Nevanlinna theory

Nevanlinna theory

Nevanlinna theory

deals with quantitative aspects of entire

holomorphic

mappings

into complex varieties. Let

ω

=

i

⁢

∑

j

,

k

ω

j

⁢

k

⁢

d

⁢

z

j

∧

d

⁢

z

¯

k

be a smooth

(

1

,

1

)

-form on a

complex manifold

X

. Suppose that

ω

is

positive definite

, i.e. the

Hermitian matrix

(

ω

j

⁢

k

⁢

(

z

)

)

is positive definite at every point. Thus

ω

can be viewed as a Hermitian metric on the tangent bundle

T

X

.

If

f

:

ℂ

→

X

is an entire curve, the

growth indicatrix

of

f

is the

function

T

f

,

ω

⁢

(

r

)

=

∫

r

0

r

t

f

,

ω

⁢

(

ρ

)

⁢

d

⁢

ϱ

ϱ

,

where

t

f

,

ω

⁢

(

ϱ

)

=

∫

D

⁢

(

0

,

ϱ

)

f

*

⁢

ω

.

It is clear that

t

f

,

ω

⁢

(

ϱ

)

is nothing more than the area with respect to

ω

of the image

f

⁢

(

D

⁢

(

0

,

ϱ

)

)

of the disk in the

complex line

centered at

0

and of ray

ϱ

.
Now let

L

be an ample holomorphic line bundle over

X

(

compact

,

connected

). Then

L

carries an Hermitian metric

h

with

positive

curvature

ω

=

Θ

h

⁢

(

L

)

and so, in this case, one can suppose that our

ω

is in fact the curvature of this line bundle. Furthermore, it is clear that if one is merely interested in the order of growth

O

⁢

(

T

f

,

ω

⁢

(

r

)

)

when

r

goes to

infinity

, then this order is

independent

of the choice of

ω

.

Let us consider an

hypersurface

H

=

{

σ

=

0

}

⊂

X

defined by a global section of

L

: one would like to “

measure

”the

intersections

of the entire curve

f

:

ℂ

→

X

with

H

. For this purpose one looks at the

holomorphic function

σ

∘

f

:

ℂ

→

L

and introduces the

enumerating of zeros function

N

f

,

σ

⁢

(

r

)

=

∫

r

0

r

n

f

,

σ

⁢

(

ϱ

)

⁢

d

⁢

ϱ

ϱ

,

where

n

f

,

σ

⁢

(

ϱ

)

=

number of zeros of

σ

∘

f

on

D

⁢

(

0

,

ϱ

)

counted with

multiplicities

. Finally one introduces the function

m

f

,

σ

, called

proximity function

, defined by

m

f

,

σ

⁢

(

r

)

=

1

2

⁢

π

⁢

∫

0

2

⁢

π

log

⁡

1

∥

(

σ

∘

f

)

⁢

(

r

⁢

e

i

⁢

ϑ

)

∥

h

⁢

d

⁢

ϑ

.

This function is non negative, once one has normalized

σ

with a

constant

in such a way that

∥

σ

∥

h

≤

1

. Morally,

m

f

,

σ

⁢

(

r

)

is bigger and bigger when

f

often goes near

H

=

{

σ

=

0

}

on the circle of ray

r

.

The

first fundamental theorem of Nevanlinna

states the following:

Let

(

L

,

h

)

be an Hermitian line bundle with curvature form

ω

=

Θ

h

⁢

(

L

)

>

0

. For all

section

σ

of

L

and all curve

f

:

C

→

X

such that

f

⁢

(

C

)

is not entirely

contained

in the hypersurface

H

=

{

σ

=

0

}

, one has

m

f

,

σ

⁢

(

r

)

+

N

f

,

σ

⁢

(

r

)

=

T

f

,

ω

⁢

(

r

)

+

O

⁢

(

1

)

.

In particular, the order of growth of the

left hand side

when

r

→

∞

does not depend on the choice of

σ

, but only on the growth indicatrix of

f

.

Classically, one introduces the

defect

of

f

with respect to

H

=

{

σ

=

0

}

, defined by

δ

σ

⁢

(

f

)

=

lim inf

r

→

∞

⁡

m

f

,

σ

⁢

(

r

)

T

f

,

ω

⁢

(

r

)

∈

[

0

,

1

]

.

In particular, the defect is equal to

1

if

σ

∘

f

is never zero, and equal to

0

if the enumerating of zeros function

N

f

,

σ

⁢

(

r

)

grows as much as possible. One of the most important results of Nevanlinna theory concerns the entire curves which map into the

Riemann sphere

ℙ

1

and states that the sum of defects

∑

a

∈

ℙ

1

δ

a

⁢

(

f

)

is at most equal to

2

. One of the essential steps for the proof of this statement is an estimate of the proximity function of the

logarithmic derivative

of a

meromorphic function

.

More precisely the following “logarithmic derivative lemma” holds:

Let

f

:

C

→

P

1

be a meromorphic function and

D

p

⁢

log

⁡

f

the

p

-th logarithmic derivative of

f

. Then, for all

ε

>

0

, there exists a set

E

of finite

Lebesgue measure

in

R

+

such that

m

D

p

⁢

log

⁡

f

,

∞

⁢

(

r

)

≤

(

p

+

1

+

ε

)

⁢

(

log

⁡

r

+

log

+

⁡

T

f

,

ω

⁢

(

r

)

)

+

O

⁢

(

1

)

,

∀

r

∈

ℝ

+

∖

E

.

An important

consequence

of the logarithmic derivative lemma is what is called the

second fundamental theorem of Nevanlinna

from which it follows immediately the estimate for the sum of the defects introduced above:

Let

f

:

C

→

P

1

be a meromorphic function. Define the

ramification

divisor

R

f

of

f

as the sum

∑

e

j

⁢

[

w

j

]

where the

w

j

’s are the points where

f

′

is zero and the

e

j

’s are the multiplicities of zero of

f

′

at

w

j

(where

f

⁢

(

w

j

)

=

∞

one looks at

1

/

f

instead of

f

). Then, for all

finite set

{

a

j

}

⊂

P

1

, there exists a subset

E

⊂

R

+

of finite Lebesgue measure such that

N

R

f

⁢

(

r

)

+

∑

j

m

f

,

a

j

⁢

(

r

)

≤

2

⁢

T

f

,

ω

⁢

(

r

)

+

O

⁢

(

log

⁡

r

+

log

+

⁡

T

f

,

ω

⁢

(

r

)

)

,

∀

r

∈

ℝ

+

∖

E

,

where

N

R

f

⁢

(

r

)

=

∫

r

0

r

n

R

f

⁢

(

ϱ

)

⁢

𝑑

ϱ

/

ϱ

is the

enumerating function

of the ramification divisor.

References

J.-P. Demailly,

Variétés projectives hyperboliques et équations différentielles algébriques

. (French)

[

Hyperbolic projective varieties and algebraic differential equations

]

Journée en l’Honneur de Henri Cartan, 3–17, SMF Journ. Annu., 1997, Soc. Math. France, Paris, 1997.

Title

Nevanlinna theory

Canonical name

NevanlinnaTheory

Date of creation

2013-03-22 15:36:32

Last modified on

2013-03-22 15:36:32

Owner

Simone (5904)

Last modified by

Simone (5904)

Numerical id

14

Author

Simone (5904)

Entry type

Topic

Classification

msc 32A22

Classification

msc 30D35

Related topic

ErnstLindelof

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