dimensional manifolds, under some assumptions for the foliations. We start with some relevant definitions and properties (and refer to [CCS97] and [Tom97] for further details).

Definition 2.5. Let F be a foliation by curves on a n-dimensional complex manifold X. We say that p

₀

∈ Sing( F ) is an absolutely isolated singularity (A.I.S.) of F if and only if the following properties are satisfied:

1. p

₀

is an isolated singularity,

2. If we consider an arbitrary sequence of blowing-up’s

(X, F ) ←−

^π1

(X

₁

, F

1

) ←− · · ·

^π2

←−

^πn

(X

_n

, F

n

)

where the center of each blow-up π

i

is a singular point p

i−1

∈ Sing F

i−1

, then #Sing( F

n

) <

+ ∞ .

Since locally the foliation F is generated by a holomorphic vector field v = P

n i=1

a

i ∂

∂zi

, we can define the algebraic multiplicity m

_p

( F ) of F at p to be the minimum of the vanishing orders ord

_p

(a

_i

). Recall that the linear part of F at p is defined by

L

v

: m

_p

/m

²_p

→ m

_p

/m

²_p

.

A singular point p ∈ Sing F is called reduced if m

_p

( F ) = 1 and the linear part of F at p has at least one non-zero eigenvalue.

We shall say that p ∈ Sing F is a non-dicritical singularity of F if π

⁻¹

(p) is invariant by F , where π is the blow-up of p. Otherwise p is called a dicritical singularity.

Let (X, F , D) be a 1-foliated triple. We assume that all singularities of F lie on D (this can be achieved after we take a log resolution of J

F

). Fix a point p ∈ Sing( F ) and denote by e = e(D, p) the number of irreducible components of D through p. Since Sing( F ) ⊂ D, we have e ≥ 1. Then the vector fields which generate F are given by

v = X

e j=1

z

_j

a

_j

∂

∂z

_j

+ X

n i=e+1

a

_i

∂

∂z

_i

, where z

₁

z

₂

. . . z

_e

= 0 is the local equation of D at p.

Definition 2.6. Assume that e = 1. Then p is a simple point iff one of the following two possibilities occurs:

(A) a

₁

(0) = 0, the curve (z

₂

= . . . = z

_n

= 0) is invariant by F (up to an adequate formal choice of coordinates) and the linear part L

v|D

of v |

D

is of rank n − 1.

(B) a

₁

(0) = λ 6 = 0, the multiplicity of the eigenvalue λ is one and if µ is another eigenvalue of

the linear part of L

v

, then

^µ_λ

∈ / Q

₊

.