Distribution of Values of Holomorphic Mappings (Translations of Mathematical Monographs)

It is clear the the differentiation operators on currents satisfy the usual a2 properties of nilpotence: d2 = a2 = = 0. It is also not difficult to see that on currents of bidegree (r, s) the operator dd' = (i/2ir)495 can be computed by the rule E`-r-I,n -1 (g) dd`T(,p) = T(dd`cp), v We now define the operation of regularization of currents. Let A : Cn -+ R+ be a compactly supported function of class C°° which depends only on [zi and

is such that f a4) = 1, where 4) is the Euclidean volume form on Cr". Let us fix e > 0 and set AE(z) = (1/c2r")A(z/e). This is a test function in 7,0; as e --> 0 it converges in the sense of currents to the 6-function, which is the functional associating to every test function its value at the point z = 0. We represent any current T of bidegree (r, s) on a domain of C' by Theorem 1

§3. CURRENTS AND SOME OF THEIR APPLICATIONS.

51

as a form with coefficients TAB which are generalized functions. We define the regularization of T to be the current TAB dzA A dzB,

TE

(10)

A,B

where TAB(z) = fcEC. TAB(Ae(c - z))4'S (the coefficient under the integral is the value of the functional TAB at the test function A j For any e > 0 the current TE is clearly a form with coefficients of class C°° (in z), and TE -+ T as a --> 0 in the weak topology on the space of currents.

That is, (TE - T)(V) -> 0 for any test function sp (the main property of regularization). It is also clear that the operation of regularization commutes with the operation of differentiation with respect to z and z; in particular, dd°(TE) = (dd`T)E.

12. The Poincare-Lelong formula. Of particular interest to us will be the differentials of currents defined by forms which have singularities on an analytic set A C M but which are smooth on M \ A. Let us begin with the example of the Poincare form a = d` In 1z12A(ddc In Iz12)m-1 in C", which has

a singularity at the point z = 0 (see (31) of §1). By definition the differential of the current satisfies

d[a] (,) = f a

A dip,

where p E $ is a function. The integral must be understood as an improper integral, i.e, as the limit as e - 0 of the integral over the exterior of the ball {IzI > e}. There is a + sign before the integral because a has odd degree. For z # 0 we have a A d

lim

f

r

IZI

>}

d(cpa) = lii o J {IzI = e}spa =

(o)

We applied Stokes' formula taking into account the fact that the boundary of the domain {Izl > E} is the sphere {Izj = e} oriented negatively; then we used a property of the form a from §1. Thus the functional d[a], acting on the function p, gives the value of the function at z = 0. That is, it coincides with the S-function or, in other words, with the current [0] determined by the Singularity of the form: d[a] = [0]. This result in essence goes back to Poincare; we will now describe a variant of it which is due to Lelong [2].

THEOREM 2. Let f be a holomorphic function on an n-dimensional comPlex manifold M and let D = D1 be its divisor. Then the following equality of currents is true: dd°InIf12 = [D] (11) (the Poincare-Lelong formula).

I. THE CHARACTERISTIC AND COUNTING FUNCTIONS

52

3n-1,n-1

4 By definition, (11) means that for any form cp E fM In IfI2ddEp=f cp,

(12)

D

Using a partition of unity, the problem can be localized so that (12) only needs to be verified for forms supported in neighborhoods on M. Moreover, in neighborhoods not containing any points of D, equality (11) is trivial, since differentiation there in the sense of currents coincides with the usual differentiation of forms, and dd` In If I2 = 0 by the holomorphy off Therefore, it is sufficient to verify (12) only for neighborhoods of points of D .

(a) If z° is a regular point of D, then in a neighborhood U of it local zn = 0 } and f has the

coordinates can be chosen on M so that D fl U form zn,, where p is the order of the divisor D. Then (12) reduces to the equality pf

In Izl2d°p = fDnU

E

t

U

The integral on the right takes account of the order of the divisor; it is p times the integral over { zn, = 0 } fl U. Therefore, the p cancels and we may take p = 1. The integral on the left is understood to be an improper integral, i.e., it is the limit as e --+ 0 of the integral over GE _ { z E C` : Izn, I > e }. Because cp has compact support, this integral can be considered as an integral over all of C n. Since in GE we have In Iznl2dd`cp = d(In Iznl2d`v) - din IznI2 A d`cp,

by Stokes' formula, using the fact that cp has compact support and recalling that { IznI = e } is negatively oriented with respect to GE, we have

f In I znl2dd`t = - lim f e-.o

Iz,.1=E}

In Iznl2d- Im J dln IzI2 6-0 G.

The first limit here equals 0, since In I zn 12 = In e2 while the integral of d` cp over { Izn I = e } has order 0(e). To compute the second limit we use the fact

that dIn Iz12Ad`cp=dcoAdc In Iznl2 =d(cpAddlnIzn12), since ddC In IznI2 = 0 in G. Then we apply Stokes' formula (again taking account of the compact support of cp and the orientation of { Izn e } with respect to GE):

f InIzl2dd°cp= lim

E-0

f

{IZn1=E}

c,Ad`InjznI2 = E-.o lim f

cpA

{IZn1=E}

d

;

27r

At the last step we set zn = eei8 and used the fact that the form do In IznI2 = dO/21r. It only remains to observe that for Izn,I = e the form cp = c° ,,=o +cp where the coefficients of cp' approach 0 as e , 0, and thus J In Iznl2dd`cp = lim E-0

.II

{z,

Jl

co

/

{IZnI-E} 2v JJ{Zn=o}

cp.

§3. CURRENTS AND SOME OF THEIR. APPLICATIONS.

53

(b) If z° is a critical point of D, then in a neighborhood U of z° we may choose coordinates z = (z1,...,zn) such that the Weierstrass preparation theorem applies to f with respect to each of the variables zj (see Shabat II, p. 114). Since forms V of bidegree (n - 1, n - 1) are sums of terms in each of which one of the products dzj Adzj is missing,(19), it is sufficient to verify

(12) for one of these summands. Here all the variables are equivalent, so without loss of generality we can assume that dz,, A dz is missing. Let us write zn = w and (z1,. .. , z,,_ 1) = z, so that the form cp can be written as = a(z, w)dz A dz, where a is a smooth function with support in U. Furthermore, by the Weierstrass Preparation Theorem f = Pg, where P(z,w) = Wk+cl(z)wk-1+ +ck(z), and g E O*(U) so that dd`In JgJ2 = 0, that is, without loss of generality f can be replaced by P. Let wj (z), j = 1, ... , k, be the roots of the polynomial P for a fixed z (some of these may perhaps coincide); then k

lnIPI2 =EInJw-wj(z)12. j=1

Using the fact that for a form
82a

dd`rp = 2-a

=2a

dw A dw A dz A dz,

we repeat for fixed z the argument in (a) and obtain the relation k

2

27r

A dw = E 0'(Z' w j (z))

f In

j=1

Finally, setting U = U1-1 x V, we conclude from this that

f

U

r

r

In

I

27r

k

f

E j=1 fu_

dz A dz

1

J

In IPI2

dw

a(z, wj(z))dz A dz,

Since the right-hand side is clearly equal to the integral of the form tip over D, everything is proved. From the theorem just proved there follows simply a variant of the PoincareLelong formula which is suited for application in value distribution theory. COROLLARY. Let L -i M be an Hermitian line bundle with metric h = { ht } and Chern form ch, and let s = { sQ } be a holomorphic section of it. (19)We

only consider here the special case of (n - 1, n - 1) forms; the general case can be reduced to this one (see, for example, Harvey [1], p. 26).

I. THE CHARACTERISTIC AND COUNTING FUNCTIONS

54

Let 11,9112 = halsal2 in Ua be Hermitian modulus squared, and let D be the divisor of s. Then the following equality of currents is true: dd` In 11 31I2 = -Ch + D.

(13)

4 In Ua we have In 11,9112 = In ha + In I sa 12. Since sa E 0 (Ua) and D,a = D fl Ua, by Theorem 2 dd` Is,, 12 = [D] in the sense of currents. The function ha is smooth and positive; and by definition of the Chern form, dd` In ha = -Ch. The following theorem generalizes Theorem 2:

THEOREM 3. Let A C M be a holomorphic chain of codimension k which is defined by the equation g(z) = 0, where g = (91, ... , gk) and g; E 0 (M). Then the following equality of currents holds: dd`(ln I912(dd, In

I912)k-1) = A,

(14)

and for any integer v < k dd`(In 1912(dd` In IgI2)i-1) = (dd` In 1912)"

(15)

i Outside the set A of singularities of the function In 1812 the second equality is obvious. The first follows from the fact that locally at least one of the gj # 0 outside A; if, for example g1 # 0, we set lnlgl2 Since dd21n 18112 = 0 by holomorphy, the form ddc In 1812 contains the differ-

entials of only k - 1 functions and consequently its kth exterior power equals 0 (cf. subsection 2).

As in the proof of the previous theorem, the problem can be localized and limited to test forms with support in neighborhoods of points of A, and

the multiplicity can be taken to be 1. If z° is a regular point of A, in a neighborhood U of zo we choose local coordinates such that A fl U = { z1 =

= zk = 0 }; we change the notation, setting z = (z1, ... , zk) and w = (zk+1, ... , zn). To prove (14) one must check that for any form V(z, w) E

Fn-k.n-k (U) C

ll

I

fC ^

f

A

cP,

(16)

where the integral on the left is understood to be the limit as e --- 0 of the integral over the set AE = { (z, w) E Cn : IzI > s }.

§3. CURRENTS AND SOME OF THEIR APPLICATIONS.

55

The remainder of the proof proceeds like the previous one. By Stokes' formula

f In Iz12(dd` In

Izl2)k-1

In IzI2(dd` In Izl2)k-1 A

A

A (dd` In

d

IzI2)k-1

A deco

a

_ - In E2f

-f

(ddC In IzI2)k-1 A drV AE

do A d` In IZ12 A 1(ddd In IzI2)k-1 As

(17)

(in the second term we interchanged factors using the evenness of the degree of the form (ddc In Iz12)k-1 and then applied Lemma 1 from the end of §1). In the first term the integral over 8AE can be replaced by an iterated integral, first over the sphere SE = { z E CA' : IzI = E } and then over Cn-k. Here the Iz12)k-1 by the appropriate part of the inner integral of the product of (dd` In form d°V has order E; since by (11) of §1 the form on SE in the integrand has Elk-1. By the smoothness and 1/elk-2 and the volume of SE has order order the fact that the form rp has compact support, the whole integral over 8AE also has order e. But the coefficient in front of it equals 2ln(1/E); therefore, the first term in (17) approaches 0 as E -+ 0. We denote this term by 77(E). To the second term of (17) we again apply Stokes' formula, noting that on AE the form (dd` In Izl2)k = 0:

f In I zI2(ddd In AE

= 71 (e) +

f

Izl2)k-1

A

ro A d` in Izl2 A (dd` 1n

IzI2)k-1.

OA'

Representing the form cp on 8AE in the form co = tplA + p', where the coefficients of co' go to 0 as E --* 0, we obtain from (18)

L

In

Iz12(ddc In Ix12)k-1

= fC-k APIA f

A dd`q

dcInlzl2A(dd`lnlzl2)k-1+1(E)=IA S9

(19)

where 171(e) --> 0 as e -+ 0. Here we used the fact that

d`InIzI2 A

(dd`Inlzl2)k-1

is the Poincare form on Ck, and its integral over SE equals 1 (see §1) and also the fact that the integral of VIA on C"-k coincides with the integral of co over A. Relation (16), and hence (14), is proved.

I. THE CHARACTERISTIC AND COUNTING FUNCTIONS

56

Relation (15) is proved analogously. Equality (17) remains true when k is replaced by any v < k for any form p E 7,m-v,m-"(U), while instead of (18) we obtain InIZ12(dd`InIzl2)v-l Add`cp

fAI

= n(E) + f

A d` In IZ12 A (dd` In I2i-1 + A E

f

o A (dd` In Iz12)". e

However, the singularity of (dd` In Iz12)"-i is weaker in this case, so the same computation as above shows that the integral over 8AE goes to 0 as e -> 0. Therefore, as E 0 we obtain the limit IA In l z12(dd` In IzI2)"-1 A dd`tip =

fA(dd` in Iz12)" A o

a relation equivalent to (15). The case of neighborhoods of critical points of A demands special consideration, which we skip (see Griffiths and King [1], pp. 159-162).

13. The relation between characteristic and counting functions. As an application of the theory of currents, we will prove here, following Shiffman [2], that a characteristic function is the average of the counting functions of the divisors of all the holomorphic sections of the line bundle under consideration or (in the case of codimension k > 1) the intersections of such divisors. Let us begin with the case of the hyperplane bundles on P. Here the average will be taken over the complex hyperplanes or their intersections, i.e., the complex planes of higher codimension. We denote by Gn the set of all complex planes in P" of codimension k (k = 1, ... , n). This is called the Grassmann manifold; if one wishes, one can represent it also as the set of (n - k + 1)-dimensional complex planes in Cn+' which pass through the origin, i.e., as a generalization of projective space: Gn = Pn. We denote by U,,+1 the group of unitary transformations of the space Cn+1 and by A a measure on Gk which is invariant with respect to

Un+1 and which is normalized so that p(Gk) = 1 (we are using here the second interpretation of Gn ). LEMMA 1. Any form a with generalized coefficients on Pn which has bide-

gree (k, k), k > 1, and which is invariant with respect to the group U.+1 (20) is proportional to the kth power of the Fubini-Study form w. Since Un+' acts transitively on P" and the forms a and wk are invariant with respect to the group, it is sufficient to verify the proportionality at some

point p c Pn. The subgroup of Un+1 which leaves fixed the line I C Cn corresponding to the point p is the group U. It only remains to verify that (20)Here we interpret P" as the set of all complex lines through the origin in Cn+,.

§3. CURRENTS AND SOME OF THEIR APPLICATIONS.

57

the space of (k, k)-forms defined in a neighborhood of p and invariant with respect to U has complex dimension one. We write a in local coordinates in a neighborhood of the point p:

a=>ajjdzjAdzj, f,J

where I and J are multi-indices of length k, dzj and dz,j have their usual meaning, and the ajj are generalized functions. If 10 J and v is an index which appears in I but not in J, then consider an element g E Un which changes the sign of the with coordinate without changing the others. From the condition of invariance ga = a (g«a is the image of a under the action of g) we find that ajj = 0. Analogously, considering the transformations in U,, which permute coordinates, we find that all = aJ j = a for all I and J; consequently, a= a Fl dzj A d1. LEMMA 2. For any nondegenerate holomorphic mapping f : Cm -+ P" and any k = 1, ... , n; the following is true in the sense of currents:

f«wk =

fI (P) d t(P),

f EC

(20)

k

where f «wk is the pullback of the kth power of the Fubini-Study form on Pn and it is the invariant measure introduced above.(21) Let us fix a point Po E Gk, i.e., a plane of codimension k passing through the origin in Cn+l, and consider the current

T=f

g(Po) da(g),

"E Un, i

where g(Po) is the image of P0 under the transformation g and A is the Haar measure on the group Un+1 normalized so that A(U,,+1) = 1. It is obtained

from the measure a on Gk under the mapping h: Un+1 - Gk defined by Tn-k,n-k(P") h(g) = g(Po). By definition the current T acts on forms p E by the rule T (AP) = f gEU,+j

dA(g) f

,c

g(Po)

(we observe that g(P0) is an (n - k)-dimensional plane in P"; hence the form W can be integrated over it).

By Theorem 1 the current T can be viewed as a form of bidegree (k, k) with generalized coefficients. It is clearly invariant with respect to the group

Un+1. By Lemma 1 we conclude that T = cwk with some constant c. To (21)The left side of (20) is a differential form while the right side is a current which may be viewed as the limit of the holomorphic chains r_ t f -'(P3) which are determined by the Riemann sums of the integral. The meaning of the right side of (20) will be made Precise in the course of the proof of the lemma.

I. THE CHARACTERISTIC AND COUNTING FUNCTIONS

58

compute the constant we apply the (compact) current T to the form wn-k; from the definition of T and the properties of the form w we find that 7'(w" -k) =

fu(Po) da(g) f

wn-k

=f

dA(g) = 1, n +1

and from the relation T = cwk that l

T(wn-k) = C[Wk](Wn-k)

= C f Wk A n-k = C. P

n

Thus, c = 1, and the following holds in the sense of currents:

gEGk

Wk = f

g(Po) d.(g) = J

P d1i(P)

Gk described above, noticing that (we have used the mapping h: Un+1 when g varies over Un+i the image g(Po) varies over Gk ). It only remains

to take this equality and apply the pullback by the mapping f to obtain (20).

Now for the hyperplane bundle on Pn it is not difficult to establish the relation which was announced at the beginning of this subsection.

4. The characteristic function Tfk) (r) of a nondegerate

THEOREM

holomorphic mapping f : Un -+ Pn is the average of the counting functions N f (P, r) of the planes P E Gk with respect to the invariant measure p: 7 (k) (r) =

f

n Gk

k = I,-, n.

Nf (P, r) dµ(P),

(21)

A By definition of the kth characteristic function T(k)(r)

f T dt t 0

Bt

Aw0 -k

f Bt

representing (f* W) k = f' (Wk) by means of Lemma 2 and applying l bini's theorem, we obtain

f =f

T(k)(r) =

" EGk

du(P)

dµ(P) k

[f-1(P) n Bt]WO -k 0

f "If 0

dt

t

wo -k = -1(P)nB,

f

Nf(Pr)dp(P)

Gk

We point out in particular the special case of the principal characteristic function. Here k = 1 and the Grassmann manifold G; of hyperplanes D C Pn is itself the n-dimensional projective space (Pn)` dual to Pn. (In fact, 0, where the z are homogeneous coordinates, is the hyperplane Eo determined by the set of coefficients [ao, ... , an] up to a complex multiple, so

§3. CURRENTS AND SOME OF THEIR APPLICATIONS.

59

it can be viewed as a point in a projective space.) Formula (21) takes the form

Tf(r) =

J

Nf(D, r) du (D)

(22)

In the general case of interest for us, M is a closed subset of a complex projective space pN and the bundle L is the restriction to M of the hyperplane bundle on PN. The divisors of the holomorphic sections of L are the intersections with M of hyperplanes D C PN. These divisors can be indexed by the

points of the dual space (PN)'. In this case, for a nondegerate holomorphic mapping f : C "I --r M, it follows from (22) that

Tf(L,r) =

f

N).

Nf(D,r)du(D),

(23)

where u is the invariant measure on PN normalized by the condition that P(PN) = 1. Analogous formulas are also true for the functions T fkl (L, r). Now let M be an arbitrary compact complex manifold on which is given a positive line bundle L. By the theorem of Kodaira cited in subsection 9, the sections of a sufficiently high power of L imbed M into a projective space of some dimension, and we have the situation just described. Sometimes, in order to increase the supply of divisors on the manifold under consideration, it is also useful to imbed the manifold in projective spaces of higher dimensions. Having this in mind, it is helpful to distinguish specifically the bundles whose sections realize such imbeddings. Namely, as we observed in subsection 7, the set H°(M, L) of holomorphic sections of a positive line bundle L on a compact complex manifold M is a finite-dimensional vector space over C; let its dimension be N + 1. If there exist linearly independent sections sl, ... , sN E H°(M, L) such that [sl, ... , sN]

realizes an imbedding of M into the space PN, then the bundle L is called ample. By the theorem of Kodaira cited above, sufficiently high powers of any Positive line bundle on a compact manifold are ample; also, powers of ample bundles are ample (see Shafarevich [1], Chapter 1, §4.4, Example 2). The divisors of the holomorphic sections of an ample bundle L -* M are clearly intersections of M with the hyperplanes of the space PN; the restriction

to M of the Fubini-Study metric for this space may serve as a metric on L. Thus in the case of ample bundles, the situation described above-of Submanifolds of projective space-obtains. Thus formula (23) can be applied was well as the analogous formulas for the higher characteristic functions. One Can prove (see Shiffman [21) that these formulas can be extended to the case what are called weakly ample bundles L - M. (A bundle is weakly ample if at every point p E M there is a section s e H°(M, L) such that s(p) 0.) In conclusion we observe that in classical integral geometry there is a formula called Crofton's formula, which says that the length of a real plane curve '7 is equal to the average of the number n(l fl -y) of points of intersection of y

60

I. THE CHARACTERISTIC AND COUNTING FUNCTIONS

with the line 1, with respect to a measure µ on the set of all lines in the plane:

length y = f n(1 fl y) dµ(1). J{t}

The averaging formulas (21)-(23) are in a certain sense analogous to this one and they are also called Crofton formulas.

CHAPTER II

The Main Theorems of Value Distribution Theory In this chapter we present the proofs of the two main theorems for the case of holomorphic mappings of C' into smooth projective manifolds, and also discuss some applications of these theorems. Our treatment relies mainly on the work of Griffiths and Carlson [1] and Griffiths and King 111-

§4. First main theorem The first main theorem of value distribution theory is also called the theorem of uniform distribution. In a certain sense it is a far-ranging generalization of the theorem that polynomials in one complex variable take on every value (counting multiplicity) equally often, and this frequency is determined by the degree of the polynomial, which characterizes its growth. In the general case the role of the number of values is assumed by the counting function, and the role of the indicator of growth by the characteristic function. In the relation linking them, additional terms appear; these will be considered below.

1. The case of divisors. Let us begin with the simplest and best-underM stood case. We consider a nondegenerate holomorphic mapping f : Cm

into a compact complex manifold M on which is given an Hermitian line bundle L M with metric h = { ha }. We understand the nondegeneracy of f to mean that the inverse image of an analytic subset of codimension 1 also has codimension 1. As the system of divisors, the distribution of whose preimages we are going to study, we take the set of divisors of the holomorphic sections of this bundle. If a = { sa } is such a section, the square of its Hermitian modulus 118112 =

hal8al2 in Ua, and its divisor D = { s = 0 }, then, by the Poincare-Lelong formula (13) in §3, dd` In 113112 = -Ch + D,

(1)

where Ch = -dd` In h,, in Ua is the Chern form of the metric h and dd` and the equality are understood in the sense of currents.

We pass now to pullbacks by the mapping f :

M In

118 0 1112 - f *(Ch) + f -1(D) 61

II. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

62

and we assume that f -1(D) does not contain the point z = 0, i.e., that s 0 f (0) # 0. Then, as seen from the right-hand side of the last equality, the current dd` In Its o f 112 is applicable to the integrable form Xt(z)wo -1, where Xt is the characteristic function of the ball Bt. Hence we obtain

f

f

dd`in1180f112Awo-1=-

'(ch)Awo-1+

f

wo-1. (2)

fB , -'(D)nB, By definition, differentiation of currents reduces to an application of Stokes' formula, and since in our case dwo = 0, one gets Bt

-'

d& inIlof112= f d`in

JB, while a repetition of the method used in the proof of Lemma 2 of §1 leads to the relation Iscf112Ae

/' d' In11s01112A=

"

L

f Inls0fIl -In110f(0)11t

,

where or = d` In lzl2 Awo -' is the Poincare form in C"`. Thus, the logarithmic

average of equality (2) has the form Js.

lnllsoflla= +

fT dt f o

t

B,

wo-1+Inllsof(0)II. f'dt 0 t J -l(D)nB,

(3)

On the right-hand side of this equality appear some familiar quantities, the characteristic function T f(L, r) and the counting function Nf(D, r), while the left side leads to the third actor in value distribution theory. DEFINITION. Let a holomorphic mapping f: C'n - M and the divisor D M be given. The of a holomorphic section s of an Hermitian bundle L proximity function of this divisor is defined to be

mf(D,r)=fYin

30-fIIa,

(4)

where IlslI is the Hermitian modulus of s. The meaning of the quantity mf (D, r) and its properties will be considered a bit later; right now we return to the argument which was interrupted. Introducing into (3) the proximity function, we rewrite it in the form of an equality

-mf(D, r) = -Tf(L, r) + Nf(D, r) + 0(1). It remains to remove the supplementary condition 0 ¢ f -1(D), which we introduced above. If this is not satisfied, then we integrate in (3) not from r = 0 but from some ro > 0; then in the final relation an additional bounded term appears. But we include this in 0(1), so the form of the relation does not change. Thus we arrive at the first main theorem of value distribution theory.

§4. FIRST MAIN THEOREM

63

THEOREM 1. Let f: C' - M be a nondegenerate holomorphic mapping into a complex manifold on which is given an Hermitian line bundle L. Then for the divisor D of any holomorphic section of L, the sum of its counting function and proximity function is the same up to the addition of a bounded term and equals the characteristic function of the mapping:

Nf(D, r) + m f(D,r) = Tf(L,r) + 0(1).

(5)

We now pass to the discussion of the concept of the proximity function. From _(4) it is evident that this function indicates how close the image of the sphere f (Sf) is to the divisor D. At points z E Sr for which the image f (z) is close to D the quantity Its o f (z) 11 is small and ln(1/Ils o f p) is large. Further, for this function the following is true: THEOREM 2. If M is a compact manifold, then the proximity function is independent, up to a bounded term, of the choice of the section s defining the divisor D and also of the choice of metric on the bundle L.

i Let D be defined by the section s' = { s } as well as by s = { s, }, and let us consider the metric h' _ { h« } as well as h = { hQ }. In the intersections Uap of the domains of the covering, by the compatibility conditions (4) of §2,

which means that A = s«/sa in U, is a globally we have s' /sp holomorphic function on M. It must be constant since M is compact, and clearly A # 0. In exactly the same way, p = h«/hc, in U,,, is a globally smooth positive function on M, which is therefore bounded and bounded away from 0 (see §2). But in U, we have JIs112 = h«Is«l2 and 115'112 = h«1s«I2, so JIs1112 = pIAI2JJsJJ2 on M and the difference In JJs' o f 11 - In Ids o f 11 = 2 ln(po f) + In IAI

is bounded on Cm. By properties of the form a we conclude that In 11811

lla-

I

in1Is0Aa=0(1).

1.

Further, in the metrics of the bundles of the respective divisors, the proximity function

mf (Di + D2: r) = m f(Di, r) + mf (D2, r)

(6)

In fact, if s' _ { s« } and s" = { s' } are sections defining Dl and D2, the divisor D = Dl + D2 is defined by the section s = { s' s" }, and the metric coefficients of the bundles of these divisors are related by ha = h«hQ. From this follows the equality JJsjI = 1Is'II Ils"11 for the Hermitian norms, which by definition of the proximity function leads to (6). We now compute the proximity function in the classical case of a meromorphic function of one variable, i.e., a holomorphic mapping f: C P, where P = C is furnished with the spherical metric. The role of divisors is here played by points a E C. In P we introduce homogeneous coordinates [Wo, wi] and local coordinate w = wl /wo in the domain Uo = {wo 54 0).

64

II. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

As was shown on p. 51, the metric on Uo is given by the function ho = Iwol2/(Iwo12 + Iw1I2) = 1/(1 + (1 + IwI2). If the finite value a is defined by the section so = (w - a)/ 1 -+1a12, then the Hermitian modulus If (z) - al = P(f (z), a), 1+If(z)12 1+Ia12

Ilso 0 f II =

which is the spherical distance from a to the point f (z). Analogously, for a = oo the function wo/wl = 11w serves as a local coordinate in the domain U1 = { w1 # 0); the metric is given by h1 = Iw112/(Iwo12+Iw112) = 1w12/(1+ IwI2), the defining section is sl = 1/w and the Hermitian modulus is 1 -If (Z) 12 = P(f (z), 00).

Ilsl c f ll = 1/

Taking into account that a = dO/27r for m = 1, we obtain from (4) the following formula for all a E C: 1

mf (a, r)

27r

2a

/c

1

In

P(f (7eie), a) de' which coincides with the classical one. Recalling that in the case under consideration

Hf (a, r) =

f ' n(a, t) t n(a, 0)

dt + n(a, 0) In r,

f

Jo

_ i

'' dt I f'(z)12 dz A dz fo t t (1 + If (x)12)2 (see (26) in §1 and (12) in §2), we arrive at the following conclusion: In the special case m = 1 and M = C with the spherical metric, Theorem 1 coincides with the classical first main theorem of the value distribution theory of meromorphic functions in the Ahlfors-Shimizu formulation (see Hayman [1], Theorem 1.4). It is also easy to compute the proximity function for mappings f : C^` -+ P" with the hyperplane bundle on Pn. If the divisor D is given by an equation in homogeneous coordinates

Tf(r)

27r

n

E a,.w,. _ (w, a) = 0,

lal = 1,

=o

then in the domain Ua = { wa # 0 } the section defining D has the form S.

n

= ,.=o

aw = wa

(w, a)

Wa

Since in U. the metric coefficient ha = IwaI2/IwI2, the Hermitian modulus of the section is IIsaO = Iwal Isal/IwI = I(w,a)I/IwI, and hence by (4) m f (D, r) =

If(z)I

- Js , In I (f (z), a) I

or.

(7)

§4. FIRST MAIN THEOREM

65

2. First applications. Here we note a number of consequences of the first main theorem. 1) Nevanlinna inequality. THEOREM 3. Let M be a compact complex manifold on which is given an Hermitian line bundle L, and let f : C" -' M be a nondegenerate holomorphic mapping. Then for the divisor D of any holomorphic section of L,

Nf(D,r)
(8)

4 If the section s defining D is replaced by the section As, where A is any (nonzero) complex number, then by what was proved above the function

mf(D,r) is changed by the addition of a bounded term and relation (5) is preserved. By the compactness of M, one can consequently assume that 1[s[l < 1 without loss of generality. Then from (4) it is clear that m f(D, r) > 0. It remains to apply (5). The Nevanlinna inequality (8) asserts that if some divisor D strongly intersects f(C-) (that is, the counting function grows rapidly), then the characteristic function of the mapping f (which was assumed nondegenerate) also grows rapidly. This generalizes the well-known "recalcitrant" property of entire and meromorphic functions: if such a function (different from a constant) is made to take on some value frequently, then it will grow rapidly. 2) Sokhotskii's theorem. We will prove this theorem following Griffiths and King [11 for the case of ample bundles, which were defined in §3. In the same section we said that the divisors of the holomorphic sections of such bundles can be viewed as the points of a projective space pN. The phrase "almost all" in this theorem is understood in the sense of the invariant measure on pN, normalized so that µ(pN) = 1.

THEOREM 4. If L -> M is an ample bundle and f : C" -+ M is a nondegenerate holomorphic mapping, then f (C") intersects almost all divisors of holomorphic sections of L.

4 Let us denote by E the set of divisors D E pN which intersect f (C"), and let us suppose by contradiction that 1(E) < 1 - s for some e > 0. By Crofton's formula (22) in §3,

Tf(L, r) =

fr' Nf(D, r) dp(D) = fB Nf(D, r) dµ(D),

since Nf(D, r) = 0 for D ¢ E. Hence by the Nevanlinna inequality

Tf(L, r) < (1- e)Tf(L, r) + 0(1), and since an ample bundle is positive, Tf(L,r) oo as r oo. Therefore, dividing the last inequality by T1 (L, r) and letting r tend to oo, we obtain a Contradiction.

66

II. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

In particular, if M is a submanifold of P', then f (C') meets intersections of M with almost all hyperplanes of Pn, almost all hypersurfaces of second order in P", etc. (see subsection 13 of Chapter I). 3) Condition for the rationality of mappings. By the theorem of Kodaira cited in subsection 9 of Chapter 1, some power of a positive line bundle L -' M is an ample bundle. The sections of the latter imbed M into some projective space pN; therefore, a mapping f : Cm , M can be viewed as a mapping

f : Cm --+ PN and be given in homogeneous coordinates [ fo.... , fN]. The mapping f is called rational if it can be written in this way with rational coordinates fo, ... , fN. We will need the following lemma: LEMMA. A holomorphic mapping f : C' -y PN is rational if the inverse image of any hyperplane H C PN is an algebraic subset of C"°.

In the special case m = N = 1 the assertion reduces to the statement that a meromorphic function f : C -* C is rational if the preimage of each point a E C is a finite set, and this follows from Picard's theorem. In fact, since f has finitely many poles, the point z = oo is an isolated singularity of f ; even if three different values are attained at a finite number of points, there is a neighborhood of z = oo in which f does not take these values. Thus z = oo cannot be an essential singularity, and f is rational. In the general case at least one of the homogeneous coordinates of f, say fo, is not identically equal to zero; introduce local coordinates in U0 and write f in terms of them. By hypothesis, the inverse images of all hyperplanes in PN, and in particular the level sets of the local coordinates of f, are algebraic subsets of C". Such a subset intersects a complex line at finitely many points

or else contains it entirely. Hence every local coordinate of f, on any line parallel to one of the coordinate axes in Cm, either takes on every value at finitely many points or is constant. By what was proved above, the function is rational in every variable z3 when the remaining coordinates are fixed; but then it is a rational function. The following theorem asserts that the minimal possible growth, i.e., logarithmic growth, of the characteristic function (see §2) is attained for rational mappings and only for them. It is a direct generalization of a classical theorem of Nevanlinna. THEOREM 5. If a bundle L , M is positive, then a holomorphic mapping

f: C' - M is rational if and only if Tf(L,r) = O(lnr). 4 Replacing the bundle L by its kth power Lk means by definition replacing the transition functions gap by (ga0)k and the metric ha by (ha)k; that is, the Chern form Ch = -dd` In ha in Ua is multiplied by k, and thus so is T f (L, r). Such a replacement changes neither the hypothesis nor the conclusion of the theorem; thus by choosing k sufficiently large, we can assume from the outset that the bundle L is ample. The divisors of holomorphic sections of L are

§4. FIRST MAIN THEOREM

67

now intersections of M with hyperplanes of some projective space pN and thus are algebraic subsets (M itself is an algebraic subset of pN by the wellknown theorem of Chow; see, for example, Gunning and Rossi [1], Chapter V, §D, Theorem 7).

Let f : On -4 M be a rational mapping. Then the inverse images of the divisors in question are algebraic subsets of C^` of no higher degree than a fixed p. By Theorem 5 of §1, for any such divisor Nf(D, r) < plnr + 0(1). Then by Crofton's formula

Tf(L,r) <J

pN

Nf(D,r)dp(D)
Conversely, let Tf (L, r) < pInr+0(1). Then by the Nevanlinna inequality, for any of the divisors under consideration, N1 (D, r) < pin r + 0(1). By the same Theorem 5 in §1, the preimage f -'(D) is an algebraic divisor. But D

is the intersection with M of a hyperplane in pN. Considering f to be a mapping into pN, we see that the inverse image of every hyperplane under this mapping is an algebraic set. Rationality of f follows from this by the lemma.

4) Jensen's formula. Let us consider the case of a holomorphic mapping

f : Cm -- Cn with the hyperplane bundle on C". Here the divisors D are defined by linear functions s = > ' b, the metric h = (1 + Iw12)-1,

and the Hermitian modulus n

Ilsofli =

E v=1

/v/1+ If 2,

b

so the proximity function of the divisor D

mf(D,r)=

f Js,

In

En1

I

1+If(z)I2 Q. avff(z) - bI

But by (13) in §2

1+If(z)I2a=Tf(r)+In 1+ ff(0)12,

f In r

Consequently, Tf in (5) cancels, and the relation takes the form(1) n

Nf(D,r) = /

a - In

In

.I S.

a f (0) - b

1

For m = n = 1 we obtain in this case the classical formula of Jensen: 1

Nf(b, r)

(2a

2 Jo

in I f(reie) - bI dO - In If (0) - bI.

(')In (5) we set 0(1) = - In 118 0 f(o)I1, using (3).

(9)

11, THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

68

3. The case of sets of codimension greater than 1. In general case, as M on a compact nbefore, we consider an Hermitian line bundle L dimensional complex manifold M. We are interested in the distribution of the inverse images under a holomorphic mapping f : Cm --+ M of a system of holomorphic chains A c M of codimension k which are intersections of k divisors of holomorphic sections s; of the bundle L: k

A=nD,; Such a system { A } of sets will be called an admissible system. The problem considered above is obtained from this one when k = 1. Let us begin with generalizing the definitions of the three main quantities of the theory. The characteristic function for codimension k was already defined in §2:

f

Tfk)(Ch,r) = 1 r

t

O

B'

(.f*ch) AW0

-k,

(10) l 1

where Ch = -dd` In ha in Ua is the Chern form of the metric h = { ha }, and wo = dd` In Iz12 is the homogeneous metric form in Cm. The counting function of the set A is defined as in §1:

Nf(A,r)_ /rdt /' 0

wo-k;

(11)

f-'(A)nBLL

t

For this to make sense, we assume that the mapping f is nondegenerate, i.e., that for any set A in the admissible system (for fixed k) the preimage f -1(A) has codimension k. We introduce the Hermitian modulus squared of the section s = (sic , ... , S' k'),

which determines the set A in Ua, by the formula haIsal2 in Ua, 118112 =

(12)

where 1sa12 = Ei Isj I2. We also introduce the forms k-i A

>2(ch + dd` In JIsII2)" A ch-v-1

In 1181

2

(13)

"=0

and

ak=d`Inlzl2AWO-k.

(14)

For k = 1 we have A = ln(1 / 11 s 112 ), and al = a, the Poincare form from §1. Now the third quantity can be defined: the proximity function for a set A in an admissible system with respect to a nondegenerate holomorphic mapping f : C'° --p M is defined to be

mf(A,r) = 2

J

f*(A) A ak.

(15)

§4. FIRST MAIN THEOREM

69

For k = 1 this clearly reduces to the old definition (4). We observe that for positive bundles (Ch > 0) the form A > 0 provided Ilall < 1. In fact, locally in Ua it follows from (12) and the definition of Isa 12

> 0 as the pullback of the positive ch that Ch + dd° In 113112 = dd` In form dd` In Iw12 by the holomorphic mapping Ua - Ck given by the function sa = (si , ... , sk ). From this it is clear that the coefficients of the form A llsII2(k-1)

become infinite of order

In 11311 on A.

Let M = P", and let A be the point w = 0 in the local

EXAMPLE.

coordinates (w1,. .. , w") on the chart Uo. Here 2

1 + Iwl2'

h0

113112 = 1

+

Iw12'

Ch = ddc In(1 + 1w12) = w

so, consequently, 112 12 1n

A = In

1

w

1

: (d& In I w 12 )" A n-,-I.

1,.

(16)

"=0

LEMMA. Under the conditions and notation just described the following equality of currents holds: (17) dd` A = ch - A. 4 Locally in Ua we substitute ln(1/118112) = - In ha - In 1sa 12 from (12) and Ch + ddc In 118112 = dd` In 1sa 12, and obtain k-1

"-1

A

"=o

-

k-1

In 1 sa 12 (dd` In 1 sa 12 )" A

ch-"-1

"=0

Now we recall that -ddc In ha = Ch in Ua and that by Theorem 3 of §3 (ddcIn 1sa12)"+1 ddc(Inlsa12(dddIn13,12)") = JlA

for0
Consequently, k-1

k-2

dd`A = E(dd In 18«12)" A ch-" - 0(dd` In Isa12)"+1 A ch-"-1 - A V=0

V=0

=ch - A. From this, as in the case of divisors, follows the first main theorem of value distribution theory in the general case.

II. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

70

THEOREM 6. Let L , M be an Hermitian line bundle on a compact complex manifold, and let { A } be an admissible system of subsets in it of codimension k. Then for any nondegenerate holomorphic mapping f : Cm M and any set A in the admissible system Nf(A, r) + m f(A, r) = Tfk) (ch, r) + Rj(A, r) + 0(1),

(18)

where Rf(Ar)=ZL,f*(A)Awo-k+1

(19)

is the supplementary term. ,4 Passing in (17) to preimages under the mapping f, we obtain an equality of currents (20) dd` f *(A) = f'(Ck) - f -1(A). Applying both sides to the integrable form Xt(z)wo -k, where Xt is the characteristic function of the ball Bt, we obtain at first, using the notation from (10) and (11)(2)

ff dd` f()A A

= Tf

c r) (h,

, t Then, after applying (34) from §1, according to which

A

=

1

J f.() n ak - f f'(A) n

t fB , and employing the notation of (15) and (19), we arrive at (18). In the case k = 1 the supplementary term R f(A, r) disappears, since wo JO

0 (see (19)), and we return to the first main theorem in the form (5). In the general case k > 1 the presence of this term seriously impedes applications, because it is rarely possible to estimate it. Unfortunately, this observation applies in particular, to the case k = n, most interesting for analysis; this is the case concerning distribution of preimages of points of the manifold M. 4. On the Nevanlinna inequality for eodimensions greater than 1. If L M is a positive line bundle, then without loss of generality we can assume that the Hermitian length of a section s defining a set in an admissible system does not exceed 1, and then, by the remark in the previous subsection, A > 0. From (15) it follows that in this case, for all sets A in the admissible system, mf(A.r) > 0. The first main theorem in the form (18) leads to an analog of the Nevanlinna inequality: Nf (A, r) < T fk) (c,,, r) + Rf(A, r) + 0(1)

(21)

for all A in an admissible system of sets of codimension k. (2)We are assuming that 0 V A; otherwise, we integrate not from 0 but from some rp > 0, which adds another 0(1) term.

§4. FIRST MAIN THEOREM

71

However, for k > 1 this inequality is less interesting than for k = 1, since it contains the additional term Rf, which is difficult to estimate. For k > 1 in the general case it is impossible to estimate this term as a quantity which is small in comparison with Tf k as r - oo. This is shown by the following example, which is due to Carlson [3] (the first example of this type was obtained by Cornalba and Shiffman [1]). EXAMPLE. It is not difficult to see (from, e.g., the proof of Weierstrass's

theorem in Shabat I, p. 257) that the infinite product 00

P(zl) _ fl

(1 - 2k }

(22)

k=1

defines an entire function of zero order, so that for any e > 0 and for all x1 E C the estimate lp(z1)l < CelzlI° holds with some constant CE. The same estimate (with the same constant Ce) is also valid for the entire function ' P k (z1)

_ll( 1 - zl) 2j

k = 1, 2, .. .

j=1 j#k

We choose further a function x: R+ -' R+ growing arbitrarily rapidly such that X(k) > k for all natural numbers k. We construct polynomials Xk

Pk(z2)

_

1

f(

j=1 \z2 - J

of degrees Xk, which equal the integer part of x(k) + 1; using them we form the series 00

1

gl(z) = E 27- k(z1)Pk(z2) k=1

This is clearly majorized by CEeI-1I` E 2-xkIz2lxk. By a well-known formula (see Shabat I, p. 266), for fixed z1 this defines an entire function in z2 of order zero, so that (g1(z)l < Cee1zlI'+Iz31' with some constant CE. Now consider the holomorphic mapping (23) g = (gl, g2): C2 -' C2, where the function g1 is the one just defined and g2(z) = p(z1) is defined in (22). This is a mapping of zero order, and so by Theorem 4 of §2 its

characteristic functions T9(r) and T92 (r) grow no faster than rE for any e > 0. On the other hand, it is clear from the construction that g(2', 1/k) = (0, 0) for any natural number j and (for fixed j) for k = 1, 2, ... , X3 . Since x is a rapidly increasing function, the preimages of 0 E C2 accumulate rapidly near the points (23, 0) on the z1-axis (see the schematic Figure 4). For fixed r these preimages belong to the ball Br if 2? < r, i.e., j < 1092 r. Therefore, the number n9(0, r) of them in BE is no less than x(log2 r+0(1)),

72

II. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

Z2 j

X; points

2

21

22

FIGURE 4

and hence the counting function Ng (0, r) grows arbitrarily rapidly, given a suitable choice of X. Thus for the mapping (23) and the point 0 E C2 (a set of codimension 2) it is impossible to estimate .'V,(0, r) from above in terms of Tg2) (r). Turning to

(21), we see that for this example the dominant term on the right-hand side is not T(2) (r) but R. (0, r). We observe that by a minor change in the construction the same phe-

nomenon can be attained not only for the preimage of 0 but also for the preimages of all the points in a countable everywhere-dense set E C C2. Indeed, let the points ak c E be indexed so that lakl < k; we form the sequence bl = a1, b2 = a1, b3 = a2, b4 = a1, b5 = a2, b6 = a3, b7 = al, ..., in which every point a3 is repeated infinitely many times. We construct a complex curve h = (h1, h2): C -> C2 by the formula

h(z1) _

00

ick(z1)bk

k=1 'Pk(2k)

Since ypk(2k) = (1 - 2k-1)(1 - 2k-2) ... (1 - 2)p(1), we have IPk(zk)I C2k(k-1)/2 > C2k2/3 with some constant C. Hence the last series is majorized by 00

M 2k2/3 2 k2/3IVk(z1)I Ibkl < Cel='l` E k=1

k=1

(we have used here the estimate for
and k = 1, 2,...,Xj. The points f -1(b) E 13, if j < 1092 r, while if j = 1(1 + 1)/2, there are + 0(1), the l distinct points a" among the points bk, k < j. Since 1 =

§4. FIRST MAIN THEOREM

73

number of preimages f -1(a") in the ball Br has order x( 2log2 r). Thus for a suitable choice of x the counting function Nf(a, r) for any point a E E grows as rapidly as desired while the characteristic function Tj") = O(r6) for an arbitrarily small E > 0 (v = 1, 2). 1 This example shows that in the case of an analytic set A of codimension k > 1 the order ord f -1(A) in the sense of §1 cannot, generally speaking, be estimated from above in terms of the order ord f in the sense of §2. It also gives a counterexample to the so-called transcendental Bezout problem. The classical theorem of Bezout in algebraic geometry asserts that the degree of the intersection of algebraic sets Aj does not exceed the product of the degrees of the sets: 4

4

degnAj
j=1

where equality is attained if the Aj intersect in general position, so that the codimension of the intersection equals the sum of the codimensions of the Aj. Since the degree of an algebraic set A is equal to its projective volume n(A, oo), in the nonalgebraic (transcendental) case the ideal analog of the Bezout theorem would be the analogous inequality for the functions n(A, r) or N(A, r):

N n Aj, r

a

< fl N(A3, r). j=1

j=1

For holomorphic mappings f : C' , C' and inverse images of divisors Dj such an estimate with the Nevanlinna inequality N f(D f, r) < Tf(r) taken into account would lead to an estimate of the counting function of the inverse image of the intersection of the divisors in terms of the characteristic function of the mapping: Q

Nf n Dj, r

< (Tf(r))

j=1

The example above shows that such an estimate is in general impossible. We note, however, that in the paper of Carlson [31 cited above it is proved that an estimate from above of N1 (a, r) in terms of T f fi (r) for holomorphic mappings f : C" - Cr, is possible for all the points a E C' outside an exceptional set which is insignificant in a certain sense (for instance, its 2n-dimensional Lebesgue measure is 0). We will present this result in Chapter V (see subsection 4).

5. Sokhotakii's theorem for codimensiona greater than 1. In the case of codimension 1, Sokhotskii's theorem that under a nondegenerate holomorphic mapping f : Cm - C' almost all divisors intersect follows from Nevanlinna's inequality. But as we have just seen, for codimension k > 1 the Nevanlinna

II. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

74

inequality is no longer true in general. This complicates the problem of generalizing Sokhotskii's theorem. The most interesting case is k = n, which deals with the distribution of preimages of points. Then Sokhotskii's theorem must assert that f takes on almost all values in C', that is, that the image f (Cm) is dense in C". Such an assertion is not-true in eneral; the well-known example of Fatou (see Shabat II, p. 62) shows that the image under a nondegenerate holomorphic mapping can fail to cover a rather large open set.

Thus theorems on the density of the image can only be true under additional conditions. We present here one such theorem, following Griffiths and King [1], for mappings into a compact complex manifold M, on which is given a line bundle L. We suppose that this bundle is ample; then its sections imbed M into some projective space PN, and the admissible system can be regarded as the set of intersections of M with planes in PN of codimension k. For the proof we need a simple corollary of the first main theorem in codimension k which is true in the case of an ample bundle L. As we saw in §3, the following averaging formula is true in this case: (ch, r) = J{A}

Nf (A, r) d1a(A),

(24)

where A is the invariant measure on the admissible system { A) of sets of codimension k normalized so that the measure of the whole system equals 1. Taking into account this formula, from (18) we immediately get

I mf (A, r) dp(A) =

Rf(A,r)d1(A)+0(1).

(25)

A}

A}

THEOREM 7. Let L -y M be an ample line bundle and let { A } be an admissible system of analytic subsets of M of codimension k. Suppose further that tk-1(ch,r) _ 0, lira (26) r-+oo T(ch,r) lk) where

tk-1(eh,r) = r

dr

=1

(f*Ch)k-1 A Wpm-k+l

(27)

Then for any nondegenerate holomorphic mapping f : On M, the image f (C'") intersects almost all (in the sense of the measure Ez) sets of the system

{A}. 4 Suppose the contrary, that the set E of those A which intersect f (C') has measure 1 - e and e > 0. By the averaging formula (24), Tfk) (ck,

r) = L Nf (A, r) dµ(A),

§4. FIRST MAIN THEOREM

75

since Nf(A, r) = 0 for A V E. Now we use the analog of the Nevanlinna inequality (21): T f(k) (Ch, r) < (1- E)Tfkl (Ch, r) +

f Rf(A, r) dA(A) + 0(1).

We assume without loss of generality that 11s < 1 for all A, so that f -(A) > 0 and (according to (19)) Rf is nonnegative. Then we replace integration over E by integration over the whole system { A }: (Ch, r) G (1 - E)T1kl (Ch, r) + f A}

Rf (A, r) dt1(A) + O(1).

(28)

But by (25) and the definition of m f (see (15)) we have Rf(A, r) dp,(A) = 1 2

A}

r

f' J Adµ(A) A ak + O(1) {A}

(we have used the local summability of A and applied Fubini's theorem). Recall that a set A is the intersection with M of a plane in projective space pN and that the admissible system {A} is obtained from one set by the action of the unitary transformations on P'. Since Ch is the restriction to M of the Fubini-Study form on pN and the measure W is invariant under the unitary transformations, the integral f{ A) A dµ(A) is a (k - 1, k - 1)-form which is invariant under these transformations. Thus by Lemma 1 of §3 it is proportional to C-1 Inserting this in rthe last equality, we find that -

Rf(A,r)d1c(A) =cJ iA}

(f*Ch)k-1

A dk+O(1)

S

or, after an application of Stokes' formula,

]A}

Rf(A,r)d{(A) = c

0(1) f (f-1 A+O(1) B.

l

= Ctk-1(Ch, r) + 0(1)

(see (14) and (27)). Now (28) takes the form Tfk)(ch,r) < (1

-

E)Tfk)(Ch,r) + ctk-I(ch,r) + 0(1);

dividing it by Tfk) (Ch, r) and taking the lower limit as r -' oo with (26) taken into account, we arrive at a contradiction. We will give an example of an application of this theorem in subsection 8

of chapter V. It is interesting that (26) almost coincides with condition (28) of §2 for (Ch, r) to be independent, up to a bounded term, of the choice of metric h; however, the reason for this coincidence is not clear. For k = n, (26) is a sufficient condition for the density in M of the image f (C"`) under a

76

11. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

holomorphic mapping f : C"' - M. In particular, for holomorphic mappings f : C" -# Cn with the Fubini-Study metric on the range, it has the form lim

to_1(r)

T---.00 Tfi"1

= 0'

(r)

(29)

where by (25) of §2 the quantity Tf(n) (r)

t

fT dt

i = n.J

Idfz)j2$= iB, (1+If(z)I2)n+1

(30)

is the logarithmic average of the volume of the image of the ball BT (counting multiplicity); and

tn_1(r)= 41 (dd'ln(l + jf(Z)j2))n- '

A ddc In

IZ12.

(31)

We observe that in the case of mappings of finite degree the volume of the image of BT in the Fubini-Study metric is a bounded quantity. Hence Ti') (r) -* oo only at the rate of In r; therefore, for such mappings condition (29) is fulfilled extremely rarely.

§5. Second main theorem In the classical formulation the second main theorem of value distribution theory for meromorphic functions relates the counting function Nf(aj,r) of a system of points aj E C to the characteristic function Tf(r). It has the form q

(q - 2)Tf(r) + Nf(S, r) = E Nf(aj, r) + R(r),

(1)

j=1

where Nf(S,r) = Nf-(0,r) + N f(oo,r) is a function which counts how stationary the mapping f is (N f, (0, r) denotes the counting function of the zeros of the derivative, i.e., the stationary points of the mapping, while N f (oo, r) is a function which counts (v - 1) times a pole of f of multiplicity v, i.e., it only counts multiple poles). The function R(r) is a supplementary term which

admits an estimate R(r) = O(1nTf(r)) outside a set E of finite logarithmic measure or everywhere, provided that f is of finite order. It is assumed that the number of points q > 2, or else the theorem becomes vacuous. Here we will describe a generalization of this theorem obtained by Griffiths and his students. As before we will consider holomorphic mappings f : C'

M to compact complex manifolds, but in addition it will be assumed that dim M = n. Only the distribution of the inverse images of divisors will be considered; the theory for sets of higher codimension is still not developed.

6. Singular volume form. Instead of a system of q points, in the general case we will consider a system of q divisors D j on the manifold M, assuming that they satisfy the following:

§5. SECOND MAIN THEOREM

77

CONDITION A. The sets Dj are manifolds (i.e., they have no critical points) and intersect in general position.

The latter means that the union e

D=ED;

(2)

can, in a neighborhood of every point, be given by an equation wl . . w,,, _

0 in local coordinates on M, where m < n. For the case of a system of hyperplanes Hj C P", this condition reduces to the usual condition of general position (at most n hyperplanes pass through each point). In order to formulate the second condition, let us recall the concept of the canonical bundle. This is the line bundle KM -> M whose transition functions are the Jacobians of the coordinate change mappings in the intersections U«p of the domains in a covering of M: 8wc" (3)

9«Q = det k

If { s« } is a section of the bundle KM, then the form defined locally on U« as s« dwa A ... A dwn is clearly a global form on M. Hence it follows that if n

-b« = f 2- dwv A dwv=1

in U«

(4)

is the local Euclidean volume form, then a positive (n, n)-form(a) which is a global form on M, i.e., a volume form of M, defined locally on U« as if and only if A$ = Ig«o12A , in U. Comparing this compatibility condition

with (5) of §2, we conclude that the collection of functions h« = 1/a« on U« can serve as an Hermitian metric on the canonical bundle KM. The Chern form of this metric c(KM) = ddd In A,,,

in U,,,

(5)

is called the Ricci form of the volume form 11 = A 4 in U« and is also denoted by Ric f2. If Aa - 1, then Ric f2 - 0, so Ric 0 indicates how twisted the form i2 is in comparison with the Euclidean volume form $. EXAMPLE 1. For a complex one-dimensional Hermitian manifold the volume form it = (i/2ir)h dw A duo, where h > 0, and the Euclidean form is $ = (i/2ir) dw A dw. Thus A = h and Ric i2 =

l 92 In h 2 -r

aw 8w

dw A dw

(3)We recall that on an orientable manifold the forms which are of maximal degree and have no zeros can be divided into two classes-the positive and negative forms-in accordance with orientation. Here the orientation on M is chosen so that for positive forms

all theA«>0.

78

II. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

(all in local coordinates). We observe that if the complex one-dimensional manifold is viewed as a real two-dimensional manifold, then the Gaussian curvature

_

27r 82 In h

K

h 8w 8w'

and its sign is opposite that of the Ricci curvature. I EXAMPLE 2. For complex projective space Pn the volume form corresponding to the Fubini-Study metric is, by (24) of §2, _ n =

n!

(1 + Iwl2)n+1

w'

where w = dd® In(1 + lwl2) and -0,,, is the Euclidean volume form (we use the

local coordinates in the domain U0 of the standard covering of Pn). Thus, here A = n!/(1 + IwI2)n+1 and

c(Kp,.) _ -(n + 1)dd` ln(1 + lwl2) _ -(n + 1)w.

1

(6)

Now one can formulate

CONDITION B. The sum of the Chern forms of the line bundle LD of the divisor D and of the canonical bundle KM is a positive form:

c(LD) + c(KM) > 0.

(7)

In particular, if M = Pn and the divisor D = Ei Hj is the union of q hyperplanes, then LDis the qth power of the hyperplane bundle, and by formula (9) of §2 we have c(LD) = qw. But c(Kp.) = -(n + 1)w, so c(LD) + c(KM) = (q - n - 1)w and (7) reduces to the inequality q > n + 1. Thus Condition B is a generalization of the classical condition q > 2 on the number of points. Let us denote by Lj = LD3 the line bundles of the divisors Dj which make up D, and let sj be sections of these bundles whose divisors D., = Dj. Since the transition functions of the bundle L = LD are products of the transition

functions of the Lj, Hermitian metrics hj can be chosen on Lj so that the product h = h1 ... hq is a metric on L whose Chern form ch = c(LD). We denote the square of the Hermitian modulus of the section sj by Ilsil12, which equals h j 13j12 locally. (In contrast to I s j l 2, this function is defined globally, see §2). Using these functions, we construct on M \ D a singular volume form with singularity on D:

0

IIi(ln

IIsjII2)2llsjll2'

(8)

where fl is the volume form on M. Conditions A and B allow us to establish the following properties of this form, which are important for the sequel:

§5. SECOND MAIN THEOREM

79

THEOREM 1. If divisor D satisfies Conditions A and B, then for a suitable choice of the form f1 the singular volume form (8) has the following properties:(') (a)

Ric I > 0,

(b)

(Ric

(c)

f

W)" > W, (9)

(Ric W)" < oo.

M\D

A From (8) we have on M \ D 4

9

Rich=Rici1-EddcInllsjll2-1: dd`ln(lnI Isj112)2, j=1

j=1

where ddc In 11 S j 112 = dd` In h3. By our choice of metric the sum e

- E ddc In 11 sj1l2 = Ch = c(LD) j=1

is the Chern form of the metric h, while Ric 11 = c(KM). Thus the previous relation can be rewritten in the form

Ric T = c(LD) + c(KM) - 2

ddc In(ln Ilsj 112).

(10)

j=1

By Condition B the sum of the first two terms on the right is a positive form, which we denote by Eo. Further, by an elementary formula ddCln(ln11sjll2) =

dd'1n11s2112

In IIsjII

- dln!IsjII2Ad21nllsiIl2 (In IIsj11 )

After multiplying the metrics hj by constants (this does not change their Chern forms), we may assume that all the Ilsjll < b, where b < 1. Thus the first term here will be a continuous form on M; from (10) we obtain

Ric 'Y > c1Eo+2q dinIIsjll2 Ad`InllsjII2 j=1

(11)

(In 118j112)2

where cl is some positive constant. Since the form dpAdcp = (i/21r)BpA5p > 0 for any real function p (the corresponding Hermitian form is 18p12), the sum in (11) and c1Eo as well are positive forms. Property (a) is proved.

To prove the second property, in a neighborhood U of an arbitrary point P E M we choose local coordinates w = (wl,... , w") with origin at p such (!)The form Ric * is defined on M \ D just as Ric 0 was above, by comparing it with a local Euclidean volume form; condition (b) means that (Ric W)" - WY is a nonnegative (n,n)-form, where (Ric iI±)n is the exterior power.

II. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

80

that Dj I u = { wj = 0) for j = 1, ..., m. This can be done on the basis of Condition A (if D n U = 0, the estimate (Ric *)n > c1Q with some positive constant c, which is proved below, is trivial). In these coordinates I13j112 = pj Iwj I2, where pj > 0 is a smooth function, so that din113j112 Ad`lnII3j112

= -01nIIsj112 Aaln113j112 =

where

i dwj A dwj + Aj 27r

(7 = 1,...,m),

Iwj12

_

- IpJ2

apj A apj + apj n dwj + 9W.7 A apj A _ wj1 Pjwj P3 W) is a smooth form which is 0 at the point p. Consequently, the first m terms in the sum in (11) admit the estimate 2

>c2idwjndwj+Aj (In IIsj 112)2113112

(In Ilsj 112)2113112

with some constant c2 > 0. The remaining terms on the right side of (11) make up a positive (1, 1)-form and thus can be bounded from below by the Euclidean metric form multiplied by some positive constant c3. Therefore,

+c3E2 dw,Adw,,, j=1

V=1

whence (Ric T) n > e4

Z

( 27)

(rm dw n

A

III (ln 113,112)2118;112

where A is a smooth (n, n)-form which is zero at the point p, and where cl > 0 is a constant. Increasing c4 slightly and shrinking the neighborhood U, we can discard A. On the other hand it can be seen from (8) that in U the following estimate is valid with some constant c5 > 0: w < C5

2

( 27r)

n (rn dw A d-,,) n III (n 113112)2113112

(recall that in U the functions sj 0 0 for j > m). Thus (Ric W)n > c6W there, with some constant cg > 0. Covering the manifold M by finitely many such neighborhoods (it is compact), we prove that (Ric W)72 > c%P on all of M with

some constant c > 0. It remains to observe that if the form 11 is replaced by ci1 (this does not change Ric W), the constant in the inequality can be taken to be c = 1; this proves property (b). For the proof of (c) we will use the same local coordinates w and observe that in these coordinates (Rip lY)n -

E)

IIT(ln

Iw?I2)2IwjI2,

95. SECOND MAIN THEOREM

81

where 9 is a smooth (n, n)-form. Therefore, the integral of (Ric W)n in a neighborhood of the divisor D can be estimated locally by the product of convergent integrals of the form

dwj A dw

i 2

U,

(Inlwwl2)2IwjI2,

where U; is a disk with center wj = 0.

We observe that in the case n = 1, when M = U is the unit disk and D = { 0 }, the form T coincides with the form defining the invariant metric on the punctured disk U" = {0 < Izi < 11

i

dz A dz

21r IzI2(ln z12)2

(see Shabat II, p. 315). The singular volume form (8) is a direct generalization of this form.

7. Preliminary formulation. As before we consider a nondegenerate(5) holomorphic mapping f : Cn - M to a compact complex manifold of the same dimension; suppose that on M a divisor D = E?=i Dj satisfying Conditions A and B is given. By Theorem 1 a singular volume form * can be constructed on M \ D; it can be used to define a singular characteristic function

Tf(r)

=T

dt

/Bt

f*(RicT) Awo-1.

(12)

where w0 is the homogeneous metric form on Cn (see §1). We will determine the relation between this function and the other quan-

tities later; for the time being, we observe only that it-like the usual characteristic function-increases as r grows, and as r -4 oc it goes to infinity no slower than In r. This follows from the fact that the form f (Ric W), which is in the inner integral in (12), is a positive form by Theorem 1. In order to formulate the second main theorem we must introduce the concept of the divisor of stationarity of the mapping f, i.e. the set of its critical points: (13) Sf = {z E Cn: Jf(z) = 0} (Jf is the Jacobian of the mapping f). In preliminary form this theorem

appears thus:

THEOREM 2. Let f : C' -+ M be a nondegenerate holomorphic mapping into an n-dimensional compact complex manifold, and let D be a divisor on Al satisfying Conditions A and B. Then

tf (r) + N(Sf, r) = Nf (D, r) + R(r),

(5)The nondegeneracy of f here means that the Jacobian Jj(z) 0- 0.

(14)

82

II. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

where Tf is the singular characteristic function, the N are the counting functions, and R is the remainder expressed by formula (16) below. 4 Let us construct by Theorem 1 a singular volume form corresponding to the divisor D, and let its pullback be

f*(T) = SIbz,

(15)

where 4% is the Euclidean volume form in Cn. The nonnegative coefficient 1; is

clearly equal to zero on the divisor of stationarity Sf and equal to infinity on the inverse image f-1(D) of D. Outside these sets, by definition of the Ricci form, Ric f * (W) = f' (Ric 1Y) = ddc In in the classical sense; consegeuntly, in the sense of currents, the following equality is true:

dd`lne = f*(Ricw)+Sf - f-1(D), This is a variant of the Poincar6-Lelong formula (see §3). If, as in the proof of the first main theorem, both sides are multiplied by the form W0`1 and averaged logarithmically over the ball Br, then we get jr

t

wno-1

ddc

= T f(r)+ N(Sf,r)-Nf(D r).

t

The le ft-hand side can be transformed using Lemma 2 of §1 into an integral over the sphere Sr, which is denoted by

J r in e a + O(1).

R(r) = 2

s

(16)

Then we arrive at (14). In order to pass from the theorem in preliminary form to a theorem which can be applied, the remainder term R(r) must be estimated. We do this in several steps. a) We introduce the quantity

T(r) _ f

t2n-1

CC1/nn p

fB,

SS

0

(17)

where a is defined by (15), po is the Euclidean metric form, and c is a positive

constant depending only on n. We will show that under the hypotheses of Theorem 2.

T(r)
(18)

In fact let f ` (Ric T) = (i/27r) E; k=1 Rfk dz,, A dzk; by Theorem 1 this is a positive form, and hence the matrix (R,k) = R is positive definite. According to (15) and the same theorem,

=

(f' Ric*Y)n = n! detR C,

§5. SECOND MAIN THEOREM

83

< n! det R. But by the well-known Hadamard inequality for positive definite matrices, (det )Z)1/n < (1/n)tr R, so whence

y/n Cn,

n

0 C n 1: Rjj o0. j=1

On the other hand, n Po-1

n-1

= (n - 1)! E

(27rdzi A dzi A ...

j=1

///

po-1

=n

whence f'(Ric 41) A

1

i ... A dzn n dzn,

J 1 Rjj<po; thus

f *(R.ic W) A

,n-1

-

c = (n!)-1/n. Furthermore, repeating the transition in §1 from (19) to (20), we can write the definition (12) in the form

Tf(r) =

f

o

dt t dt 2

1

L f*(Ric'&) A

,0n-1

:

It remains to compare this with (17). b) For the next estimate we will need the property of the convexity of the logarithm,(6) which says that for any positive measure dµ and any positive integrable function h

µ(E)

fE In h dµ < in I µ(E) fE h dµ }

(19)

Applying this property to the definition (16) and using the fact that the integral of the form o over the sphere Sr is equal to 1, we obtain

k(r) =

fr

ln(ce1/n)o + 0(1) <

hi

f, CeIjno + 0(1).

(20)

2 2 Now we use the fact that the Euclidean volume form is 4 = r2n-1o A dr,

since or is the homogeneous area form of the sphere; thus CC1/no

fs

r2n_1 dr

f

B>

(6)The discrete analog of (19) is obtained by taking the logarithm of the inequality between the geometric and the arithmetic means. For the proof of (19) let us write c = (1/µ(e)) fE hdµ and g = h - c. We clearly have fE gdµ = 0 and ln(1 + g/c) < g/c. since 9/c > -1. Therefore,

p(E)lEInhdµ=k' f l.(g+c)dµ


9dµ=Inc.

R c

84

IL THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

But according to (17)

1/nz -= nl 1

1 r2n-1 dT W!

B.

dr '

so (20) can be rewritten in the form

R(r) < n In

1

d

ran-1 dr

2

r2n-1 d7' dr

+ 0(1).

(21)

c) In order to avoid having derivatives in the estimate, we must use the following lemma in the style of classical Nevanlinna theory: LEMMA 1. Let g and h be continuous positive functions on R+ _ (0, oo).

Then for any increasing function F: R+ - R+ which is of class C' outside an open set E C R+ and for which fE g(r) dr <

j h(r)

for some ro > 0,

the following inequality is true:

4 Let E

F'(r) < g(r)h(F(r)). (22) r E R+ : F'(r) > g(r)h(F(r)) } be the set for which (22) is

false. It is open by the continuity of the functions in the inequality, and O° dr F'(r)dr dr Ze g(r) dr < fE f(E) h(r) < I

h(F(r)) o where ro = inf F(E) is a number which without loss of generality can be h(r),

assumed to be positive.

Supposing in particular that g(r) = rb and h(r) = ra, where 6 and a are constants greater than 1, we obtain that

F'(r)
(23)

outside the set E = E(a, b), for which

L We will call sets which satisfy the last condition sets of finite 6-measure (for 6 = -1, sets of finite logarithmic measure). Applying (23) to the increasing function

ran-I dT C1/n,, C ar =,fB, S (see (17)), we obtain

(dT ) dr Cr2n-1 di'} < rb+(2n-1)a

a

outside E.

§5. SECOND MAIN THEOREM

85

The same inequality (23) applied to the increasing function t gives dT < rb(T(r))"

outside E.

If these estimates are substituted into (21), we find that In{r(a+1)b+(2n-1)(a-1) (t(r)),21+0(1)

R(r) < 2

outside E.

Now let us fix s > 0 and choose b > 0lsuch that 8(2+b/2)+ (2n - 1)b/2 = 2e/n (then b = b(e) -+ 0 ass--> 0). Set a = 1 + 6/2. Then the previous

inequality takes the form 2

R(r) < eInr+ n2 lnT(r)+O(1) outside E. Using (18) again, we obtain the following result which supplements Theorem 2: THEOREM 3. Under the hypotheses of Theorem 2, for any e > 0 there are

a number 6 = b(e) such that b(e) - 0 as a -> 0 and a set E = E(e) of values of r, of finite 6-measure, such that

R(r) <eInr+O(111Tf(r))

outside E.

(25)

8. Main formulation. Here we will produce a formulation of the second main theorem closer to the classical one. For this, first of all we will change the set-up slightly: instead of a divisor D = r9 Dj satisfying Conditions

A and B, we consider a positive line bundle L on M and q divisors Dj of holomorphic sections s j of the bundle such that: a) D1,.. . , D. are manifolds intersecting in general position, and b) qc(L) + c(KM) > 0, where KM is the canonical bundle. The new formulation is based on a lemma which expresses the relationship between the singular characteristic function T f and the characteristic functions of the bundle L and the canonical bundle KM. LEMMA 2. Under these conditions the following inequality is true for any nondegenerate holomorphic mapping f: Cn -> M:

0 < gTf(L,r) +Tf(KM.r) -Tf(r) < gInTf(L.r) +O(1).

(26)

A The divisor D = Fi Dj is defined by the section s = s1 . sy of the bundle LD = L9, whose characteristic function is Tf (LD, r) = qT f (L. r). This divisor clearly satisfies Conditions A and B; therefore relation (10) is true for it: Q

Ric 1I = c(LD) + c(KM) - >2 dd` ln(ln Ilsj II2)2. j=1

86

11. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

Taking the pullback by the mapping f of the terms in this relation and then multiplying both sides by wo -' , we average logarithmically over the ball Br to obtain

-

Tf(r) = qTf(L, r) +Tfr(KM, r)

dtJ

dd`ln(lnllsf°fll2)2Aw0-i=1

fo t B` As above, we assume all the moduli llsjll are small enough that ln(ln llsj 0 (112)2 = 2Inln(1/ Ils.i ° f 112).

Then by Lemma 2 of §1 the last equality can be rewritten in the form qTf (L, r) + Tf (KM, r) - t f (r)

_

JST In In

or.

(27)

li sj ° f112

For the same reason all the functions in the integrand here are positive, so this proves the left-hand inequality in (26). To prove the right-hand inequality we use the convexity property of the logarithm (19) and obtain from (27)

gTf(L,r)+Tf(Km,r) - Tf(r) < ln 7=1

In In

llsf ol112a

11

q

_>Inmf(D3,r)+gln2, j=1

where m f(Dj, r) is the proximity function of the divisor D, (see formula (4) in §4). Applying the first main theorem to the bundle L and using the nonnegativity of the counting functions N1(Dj, r), we obtain m f (D3 , r) < Tf(L, r) + 0(1). Then the right side of the last inequality can be estimated from above by the quantity

gln(Tf(L,r)+O(1))+gln2 = glnTf(L,r)+O(1). This lemma implies the following asymptotic formula for the singular characteristic function:

Tf(r) = gTf(L,r)+Tf(Km,r) +O(InTf(L,r)).

(28)

Now it is not difficult to prove the second main theorem in a formulation closer to the classical one. THEOREM 4. Let L be a positive line bundle on an n-dimensional compact manifold M, and let D1,. .. , DQ be divisors of holomorphic sections of it satisfying conditions (a) and (b). Then for any nondegenerate holomorphic

§5. SECOND MAIN THEOREM

87

mapping f : C" -+ M

gTf(L,r) +Tf(KM,r) + N(Sf, r) _

Nf(Dj, r) + R(r),

(29)

j=1

where KM is the canonical bundle on M, Sf is the divisor of stationarity of f and the remainder term R admits the following estimate: for any e > 0 there exist a S = S(e) --+ 0 as a , 0 and a set E of finite 6-measure of the numbers r such that R(r) < e In r + O(lnTf(L, r)) outside E. (30)

4 The divisor D = >i Dj of a holomorphic section of the positive bundle Lq satisfies Conditions A and B of Theorem 2. By Lemma 2. whose conditions

are also clearly satisfied, (28) can be applied. Thus in (14) we can replace Tf(r) by gTf(L, r) + Tf(KM, r) with an error of order O(ln T f(L, r)). If we combine this error with the remainder R(r), replacing it by R(r) = R(r) + O(ln T f (L, r)),

(31)

and if we observe that Nf(D, r) Nf(Dj,r), then (14) can be rewritten in form (29). In order to estimate R(r), we use the compactness of M. The forms c(L) and c(KM) have continuous coefficients, and the first one is positive by hypothesis. Therefore, there exists a constant 'y > 0 such that everywhere on M we have --yc(L) < c(KM) < -ye(L). From this in the usual way we obtain

'1Tf(L,r) < Tf(KM,r) < 'yTf(L,r). From this and the inequality T f (r) < qTf (L, r) + T f (KM, r), which appears in Lemma 2, we conclude that

0 < Tf(r) < (q + y)Tf(L, r).

(32)

This inequality lets us replace the estimate for f? obtained in Theorem 3 by R(r) < e In r + O(ln Tf (L, r)) outside E, Now (31) reduces to (30).

In the case where M equals P" and L is the hyperplane bundle, by (6) we have Tf(KM, r) = -(n + 1)Tf(r). For any q > n + 1 hyperplanes Hj in general position, (29) now takes the form q

(q - n - 1)Tf(r) + N(Sf, r) = E Nf(Hj, r) + R(r).

(33)

j=1

For n = 1 this is the classical second main theorem (see (1)) with a different interpretation of the divisor of stationarity.

88

H. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

REMARK 1. As in the classical case, for mappings of finite order the remainder admits the estimate

R(r) = O(ln r),

(34)

which is true everywhere, not only outside an exceptional set. In fact, for a mapping f of finite order p we have by definition T f(L, r) = O(rP), so the error when Tf is replaced by qTf (L) + Tf (KM) is of order In r by Lemma 2, and all that remains is to estimate R. For this we first choose b > p - 1 in Lemma 1; then we do what was done in the proof of Theorem 3 to obtain instead of (25) an estimate that is true outside some open set E of finite b-measure:()

R(r) < c.In r + O(InTf(r)) = O(ln r),

(35)

since according to (32), T f(r) = O(rP) in our case. Now let r E E and ai < r < b;, where (aj, b3) is one of the intervals which comprise E. By (14), taking account of the growth of the functions Tf and N,

R(r) = T1(r) + N(Sf, r) - Nf(D. r) < T f(bf) + N(Sf, b;) -1V f(D, af)

(36)

= R(bf) +-'Vf (D, b;) -1'Vf (Dj ai). Since b;

E, by (35) R(bf) = O(lnbf), and

)

)

N D, ( b. - 1Vt(D, a j)

f

i' n f (D, t) dt < t

aj

n f (D, t)

fE

t

dt

( 37 )

(for the definition of n f, see §2). Further, by the Nevanlinna inequality Nf(D, r) < Tf(L9, r) + 0(1), we have Nf(D, r) = O(rP), and thus, by the growth property of n f, 2t

of (D, t) <

If

n f(D. r)

l n.2

dr <

- in 2 Vf (D, 20 <_ city

with some constant cl > 0. Substituting this into (37) and using the facts that the set E has finite 6-measure and that p - 1 < 6, we obtain

N1(D,bf)-Nf(D,a.)
Ltt =0(1);

Now f rom (36) it is evident that R.(r) = O(/In b3 ). It remains to observe that In

=lnr+ Jb, d <_1nr+J tbdt.=Inr+O(1),

I

t

E

(7)In contrast to the proof of Theorem 3. the number 6 is not selected but is given, so that the coefficient of In r is c > 0 instead of e.

§6. PICARD'S THEOREM. DEFECT RELATION

89

and we obtain the required estimate R(r) = O(Inr). I REMARK 2. In our proof of the second main theorem, condition (b) on the divisor Ei D, was used in an essential way; this condition says that qc(L) + c(KM) = c(LD) + c(KM) > 0. Precisely this condition permits the construction of the singular volume form which lies at the basis of the proof. In the case of the hyperplane bundle

on Pl this condition reduces to the inequality q > n + 1. But from (33) it is clear that if this is not fulfilled, then the assertion of the theorem is not violated but rather becomes trivial. We observe that in the general case of a positive line bundle L --4M the assertion of the theorem remains true if only condition (a) is fulfilled.

In fact one can add to D a divisor D' consisting of the intersection of M with sufficiently many hyperplanes of the space p N which contains M so that the divisor A = D + D' satisfies both conditions (a) and (b). Then by Theorem 4 T f(LA, r) + T f(KM, r) + N(Sf, r) = Nj (A, r) + R(r)

(38)

with the estimate R(r) _< elnr + O(lnTf(LA, r)) outside a set of finite 6-measure. But by the first main theorem for the bundle LA

T f(LA, r) - N1 (A, r) = m f(A, r) + 0(1), Since m f (A, r) = m f (D, r) + m f (D', r) > m f (D, r) by (7) of §4, and since m f (D', r) can be taken to be nonnegative, we have

T f(LA, r) - N1(, r) > m f(D, r) + 0(1) = Tf(LD, r) - N f(D, r) + 0(1) (here we have used the first main theorem for LD). Therefore in (38) one can replace Tf(LA, r) and Nf(A, r), respectively, by T f(LD, r) and N f(D, r), combining the nonpositive term which arises on

the right side with the remainder R(r). It only remains to show that in the estimate of this term Tf(LA,r) can be replaced by Tf(LD,r). But since the manifold M is compact and the Chern forms c(LD) and c(LA) are positive and smooth, there exists a constant A > 0 such that c(LA) < Ac(LD). From this and from the definition of the characteristic function it follows that Tf(LA,r) < ATf(LD,r)+O(1), from which it is clear that the quantity O(InTf(La,r) is also O(In Tf (LD, r)) as well. I

§6. Picard's theorem. Defect relation 9. Picard'a theorem. We have already mentioned that the second main theorem of value distribution theory leads to results of the type of Picard's theorem while the first main theorem leads only to results like Sokhotskii's theorem. As the first corollary of Picard type, we mention this result:

90

II. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

THEOREM 1. On an n-dimensional compact complex manifold M let a divisor D be given which is the union of q manifolds Dj in general position and let LD be the bundle of this divisor. If the Chern form of LD satisfies the condition (1) c(LD) + C(KM) > 0, where KM is the canonical bundle, then any holomorphic mapping f : C" M \ D is degenerate.

4 If f is nondegenerate, then under these hypotheses the second main theorem in the first formulation is true. Thus t f (r) + N(Sf, r) = Nf (D, r) + R(r), where, by (25) of §5, for any e > 0

R(r) < elnr+O(InTf(r)) outside a set E C RT of finite 6-measure. But N(S f, r) > 0 and by hypothesis Nf(D, r) = 0, since f takes on no values in the divisor D. Therefore, outside

E

Tf(r) < rlnr + o(T f(r)). Since a is arbitrary, such an inequality is impossible. In particular, for the hyperplane bundle on Pn condition (1) is satisfied if the divisor D consists of n + 2 hyperplanes in general position, since, as we saw in §5, in this case c(LD) = (n + 2)w and c(KM) = -(n + 1)w. Therefore this is true: COROLLARY. A holomorphic mapping f : Cn -+ Pn which takes on no values in a set of (n + 2) hyperplanes in general position is degenerate.

For n = 1 the role of the complex hyperplanes is played by the points of P1 = C. and the corollary asserts that meromorphic functions which omit three distinct values degenerate to constants. This is the small Picard theorem, so Theorem 1 can be considered a higher-dimensional generalization of this theorem.

10. Examples. We now cite some examples which indicate the precision of the result just obtained. be the union of q EXAMPLE 1. Let M = Pn and let the divisor D hyperplanes in general position. As we just pointed out, here c(LD)+c(KM) = (q - n -1)w and condition (1) reduces to the inequality q > n+ 1. For q = n+ 1 the assertion of Theorem 1 can be untrue: the nondegenerate holomorphic mapping f : Cn Pn defined in the homogeneous coordinates [wo, ... , wn] by

f(z) = [1,ez1,...,ez°], where z = (zl,

.

.

.

,

zn), does not take on values in the divisor

=0}, 1

(2)

§6. PICARD'S THEOREM. DEFECT RELATION

91

consisting of (n+1) hyperplanes in general position. This example shows that condition (1) is essential. 1 EXAMPLE 2 (B. SHIFFMAN [31). Let M = P2 and D = { [w] E P2 wo - Wi w2-1 = 0 }. For any q > 1 there exists a nondegenerate holomorphic mapping

f (z) = [1 e+ ezs ez, ]

(3)

from C2 to p2 such that 1(C2) does not intersect the divisor D (in fact wo - wlw2-1 = 1 - (1 + ezj+(Q-1)z,) # 0). The divisor D is equivalent to the divisor consisting of the line at infinity H. with multiplicity q since the ratio (wo - wlw2-1)/wo is a meromorphic function on p2. Consequently, C(LD) = qW.

For q < 3 condition (1) does not hold, while for q > 3 critical points appear in the divisor D and it ceases to be a manifold. For example, in local coordinates x = wo/wl, y = w2/w1 on the domain U1 = { [w] E P2 : w1 54 01 the equation for D takes the form V(x, y) = x4 - y9 -1 = 0 and the gradient V

3 at the point x = y = 0 (in particular for q = 3 the divisor D is the semicubical parabola x3 = y2). Thus for q > 3 this example shows that the condition that the divisor D consist of manifolds is essential. I EXAMPLE 3 (M. GREEN [3]). As before, let M = P2 and let the divisor

D={wo=0}U{wl =0}U{(wo-wl)w2+(wo+w1)2 =0} be the union of three manifolds Dj-a curve of second order without critical points and two complex lines which are not tangent to the curve. The nondegenerate mapping f (z)

[1,ezl,ezi+3+4 1- e(e" -1)Z2 1

ez'-1

( 4)

J

-

from C2 into p2 is holomorphic (since 1 5V' -1)z2 = -(ezl - 1)x22, (ezl - 1)222 is divisible by ezi - 1) and takes no values in D (since wo

0, w1 A 0, and (WO - w1)w2 + (wo + wl)2 = 4e(e ' -1)z2 # 0).

Here the divisor D is equivalent to the divisor consisting of the line at infinity with multiplicity 4, so c(LD) + c(KM) = w > 0 and condition (1) is satisfied. The hypothesis of Theorem 1 is violated because the manifolds D3

do not intersect in general position. They intersect in three points; two of these, [1, 0, -1] and [0, 1, 1], are the intersection points of two manifolds (as they should be for general position in P2), but the third point [0,0,1] is a triple intersection point (Figure 5), which violates the requirement of general Position. Thus this condition also is essential. 1 REMARK. The mapping of (4) is of infinite order and a nondegenerate mapping f : C2 -+ P2 \ D does not exist. In fact let such a mapping have the form f = [l, fl, f2]. Since fl 0 0 and is of finite order, then fl = e1', where P

92

II. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

FIGURE 5

is a polynomial in z = (z1. z2). From the condition (wo-wl)w2+(wo+wl)2 # 0 it follows that (1 - ep) f2 + (1 + e')2 = eQ, where Q is also a polynomial since f2 is of finite order. Hence

cQ-(1+e")2 r= 2 1-ep , and since f2 is entire, eQ = 4 whenever a j' = 1. Thus, on the level curves P = 2kzri (k = 0, ±1, ...) the polynomial Q takes on constant values. This implies

that on these curves the Jacobian 3(P, Q)/a(zl, z2) = 0. This Jacobian is a polynomial and equals zero on an infinite set of complex curves; consequently,

it is identically equal to 0. But then also the Jacobian

a(fi, f2) = a(fi, f2) a(P, Q) = a(P, Q) a(zl, z2) - 0, a(zi, z2) i.e., the mapping f is degenerate. Thus, the divisor D from Example 3 has the interesting property that a holomorphic mapping from C2 into the complement p2 \D is either degenerate or has infinite order. There are not yet any general results of this nature. I

11. Defect relation. Let a holomorphic mapping f be given from C' to an n-dimensional compact complex manifold M, on which is defined a positive Hermitian line bundle L. According to the first main theorem applied to the divisor D of any holomorphic section of this bundle, the sum of d;c counting

function Nf(D,r) and the proximity function mf(D,r) is the same up to the addition of a bounded term, and is equal to the characteristic function Tf (D, r). As we shall soon see, for the "majority" of divisors the second term of this sum is small in comparison with the first, i.e., the quantity

bf(D)

N D, r T1( ,r)) = 1 r-0 Tf(L,r) - rli .

(5)

§6. PICARD'S THEOREM. DEFECT RELATION

93

which is called the defect of the divisor D under the mapping f, is equal to 0 are called zero. In accordance with this, divisors D for which b f(D) exceptional divisors.

From the Nevanlinna inequality Nf (D, r) < Tf (L, r) + 0(1) it follows that

for all these divisors bf(D) > 0, and from the positivity of Nf and Tf that

bf(D) < 1. If the image f(C) does not meet the divisor D at all, then Nf(D, r) = 0, and thus the defect of such a divisor is the maximum value 1. Further, for ample bundles the averaged defect turns out to be zero: THEOREM 2. If L M is an ample bundle and f : Cn -> M is a nondegenerate holomorphic mapping, then

JPN bf(D) du(D) = 0,

(6)

where PN is the projective space of the divisors of the holomorphic sections of

L and p is the invariant measure on this space with p(PN) = 1. 4 By Crofton's formula (23) in §3, T f (L, r) =

Nf ( D, r) dp(D)

or

f

N

C1

TfI (L 'r)

0. I du(D) =

From this, by Fatou's lemma on passing to the limit inside an integral sign, we have

fP r

bf(D)dp(D) < rhm fp.

(1

7,f ( D'r) ) dp(D)

and by the nonnegativity of the defect we obtain (6). From this it follows that for ample bundles the exceptional divisors (for which bf(D) > 0) form a set of measure 0. Consequently there are relatively few of these divisors and the intersection of each of them with the image f (Cn) is less than usual. This result is a strengthening of Sokhotskii's theorem (Theorem 4 in §4), since the divisors which do not intersect f (Cn) have maximal defect and thus are exceptional. The proof of Theorem 2 only uses the first main theorem and this leads to a result of the type of Sokhotskii's theorem. In the proof of the following theorem the second main theorem will be used and this will give a stronger result of the Picard type.

THEOREM 3 (defect relation). Let there be given a nondegenerate holomorphic mapping f : C" M into an n-dimensional compact complex mani-

fold M, and let L

M be a positive Hermitian bundle. Then for any set of

H. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

94

divisors D) of holomorphic sections of L, where the D, are manifolds intersecting in general position, q

Ebf(D,)+Of 0},

(7)

i=1

where Of = limr N(S f, r)/T f (L, r) is the index of stationarity of f . t Let us denote the right-hand side of (7) by Ao; then Ac(L) + c(KM) > 0 for A > Ao (we have by hypothesis that c(L) > 0). Multiplying this by the form wo-1 and averaging logarithmically over the ball Br, we find that the characteristic functions of the bundles L and KM satisfy the inequality

ATf(L,r) +Tf(KM,r) > c

(8)

with some constant c > 0. Further, since some power of a positive bundle is ample, by Theorem 2 there exist sufficiently many divisors of holomorphic sections of L with defect zero. Adding these to D = Ei Df, we increase the number q without changing the value of the sum on the left side of (7); as before it can be assumed that all the Dj are manifolds in general position. Thus without loss of generality we can suppose that qc(L) + c(LM) > 0. Then the hypotheses of the second main theorem in the classical formulation are fulfilled (Theorem 4 §3). By this theorem, outside some set E of finite b-measure

gTf(L, r) - Nf(D, r) < -Tf(KM, r) - N(S f, r) + E In r + O(lnT f(L, r)). (9) By the definition of the defect,

bf(D;)=q- rm q

oo

i=1

Nf (D, r)

Tf(L,r)

_-lim qTf (L, r) - Nf (D, r) Tf(L,r)

Since the set E is of finite 6-measure, it cannot contain any ray (ro, oo), so there exists a sequence of numbers rk -+ oo with rk ¢ E. Inequality (9) can be applied to this sequence to obtain that 9

E bf (Di) < lim

j=1

(since Tf(L,r)

- r--oo

Tf (Km, r) Tf(L,r)

N(Sf , r)

r-.oo Tf(L,r)

oo, we have

lim

r-oo

O(lnTf(L,r)) =0). Tf(L,r)

It remains to observe that by (8)

Tf(KM,r) < A Tf(L,r)

- Tf(L,r) c

+ E lim

In r

r-oo Tf(L,r)

§6. PICARD'S THEOREM. DEFECT RELATION

95

and hence the first term on the right in (10) does not exceed A. The second

term is equal to -Of. The third is always finite (it is different from zero only for rational mappings), and it may be discarded since e is arbitrary. We conclude from this that 9

1: S(D.i)+Of
In particular, for the hyperplane bundle on P" the Chern form c(L) = w and c(KpN) = -(n + 1)w, where w is the Fubini-Study form (see §5), so the right-hand side of (7) equals n + 1 and we obtain a corollary (discarding from the left-hand side the nonnegative term Of): COROLLARY 1. For any nondegenerate holomorphic mapping f : Cn

Pn

and any set of hyperplanes H; in general position, the sum of the defects satisfies a

Ebf(Hf)
The little Picard theorem follows from this inequality as a special case: any meromorphic function which omits three distinct values in C is constant. COROLLARY 2. Let D be a divisor in Pn which is an algebraic manifold (without singularities) of degree q. Then for any nondegenerate holomorphic mapping f : C'" -+ Pn, the defect of the divisor D satisfies

bf(D) < n 9

(12)

4 For the bundle L of D we clearly have c(L) = qw, and for the canonical bundle c(K) = -(n + 1)w as before. Therefore, the right side of (7) is equal to (n + 1)/q. If the degree of the divisor q > n + 1, then from (12) it is evident that. its defect bf(D) < 1, and the nondegenerate mapping f cannot omit such a divisor. The requirement that D be nonsingular is essential: Example 2 in subsection 10 shows that a nondegenerate mapping can omit a divisor of arbitrarily high degree if the divisor has singularities.

11. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

96

COROLLARY 3. Under the hypotheses of Theorem 3, any set of exceptional divisors in general position is at most countable.

Let E be such a set and let A0 be the right side of (7) with k = 1, 2, .... From (7) it follows that the number of divisors D E E for which bf (D) > 1/k is no greater than kA0 and so is finite. From this follows the countability of E.

12. Example. We need a formula relating the characteristic function of M and that of its restriction to a complex a holomorphic mapping f : C" plane P C C' passing through the origin. If dp is the measure on the Grassmann manifold Gk of such planes of codimension k which is invariant under the unitary transformations of C" an which satisfies 1 (G) = 1, then

Tf(L,r)

(13)

fG'

kk

where fp = f lP is the restriction of f to the plane P E G. An analogous formula is valid for the counting functions of the divisors D of holomorphic section of the bundle L:

Nf(D,r)= fGn NVf,(D,r)dp(P).

(14)

k

T he derivation of these formulas is based on the equality of currents

k= J Pdp(P),

(15)

WO

ck

which is proved exactly as Lemma 2 in §3, so we will not repeat this proof. By the definition of the characteristic function for a fixed plane P E Gk,

r dt fil(ch) A wo -k-', t ,nP where ch, is the Chem form of the bundle L. By Fubini's theorem

Tf(L, r) =

f Tf, (L, r) dµ(P) = k

f f

f T dt f t

0

BsnP

f

Gk

fil(ch) dµ(P)

A wa -k-1;

representing fil(ch) = f *(ch)I P as the product of currents fil(ch) A P and using (15), we rewrite the last integral in the form

fr t f dt

t

and thus prove (13). Relation (14) is proved in an analogous manner. EXAMPLE (SHIFFMAN [21). Let us consider the mapping f: C2 --+ P2 defined in homogeneous coordinates on p2 by the formula f(z) = 11,ez',ell.

(16)

q6. PICARD'S THEOREM. DEFECT RELATION

97

We fix the complex line 1: z = AS where A E C2 and S E C, and consider the restriction of the mapping (16) to it, i.e., the holomorphic curve (17)

ft (S) = By Ahlfors' formula (19) of §2, its characteristic function

Tf,(r) = r (IA1l + IA21 +

IAl - A21)

+O(1).

(18)

The characteristic function of the mapping f can now be found by (13): Tf (r) =

JP1

(19)

Tf, (r) dµ(1),

where P1 is the set of all lines l and dµ is the normalized invariant measure on this set. Without loss of generality we will assume that JAI = 1, and we will identify dµ with the (normalized) Euclidean volume element on the sphere S1 C C2, which is clearly defined by the Poincare form a = dr In IAl2 A

dd`lnlll2 (see (31) in §1). Then from (19) and (18) we find that

rf

IA1 - A21)a + O(1).

(IA1I + IA21 +

(20)

Tf (r) = , To calculate this integral we introduce on S1 the real parameter 0 E (0, ir/2) and T1, r2 E (0, 21r) by the formulas A I = sin 0 e2T', A2 = cos 0 e2T2. Then

a= 2 r2 sin 0 cos 0 dO A dr1 A dr2 and

f

2"

2

/

lalla =

f71/2

r2,r dr2

o

J0

dr1

sine

ocos0d0 =

The integral of IA21 has the same value. To compute the integral of JAI - A21, using the invariance of a under unitary transformations, we make the change of variables Al - )12 = fµ1, Al + A2 = fµ2, so that

r /

Js,

rf /

rf

IA1- A2 10' = 2,, J lµl la = 37 s,

Then from (20) we get

Tf(r) =

2

0- r + O(1).

(21)

3 The divisors of the hyperplane bundle on p2 are the complex lines with homogeneous equations aowo + a1w1 + a2w2 = 0. By (20) of §2 the only ones which can have defects are those which have zero coefficients in their equations. There exist three continuous series of divisors with one coefficient zero:

Do = {wo + awl = 0}, DQ = {w2 + awo = 0},

DQ = {w1 + awe = 01,

a c C\{0},

98

Ii. THE MAIN THEOREMS OF VALUE DISTRIBUTION THEORY

and three divisors with two coefficients zero: Do = { wf = 0 }, j = 0, 1, 2. For the restriction fl to the complex line 1: z = ))5, by (20) of §2 for a # 0,(8) the counting functions are

Nf,(D°,r) = Nf,(DQ,r)

'A' 1

r + 0(l),

JAI - A2Jr+0(1),

Nf(Da,r)= I\2Ir+0(1). Then by (14), proceeding as above, we find that Nf (D r) = Nf (Da, r) = 2r +0(1)7

Nf(DQ, r) =

V'

+ 0(1).

By the definition (5) the defects of these divisors are therefore given by

bf(DO)=of(D2)=V-1, bf(Da)=3-2f

(a#0).

(23)

The divisors Do (j = 0, 1, 2) do not intersect the image f (C2); consequently, they have maximal defect: bf(1 0) = 1. According to the defect relation (11), the sum of the defects of complex lines in general position for this mapping is no greater than 3. This value is attained for the set of the Dv (j = 0,1, 2); other sets of lines in general position give a smaller total defect. By giving up the requirement of general position, a total defect larger than 3 can be attained; this shows that this condition is essential. We also observe that for n = 1 the condition of general position

is not required, and the Df can be arbitrary distinct points. From this for n = 1 it follows easily that the set of exceptional values of a meromorphic function is at most countable. From the example just considered, it is clear that for n > 1 such an assertion is not true: the set of exceptional lines for the mapping (16) is uncountable (although certainly any set of exceptional lines in general position is countable by Corollary 3).

1

(8)In the case under consideration here, the polygons in (20) of §2 degenerate into

segments, and the perimeter P' is equal to twice the length of a segment.

CHAPTER III

Holomorphic Curves In this chapter we consider the foundations of the theory of holomorphic curves in complex projective space. As was explained in the Introduction, this was historically the first theme in the multidimensional theory of value distribution. The initial period of its development is reflected in the book of H. and J. Weyl [1] and in the classical paper of Ahlfors [2]. Twenty years later Wu gave a modernized exposition of the theory in his monograph [3].

The exposition in this chapter mostly follows the recent paper of Cowen and Griffiths [1], which contains yet another interpretation of the work of Ahlfors. Since the first main theorem of value distribution theory in the preceding chapter was considered in sufficient generality to include the case of curves, we will concentrate our attention on the second main theorem, which has only been proved so far for the case of mappings which preserve dimension.

§7. Associated curves The passage to the associated curves is one of the leading ideas in the Ahlfors approach. We will define this concept after first pausing to examine the concept of a holomorphic curve itself and introducing the necessary algebraic apparatus. 1. Holomorphic curves and their representation. Let us consider a holomorphic mapping of the disk BR = { z E C : IzI < R } to the space Cn+1 (1)

where the vector f $ 0. Outside the set E

z E BR : f (z) = 0 } it defines

a holomorphic mapping

f = [fo, ... , fn] : BR\E -+ Pn,

(2)

where [fo, ... , fn] are the homogeneous coordinates of the point f (z) E P. We denote by p: Cn+i \ {0} Pn the standard mapping which assigns to a 99

100

III. HOLOMORPHIC CURVES

point to = (w0,...,

E Cn+1 \ {0} the line 1,,, C Cr+' passing through 0 and w viewed as a point [w] E Pn. Then the identity p o f (z) = f (z) is true on BR \ E. The mapping f can be extended holomorphically to the set E. Indeed, by the uniqueness theorem E is a discrete set, and for every point a E E there is a punctured neighborhood U' = {0 < Iz - al < r } C BR \ E. Let v be the smallest of the orders of the zeros of the functions fo, ... , fn at a. Without loss of generality it can be assumed that this is the order of fo and that fo(z) 0 0. Then f3(z) _ (z - a)"gj (z), j = 0,.. . , n, where the gj are holomorphic in

U = { Iz - al < r } and go is, in addition, different from zero, and f (z) = Pn is (z - a)" [9o (z), - - . , 9 n (z)] in P. The mapping 9 = [go, ... , 9n ]: U holomorphic since g(U) belongs to the domain Uo = { [w] E Pn : wo 01 of the standard covering of Pn, and in the local coordinates on this domain the mapping (gi/go, ... , gn/9o) is holomorphic. Clearly g(z) = j (z) for z E U', so g gives a holomorphic extension of f to the point a.

After being extended in this way to every point a E E, the mapping f : BR _ Pn is called a holomorphic curve on BR. Any holomorphic mapping f : BR -4 Cn+1 such that p of (z) = f (z) in BR is called a representation of the curve; if in addition f (z) # 0, i.e, f (BR) C Cn+1 \ {0}, then the representation is said to be reduced. Clearly, for every holomorphic curve a reduced representation exists and is defined up to a holomorphic function which does not vanish on BR. In fact the set E _ { z E BR : f (z) = 0 } is at most countable; and if vk is the smallest of the orders of the zeros of the f j (j = 0, . . . , n) at the point ak c E, then by Weierstrass's theorem a holomorphic function g can be constructed in the disk with zeros of orders vk at all the points ak E E

but with no other zeros. Dividing the vector f by this function, we obtain a reduced representation of the curve f. Observe that instead of the holomorphic mapping in (1) we could take a collection f = (fo,... , f,) of functions which are meromorphic in BR. Multiplying the vector f by an appropriate function g which is holomorphic in BR, we obtain another representation of the curve f : BR _ Pn corresponding to f ; this representation consists only of holomorphic functions (the existence of the function g is guaranteed by the same theorem of Weierstrass on the construction of holomorphic functions with given zeros). Thus for holomorphic curves one can also choose meromorphic representations along with holomorphic ones. Moreover, in either case (holomorphic or meromorphic representations), in the local charts U, [w] E Pn : wj # 0 } of the standard covering of P", the curve f is globally represented by the vector of meromorphic functions (fo/fj, ..., fj_1/fJ, ff+l/fj,...,fn/f,). Therefore, holomorphic curves are often called meromorphic curves. However, such curves realize holomorphic mappings between the complex manifolds BR and P' (in appropriate local coordinates); therefore, we will use the first term.

§7. ASSOCIATED CURVES

101

2. Grassmann algebra. To define the associated curves one needs the concept of a multivector on Cn+1, so we will pause briefly to introduce this concept. After choosing some basis eo...... en in Cn+1, we form 2n+1 formal products A ejk , where 0 < jo < .. < jk < n and k = 0, 1, ... , n, where the e j- A empty product (for k = 0) is denoted by e. Then we consider the vector space An+1 over C whose basis consists of these products. The elements of the space An+1 consequently have the form

aj°j, e A e'- + ... + ao...neo A ... A en.

aj0e?0 +

ae + jo

(3)

jo <ji

where aj0... jk E C. Next we define the product of an arbitrary set of basis vectors ej. It is defined to be equal to zero if any two of the factors are the same, and to sgn a ej0 A .

where 0 < jo <

- A e2^

if the factors are in the order a(jo ), ... a(jk ),

< jk < n and a is a permutation of the set (jo,... , jk )

whose sign is sgn a. Thus this product is skew-symmetric with respect to the indices, which justifies the use of the symbol of the exterior product. Finally, we define the product of basis vectors of An+' by the rule (eio A ... A eik) A (ejo A ... A ej`) = ei0 A ... A eZ A eJ0 A ... A ej, and agree to multiply out elements (3) in An+' by multiplying the individual terms and adding up the resulting products. Thus the space An+1 has been endowed with the structure of an algebra with identity e. This algebra is called the Grassmann algebra, and its elements are called multivectors. It can be proved(') that the operations in the Grassmann algebra are independent of the choice of basis in Cn+1 As a first application of the Grassmann algebra, we prove the following simple assertion: THEOREM 1. The vectors z°, ... , zk E Cn+1 are linearly dependent if and only if

z0A...Azk = 0.

(4)

4 If these vectors are linearly dependent, then one of them can be expressed as a linear combination of the others; let zk = 0-1 aj zj. Then the product (4) is equal to k-1 k-1

zoA...Azk-1rajz3 =EajzoA...Azk-1 Azj =0,

j[=Jo j=a since in every term there is the exterior product of identical vectors. If on the o t h e r hand the vectors z 0 . . . . . zk are linearly independent, then they can be

Completed to a basis of Cn+' by vectors zk+' .... z'. Then zo A ... A zn # 0 ('since this product is a basis vector of An+1), and thus a fortiori zo A ... Azk 0.

No.

(')See, for example, van der Waerden [1j, §93.4.

102

111. HOLOMORPHIC CURVES

The linear span of the products of exactly k basis vectors in C'z+1 forms a linear subspace of A"`+1 which we denote by Ak+1; its elements are called k-vectors. In accordance with (3) the space Ar}1 is the direct sum Ao+1 + + An+i, where Ao+1 = C and Al+1 = C+1 Those k-vectors Al+1 + which are products of vectors in Cfz+1 are called decomposable, and any kvector is a linear combination of decomposable k-vectors. In particular, if e° = (1, 0, ... , 0), ..., e' = (0, ... , 0, 1) is the standard basis of C"+1, then any (k + 1)-vector

A = E ajo...;k e7o A ... A e.ik,

(5)

jo<...<Jk

so that the space Ak+i can be naturally identified with CN, where N = (k+1) is the binomial coefficient. Let us note still another correspondence which exists between the decomposable multivectors over Cs}1 and the subspaces of C'a+1, i.e, the complex planes passing through the origin. Every decomposable (k + 1)-vector A = a° A - A ak which is different from zero uniquely defines the (k + 1)dimensional plane 11 spanned by the vectors a°, ... , ak (which are linearly independent by Theorem 1); the equation of this plane (by the same Theorem 1) is A A z = 0.

(6)

Conversely, if n is some (k+l)-dimensional subspace of C"+1 and a°,

is a basis of it, then we set A = a° A

,

ak

A ak. Then (6) by Theorem 1 is equivalent to the condition that the vector z is a linear combination of the vectors a0,. .. , ak, that is, it belongs to U. If b°.... , bk is another basis of n, then b3 = Eo ai jai (j = 0.... , k), and then clearly b° A .. A bk = -

A ak, so that the (k + 1)-vector corresponding to II is defined up to a (nonzero) complex multiple. In the space CN which is identified with Ak+i, to the plane II there consequently corresponds a punctured complex line passing through the origin. It is often convenient to consider the collection of projective k-dimensional subspaces of Pn, i.e., the Grassmann manifold G(n, k) (see §3, where it is det(ai3 )a° A

denoted by Gn_k). Using the standard mapping p: C"+1 \ {0} -+ P", it is identified with the set of (k + 1)-dimensional planes in C`+1 passing through the origin. The correspondence described above allows one to associate to a point A E G(n, k) a decomposable (k + 1)-vector over C"+1 a° A ... A ak =

F,

ado---,Jk ei°

A ... A e&k

(7)

.1O <... <Jk

where the coefficients a.,b ...,k are defined up to a common (nonzero) factor.

§7. ASSOCIATED CURVES

103

These are called the homogeneous Plucker coordinates of the point A; if ao = ( a , . .. , a;,), then clearly 0

a3°

...

aj0...j,, = det

a

...

0 a3k

...

(8)

aA

In particular, for the bivector ao A a1 these coordinates area ° ai - aPl a , so the exterior product is a generalization of the vector product. The identification Ak+1 = CN+1 described above, where now N + 1 = (k+i), permits G(n, k) to be considered a submanifold of the projective space PN.

3. Associated curves. Returning to the meromorphic curve f : BR -+ pn,

we choose a representation f = (fo, . . . , fn): BR --' Cn+l, and for k = 0, 1, ... , n and for fixed z E BR we consider the decomposable (k + 1)-vector over Cn+1 (k) Fk (z) = f (z) A f'(z) A ... A f (z),

(9)

where f (j) _ (fo3), ... , fnj)) are the derivatives of f. By (8) we have

h. Fk(z) _

det 0<jo<...<jn
...

fj°(k)

... h" ... ... ... fj,(k)

e3o A ... A ejk.

(10)

Since the exterior product of collinear vectors is equal to zero, whenever the representative f is replaced by cd f , where ,p is a holomorphic function, the multivector Fk is multiplied by the scalar function Pk+l, i.e., Fk = p(Fk) is unchanged. Consequently, for fixed k, to the curve f there is associated a definite curve in the Grassmann manifold

Fk: BR->G(n,k) or, if it is more convenient, a curve in the space pN, where N = (k+i)

(11)

- I.

These curves are called the associated curves of f.

For k = 0 this is the curve itself: Fo = f. For k = 1 it is the curve p(f A f') corresponding to the tangent lines to f; for k = 2 it is the curve p(f A f' A f") corresponding to the osculating planes (see the schematic Figure 6), etc. For k = n the associated curve is trivial; by (10)

Fn(z) = W (fo, ... , fn)e0 A ... A en,

(12)

where W is the Wronskian of the functions A,-, fn. The associated curves permit one to determine the degree of degeneracy of a curve f . Namely, the following is true:

104

M. HOLOMORPHIC CURVES

FIGURE 6

THEOREM 2. The image f(BR) lies in a k-dimensional subspace of P" but does not lie in any (k - 1)-dimensional subspace if and only if for any representative f of this curve Fk(z) 0 0 but Fk+I(z) = 0.

The assertion of the theorem is equivalent to saying that f (BR) lies in a (k + 1)-dimensional subspace of C"+1 but does not lie in any k-dimensional subspace of it. Let this be true; without loss of generality we can suppose that f (BR) lies in the plane { wk+I = = w" = 0 } which we identify with the space Ck+1 of variables (wo, ... , wk). Then all the functions f k+1, ... , f" are identically equal to zero. From the expansion (10) written for k + 1 instead of k, it is clear that Fk+1(z) - 0, since the coefficients of this expansion are Wronskians W (f3o, , f?, ,,) of sets of k + 2 functions f3 in each of which there is a function identically equal to zero. But W (f 0i ... , fk) 0 0; because if this Wronskian were identically equal to zero, then by the holomorphy of the functions fo.... , fk it would follow that they were linearly dependent, i.e., that the image would lie in some hyperplane of But this is the same as a k-dimensional subspace of C"+1. Thus Fk(z) 0. Conversely, let Fk(z) $ 0 while Fk+1(z) - 0. Then one can find at least one coefficient in (10), say, W (f0.... , fk) $ 0 but with W (fo, .... fk, f3) 0 f o r all j = k + 1, ... , n. From this it follows that fo, . . . , fk are linearly independent, while the remaining functions are linear expressions in them; let f3(z) = Ek b3 f. (Z), for j = k + 1, ... , n. But this means that f (BR) lies in the (k + 1)-dimensional plane w3 = r_o j = k + 1, ... , n, passing through the origin of the space C"+1 That is, it lies in a (k + 1)-dimensional Ck+1.

subspace of C"+1. and by what was shown before, it does not lie in any k-dimensional subspace of Cn+t A curve f : BR - Pn is called nondegenerate if I (BR) does not lie in any

proper subspace of P". By Theorem 2 the curve f is nondegenerate if and P" only if F,, (z) A 0. By the same theorem a degenerate curve f: BR

f7. ASSOCIATED CURVES

105

is nondegenerate in some subspace pk C P" if and only if Fk (z) jt 0 but Fk+1(z) = 0. The degeneracy of f to a constant is the strongest form; the condition for this is Fj(z) = 0. To conclude this section, let us consider the points of stationarity of holomorphic curves and their associated curves. Let f : BR P' be a nondegenerate holomorphic curve; a point zo E BR is called a point of stationarity of the curve if the differential of f is zero at the point, where f is viewed as a mapping of complex manifolds. Without loss of generality we may assume that z o = 0, that f (zo) = [1, 0, ... , 0] and that the representation of f in a neighborhood of 0 has the form

zvn +...),

f(z) =

(13)

where 1 < v1 < ... < v,. Expressing fin local coordinates wl

wn

wo

WO

we see that zo = 0 is a point of stationarity if and only if vl > 0. The number µo = vl - 1 is called the index of stationarity of f at this point; it is clearly independent of the choice of representation but is determined by the curve itself.

Further, after performing an additional unitary transformation of Cn+1 if necessary (which corresponds to a projective motion), it can be assumed that

in (13) all the inequalities are strict: v1 < ... < v,,. Indeed, let us suppose for example that v1 = v2 < v3i choosing instead of (w1, w2), the coordinates

wi = (w1 +w2)/f, w2 = (wl - w2)/f, we find that vi = v1 but v2 > v2 in these coordinates. One must proceed analogously in the general case. This

remark will be needed when we compute the index of stationarity of the associated curves. We choose for the kth associated curve Fk: BR -F G(n, k), k = 1, ... , n - 1, the representation Fk = f A - - - A f('), where f locally has the form (13) with

v1 < ... < vn; then we write out the terms of the lowest degree in z: Fk(z) = A ... A ek + xv' +...+Yk -+vk+,

-(1+...+k) eo

A ... A ek-1 A ek+1 + .. .

(we omit the coefficients of these terms, only observing that they are nonzero). Introducing the notation

mk=v1+---+vk-(1+2+...+k) and representing Fk as a curve in local expansion in the form

(14)

where N + 1 = (k+i), we rewrite its

Fk(z) =zrnk(1+...,z1k+1-1k +...,...).

(15)

106

III. HOLOMORPHIC CURVES

Comparing this with (13), we see that the index of stationarity of the curve Fk at the point zo = 0 is Irk = Uk+1 - Vk - 1

(k = 1, ... , n - 1).

(16)

This index can be expressed also in terms of the degrees Mk of lower powers of z which appear in the expansion (15). Taking (14) into account, we conclude

that the index of stationarity of Fk is equal to the second difference of these degrees:

Ilk=mk+1 -2mk+mk_1

(k= 1,...,n- 1).

(17)

We observe that this formula remains true also for k = 0 if we formally set mo = m_1 = 0 (it then takes the form µo = m1 = v1 - 1). Above we noted that Fn = We°n Aen, where W is the Wronskian of the functions fo, . . . , fn. For a nondegenerate curve W 0, so it is natural to assume un = 0. In order to preserve (17) for k = n also, we set mn+1 = 2m,, - m_ 1 (or equivalently, Vn+1 =Un+1).

§8. Characteristic functions Here we introduce the characteristic functions of a holomorphic curve and its associated curves and obtain a relation among them. These functions are defined using forms which are the pullbacks under the mappings Fk: BR G(n, k) of the metric forms of the Grassmann manifolds for various k.

4. Metric forms. We define the inner product of multivectors, setting for decomposable (k + i)-vectors A = a° A A ak and B = b° A A bk

-

(a°, b°)

...

(ak, bo)

...

(A, B) = det

(a° bk) ,

(1)

(ak,bk)

where (a', bk) is the usual (Hermitian) inner product; we continue it by linearity to all of Ak+i . The modulus of a multivector is defined as usual: I Al = (A, A). In particular, for a decomposable multivector A = a° A . Aak it can be seen from (1) with B = A that IAI is the volume of the parallelepiped spanned by the vectors a°, ... , ak. It is not difficult to see that the inner product defined in this way on Ak+i is the same as the usual inner product of vectors in the space CN+1 in which, as we indicated in the previous section, Ak+i can be imbedded (it is sufficient to verify this using formula (1) for products of vectors in an orthonormal basis of Ak+i and then use linearity). Hence it follows that for the inner product of k-vectors the Bunyakovskii-Schwarz inequality holds:

i(A,B)I S IAIIBI

(2)

§8. CHARACTERISTIC FUNCTIONS

107

Such an inequality is also true for the exterior product of multivectors A E Ak+1 and B E Al+11: IA A BI < IAIIBI.

(3)

T o prove this we choose in C"+I an orthonormal basis a°, ... , e" and form the formal products ej = eto ®...®eik for arbitrary sets of indices I = (io, . . . , ik). We denote by Ak+i the vector space over C spanned by these products. Let ir:Ak+i -' Ak+i be the projection sending the element A = >ajej to the multivector A = > a jet, where e1 = eio A . A eik. In the space Ak+i an inner

product can be introduced by declaring the elements ej to be orthonormal and then using linearity. Then Iir(A)I < Iwith Al, equality for elements A = ajej, where I = (i0,.. . , ik) and io < < ik. Further, one can formally introduce the product A ® B of elements A E Ak+i and B E Ai+11. For this product, clearly ir(A (9 B) = ir(A) A ir(B) and IA ® BI = IAI IBI. Now let multivectors A E Ak+i and B E Ai+l1 be given; we choose A E it-1(A) and BE 7r-1(B)such that JAI JAI and IBI=IBI. Then IA A BI = 17r(A) A ir(A)I = 17r(A (&B)I

< IA ® Al = IAIIBI = IAIIBI

Inequality (3) is proved.

By means of the inner product we can define for (k + 1)-vectors Z = z° A

A zk and dZ = dz° A Wk -

A dzk the Fubini-Study Hermitian form

(Z, Z) (dZ, dZ) - (Z, dZ) (dZ, Z)

(Z Z)2

to which corresponds the differential form

Wk _ i f a1ZZI2

- aIZI ZI aIZI2 I = ddc In IZI2

(4)

which is the natural metric form on the Grassmann manifold G(n, k). For k = 0 we obtain the usual Fubini-Study form on P", which was denoted in Chapter I by w. As in the general theory we will be interested in the pullback of the form wk by the mapping Fk: BR - G(n, k) realized by the kth associated curve of a nondegenerate holomorphic curve f :

k=0,...,n, Ilk=FkWk=dd`1nIFk(z)I2, (5) where Fk = f A f' A . . . A f (k) and f is a reduced representation of f . If f is replaced by V f , where V is a holomorphic function without zeros, then I Fk 12 is multiplied by Icpl2k. Since dd` In Icp12k = 0, f1k does not change. Thus f1k does not depend on the choice of reduced representation of f and is determined by the curve itself.

III. HOLOMORPHIC CURVES

108

For k = n we have Fn = We° A

.

A en, where W is the Wronskian of the

functions fo,... , fn : W is a holomorphic function and is not identically zero by the nondegeneracy of j, so fln = dd° In I W 12 = 0. For k = 0, ... , n - 1 the form wk is positive by the Bunyakovskii-Schwarz inequality for multivectors (cf. §1). Therefore, its pullback 11k is a positive (1, 1)-form in all of BR

except for the points of stationarity of the associated curve Fk. Thus the forms f2k define in BR pseudometrics induced by the natural metrics of the Grassmannians G(n, k); these forms are called the metric forms. If we set

n-1

k=U

zdzAdz

(6)

then by what has been said, the coefficient hk(z) > 0, and it is 0 at the points of stationarity of Fk and only at them. We will call the hk the metric coefficients.

LEMMA. For k = 1, ... , n - 1 the metric coefficient hk = IFk-1I2IF'k+1j2/IF'ki4

(7)

-4 At the points of stationarity of Fk we will use the local coordinates described at the end of the preceding section. Since by (17) of §7 we have Mk-1 + mk+l - 2mk > 0 at these points, the term Izl appears on the right side of (7) with positive degree, i.e., this side is zero there, as is the left side. It remains to check (7) at the nonstationary points. Let us consider any such point and again suppose that the point is z = 0. We will show that instead of f one can choose a reduced representation + ak+lzk+l) f(z)

g(z) = (1 + a1z +

of the curve f such that the inner products (g(i)(0), g(k+1)(0)) = 0

for j = 0, ... , k.

(8)

In fact from Leibniz's formula for the differentiation of products g(i) (0)

i = vr=-O

( vj )

v!avf(? ' (0)

(ao = 1)

it is clear that condition (8) is fulfilled if (f(i)(o),g(k+1)(0))=0

forj=0,...,k,

and this, by Leibniz's formula again,. is a linear nonhomogeneous system in (kv 1) v!a unknowns (v = 1, ... , k + 1) with determinant det

(f, f) ... ... ... (f(k) f) ...

(f,f(k))

... (f(k),f(k))

= F'k(0)I2

§8. CHARACTERISTIC FUNCTIONS

109

by (1); it is not equal to zero since z = 0 is not a point of stationarity. Thus a choice of al, ... , ak+1 for which condition (8) is fulfilled is possible. Without loss of generality we will assume that (8) is satisfied for the representation f itself. Taking this into account, we obtain from (1) that at

z=0 (f A ... A f(k1' f A ... A f(k-1)

f(k+1))

= a, (9) since this inner product is expressed by a determinant with last column zero. In exactly the same way f(k-1)

if A ... A

A f(k+1) 12

(f, f) = det

A

(f,f(k-1))

(f(k-1), f) 0

lf(k-1f(k-1))

.. .

0

...

J

= I f(k+1)12 IFk_112

and analogously IFk+112 = I f(k+1)I2 1Fk12, so that

If A ... A

f(k-1)

A

f(k+1)12 = IFk_112

IFk+112/IFkI2.

(10)

On the other hand, from (5) and (6) it follows that

hk(0)=

a2ln IfA...A f(k)I2I azaz

Iz=o

a (f A ... A f(k), f A ... A f(k-1) A f(k+l)) IFkI2

09Z

Here we have used the usual rule for differentiation of products and the fact that 9f (j)/az = 0 by holomorphy (hence it is only necessary to differentiate the second argument in the Hermitian product with respect to z). We have also used the fact that the exterior product of identical vectors equals 0. By the same reasoning, and with (9) taken into account, we further obtain

hk(0) = If A ... A f(k-1) A f(k+1)12/IFkI2, and substituting (10) we obtain the required result. We observe that (7) remains true also for k = 0 if we formally set IF-11 = 1. Then condition (8) for f when k = 0 has the form (f(0), f`(0)) = 0; taking this into account, ho(0) =

a2 In If I2

8z8z

L=0

-

If'(0)12

If(0)12

_

IF112

IFol4

III. HOLOMORPHIC CURVES

110

5. Characteristic functions. A basic definition for the theory is made using the metric forms f1k. DEFINITION. The quantity

=rdt f t

11

Bt

where Bt is the disk { Isi < t }, is called the kth characteristic function of the holomorphic curve f : BR

P.

For k = 0 this quantity is the same as the characteristic function Tf(r) studied in Chapter I, since no = f (w), and we have m = 1. However, for k > 0 this is not the function T7 (r) introduced in §2 but is the characteristic function of the kth associated curve F: BR -> G(n, k). We observe also that

Tn (r) - 0, since 1l - 0. We wish to obtain a formula relating the characteristic function of the kth associated curve with its divisor of stationarity and with other characteristic functions, basing its derivation on the Poincare-Lelong formula for the functions Inhk(z). Above it was observed that at nonstationary points of Fk the function hk(z) > 0 and the form S1k defines a metric whose Ricci form, according to (6), equals Ric S1k = dd` In hk

(12)

(see §5). In a neighborhood of a point of stationarity zo, by the lemma proved above, hk(z) = Iz - zol2µk So(z), where (13) P k = mk-1 + mk+1 - 2mk and V is a smooth positive function (the set of such points is clearly discrete in BR). Therefore, the form dd` In hk, where differentiation is understood in the sense of currents, can be represented as a sum of two terms-the form (12) with differentiation in the classical sense and the current defined by the sin-

gularities of In hk, i.e., by the points of stationarity of the curve Fk. In a neighborhood of a point of stationarity zo, as we just pointed out, ddc In hk = /kdd` In Iz - z012 + dd` In gyp; by the calculations carried out in §3, the contribution of this point to the singular part equals the current Itk[zo], consisting of the point zo with multiplicity µk. In this case, according to (17) of §7, {Lk is the index of stationarity of Fk at zo. Thus the singular part of the current ddC In hk is the divisor [Sk], which consists of all the points of stationarity of the curve Fk counting multiplicity; it is called the divisor of stationarity of the curve. Thus the Poincar6-Lelong formula in our case has the form

dd`In hk =Ricflk+[Sk], But, by the lemma, Ric S1k = dd° In IFk_ 112 + dd° In IFk+1 I2 - 2dd° In I Fk 12

(14)

§8. CHARACTERISTIC FUNCTIONS

111

or, using (5),

Ricilk=Ilk- 1+ftk+1-21lk,

(15)

where 1l_1 - 0 in accordance with the convention IF-11 = 1 adopted above.

If we substitute this in (14) and integrate over the disk Bt, then take the logarithmic average and introduce the characteristic function as in (11), we arrive at the relation

f

r dt

o

t

fB d dclnhk =Tk_1(r)+Tk+1(r) =2Tk(r)+ f

n(Sk,t) dt, o

t

t

(16)

where n(Sk, t) is the number of points of [Sk] in the disk Bt counting multiplicity.

The last integral is the characteristic function of the divisor of stationarity

of the curve Fk and is denoted by the symbol N(Sk, r) (see §2). In order for this to converge, one must assume that 0 V [Sk]; in the general case it is assumed that f r n(Sk, t) n(Sk, 0) dt + n(Sk, 0) In r. (17) N(Sk, r) = t

Finally, the left side of (16), in which the differentiation is understood in the sense of currents, can be transformed by Lemma 2 of §1:

/ T dt o

t

f

dd` In hk = 1 J 2 In hk (reie) d8 - 1 ,

41r

2

00

In hk (0)

(we used the fact that for m = 1 the form or = dd In Iz12 = dO/2a). Now (16) leads to a theorem expressing the relation which we wished to obtain: THEOREM 1. The characteristic functions of a nondegenerate holomorphic curve f : BR -+ P' are linked by the relation

f21r 0

Tk_1(r) -2Tk(r)+Tk+1(r)+N(Sk,r) =

47r

lnhk(reie)d9+C, (18)

where k = 0.... , n - 1 and N(Sk, r) is the counting function of the divisor of stationarity of the kth associated curve; hk is the metric coefficient and C is a constant term. We do not mention the condition 0 ¢ [Sk], since, as in Chapter II, one can get rid of it at the expense of a change in the constant term. In accordance with the convention above that fl_ 1 - 0, we assume T_ 1 to be identically 0; recall that we also have 0.

III. HOLOMORPHIC CURVES

112

6. The case of entire curves. For entire curves, i.e., holomorphic mappings f : C -+ P", a useful supplement to Theorem 1 is THEOREM 2. For entire curves, outside some set E C R+ of finite logarithmic measure 2n In hk(re'e) dO < c'ln Tk(r),

(19)

where k = 0, ... , n -1 and c' is a constant.

4 From the definition of the characteristic function, which according to (6) can be rewritten in the form Tk(r) __ r dt o

hk

t

at

i

dz l1 dz,

21r

it follows that r drk

f2'

rr

dTk

1aehk 2 dz A dz = . Jo

d In r

t dt

hk (tes8) dO

(20)

(we passed to polar coordinates z = te'B and replaced the element of area (i/2)dz A dz = t dt A do). Differentiating once more, we obtain

ir

fp2

r2 (d In r)21

0

from which, using the convexity of the logarithm (see (19) in §5), we find that 1

2

f2w

In hk dB < In

27r

hk dB)

27r

/

2n

=1n `

\ 2r2 (d I

0

r)2(21)

Now we use Lemma 1 from §5, setting g(r) = 1/r and h(r) = rl+e Applying it first to the increasing function Tk, we find that outside a set E of finite logarithmic measure dTk

dTk

1+£ dlnr = r dr < [Tk(r)].

By the same lemma, applied to the function dTk/d In r, which is increasing by (20), and by the nonnegativity of hk, we then obtain that

Tk dlnr)2

( d2

- r d / dTk dr

1+E

dlnr )

drk < 7 dT

)

outside E.

Substituting the previous inequality in this and the result in (21), we obtain the needed estimate:

jInhk(re'°)dO <21n {

l

[Tk (r)]

11 < c' In Tk (r)

outside E.

JJJ

The result just obtained can be given a somewhat different form. For this we set T (r) = max Tk (r) for k = 0, . . . , n - 1, and we let 77 be any function

§9. SECOND MAIN THEOREM

113

admitting the estimate rl(T(r)) < c' lnT(r) + c outside E, where c and c' are constants and E is a set of finite logarithmic measure.(') If we substitute (19) into (18) and discard the positive term N(Sk, r), we obtain this COROLLARY. For a nondegenerate entire curve and k = 0, . . . , n - 1 Tk_1(r) - 2Tk(r) +Tk+1(r) = n(T(r)) (22) This corollary permits the comparison of characteristic functions with different numbers. First we observe that for k = 1, . . . , n - 1 kTk(r) = (k + 1)Tk-1(r) + TI(T), (n - k)Tk_1(r) = (n + 1 - k)Tk(r) + 77(T).

(23)

Indeed, for k = 1 the first relation (23) takes the form T1 = 2To +,q(T); it is true since it coincides with (22) for k = 0. Let us suppose by induction that it is true for some k. From (22) and this assumption it follows that (k + 1)Tk+l = 2(k + 1)Tk - (k + 1)Tk_1 + 77(T)

=(k+2)Tk+i1(T), i.e., the relation is true also for k + 1. The second relation (23) is proved analogously: for k = n - 1 it coincides with (22); next, one carries out an induction on decreasing k. Further, applying (22) sequentially, we conclude that for all k, l = 0, ... ,

n-1

(k + 1)TI(r) = (1 + 1)Tk(r) +, (T), if 1 > k, (24) (n - l)Tk(r) = (n - k)Tk(r) +7J(T), if l < k. Thus the growth of the characteristic functions of an entire curve and all of its associated curves is essentially the same (if we neglect a set of finite logarithmic measure and quantities of order inT(r)). Since the logarithmic growth of To(r) characterizes the rationality of a curve, by what was proved in §4, we obtain in particular that for rational curves, and only for them, Tk(r) = O(lnr).

§9. Second main theorem At the basis of the Ahifors approach to the proof of the second main theorem for holomorphic curves lies the use of a singular metric form. This approach is also used in §5, but to construct such forms there it was necessary to limit oneself to mappings which preserve dimension. In the case of curves the singular forms are constructed differently, using the so-called contact functions. (2)We observe that the inequality defining the symbol +1 is one-sided (in contrast with the similar inequalities with moduli). Therefore, two relations containing this symbol can be added but not subtracted.

III. HOLOMORPHIC CURVES

114

7. Contact functions. We will need the definition of interior multiplication of multivectors over C"+1, which is in a certain sense the opposite of exterior multiplication. Let us suppose for definiteness that I < k; the interior product of the (k + 1)-vector A by the (I + 1)-vector B is defined to be the (k - I)-vector A V B such that for all (k - I)-vectors C the inner product (A V B, C) = (A, B A C).

(1)

The idea of this definition will be understood, by virtue of its linearity, after we find out how to multiply products of the vectors e0, ... , en of an orthonormal basis of Cn+l. Let el = e'O A . . . A e`k and eJ = e&O A A e2' where I = (i0,. .. , ik) and J are ordered sets of indices 0,...,n; by definition from,

(e' V eJ, C) = (er, eJ A C).

The inner product on the right consists of one term, the complex conjugate of the coefficient of er in the expansion of the (k + 1)-vector eJ A C. From this it is clear that eI V eJ = 0 if among the indices j there is even one not belonging to I. If on the other hand J C I, then (ei, eJ A C) = QCI\J, where I \ J is the ordered set obtained from I by removing all the indices belonging to J, CI\J is the coefficient of er\J in the expansion of C, and or is the sign of the permutation (J, I \ J) with respect to I. Thus

J0,

e V e J= l oeI\J, I

ifJ¢I, if J C I.

(2)

This argument also shows the uniqueness of the interior product. For k = I the interior product clearly reduces to the inner product of the multivectors. We note a simple consequence of (2). Let A = ao A A ak be a decomposable (k+1)-vector and let b E Cn+1 be any vector; from (2) and considerations of linearity it is clear that

AVb=O

(3)

if and only if b belongs to the orthogonal complement to the plane spanned

by the vectors a°,...,ak. We will show that for interior multiplication the Bunyakovskii-Schwarz inequality is preserved: IA V BI < IAI IBI.

(4)

First take the case where E is a unit (k-I)-vector. By (1) we have (AVB, E) _

(A, B A E); but by this inequality for inner and exterior multiplication of multivectors (see subsection 2), I(A, B A E)I <_ IAI IB A El < JAI IBI.

Choosing E = (A V B)/ I A V BI, we obtain (4).

We now return to our main theme. In accordance with the ideology of multidimensional value distribution theory, we will study the distribution of

§9. SECOND MAIN THEOREM

115

inverse images of hyperplanes D C Pn, i.e., points z E BR where the holomorphic curve f : BR P' intersects D. We will agree to specify the hyperplanes by equations in homogeneous coordinates F_o 0, or (w, a) = 0, where the vector a = (a°, ... , an) will always be assumed to be a unit vector so that it is determined by the hyperplane up to a factor e29, with 0 E R. The kth contact function (k = 0, ... , n) of a nondegenerate holomorphic curve f : 3R -> Pn with respect to a hyperplane D determined by a vector a is defined to be the function

'Pk(D,z) =

IFk(z) V a12

_

If A ... A f (k) V a12

If A... A f(k)12

IFk(z)I2

.

(5)

This function is defined outside the set of points of stationarity of the curve Fk, which is a discrete set in BR by the nondegeneracy of f. However, the analysis of the behavior of Fk in a neighborhood of a point of stationarity carried out in subsection 3 shows that PPk can be extended smoothly to these points. From (4) it follows that Sok(D, z) < 1. We will give a somewhat different formulation to this definition. To do this, for a fixed z which is not a point of stationarity, represent the decomposable (k + 1)-vector Fk(z) as the (k + 1)-dimensional plane spanned by the vectors f (Z), ... , f (k) (z), and choose in it an orthonormal basis E°, ... , Ek. If this is extended by vectors Ek+l, ... , En to an orthonormal basis of C"+1, which is called a moving frame, the vector a determining the plane D can be written in the form n

a, E', where a = (a, E").

a= 0

Since Fk = eit IFkI E°.. k, where t E R and

E°...k

= E° A ... A Ek, by (2) we

get

n

k

0

0

Fk V a = e at E a,,Eo...k V E" = ett E(_ IFkI

and thus

=

k

k

(6) E Ia. I2 = '=0 v=0 The functions and the moving frame clearly extend smoothly to the points of stationarity of Fk. From (6) it can be seen that the function Pk is zero at precisely those points where the vector a is orthogonal to all the vectors E°, ... , Ek, i.e., the plane Fk(z) belongs to the plane p-1(D) C But as we saw in subsection 3, the plane Fk(z) has contact of order k with the curve f at the point z; therefore the equality Ppk(D,z) = 0 means that at z the curve f has contact of order at least k with the hyperplane D. The name of the function

cok(D,z)

I(a,EY)12.

Cn+i.

SPk is thus justified.

III. HOLOMORPHIC CURVES

116

We make special mention of the extreme cases k = 0 and k = n. Since Fo(z) - f (z) is a vector and interior multiplication of vectors reduces to inner multiplication, by (5)

po(D,z) = I(f,a)I2/jf(z)I2. For k = n we have Fn (z) = W (z)e° l1. A en, where W is the Wronskian of the functions foi ... , fn and the e" are the vectors of the standard basis of Cn+1

Since a can be represented as a linear combination > a"e" in this case with constant coefficients a", analogously to (6) we find that Pn(D, z) _ Eo la"12 is a constant equal to 1. 8. Two relations. For the proof of the second main theorem we will need two relations linking the contact functions with the forms flk: dcok AdcPk = (Pk+l -'Pk)(Pk - cok-1)f1k, dd` In cPk =

iPk-1(0kk+1 - Pk Dk.

(8) (9)

1Pk

here k = 0, ... , n - 1 and it is assumed that cP-1 - 0. To prove these, following Cowen and Griffiths [1], we proceed by the method of moving frames. Recall that a moving frame attached to a holomorphic curve f : BR --+ Cn+1

is an orthonormal system of vectors E°, ... , En in Cn+1 such that at every point z E BR for any k = 0, ... , n - 1 the vectors E°, ... , Ek define the same A f (k), i.e., complex plane as Fk(z) = f (z) A ettk lFkFk (Z)

(z)I _ EO A ... A Ek,

k = 0, ... , n.

(10)

The analysis of the behavior of Fk at the points of stationarity carried out in §7 shows that for a nondegenerate curve f a family of such frames can be constructed which depends smoothly on z in the disk BR. Decomposing dEJ' relative to unit vectors of the moving frame, we get n

u=0,...,n,

dEJA _>BA"E

(11)

"=0

where B,A" = (dEµ, E") are forms of degree 1 with smooth coefficients. Dif-

ferentiating the condition of orthonormality (Eµ, E") = Sµ" we get that (dEl, E") + (ElL, dE") = 0, whence by the Hermitian property of the inner product it follows that

9µ"+B",,=0.

(12)

Further, taking the exterior differential of (11) we obtain n

n

n

0 = E d9µ"E" - > 0,," n E 0"1 E' ,

§9. SECOND MAIN THEOREM

117

from which, after regrouping the terms of the second sum, we find that n

d9µ = E 9µj A 8j , .

(13)

j=0

We observe that in our case the vector Eµ is a linear combination of the vectors f, ...J(11), and hence dEµ can be expressed in terms of only f, .. , f(µ+l); that is, in terms of E°, ... , Eµ+1 From this it follows that in (11) actually 9,,, = 0 for v > p + 1, while it follows then from (12) that. 9µ = 0 also for it > v + 1. Thus we have 9µ = 0 for I u - vI > 1, and in particular, (11) can be rewritten in the form (14) + 9µ,µE" + 9µ,µ+1Eµ+1, dEµ = 0,.. . , n (in the extreme cases p = 0, n one must set Bo,-1 = Bn.,n.+1 = 0), and (13) can be rewritten for v = p in the form 0µ,µ-1Eµ-1

d9µµ = eµ,µ-1 A 9µ-l,µ + Bµµ A gµµ + 8µ,µ+1 A 9µ+1.µ

(15)

= Bµ-l,µ A 9µ-l,µ - 8µ,µ+1 A gµ,µ+1 (we have used (12) and the fact that gµµ A 901, = 0 since gµ.µ is a form of degree 1).

We will need an expression for the form 11k in terms of the G. In order to get it, we observe that on the basis of (14) and properties of the exterior product k

µ=0 k

_

9µµA...AEk+9k,k+lE°A...AEk-1 nEk+1

µ=o

and consequently differentiation of (10) leads to the relation k

E9µµE°A...AEk+9k,k+IE0 A... A Ek-'AE k+1 µ=0

=

dFkl eitk

-

kI2dlFkle'tk + IFkI ie2tk dt I

k

We take the inner product of this with the unit (k + 1)-vector E° A / (Fk/ IE'kl)e'tk, getting k

> Bµµ =

µ=0

(dlk , Fk) IFkl2 2

21Fk 1

- dlFk IFkI

[(dF'k, Fk)

AEk =

+ x dtk

- (Fk, dFk+ i dtk(3)),

(3) We have used the equality 2 IFkI d IFkI = (dFk, Fk) + (Fk, dFk), which follows from IFkl2 = (Fk,Fk)-

III. HOLOMORPHIC CURVES

118

and since by the holomorphy of Fk and the Hermitian property of the inner product (dFk, Fk) - (Fk, dFk)

a - a in I

21Fkl2

= 27rid` In IFk12,

2

the last equality can be rewritten in the form 1

k

E 0µµ = µ=0

d` In IFkI2 + 2r

Differentiating once more, we have k

flk = dd` In IFk 12

=

2

. E d9µµ, µ=0

and using (15) and obvious simplifications we obtain the desired expression h ek,k+1,

12k =

k = 0,... , n -

1

(16)

(we have used the fact that 9_1,0 = 0). Now we can begin to derive (8) and (9). To derive the first of these from (10) we represent the contact function in the form Sok(D, z) = IFk V aI2

(17)

IFk12

=(E°A...AEkVa,E) A

AEkVa)

and compute arpk A8Pk at an arbitrary point z° E BR, assuming for simplicity

that at this point all the O. = 0. This can be arranged using the fact that the vectors Eµ are defined up to a factor e",,; replacing Eµ by e'T- E'` in (14), we get

dEµ =

eµ,µ-1e_iT'E"`-1

+ (0µµ - idr)E'` +

eµ,µ+Ie-"-Eµ+1

Consequently, it is sufficient to set 9µµ = MT. (since, as can be seen from (12), the form 9µµ is pure imaginary). We write 8µ as a sum of forms of bidegree (1,0) and (0,1); then at the point z° we have 9µµ-9µµ = 0 and thus 8Eµ = _ Further, from (10) by the holomorphy of Fk we have 5(E° A ... A Ek) 8(e`tk/ IFk1)Fk, from which it can be seen that 8-differentiation does not lead out of the plane Fk, i.e., that 0µ,µ+1 = 0 and 0,',,,,,+1 = 9µ,µ+1 In particular, at the point zd 9µ,µEµ-1+9µEµ+1.

27E't = 9µµ_1Eµ-1

§9. SECOND MAIN THEOREM

119

and analogously MIA By this remark and the properties of exterior multiplication we get from (17) that

afPk = Ok,k+1(E° A ... A Ek-'AEk+1 V a, E° A ... A Ek V a).

If the vector a is decomposed relative to unit vectors of our frame, repfor v = 0,... , n, then by resenting it in the form of a sum of terms using the properties of the interior product and the orthogonality of the unit vectors, the last relation/ can be rewritten as d Pk = Ok,k+l(ak+lE° A ... AEk-1 , akE° A ... A Ek-1)1 = ak+lak9k,k+1-

Since the Pk are real, it follows from this that acok = akak+l Ok,k+l and thus aPk A a'Pk = IakI2 Iak+1I2 0k,k+1 A Ok,k+1.

Now (16) gives us a

dcPk A d`P = -09Vk A alPk = IakI2 Iak+112 Ilk,

To obtain (8) it remains to observe that by (6) Iak12

Iak+112 = cPk+l -'Pk= Cn+1 by the formula

fa(z) = Fl(z) V a= f(z) A f'(a) V a

(19)

a n d prove that f o r a n y k = 1, ... , n its associated curves (Fa )k_ 1 satisfy the relation (f(z),a)k-1Fk(z)

(Fa)k-1(z) =

(f,

a)k-l If A

Va

... A f (k) V a.

(20)

We will prove this by induction on k. For k = 1 the relation is true since it reduces to (19). Now we suppose it is true for some k. Without loss of generality we can assume that at the point under consideration, z° E BR, (f(.i)(zo),a)=0, j=1,.--,n (f(zo),a) 0, (we replace, if necessary, the representation of fa as in the proof of the lemma in §8). Then by the properties of the interior product, formula (20) at z° can be rewritten in the form

(Fa)k-1 = (f,a)kf' A ... A f(k),

III. HOLOMORPHIC CURVES

120

and since at this point the kth derivative fak) = f A f(k+1) V a = (f, a) f (k+1}

then at this point (Fa)k = (Fa)k-1 A 1(k) = (f, a)k+l fi A ... A f (k+1) = (f, a)kf A ... A f' (k+1) V a

we have obtained (20) with k replaced by k + 1. Thus (20) is proved, and (18) is derived from it by applying the lemma of §8 to the auxiliary curve fa. Indeed, from (20) we obtain by the holomorphy of f and by this lemma

dd`1nIFkVal2 =dddlnl(Fa)k-112 21r

l(Fa)k-212 I(Fa)k12 dz Adz, I(Fa)k-114

or, again applying (20) and then (17),

dd`InIFkVaI2=

IFk-1 V a12

IF

27r

k

1VaI2dzAdz

i Pk-1Vk+1 kV IFk-112 IFk+1l2 27r

`Pk

dz n dz.

IFkJ4

It remains to apply again the lemma from §8, by which IFk_ 1I2 1Fk+1 I2 21r

IFkl4

dz n dz = S1k

and we arrive at the desired relation (18). Thus (9) is also proved.

9. Second main theorem. Let a nondegenerate holomorphic curve f : BR PR be given as well as q > n + 2 hyperplanes in general position. With the help of the contact functions 1Pk (D2) (4) we define the singular metric forms 9 1/(n-k) k+1(D,) l

f2k=ckf

it

/

12k,

k=0,...,n-1, (21)

with singularities on the (discrete) set of points z E BR at which the curve f has contact of order at least > k with at least one of the D? . Here Ck and it are constants which will be chosen later. The use of these forms is decisive in the approach of Ahlfors to the proof of the second main theorem of value distribution theory for the case of holomorphic curves; this theorem is formulated in the following way: THEOREM 1. Let an entire nondegenerate holomorphic curve f : C , P" be given and let there be given q > n + 2 hyperplanes Dj C P" in general (4)To simplify the notation, we will not indicate the dependence of pk(D3) on z.

§9. SECOND MAIN THEOREM

121

position. Then for any e > 0 there is a set E C R+ of finite logarithmic

measure such that for r V E n-1

q

(q-n-1-e)Tf(r)+E(n-k)N(Sk,r)+C
(22)

j=1

k=0

Here Nf(Dj, r) is the counting function of the hyperplane Dj (it was defined in §2), N(Sk, r) is the counting function of the divisor of stationarity of the kth associated curve F: C G(n, k), Tf(r) = To (r) is the characteristic function

of the curve, and C is a constant. The proof of this theorem will be carried out in several stages. We set Ilk = (i/21r)hk dz A dz; then 1 Pk+I A) hk = Ck j-1 Pk(Dj) (µlPk(Dj))1

f

hk,

where hk is the metric coefficient of the form ilk, while the constants ck and tt will be chosen later. From this, by the Poincare-Lelong formula, we get an equality of currents ddr In hk = Ric Slk +

1

n

q

k E { ['Pk+1(Dj )]

-

(Dj )]} + [Sk],

(23)

j=1

where Ric Ok is the regular part of the current dd` In hk. We have used the fact that the singular part of this current consists of two parts-the current caused by the zeros of the contact functions (the term with ln2 (j (D j) ) does not make a contribution because of the weakness of its singularity) and the singular part of the current dd` In hk, which is the divisor of stationarity of the curve Fk (see (14) in §8). From (23) in the standard manner we get the equality r2,r 1

47r

Jo

pr

In hk (rese) d8 = J d 0

+

t

f

Ric flk

B,

E{Nk+1(Dj, r) - Nk(Dj, r)} + N(Sk, r), j=1

(24)

where Nk(Dj,r) is the counting function of the zeros of pk(Dk,r). We will not calculate these functions, but instead we multiply (24) by n - k and add over k = 0, ... , n - 1. We will first take care of the middle term on the

III. HOLOMORPHIC CURVES

122

right-hand side. After obvious rearrangements and cancellations, we obtain n-1 q

4

E E{Nk+1(Dj) - Nk(Dj)} _ k=0j=1

>{Nn(Dj)

j=1 4

E N1(DI, r), j=1

since Nn(Dj, r) = 0 and No(Dj, r) = Nf(Dj, r). Further, by Theorem 2 of the previous section, with f2k replaced by the singular form 11k, we get that outside a set of finite logarithmic measure 2n

In hk(rexe) d8 < ck In tk (r),

Jo

where Tk is defined using 11k as Tk is using Ilk (see (11) in §8). Therefore, outside such a set E, n-1 n - k k=0

41r

p2n J/j 0

n-1 In hk(rexe) d8 = E 0(InTk(r)). k=0

Taking account of these remarks, we obtain from (24) that outside E 14'f(Dj, r) =

f Tf

(n - k)Ric flk

B, k=0

j=1

(25)

n-1

n-1

+ E(n - k)N(Sk, r) + k=0

O(In Tk(r)). k=0

At the second stage of the proof it is necessary to estimate this integral from below. For this we need LEMMA. For any E > 0 there exists a p(E) > 1 such that for all E.t > µo(E) and for any hyperplane D C Pn 1

dd` In ln2(ILI

Sok)

2Vk+1 > Pk In2(ILI Pk) Ilk - Ellk.

4 Calculations reduce the left side of (26) to the form

dd In

1

In

dd° In'Pk

dpk A d`
2ln(µIPk) +2 `o In2(/iI'Pk).

(26)

§9. SECOND MAIN THEOREM

123

Substituting (8) and (9), which were obtained in the preceding subsection, we rewrite this in the form dd` In

2'Pk+1

=

1

In2(/1./'Pk)

cPk

ln2(u/cPk)

ilk

+ 2 Pk-1iPk+i gk

1

1

ln(p/cPk)

ln2 (u/cok )

-(

cPk-1

1

1

(ink) + ln2(/U/Pk)/ --

+ Pkln2(IU/cPk)

Ilk;

then, using the fact that cPk < 1, we choose a number µ so large that ln(µ/cpk) > 1 and 1

1

ln(A/Pk)

< cok)

E

2

for the givens > 0. Then dd In

2c 'k+1

1

ln2(tlI(Pk) > which was to be proved.

cPk In2(1I

Slk

cPk)

-

sfZk

We formulate the estimate we need as a separate proposition: THEOREM 2. Let f : C -+ Pn be a nondegenerate entire holomorphic curve and let there be q > n + 2 hyperplanes Dj C Pn in general position. Then for a givens > 0 and for a suitable choice of constants ck and A, the singular forms 11k satisfy the inequality n-1

n-1

n-1

E(n-k)RiCOk> (q-n-1)flo+1:Ok-e1: f1k.

(27)

k=0

k=0

k=0

4 By the definition of the Ricci forms, we obtain from (21) that n-1 n-1 q

E (n - k)Ricf2k = E E{dd`In PPk+1(Dj) - dd`InVk(Dj)} k=0

k=0j=1

(28)

n-1

n-1 q

+

dd` In k=0j=1

n2 (µlcok)

I

+ E (n - k) Ric flk. k=o

Here the first sum on the right, after a change in the order of summation and some obvious cancellations, takes the form

{dd`1nVn(Dj) - dd`In po(Dj)} = gf1o, j=1

since rPn(D1) is a constant, and by (7), taking into account that dd` In = 0 by the holomorphy of f , we have dd` In cpo(Dj) = _ddc In If I2

= -no

a)12

III. HOLOMORPHIC CURVES

124

(cf. (5) in §8). The third sum is easy to compute by means of (15) in §8 and the equality fZn = 0: n-1

n-1

E (n - k)Ric flk =

(n - k)(flk_ 1 - 211k + Ilk+1) k=0

k=0

_ -(n + 1)fl

.

Substituting these calculations into (28) and estimating there the second sum on the right by the lemma, we obtain n-1 F, (n - k)Ric 11k > (q - n - 1)SZo k=0 (29)

9 n-1 +2

Pk+

Dj )Qk

2 j=1 k=0 Pk(D,) In (µ/Pk(D311

- 6 E Qk k=0

(we assume that p > p.o(e) is taken as in the lemma). It remains to estimate the double sum. By the remark at the beginning of this section, Fk V a = 0 only when the vector a belongs to the orthogonal complement of the (k + 1)-dimensional

plane defined by Fk. The dimension of this complement is equal to n - k, and the hyperplanes Dj are in general position. Since to such planes (passing through a given point) correspond linearly independent vectors aj, we have

that Fk(z) V aj and hence Pk(Dj, z) can be zero for no more than n - k hyperplanes. For the remaining hyperplanes, tOk(Dj, z) > 0, and by the continuity of these functions and the compactness of the Grassmann manifold, there is a constant m > 0 such that pk(Dj, z) > m for all z and all Dj, except

for at most n - k of them. We denote for brevity SPk 21(Dj)

jk =

,

Pk(Dj)ln (P1Vk(Dj)) Then, according to what just has been said, there is a constant M > 0 such that '13k < M except for at most n-k values of j. Reindexing the hyperplanes

and4bjk <Mforj=l+l,...,q,

Djsothat we have q

1

1

1: (Ijk4)jk>M1[I M) j=1

j=1

1/1

1

>M,

j=1

?k

1/(n-k)

"Ijk j=1

(we have used the inequality between the arithmetic mean and the geometric mean and the f a c t that 1/1 > 1/(n - k)). Since f o r j = I + 1, ... , q we have 'Ojk/M < 1, then a fortiori 9

q 4

2 F, 4jk > Ck j=1

j=1

//(n-k)

910. DEFECT RELATION AND PICARD'S THEOREM

125

where ck is some constant. Thus n-1

n-1

$jkllk

2

n-1

q

1k,

ck k=0

if the constants in the definition (21) of the forms 11k are chosen equal to ek. Substituting this into (29), we get the desired estimate (27). Now the proof of the second main theorem can be concluded very simply. From (27) we get n-1 n-1 fr n-1 (n - k)Ric SZk > (q - n - 1)TI(r) + Tk(r) - e E Tk(r),

t

$° k=0

k=0

k=O

Substituting this into (25), we find that outside a set E of finite logarithmic measure n-1

q

N I (Dj, r) > (q - n - 1)TI (r) - e

Tk (r) k=0

j=1

n-1

n-1

(30)

+ > (n - k)N(Sk, r) + > {Tk(r) + O(ln Tk(r))}. k=0

k=0

It remains to observe first that, in accordance with the remark at the end of §8, outside such a set the growth of all the Tk(r) is the same as the growth of To(r) = TI (r) up to the addition of a logarithmic term; therefore, it can be assumed that outside E n-1

?0Tk(r) < c'TI(r) with some positive constant c'. Second, we observe that the last sum in (30) can be estimated from below by a constant C. We obtain the required estimate (22), where a is replaced by c'e, which is not essential. The second main theorem of the theory of holomorphic curves has now been completely proved.

§10. Defect relation and Picard's theorem Here we will consider several applications of the second main theorem for holomorphic curves.

10. Defect relation and Borel's theorem. The defect of a hyperplane D C Pn for an entire curve f : C P'i, as in §6, will be defined to be the quantity

bt(D) = 1 - lim NI(D, r)

raa Tf (r)

(1)

where NI(D,r) is the counting function of D and TI (r) is the characteristic function of the curve. As in §6 it can be proved that always 0 < 6f(D) < 1,

III. HOLOMORPHIC CURVES

126

and that the hyperplanes with a positive defect are precisely those which intersect the curve less often than usual. In particular, those hyperplanes which do not intersect the curve at all have the maximal defect, which equals 1.

The main consequence of the second main theorem is the following defect relation: THEOREM 1. For any nondegenerate entire curve f : C - P" and any system of q hyperplanes Dj C P" in general position q

Ebf(Dj) < n+

(2)

j=1

4 By the second main theorem (Theorem 1 of §9), outside a set E of values r of finite logarithmic measure we have q

E Nf (Dj, r) > (q - n - 1- e)Tf (r) + C

(3)

j=1

(we have discarded the nonnegative sum (n - k)N(Sk, r) on the left-hand side of (22) of §9). Since E cannot contain any ray (ro, oo), there is a sequence of

numbers r - oo with r, ¢ E; consequently, by (3) lim

r-. oo

q Nf(D r) j=1

>q-n-1-E.

Tf(T)

Therefore,

bf(Dj) j=1

1- lim =E( r-j=1 lIM

q- r-00

q

Nf(Dj,r) Tf(r)

Nf(D r) Ti (T)

I


(we have used the fact that the upper limit of a sum cannot exceed the sum of the upper limits). Since e > 0 is arbitrary and the left side is independent of it, we get (2). For n = 1 this theorem, as in the equidimensional case (§5), gives the classical defect relation; for any nonconstant meromorphic function f and any q distinct values aj E C,

sf(aj) < 2.

(4)

Since the hyperplanes which do not intersect the curve have defect equal to 1, Theorem 1 leads immediately to a generalization of the little Picard theorem:

§10. DEFECT RELATION AND PICARD'S THEOREM

127

THEOREM 2. Any entire curve f : C - Pn which does not intersect n + 2 hyperplanes in general position is degenerate.

We recall that for n > 1 the degeneracy of a curve does not necessarily reduce to degeneracy to a constant; it only means that I (C) lies in some subspace of Pn. As an example of the application of Theorem 2 we derive the well-known theorem of Borel on entire functions of one variable:

THEOREM 3. An entire function without zeros cannot be a linear combination of linearly independent entire functions without zeros.

-4 Assume on the contrary that the entire function f without zeros can be represented in the form f (z) = aofo(z) + ... + anfn(z),

(5)

where the f j are linearly independent entire functions without zeros and the aj are constants not equal to zero. Since the f j are linearly independent, 0, this curve h o , . . . , fn] is a nondegenerate entire curve in P. Since the f j does not intersect the n + 1 hyperplanes { wj # 0 }, j = 0, ... , n. Since also f 0 0, by (5) the curve does not intersect the hyperplane aowo + +anwn = 0. We have arrived at a contradiction to Theorem 2. The theorem of Borel can be given a somewhat different form: If entire functions gj satisfy the relation e9o(z) +

... +

1,

(6)

then at least one of them is constant. t Assume on the contrary that none of the gj are constant. Choose among the functions e9j the maximal number of linearly independent ones; let these be the functions e90, ... , egm (m < n). Then e911 = >o ajkeg' for k = m + 1, .... n, and (6) takes the form ao ego(z) +

... + ameg-(z)

n

aj = 1 +

E ajk k=m+1

It is clear that not all the a j can equal 0; and since gj is not constant, at least t w o of the aj are d i ff e r e n t f r o m 0. Let a 0 , . . . , al (1 < I < m) be different from 0 and the rest of the aj = 0. Then (6) reduces to the identity aoe90(z) + + a jeg, (z) - 1. Since the f j = e91, j = 0, ... ,1, are linearly independent entire functions without zeros, this identity contradicts Theorem 3. REMARK. From an identity of the form eg+eg+" - 0 one sees that in (6) it cannot be asserted that all the g; are constant. We mention still another form of Borel's theorem: From the relation e90(z) + ... + ega(z) = 0

(7)

III. HOLOMORPHIC CURVES

128

it follows that at least one of the differences g3 - gk, for j # k, is constant.

This form reduces to the previous one if (7) is rewritten in the form ... + egn- l -gn+ai = 0. I

ego-gn+iri +

In conclusion we note without proof that the defect relation generalizes to G(n, k) with planes A C Pn the intersections of the associated curves Fk: C of codimension k + 1. To formulate this generalization, we associate to every

A ak+l, where (w, a3) = 0, for such plane the (k + 1)-vector Al = a1 A j = 1, ... , k + 1, are the equations of hyperplanes in C'a+1 whose intersection is projected to A by the standard mapping p: Cn+1 \ {0} - P" (such a (k+ 1)vector is defined uniquely up to a scalar factor). We denote by nk(A, t) the number of zeros in the disk Bt (counting multiplicity) of the inner product (Fk, A1), where Fk = f A A.

A f(k) is a reduced representation of Fk, and we

call the quantity Nk (A, r) =

f r nk(A, t) dt t o

the counting function of the plane A. The defect of A for the curve Fk is defined to be the quantity Nk(A,r) Sfk) (A) = 1 - lim r00 Tk(r)

'

where Tk is the characteristic function defined in (11) of 98. For k = 0 this definition coincides with (1). The defect relation for the associated curves is formulated in the following way:

THEOREM. For any nondegenerate entire curve f : C - Pn and any system

of A;, j = 1, ... , q, planes of codimension k + 1 in P" which are in general position a

j=1

s(k)(A3) <

k+1

k=0,...,n-1.

This theorem reduces to Theorem 1 when k = 0. Its proof proceeds along the same lines but requires rather awkward arguments. The details, which are related to the definitions introduced here, and the proof of the theorem can be found in the book of Wu [3]. 11. Big Picard theorem. The classical formulation of this theorem is this: if a function which is meromorphic in the punctured disk BR = { 0 < Izi < R } omits three distinct values, then it can be extended meromorphically to the disk BR (z = 0 cannot be an essential singular point of the function). A direct generalization of this to the case of holomorphic curves is

THEOREM 4. If a nondegenerate holomorphic curve f : BR --1 Pn does not intersect n + 2 hyperplanes in general position, then it can be extended holomorphically to the disk BR.

§10. DEFECT RELATION AND PICARD'S THEOREM .

129

Without loss of generality we assume that R = 1 and that f extends

holomorphically to some neighborhood of the circle { Izi = 1). The proof will consist of two stages. First we will prove that the characteristic function of the curve, which is understood here to be

"dtf

1t

Tf (r) = f

Ilo=

i

f f ' dt

1

t

(8)

A,,

11o,

has logarithmic growth as r -> oo under the hypotheses of the theorem. Here

At = { t < I zi < 1 } is an annulus, and no = j* (wo) as before is the pullback of the Fubini-Study form. From this we will deduce the holomorphic extendability of f to the disk B1. The first stage reproduces with some changes the proof of the second main theorem in §9. Having chosen as the D3 the hyperplanes which do not in-

tersect the curve, we construct by (21) of §9 with q = n + 2 the forms itk = (i/21r)hkdz A dz. But these forms are not singular since we have 0; therefore, instead of an equality of currents in (23) of §9, we

(pk(Dj) have

dd` III hk = Ric (k + [Sk],

(9)

where [Sk] is the divisor of stationarity of the associated curve Fk. Next we take the logarithmic average of (9), not over the disk as in §9, but over the annulus Al/r, and in accordance with this the left-hand side will be different from that in (24) of §9. Namely, by Stokes' formula

f1t Ae/ dd` In hk = f 1

dt

J

1

J z1=1

1/r

f

= In r

d` In hk -

dc In hk -

1zI_1

f

f1d 1/r t

d` 1n hk

dt

t

r d` In hk ./ zI=t

If we use the expression for the operator d` in polar coordinates d`

4ir t 8t

d9

- 4irt

j o-

which on the circle Izj = t has the form d`

47r

t

it d8,

so that

f

1

11*

1J

dt d` In h.k = t X- l=t 4a 49r

a{/

8t

Iz{=1

In hk d9 } dt JJJ

1 JzJ=1/r In hk d9, 47r

III. HOLOMORPHIC CURVES

130 then we get

dt

f/r t f

ddcIn

'

hk = 1nrJ

d`Inhk

I=1p=1

1J 41r

In hk de + 1

47r

I'1=1

r

(10)

In hk dB.

J

Thus, instead of (24) of §9 we have

r 1

r1

fzI=1/r

InhkdB+Klnr+L= /

f

ddd1/r

dt 1

J1/r t

f

tA,

Ric f)k ,

(we have denoted by K and L expressions in (10) which do not depend on r, and have discarded the nonnegative term on the right-hand side). We will proceed further exactly as in §9, and then instead of (30) of §9 we obtain that outside a set E of values r of finite logarithmic measure n-1

n-1

K` In r + L' > Tf (r) - s E Tk(r) + F {Tk(r) + O(1n Tk (r))}, k=0

k=0

where K' and L' are constants and while Tk and Tk are defined as in §9 with the disk Bt replaced by the annulus At (we have taken into account the fact

that we have Nf(D,, r) = 0 and q = n + 2, and discarded the terms with N(Sk, r)). By the reasoning carried out at the end of §8, it follows from this that outside the set E the function Tf(r) has logarithmic growth: outside E

Tf(r)
(11)

with some constants K and L. It remains to observe that, by the nonnegativity of the form 11o, from this we can deduce the inequality

f

0K

(12)

11,.

for all values of r. In fact, if for some ro the opposite inequality were true. then for all r > ro we would have by (8)

Tf (r) = / r ro

f

dt /

flo + const > K In r + const,

t

which contradicts (11), since E cannot contain any ray (ro, oo). From (12) and (8) we conclude that (11) holds for all values r > 1. We begin the second stage with a variant of the first main theorem for curves which are holomorphic on Bi . If s is a section of the hyperplane

§10. DEFECT RELATION AND PICARD'S THEOREM

131

bundle on P" defining the hyperplane D and h(z) = 1111s o III (see §4), then the corresponding Poincare-Lelong formula is written in the form dale In h = flo - [f -1(D)].

Averaging this relation logarithmically over the annulus Al/,. and transforming the left side using (10), we get the following analog of the first main theorem for a curve holomorphic in the punctured disk Bi : for any hyperplane

DC Pn Nf(D, r) + m f(D, r) = Tf(r) - lnr fl-1=1

d` lnh +47rI

1nhd8,

(13)

1

where Nf(D, r) is the counting function of D in the annulus Al/

mf(D,r) =

1

47c

JIzI=1/r

In

1

dO

(14)

11301112

is the proximity function (cf. (4) in §4) and Tf is the characteristic function of the curve defined by (8). Since the image of the circle { IzI = 1 } under the mapping f is a set in Pn of real dimension one, in the space (Pn)* of hyperplanes there exists an open set U which does not intersect this image. Thus for all D E U the function h = 1/11s o ill is bounded on { IzI = 1 }, and hence both the integrals in (13) are bounded. It may be assumed that the function m f(D, r) is nonnegative (see §4); then (13) leads to the following variant of the Nevanlinna inequality for all hyperplanes D E U:

Nf(D,r)
(15)

where K and L are constants depending on the choice of U. Together with what was proved in the first stage, it can now be asserted that under the hypotheses of the theorem N f (D, r) = O(ln r) for all D E U. From this, as we saw in Theorem 5 of §1, follows the boundedness of the function n f (D, r), i.e., the boundedness of the number of points of intersection of the curve f with the hyperplanes D E U in the punctured disk Bi B. From this it is not difficult to deduce the assertion of the theorem. First we observe that a holomorphic function -0 in the punctured disk Bi extends meromorphically to the disk B1 if it has the following property: for all points a of some open set V C C the equation 1/i(z) = a has finitely many solutions in Bi . In fact, suppose that for a° E U this equation has the maximum number

of solutions zl,...,z,,, and p = min(IzjI,...,I z, 1). Then in Bp1 = {0 < IzI < pi }, where pi < p, the function 1(i takes no values in some neighborhood

of a°, and thus the function 1/(b(z) - a°) is bounded in BP, and therefore extends holomorphically to the point z = 0. Passing to the case of a holomorphic curve f : Bl Pn, we consider its representation f = (fo,... , fn) and the functions i = fo, v = 1, ... , n,

132

III. HOLOMORPHIC CURVES

which are meromorphic in Bi (without loss of generality it can be assumed that fo 0 0). The existence of an open set U C (Pf)* of hyperplanes which intersect the curve at finitely many points allows one to apply the preceding remark to each of the functions zb,,. Thus every V), extends meromorphically to the disk B1, and this is equivalent to the holomorphic extendability of f to B1. REMARK. The condition of nondegeneracy of the curve in the preceding theorem is essential. For example, the curve f (z) = [1, e11z, -eh/z], which is holomorphic in Bi, does not extend meromorphically to B1 although it fails to intersect with four hyperplanes in general position in P2: wo = 0, wl = 0,

w2=0,andwo+wl+w2=0.

1

As an example of the application of this theorem, we give a proof of a local variant of Borel's theorem in its second formulation.

THEOREM 5. If the functions fo, . . . , fn are holomorphic in the punctured disk Bi, never vanish there and satisfy the relation

fo(z) + ... + fn(z) = 1,

(16)

then at least one of them extends meromorphically to the disk B1. 1 As in the proof of Borel's theorem in its second formulation (see subsection 10) we can pass from (16) to a shorter relation go(z) + + 91 (z) - 1, where the functions gj as before are holomorphic and nonvanishing in Bi , but are linearly independent. Then g = (go,..., 91] is a nondegenerate holomorphic curve BI P1 which does not intersect 1+ 2 hyperplanes in general position: wo = 0, ... , w, = 0, and wo + + wi = 0. By Theorem 4 this curve extends holomorphically to B1, and this is equivalent to the meromorphic extendability of all the functions g3. Hence, as in subsection 10, it follows that at least one fk can be extended meromorphically.

12. More theorems of Picard type. We present the generalization of the big Picard theorem (Theorem 4) obtained by Green [2] in 1975. Here instead of the disk with a point removed one considers an arbitrary complex manifold with an analytic subset A removed and considers the problem of extending a holomorphic mapping f : M \ A P" to A. We will see below that this reduces to the problem of extending functions. We observe that since by a well-known theorem holomorphic functions always extend to analytic subsets

of codimension greater than 1, the problem is only interesting in the case when codim A = 1. For the same reason it can be assumed that A has no critical points; that is, it is a complex manifold. Indeed, if the extendability of functions to the set AReg of noncritical points has already been proved, then the extension to A \ AReg is automatically ensured since the codimension of the latter set is no less than 2. We write the mapping in homogeneous coordinates: f = [ fo, ... , fn]. Following Green, we will consider the indices J and k equivalent if f;/fk extends

§10. DEFECT RELATION AND PICARD'S THEOREM

133

meromorphically to the set A (clearly the relation j - k satisfies the axioms of an equivalence relation). The number s of equivalence classes into which the relation partitions the set { 0, ... , n } will be called the transcendence degree of the mapping f (for the given homogeneous coordinates in Pn). For s = 1 the mapping f extends meromorphically to the set A. Since the problem of meromorphic extension is a local problem and M and A are complex manifolds, with a suitable choice of local coordinates one can suppose that M = U"`, a unit polydisk, and A = Usi-1 x { 0 } (the case m = 1 was considered in the previous subsection). We need a generalization of Borel's theorem from the preceding subsection (Theorem 5) which is formulated the same way, but with Bi replaced by M \ A and B1 by M. As in Theorem 5, it is sufficient to consider the case where the functions satisfying the relation U'-1 x U" are linearly independent and to fo + - + fn = 1 on M \ A = prove that each of them extends meromorphically to Um. Let us suppose that this does not hold and, say, the function fo does not

extend; that is, it has essential singular points in U'. By the well-known theorem of E. Levi, the set E of essential singularities is an analytic subset of Um of codimension 1 (see Shabat 0, p. 526), which under our assumptions is clearly the plane { z,,, = 0 }. But by Theorem 5, applied to functions in the variable z,,, for fixed z' = (z1i... , z,,,_ 1), fo can be nonextendable only when for fixed z' the function fo is a linear expression in the others. Using the identity fo + + fn = 1 and the same Theorem 5, it can be shown (by induction on n) that such expressions must have the form fo + >jEs f j = 0, where S is some set of indices from the set { 1, ... , n }, and consequently that there can be only finitely many different expressions. Therefore, at least one of them is true on an open subset of Usi-1, and by the uniqueness theorem it must be true on all of Um-'. Thus the functions fj turn out to be linearly dependent in contradiction to the assumption. The generalized Picard theorem proved by Green is formulated thus:

THEOREM 6. Let M be a complex m-dimensional manifold and let A be an analytic subset of it. If a holomorphic mapping f : M \ A -; Pn omits n + k hyperplanes in general position (k > 1), then its transcendence degree s (with respect to the homogeneous coordinates in Pn defined by the first n+1 omitted hyperplanes) satisfies the inequality

s
(17)

where the brackets denote the integer` part.

A Let H. = { >o a w = 0 } be the hyperplanes which are omitted (j = 0, ..., n + k - 1). The first n+1 of these are coordinate hyperplanes, i.e., Hj = { wj = 0 } f o r j = 0, ... , n. Let f = [fo, ... , fn] be a representation of the

mapping in these coordinates and let gj = >o these are nonvanishing holomorphic functions on M \ A, the first n + 1 of which coincide with the

III. HOLOMORPHIC CURVES

134

f3. { 0,

.

We introduce the following equivalence relation on the set of indices 32 if gj, /g7z extends meromorphically to A. We . , n + k - 1): j1

.

denote the equivalence classes by J 1 ,. .. , Jt.

The key assertion is the following: every class J. contains at least k indices. (5) Suppose that this is not true and that the complement of some Ju contains at least n + 1 indices, so that in this complement there exist distinct indices j1, ..., jn+i We choose still another index jo E Jµ; since the n + 2

vectors aj- = (a° , ... , a ) in Cn+1 formed by the coefficients of the Hj,

(v = 0.... , n + 1) are linearly dependent, there exist constants a such that A0a3O + ... + an,+1a),.+1 = 0,

(18)

But since the Hj are in general position, any n + 1 of the vectors aj' are linearly independent; consequently, all the A, # 0. Relation (18) generates a dependency relation Ao9j0 + ' ' ' + An+19,1;,+ = 0, which can be rewritten in the form Al 9ji

An+1 9j.41 )1o

ao 9jo

(19)

gjo

(we have gjo # 0 on M \ A). By the generalized Borel theorem, at least one of the ratios gj /gjo (v = 1, . . . , n + 1) extends meromorphically to A, and this contradicts the fact that jo and j belong to different equivalence classes. This proves the assertion. From this it follows that the number of equivalence classes

t<

[fl]

= Lk] +

1.

(20)

Further, since the complement of every J,, contains at most n elements, among the indices of the coordinate hyperplanes Ho,..., H,, there is a representative of every class Jµ. From this it follows that the number s of classes J. defining the transcendence degree of the mapping f in the chosen coordinates is equal to t, and estimate (20) proves the theorem.

COROLLARY. If under the conditions of the theorem the mapping f : P' omits 2n + 1 hyperplanes in general position, then it extends

M\A

meromorphically to the set A.

4 From (17) for k = n. + 1 it follows that s = 1, and this implies the meromorphic extendability of f.

For n = 1 we obtain the classical big Picard theorem (observing that a holomorphic mapping into P' is a meromorphic function and the hyperplanes are points). Let us now observe that in the global variant of Borel's theorem (for mappings of the whole space Cu' ), in contrast to the local one, the assertion of meromorphic extendability of the ratios f j / f k is replaced by the assertion (5)We are assuming that k > 1; for k = 1 the theorem is trivial.

§10. DEFECT RELATION AND PICARD'S THEOREM

135

that they are constant (see the third formulation of Theorem 5 in the previous subsection). Repeating for holomorphic mappings f : Cm Pn the proof of Theorem 6, we obtain as a consequence that the functions fj from one equivalence class are all proportional to one of them, i.e., that the set of them maps C' into a complex line through the origin in the space Cn+1 Therefore, the image f(C") lies in an s-dimensional subspace of Cn+1, or what is the same thing, lies in an (s - 1)-dimensional subspace of Pn, where s is the number of equivalence classes J,,. The estimate obtained in Theorem 6 therefore leads to the following result: THEOREM 7. If a holomorphic mapping f : C'" _ Pn omits n + k hyperplanes in general position (k > 1), then the image f (Cm) lies in a subspace of P" of dimension no greater than the integer part of n/k.

This theorem was proved by Green by yet another method (see Green [1] and also Shabat II, p. 328 ), which also makes it possible to verify that the estimate obtained is sharp. In particular, it asserts that a holomorphic mapping f : Cm - Pn which omits n + 2 hyperplanes in general position is necessarily degenerate, while a mapping omitting 2n + 1 such hyperplanes degenerates into a constant. For m = n = 1, both results reduce to the small Picard theorem. REMARK. In the first result it is asserted that for any holomorphic mapping f : Cm -> Pn omitting n + 2 hyperplanes H3 C Pn in general position, the image f (C'n) belongs to some proper subspace of Pn, so that f can be viewed as a mapping C' - Pn-1. It certainly omits the n + 2 hyperplanes HH C Pn-1 which are obtained by intersecting the omitted hyperplanes H;

with P". It would seem that one could apply to f the same result with n replaced by n - 1, i.e., assert that f (C-) lies in a proper subspace of Pn-1 and then reasoning further arrive at the conclusion that f degenerates to a constant. However, the hyperplanes HH may not be in general position, and therefore such reasoning would not be valid. I

CHAPTER IV

Generalization of the Main Theorems §11. Mappings of complex manifolds Up to now the mappings which have been considered have mainly been holomorphic mappings from the space Cm into compact complex manifolds M. Here we describe a generalization of the main theorems of value distribution theory to the case of mappings A --+ M where A is an affine algebraic manifold

of dimension m. Such a generalization was given by Griffiths and King [1]; besides their work, we will also use ideas developed in the paper of B. Shiffman

[3]. We will also consider a generalization of the theory developed in the preceding chapter to holomorphic curves defined on Riemann surfaces.

1. Exhaustion functions. We will consider holomorphic mappings from m-dimensional complex manifolds A C CN which are described by finitely many polynomial equations (in algebraic geometry they are called smooth of lne algebraic varieties). As is well known (see, for example, Shafarevich [1], Chapter I, §5.4, Theorem 10), for such manifolds it is possible to find a subspace of C"` (of the same dimension as A) such that the projection 7r:A-+ Cm

(1)

will realize A as a finite-sheeted branched covering with multiplicity equal to the degree of A. Without loss of generality we can assume that 7r is the projection of CN on the first m coordinates (for this it is sufficient to make a linear change of coordinates in CN). We will need some other properties of the projection (1). First of all we will prove a lemma which generalizes the lemma in subsection 4 of §1; it is the analog of Sokhotskii's theorem for analytic sets. LEMMA 1. A k-dimensional analytic set V C C^° is algebraic if its closure

V C P' omits in the hyperplane at infinity Pm-1 = P"" \ C"" a projective subspace II =

Pm-k-i whose codimension is equal to the dimension of V.

(For k = m - 1 this becomes the lemma from §1; V omits a point in and, since it is closed, omits some neighborhood of it as well.) 137

Pm-1

138

IV_ GENERALIZATION OF THE MAIN THEOREMS

0

Pte'

p

p,

Op (V)

FIGURE 7

The proof will be by induction on the codimension r = m - k of the set V. For r = 1 the lemma was proved in §1; suppose that it is true for some codimension r - 1 > 1, while

codimV=r=m-k.

In P' 1 there exists an open set U free of points of V. We choose an arbitrary -point p E U and a subspace Psi-1 C P"° different from00 P- and consider the projection ap: P- \ { p } --> P"`-1 from the point p. Since the restriction of up to V is a proper mapping, by the well-known theorem of Remmert (see, for example, Gunning and Rossi [1], Chapter V, §C, Theorem 5) its image ap(V) is an analytic set in the space Csi-1 = ap(C"') C Prn-1. The closure of this set ap(V) C (Pii-1)\ap(II). The projection up transforms complex lines passing through p into points, but the rest of the complex lines remain lines. Consequently, ap(P--1) = P 2, the image ap(II) is a 00 projective subspace of P'-1 of dimension m - k - 1, and dim ap(V) = k. Thus, ap(V) satisfies the conditions of the lemma for codimension r - 1, so by the induction hypothesis it is an algebraic set. But then up 1(ap(V)), the cone of complex lines spanned by ap(V) is also an algebraic set containing V (see the schematic Figure 7). It remains to show that V itself is algebraic itself. For this we consider the set V = this is algebraic since it is the intersection of algebraic sets (see Shafarevich [1], Chapter I, §2.1 ), and it clearly contains V. We observe however, that for z E C"` \ V the set { p E U : ap(z) E ap(V) } is the intersection with P 1 of the cone spanned by V with vertex z, and has dimension equal to dim V = k, where k = m - r < m -1 = dim U, since r > 2.

§11. MAPPINGS OF COMPLEX MANIFOLDS

139

Therefore, for any z E C'n \ V there exists p E U such that z ¢ Qp 1(Qp(V) ) and hence z V V Thus V = V is an algebraic set. .

COROLLARY. Let A C CN be an algebraic set and let it be a projection of

it as in (1). An analytic subset V C A is algebraic if and only if ir(V) is an algebraic subset of C'n. A The algebraic mapping ir: CN --> C'n extends to a meromorphic mapping

PN -> P'n. If V is an algebraic set, then it is proper on V and by the theorem of Remmert cited above 7r(V) is also an algebraic set. Now let 7r(V)

be an algebraic subset of C"` of dimension k. By the lemma there exists a II = P'n_k-1 C PO 1 such that ir(V) f111 = 0. The closure of the inverse image ;r (H) in pN is a projective space PN-k-1 C pN \ CN, and V does not intersect it. By the same lemma V is an algebraic set. Our next goal is to construct an exhaustion of the manifold A by compact sets similar to the way in which the space C'n is exhausted by the balls Br. Then using the functions decribing this exhaustion, we will define a form replacing wo = dd` In 1z12. We will call a parabolic exhaustion function of A any function r: A -+ R of class C°° which has the following properties: 1) The sets { p E A : r(p) < t } are compact for all t E R. 2) Beginning at some level t = to, i.e., on the set { p E A : r(p) > to }, the function r is plurisubharmonic:

dd`r>0.

(2)

3) Beginning at this same level, the exterior power (dd'r)m = 0.

(3)

For the case of A = C'n studied in the preceding chapters, one can take as exhaustion function r = InIzI2 for IzI > ro, i.e., for r > Inr2 = to, and any smoothing of the function in the ball B,.0. Such a smoothing changes the characteristic function and the counting function by bounded terms and does not affect the main theorems of value distribution theory, which have an asymptotic character. In the general case of an affine algebraic set A under consideration here, an exhaustion function can be defined using the projection (1), by setting rA(p) = r o ir(p). (4) Since the mapping (1) is proper, property 1) is fulfilled, and the form WA = d&TA,

(5)

which is equal to lr'c. o = dd` In Iir(p)12 for rA > ro, clearly has properties 2) and 3). Let us denote by B,. p E A : rA (p) < In r2 } for r > ro the "ball" of radius r (for the case of A = C"` this is the same as the usual ball). If r > ro

140

IV. GENERALIZATION OF THE MAIN THEOREMS

and In r2 is not a critical value of the function TA, then aBr = Sr is a smooth real hypersurface. Let us now consider a holomorphic mapping

f:A--+M

(6)

of the manifold A to an n-dimensional compact complex manifold M on which

is given a positive line bundle L -+ M with Chem form CL. We can define the characteristic function of this mapping by the formula

Tf(L,r) =

L:tfBtA-

( 7)

in the case of A = Cm this clearly differs from the similar function introduced in §2 only by the addition of a constant term. For a divisor D of a holomorphic section of the bundle L, the counting function can be defined as

dt f Nf(Df , rJru f 1(D)nR t

m-1.

wA

(8)

here it is assumed that the mapping f preserves sets of codimension 1, i.e.,

that dim f -1(D) = m - 1. Since we have that WA and f *(cL) A w` are positive forms, both functions Nf and T f grow no slower than In r. Making the change of variable z = 7r(Z) in the inner integral of (8), we reduce it to an integral which only differs by the addition of a constant term from the counting function of the set a(f -1(D)) C Cm. Since by what was proved above this set is algebraic if and only if f -1(D) is algebraic, we conclude. recalling Theorem 5 of §1, that logarithmic growth of Nf (D, r) characterizes

the case where f -1(D) is algebraic. From this, as in §4, it is derived that logarithmic growth of Tf (L, r) characterizes rationality of the mapping f.

If r > ro and In r2 is not a critical value of the function rA, then on the "sphere" Sr one can consider the form oa = dcrA A

(ddcr4)m-1 = dcrA

A wA 1,

(9)

which generalizes the Poincare form of subsection 5 in §1. It is closed by

condition 3) on the exhaustion function (daA = wA = 0 for r > ro). By Stokes' formula it follows from this that the integral of aA over Sr does not depend on r for r > ro. Repeating the reasoning in §2, we can now conclude that if the Chern form CL is replaced by another representative ;,f the Chem class, then the characteristic function is changed by a bounded term. Thus the characteristic function T f(L, r) is essentially independent of the choice of metric on the bundle L and is defined by the bundle itself (see §2).

2. Generalization of the main theorems. The first main theorem can be carried over to the case under consideration without any essential changes. Having chosen a metric on the bundle L which equals h = ha in a domain U. of a covering of M, we consider for an arbitrary divisor D of a holomorphic

§11. MAPPINGS OF COMPLEX MANIFOLDS

141

section s = { s,, } the square of its Hermitian modulus 118112 = hQ 13" 12 in U.. Then by the Poincare-Lelong formula on M, the following equality of currents is true: dd° In 118112 = -CL + D,

where CL = -dd` In ha in U,, is the Chern form of the metric h. Passing to the pullbacks under the holomorphic mapping f: A --> M, which preserves analytic sets of codimension 1, we multiply the resulting relation by the form

wA-1. After applying Lemma 2 of §1, we obtain that, for any r > ro for which In r2 is not a critical value of the function TA,

'

In 1Is o f IIaA = - J

r dt f

ot L.

f* (CL)

dtf

+fr.

t

n

WA-1

WA-t+C

(10)

-I (D)nBt

(cf. (2) in §4). Now we introduce the proximity function of the divisor D by the formula

mf(D,r)= fin 1111 A,

(11)

which generalizes the function of the same name in §4. It is defined for In r2

which are not critical values of the function TA, but as can be seen from (10), it extends continuously to these values, too (since the right-hand side of the formula is continuous). We will assume that it is defined for all r > ro. Finally we use the definitions (7) and (8) of the characteristic function Tf (L, r)

of the mapping and the counting function N1 (D, r) of the divisor, and (10) leads to the first main theorem of value distribution theory in the following formulation: THEOREM 1. Let A C CN be an m-dimensional affine algebraic manifold and let M be an n-dimensional compact complex manifold on which is given a positive line bundle L. Then for any holomorphic mapping f : A - M which preserves sets of codimension 1 and for any divisor D of a holomorphic section of L

Nf(D, r) + m f(D, r) = T f(L, r) +C for r > ra.

(12)

The corollaries of the first main theorem set down in subsection 2 of Chap-

ter H also carry over to this case without changes. We will not consider any generalization of the theorem to the case of analytic sets of codimension greater than 1. The second main theorem in the case under consideration here is proved by the pattern of §5 with some small additions. Let A C PN be an algebraic

manifold of the same dimension n as the compact manifold M, and let a divisor D be given on M which is the union of q manifolds D. intersecting in general position. We suppose that the bundle LD of this divisor is positive

IV. GENERALIZATION OF THE MAIN THEOREMS

142

and that the sum of the Chern classes of LD and of the canonical bundle KM is also positive: (13) c(LD) + c(KM) > 0.

Then, by Theorem 1 of §5, a singular volume form >I can be constructed on M \ D which has the properties indicated by this theorem.

Let us now consider a nondegenerate holomorphic mapping f : A - M (nondegeneracy here means that in local coordinates on A and M the Jacowith the form 7r* (4)) _ bian J f 0 0 on A) and compare the pullback f* (ddc lir(Z) 12),,, which is the pullback of the Euclidean volume form 4) = (ddc In Jz12)n under the projection 7r: A --; C'a. Let (14)

f*(W) = C7r*(4)),

where the nonnegative coefficient a becomes oo on the inverse image of the divisor f -1(D) and on the divisor of stationarity S, of the projection 7r, and vanishes on the divisor of stationarity S f of the mapping.(') The Poincare-Lelong formula in this situation is expressed by an equality of currents

dd`In l = f*(Ric,@)+Sf-S,r- f-1(D).

(15)

Indeed, at the points of A\ (Sf US,r U f -1(D)) the form 7r* (4)) is a volume form on A, and f *(111) is a smooth positive (n, n)-form, so that dd` In = f *(Ric q1) in the classical sense; this is the regular part of the current dd` In C. the

remaining terms on the right in (15) make up its singular part. Repeating the device used in the proof of Theorem 1, we get, for r > ro and In r2 which is not a critical value of the function TA,f1B, the relation

'f

2

1n t;oA 1+ C= J dt ro

t

f* (Ric T) A

wA-

+ N(Sf, r) - N(SIr, r) - Nf(D, r) where N(S f, r) and N(S,r, r) are the counting functions of the divisors S f and SR on A and C is a constant. If we denote the left-hand side of the last equality by R(r) (the remainder) and the first term on the right by Tf(r) (the singular characteristic function), then we obtain the second main theorem in the preliminary formulation: for

r>ro

Tf (r) + N(Sf, r) - N(S,r, r) = Nf (D, r) + R(r)

(16)

(we are assuming that R(r) has been extended to the critical values of rA by continuity). (')On f -1(D) the left-hand side of (14) becomes oo, while the right-hand side is finite. The form 7r*(fi) vanishes on S,r, while f *(4) in general does not. The left-hand side of (14) equals zero on S1, while lr* (0), in general, does not. At the points of intersection S,r n S f, if there are any, the function is not defined.

§11. MAPPINGS OF COMPLEX MANIFOLDS

143

The estimation of the remainder R proceeds as in §5 with ioo = dd` In Iz12 replaced by the form dd` k7r(Z)12 and with u replaced by vA; inequality (25) of §5 remains true. The proof of Lemma 2 in §5 proceeds without change, and we arrive at the second main theorem in the following formulation: THEOREM 2. Let L -+ M be a positive line bundle on a compact complex

manifold M and let Dl,..., D. be divisors of holomorphic sections of the bundle which are manifolds intersecting in general position, where qc(L) + c(KM) > 0. (17) Then for any nondegenerate holomorphic mapping f :A -+ M, where A is an affine algebraic manifold and dim A = dim M, q

gTf(L, r) +Tf(KM, r) + N(Sf, r) - N(SS, r) = E Nf(Dj, r) + R(r)

(18)

1-1

and for any E > 0 there exist a b > 0 and a set E of finite b-measure such that the remainder admits the estimate R(r) < cInr+O(lnTf(L,r)) outside E. (19) REMARK. As in subsection 8 of §5, it can be proved that the theorem remains true without condition (17). It can turn out to be trivial, since in this case qTf(L, r) +Tf(KM, r) does not necessarily go to +oo as r -4oo. I Since S, is an algebraic divisor on A as well as its projection S' =7r (S), after a change of variable n(Z) = z and an application of Theorem 5 of §1, we obtain

N(S, r) =

rr Jru

dt r

t JBnS

Wo -1 < c1 In r, WA-1 = f r dt /' Jrp t f{JzJ
where c1 > 0 is some constant. On the other hand, because the form c(L) is positive, there exists a constant c2 > 0 such that T f (L, r) > C2 In r. From this it follows that Cl rl'm Tf((L r) = ,c < , (20)

where, if A = C" or f is a transcendental mapping (see Theorem 5 of §4), then r. = 0. Taking this remark into account, we deduce from Theorem 2, as in §6, the defect relation for this case:

THEOREM 3. Let a nondegenerate holomorphic mapping f :A , M of manifolds be given as in Theorem 2. Then for any collection of divisors D, of holomorphic sections of the positive line bundle L -p M, where the D3 are manifolds intersecting in general position, the sum of the defects of the divisors and the index of stationarity(2) satisfies q

1: bf(D;) +Of < inf{A E R: \c(L) +c(KM) > 0} +n, -1 (2)For the definitions of 51(Dj) and 8 see §5.

(21)

144

IV. GENERALIZATION OF THE MAIN THEOREMS

where is is the quantity defined in (20); it is equal to 0 if A = C" or if f is a transcendental mapping.

3. The case of holomorphic curves. Instead of the holomorphic curves f : C - P" which were studied in the previous chapter, we consider here curves which are defined on open Riemann surfaces, i.e., on one-dimensional complex manifolds. We will assume that the surfaces G under consideration possess parabolic exhaustion functions. These are functions r: G -r R of class C°° for which the sets { p E G : r(p) < t } are compact for all t E R and which are harmonic, beginning at some level to:

dd`r(p) = 0 for r(p) > to

(22)

(conditions 1)-3) from subsection I for m = 1). We observe that the critical points of the function r are isolated for r(p) > to, since if z = x+iy is a local parameter acting in a neighborhood of a critical point p c G, then at this point

dr = and thus the function ar

1

ar

Or

az

2

ax

ay

(which is holomorphic because r is harmonic) is zero at the point also. Then by the uniqueness theorem for holomorphic functions these points form a discrete subset of G. To generalize the theory to this case, it would be possible to proceed as in the previous section, but we prefer another method which is developed in the book of Wu [3] and is based on the application of the Gauss-Bonnet formula. The formula is applied to the domains B,. = { p E G : r(p) < In r2 } which exhaust the surface G; it contains a term involving the Euler characteristic of B, which reflects the geometric structure of G (asymptotically as r -+ oo). Below we will see that it is precisely this term which leads to additional terms in the second main theorem, distinguishing this case from the one considered before.

Let a nondegenerate holomorphic curve f : G P" be given along with its kth associated curve Fk: G - G(n, k). As in the preceding chapter, we consider the Fubini-Study metric wk on the Grassmann manifold; we will denote its pullback F'(wk) by ilk. The form ilk is positive everywhere on G except for the critical points of the mapping Fk, which as we know make up a discrete (and hence at most countable) set. The points of this set, together with their multiplicities, form the divisor of stationarity Sk of the associated curve Fk. On G \ Sk the form f1k defines a Hermitian metric for which we will write down the Gauss-Bonnet formula.

911. MAPPINGS OF COMPLEX MANIFOLDS

145

We denote by ro the number such that In ro = to, where to is the level at which the function r begins to be harmonic. If r > ro and In r2 is not a critical value of r, then the boundary 8Br consists of finitely many smooth curves. We assume further that none of the points of Sk are on this boundary. Inside Br are finitely many points p j E Sk; we write Uj' = { p E Br : I z j (p) I < e }, where the zj are local coordinates such that zj (pj) = 0 and a is sufficiently

small. Then we remove from Br the union UE = U U; over all the pj E B. The Gauss-Bonnet formula can be applied to the domain Br \ UE (using the metric ilk); this has the form

X(Br\UE)- i

f

8B, Kk+

2r JaUE

1ck=

21r

f \u,

KkIlk,

where X is the Euler characteristic, rck is the form of geodesic curvature, and

Kk is the Gaussian curvature.(3) It is clear that X(B, \ UE) = X(Br) °m(r), where m(r) is the number of points pj c Sk in Br (without counting multiplicity), and that there exists a limit: limE-.o fB,\U, Kk1k = fB, Kkf2kThus in the limit as e - 0 we obtain the relation

f ck 1 = U) f E J

m(r)

X(Br) - I27rfaBr Kk + lim E-.o j-1

1 27r

1

.7

27r

k1lk.

(23)

We wish to transform this formula into a form which makes clear its link with the second main theorem. We begin with the right side; at the points of G\Sk the Gaussian curvature of the metric Ilk is expressed in local coordinates in which SIk = (hk/21ri)dz A dz by the formula Kk

_

21r 82 In hk

hk 8z8z

Hence Kkf2k

_

82 In In hk

-L 8

h dz A dz = -27rdd` In hk = -21r Ric flk,

where the Ricci form Ric 11k, like KkIlk, is defined globally on G \ Sk (see subsection 6 of Chapter II). By (15) of §8 the right side of (23) can consequently

be rewritten in the form 2x

f

B

Kk1k

-

f

B,

(Bk+1 - 211k + Iik-1).

(24)

Let us now turn to the terms with the geodesic curvature. Let -y be a smooth curve on a real two-dimensional manifold G with inner product (u, v), (3)See, for example, Wu [3] , Chapter II, §3; the + sign in front of the third term is explained by the fact that the 8UE consist of counterclockwise curves, which is the negative direction in relation to B, \ UE .

IV. GENERALIZATION OF THE MAIN THEOREMS

146

and let r and v be the unit tangent and normal vectors to y. The geodesic curvature of this curve is the quantity

r.g = (Vrr, v),

(25)

where Vr is the covariant derivative, and the geodesic curvature form is the form tcgds, where ds is the differential of the arc-length of y. We recall that the covariant derivative of a vector field v on G with respect to a field u is the vector field V .v with the properties Vuli-u2v = Vu,v + V, 2v,

VWUv = PVuv,

Vu(vl + V2) = Vuv + Vuv2,

(26)

Vu(ov) = u(cp)v + PVuv, where u1i u2, v1 and v2 are vector fields and ep is a scalar function. (These properties say that Vuv is linear in u, while it behaves like a derivation with respect to v.) Moreover, it is assumed that this differentiation is compatible with the metric, that is, for any three vector fields u, v, w on G and for basis vectors a/a and a/ark, the following identities are true:

w((u,v)) = (V,,u,v) + (u, Vu,v),

Dalaf

_ Va/a,?

(27)

Such a differentiation exists and is uniquely defined on any Riemannian manifold. (4)

We need a formula for the computation of the geodesic curvature. Let us introduce local coordinates S = C+iri on G, in which y is given by the equation

= const and the metric form ds2 = d172). Then r = (1/ f)a/ay and v and by (25) the geodesic curvature of y is

1j

a a\

(we have used (26) and the orthogonality of the vectors a/aC and a/an). Applying the first relation in (27), where we set u = w = a/ai, and v = a/ae, then the second relation, and then again the first with w = a/aC and u = v = a/ark, we obtain Kg

g 77' valaF

an,

1977

29

an

Finally, taking into account that by the choice of the vector v the inner product (a/a1,, a/a17) = g and that on -y we have ds = Vrg dry, we get an expression for the form of the geodesic curvature in our local coordinates: K = Kgds = 2g

ag

drl - 1 a i g dn.

Let us continue the transformation of (23). (4)See, for example, Gromoll, Klingenberg and Meyer ]1], §3.4.

(28)

111. MAPPINGS OF COMPLEX MANIFOLDS

147

LEMMA 2. The third term on the left-hand side of (28) is equal to the number of points of the divisor of stationarity Sk of the curve Fk inside the domain Br (counting multiplicity): m(r)

lim

i=1

tar

a°

Kk - 1

- n(Sk, r.

(29)

4 For the proof it suffices to verify that for every point p E Sk lim

1

e-.o tar

(30)

lck = /1k + 1, 8UE

where Ak is the index of stationarity of the curve Fk at this point (see subsection 3 of Chapter III; for simplicity of notation we will drop the index j when using pj and U1). Let z = peze be a local coordinate on G in a neighborhood

of the point p such that z(p) = 0 and 13U- = {p = e). Then, according to (7) in §8, the form ik = (i/21r)hkdz A dz with coefficient hk(z) = p2A'ktip(z), where cp is a smooth positive function. In order to arrive at a situation where formula (28) for the geodesic curvature is applicable, we set z = eS, where S = t; + 177, so that t; = In p and n = 0. The form ds2 = (1/7r)hk jdzI2 corresponding to 11k has in the new coordinates the form ds2

= 1 hkp2(de2 + d772).

Consequently, the coefficient g in (28) is equal to (1/ar)p20Ak+1)tp(es), and by this formula on the boundary 8Ue, where C = In e, Kk =

laIng dr< = (1.1k + 1) dB + p81np dB.

2a

2

8p

From this (30) follows, since the integral over 9Ue of the second term goes to 0 as e -* 0 by the boundedness of a In p/8p. Now let the number r be chosen as before; we choose r1 and r2 sufficiently

close to r so that ro < rl < r < r2 and the closure of the set Br2 \ Brl does not contain critical points of T. This set consists of finitely many annular components, each of which surrounds one component of the boundary aBr. Let V be one of these components, and let ry = aBr n V; we set I' = f.. d`7-. Because the function r is harmonic in the domain Go = { p E G : r(p) > to }, the form dcr is closed in V; and consequently the integral v(p) = 47r fp d`r, where po is a fixed point of ry, is defined in V up to the addition of an integer multiple of r. Thus Q(p) is a multiple-valued function in V with a single-valued differential

satisfying the relation da = 47r d°-r. From this it follows that the branches

148

IV. GENERALIZATION OF THE MAIN THEOREMS

of o(p) are harmonic functions conjugate(5) to r(p). The function S(p) = r(p) + io (p) is therefore a multiple-valued analytic function in V with a singlevalued differential dS which is nondegenerate because dr 0. The branches of S(p) can serve as local coordinates in V, and the relation Ok =

2'

hkdS A 4

uniquely defines the metric coefficient hk in V which is a smooth positive function outside the divisor Sk.

LEMMA 3. If r > ro and the number Inr2 is not a critical value of the function r, and if 8Br fl Sk = 0, then

f

r2 drd J

In hkd`r, aB, where hk is the metric coefficient defined above. 27r

B.

Kk =

(31)

4 It is sufficient to prove (31) for each connected component of BBr. Let y be one of the components and let V be the neighborhood of it described above. In the local coordinates formed by the branches of S(p), the curve ry is the coordinate line r = Inr2; consequently, its geodesic curvature can be calculated from (28) and

1 f r a In hk

rck = 2

o

r

-

dv.

r-Since a/ar = (r/2)a/ar and r = coast on y while do = 47rd`r, the last equality can be rewritten in the form

f

Kk = 7rr

d f In hk d`r.

Adding these equalities for all the components of aBr, we obtain (31). The transformation of (23) is concluded. Taking (24),(29), and (31) into account, we can rewrite (23) in the form X(Br) +

()k+1 - 211k + Ilk- I) + n(Sk, r) = Br

rd 2 dr J9 Br

In hkd`r,

(32)

This relation has been obtained for r > ro for which In r2 is not a critical value of the function r and OBR does not contain points of Sk. The set of values r > ro which do not satisfy these conditions is discrete. Between such values the terms of (32) are continuous, and at transitions across these values (5) We recall that in the local coordinates z = x + iy

4ad`r = -a 3Irdx+ azdy

and consequently as/ax = -ar/ay and 8a/8y = 8r/ax.

§11. MAPPINGS OF COMPLEX MANIFOLDS

149

the terms have bounded jumps. Moreover, the metric coefficient hk defined above in neighborhoods of the components of C3Br can he defined globally on the whole domain Go = { r(p) > to } by the formula 1 k = 4hk dT A d`T.(°) Therefore (32) can be averaged logarithmically on (ro, r), and we get r

I

X(Bt) dt + Tk+1(r) - 2Tk(r) + Tk_ 1(r) + N(Sk, r)

ro

(33) 1

J

2 aB,

In hkd`r + C,

where C is a constant. Formula (33) generalizes (18) of §8, which constitutes the content of Theorem 1. Indeed, when G = C, then Br \ Bra is a circular annulus and its Euler characteristic equals zero; since r = In Iz12, then d'7- = dO/27r. Thus in the transition from a holomorphic curve on C to a curve on an arbitrary Riemann surface the only additional term which appears in Theorem 1 of §8 is the term with the Euler characteristic: E(r) = (34) X(Br) dt, 1ror which reflects the geometric structure of G. The subsequent path toward the second main theorem is the same as in Chapter III, and we will confine ourselves to a short description of the changes which are introduced in the case of Riemann surfaces; some of the details can be found in Wu's book 13]. First it is proved that for all r > ro the integral f5 ,. d`7- = L is constant. Hence, repeating the proof of Theorem 2 of §8, we conclude that outside some set E C (ro, oo) of finite logarithmic measure

lnhkdrr < CInTk(r) foB,.

and then instead of (22) of §8 we get the asymptotic relation E(r) + Tk+1(r) - 2Tk(r) + Tk-1(r) = 77 (T) (the meaning of the notation r7(T) is explained in §8). Next we construct singular metric forms ilk by (21) of §9, but instead of the Poincare-Lelong formula we write the Gauss-Bonnet formula for these metrics:

X(Br)+n(Sk,r)+ J RicIlk+ n 1- k E{nk+1(Di) - nk(Di)} a Jrd Y

J

2 dr ag,

In itkd"T.

(6)In neighborhoods of boundary components this definition coincides with the old one,

since there ds A dS = -2i dr A da = -siri dr A d`r, while on G \ Go the form d`r is not closed and the function a is not defined.

IV. GENERALIZATION OF THE MAIN THEOREMS

150

Here the sum on the left-hand side arose from the singularities of the metrics caused by the zeros of the contact functions. From this formula, just as in §9. one deduces a generalization of the second main theorem of value distribution theory: THEOREM 4. Let G be a Riemann surface which has a parabolic exhaustion function -r.. and let f : G - P" be a nondegenerate holomorphic curve. Then for any system of q > n + 2 hyperplanes D. C P71 in general position and any e > 0, there exists a set E C (r0, x) of finite logarithmic measure, such that for r E (ro, ac) \ E "-1 (n(n + 1)

E(r) + (q - n - 1 - E) Tf(r) +

+ E

2

(n. - k)N(Sk.,r) + C

k=0

(35)

This theorem differs from the case studied in the preceding chapter only by the term with the Euler characteristic, which reflects the geometric structure of the R.iemann surface on which the holomorphic curve is defined. A similar additional term appears also in the generalized defect relation, which can be obtained from Theorem 3 in the usual way:

THEOREM 5. For any nondegenerate holomorphic curve f:G - P76 and any system of q hyperplanes Di C P" in general position q

fif(D1) < n 1 1 + n(n2

1) rlirn

Tf(r) f(

_

(36)

i=t We observe that the Euler characteristic X(Bt) is nonnegative only in the case where Bf is a topological disk (then it is equal to 1) or an annulus (then it is equal to 0). If the number of components of the boundary iBt stabilizes for large t, then X(B1) for large t is constant, equal to X(G) and negative. unless G is the same as C or as C with one point removed. In this case the term -E(r) has at most logarithmic growth, and the additional term on the right-hand side of (36) can appear only when the characteristic function Tf (r) has logarithmic growth, i_e., when the curve f is algebraic. If the number of components of dBt grows without bound, then X(Bt) -

-x, and the function -E(r) can grow arbitrarily rapidly (the condition for the existence on G of a parabolic exhaustion function does not contradict this, since an arbitrarily large number of points can he punctured out of G.). In this case the additional term in (36) can also appear for transcendental curves.(') (7)The properties of the Euler characteristics of Riemann surfaces cited here can be found. for example, in Schiffer and Spencer ;ij.

§11. MAPPINGS OF COMPLEX MANIFOLDS

151

4. The hyperbolic case. The essential property of the exhaustion function r considered above, the property for which the exhaustion received the name parabolic, is property 1) of subsection 1: the sets {p E A : r(p) < t } are compact for all t E R. This property means that the function r tends to +x: uniformly as one approaches all the boundary points of the manifold A. We will call a function r E C'° (A) a hyperbolic exhaustion function if instead of property 1) of subsection 1 it has the property: 1') there exists a number tt < oc such that for all I < t1 the sets { p E A r(p) < t } are compact and r(p) < ti everywhere on A, while properties 2) and 3) remain unchanged (for complex one-dimensional manifolds A, they reduce to the requirement that r be harmonic beginning at some level). The simplest example of a manifold which has a hyperbolic exhaustion function is the ball B C C' and for m = 1 is the disk or a Rielnann surface of hyperbolic type which is conformally equivalent to it. (s) Therefore, value distribution theory for holomorphic mappings f : A M, where A is a manifold with hyperbolic exhaustion function, generalizes the Nevanlinna theory of value distribution of functions meromorphic in the disk. The first main theorem extends to the hyperbolic case without any changes while the changes in the second main theorem only involve the estimate of the remainder term R(r). In the parabolic case for mappings preserving dimension, estimate (30) of §5 is true, according to which for any E > 0 there exist, a

number b=b(s)- 0as s->0and aset EC (ro, oo) such that f, r°dr
R(r) < Elnr+O(lnTf(L,r)). In the hyperbolic case we set rl = el,12. where tl is the number from condition 1'), and the previous estimate is replaced by the following: there exists a set E c (ro, rl) such that dr

fE (rl -- r)1+b

< 0C

and for r e (ro, rl) \ E

R(r) <eln

rl 1- r +O(lnTf(L.r)).

(37)

The meaning of the condition imposed on the set E in the parabolic and the hyperbolic cases is the same: E cannot contain intervals of the form (r', :z:)

in the first case and (r', r1) in the second case, so that outside of E there is a sequence converging respectively to Do or to rl. The proof of (37) goes along the same lines as in the parabolic case. An analogous estimate is also true for holomorphic curves on Riemann surfaces having parabolic exhaustion functions. (8)The name "parabolic exhaustion" arose in the theory of Rieuiann surfaces; surfaces conformally equivalent to C are of parabolic type.

152

IV. GENERALIZATION OF THE MAIN THEOREMS

As in the parabolic case, the defect relation can be deduced from the second main theorem. However, in the hyperbolic case it makes sense only for mappings with characteristic fimctions which grow sufficiently rapidly as r -+ rl. Namely, this quantity

k = lira r-.ri

ln(r1

- r)-1

Tf (L, r)

(38)

must be finite: In the classical case of functions meromorphic in a disk of finite radius, there is a distinguished class of functions with bounded characteristic Tf(r) called

the functions of bounded type. They have been well studied; in particular R. Nevanlinna proved that every function of bounded type can be represented as the quotient of two functions which are holomorphic and bounded in the same disk. (9)

The multidimensional case has been studied much less well. We point out the research of Malliavin [1] and Khenkin [1]-[3] on zero sets of functions of bounded type in the ball B C Cn, and also the work of Favorov [2], in which an analog of Nevanlinna's theorem is established for holomorphic curves of bounded type.

§12. Divisors with singularities The second main theorem and its corollary, the defect relation, were studied above only for divisors which are unions of complex manifolds intersecting in general position. Here, following Shiffman [3] and Griffiths [5], we will free

ourselves of these constraints. At the basis of such a generalization is the method of resolution of singularities, widely applied in algebraic geometry. We will begin with a description of this method.

5. Quadratic transformation. First we consider the ball B in the space C" with coordinates z = (zl,... , zn). Let w be a point of Pn-1 with homogeneous coordinates [w1, .... wn], and let b be the submanifold of B X Pn-1 defined by

z.,Wk-zkW)=0,

1<j, k
(1)

The mapping a: B --: B, which is the restriction to B of the natural projection B x Pn-1 -a B, is called the quadratic transformation with center 0 E B; the names monoidal transformation or a-process are also used for the mapping. If the point z # 0, then it follows from (1) that the n-tuple [wl, ... , wn] is proportional to z, and hence the preimage a-1(z) = z x [z] is a specific point

of B. If z = 0, then all the values w satisfy (1), i.e., a-1(0) = 0 x [Pn-1] is a set. Thus the quadratic transformation, or more precisely its inverse or(9)See, for example, Privalov's book [1], where these functions are called function of class A.

§12. DIVISORS WITH SINGULARITIES

153

FIGURE 8

maps B\{0) biholomorphically onto B \ (O x Pn-1) and at the point 0 attaches an entire projective space Pi-1 Let us consider in more detail the behavior of a-1 at 0. It is not difficult to see that the restriction a-111 to any complex line 1 C B passing through 0 extends holomorphically to this point. In fact, a point z on such a line equals At, where A E C" and t is a complex parameter. For t # 0 the n-tuple

... , w"] corresponding to z by (1) can be taken equal to A. Therefore, for z E 1 \ 0 we have a-'11(z) = z x [A], from which it can be seen that a-1It can be holomorphically extended to the point 0 if we set a-111(0) = 0 x [a]. This value certainly depends on the choice of line, and by changing this line we can obtain any point in the set o,-'(0) = 0 x Pn-1 [w1,

The property of the transformation a-1 just described can be used to eliminate points of self-intersection of complex curves. Suppose that a curve 'I C B has a point of self-intersection at the point 0 with different tangents 1' and 1". We will denote by 7 = a-1 o ry the lifting of -f \ 0 to the manifold B. As points approach 0 along the two different branches of 1, the corresponding points on y clearly converge to two different limits, a-1 Ip (0) = 0 x [A'] and a-1Ii,,(0) = Ox [A"]. Thus y can be holomorphically extended above the point 0 without self-intersections there (Figure 8). A quadratic transformation can also be defined on any n-dimensional complex manifold. Let M be such a manifold and let p be any of its points,

with z: U - B being a coordinate mapping of a neighborhood U C M of p such that z(p) = 0. If we identify points q E M with the corresponding points z(q) E B, then the manifold b described above can be identified with a manifold MP C U X P"-1, and we can consider the quadratic transformation

a:Mp, M

(2)

with center at p. Its inverse a-1 realizes a biholomorphic mapping of U\p onto Mp\a-1(p), and it attaches a projective space at the point p. The restrictions

154

IV. GENERALIZATION OF THE MAIN THEOREMS

01-'j, to complex curves y C M passing through p extend holomorphically to this point, and to curves with different tangents at p there correspond different values of o-'IAP). We observe that under a coordinate change z' = g(z), g(O) = 0, the obvious biholomorphic equivalence g: Mp \ o-1(p) Mp \ (0')-1(p) can also be extended to o-1(p) = p x Pr-1 if we set g(p x [A]) = p x [A'], where A' = g'(0) A (here g' = (8gj/8zk) is the Jacobian matrix of the coordinate change, and A' and A", as above, are vectors in C' representing points in Pr-1). Thus, up to biholomorphic equivalence, the quadratic transformation o: Mp - M does not depend on the choice of coordinates in a neighborhood of p. W e cover M p by open sets U = { z x [w] E M p : w # 0 }, v = 1, ... , n, and on every U" we introduce local coordinates S" = (Si, . . . , Snv), where

Sj =

wj-zj -

z"

W,

for j # v and S"" = z"-

(3)

In the intersections U,,, fl U", the coordinate change Sµ -+ S" is clearly biholomorphic, so Mp is an n-dimensional complex manifold. The set t = 01 _'(p) is defined in every neighborhood U, by the single equation !ff = 0, so it is a divisor on the manifold Mp; it is called the exceptional divisor of the quadratic transformation. The transformation itself can be written within U" in terms of the local coordinates (3) in the form

o(S") _

---,

,

,...,5L(Sn)

(4)

where S,' is located in the with coordinate position. Let N be any analytic subset of M containing the point p; we will denote by the closure in Mp of the inverse image a-1(.N \ p). In particular, let N = D in the neighborhood U be a divisor of a holomorphic function represented as a series in homogeneous polynomials cc

f(z) _ E P0 (z),

(5)

Z=;10

where Pµ(z) = FIkl-µ ckzk and P,(z) 0 (as always, z is the local coordinate in U with origin at p and k = (k1, ... , k,,) is a multi-index with I k I = k1 + - + k,,). Then o-1(D \ p) in the neighborhood U" is a divisor of the function -

00

f(s") = f o

(S")

(6)

µ=1+o

where according to (4) Pu o o(S") ck(Sl )k3 ... (Sv-1)ky

(Sv )µ

jkj=µ

,v

()"+1)kv+i ... (Sn)k

(7)

§12. DIVISORS WITH SINGULARITIES

155

Since in U there is the exceptional divisor t = { S,` = 0 }, from (5) and (7) it is clear that the intersection b fl k is described by Ck( 1

)kt ... (Sv

(Sn)k"

_

Ikl=µo

which by (3) can be rewritten in the form

()ki ...

kl=No

=.

Ck

(I

=0.

(8)

But the equation Pa(z) = 0 describes the tangent cone Cp(D) of the divisor D at the point p (see subsection 3 of Chapter I), and z? /z can be considered to be affine local coordinates in the neighborhood U = { z E U : z v j 0 }, if [z1, ... , zn] are considered to be homogeneous coordinates in P'-1 Therefore, the set D fl k, which is given by (8), is the projectivization of the

tangent cone C,(D). Further, it can be seen from (7) that all the terms of (6) contain the factor (C'?. Consequently, the divisor a-1(D \ p) fl U is described by the equation 0. Since the left-hand side is holomorphic when S, = 0, the (1/(S, )µ) f same equation also describes the closure of the last divisor, i.e., the divisor Df U,,. But the pullback a*(D) of the divisor D intersected with UL includes also the divisor of the function (S,v)µo, i.e., the µo-fold exceptional divisor E fl U,,. Since this is true for any v = 1, ... , n, the following relation with divisors is true:

a'(D) = D+ poE,

(9)

where po is the degree of the tangent cone Cp(D), which is the Lelong number of the set D at p (see subsection 3 of Chapter I). We observe further that the

set h = a-1(D), made up of the union of a-' (D \ p) and a-' (p), is clearly equal to b + E and therefore (9) can be written in the form a` (D) = D + (po - 1)E. EXAMPLE. Let M = P2, and let the divisor D be the union of four distinct complex lines DD, three of which intersect at the point p, while the fourth, D4, intersects the other three in general position (Figure 9). Performing a quadratic transformation a with center at p, we arrive at a manifold M on which to the point p there corresponds the exceptional divisor k = P'.

Corresponding to the lines Dj (j = 1, 2,3) are lines Dj which intersect k at distinct points. The set b consists of these three lines and the inverse image of D4, and a-' (D) = D = D + E has only general position selfintersections. Thus the quadratic transformation has rid the divisor D of the self-intersection which was not in general position at p. We observe that the degree M of the tangent cone Tp(D) equals 3; consequently, by (8), the divisor

a'(D)=D+3E=D+2E.

156

IV. GENERALIZATION OF THE MAIN THEOREMS

FIGURE 9

Up to this point we have considered the manifold MM only over the neigh-

borhood U of the point p E M, but it can be continued, identifying it with M outside of U and setting a equal to the identity there. Then still another relation between M and k. can be pointed out. LEMMA 1. Let M be an n-dimensional complex algebraic variety and let M be its quadratic transformation with center at the point p E M. Then the

canonical bundles KM and KM of these varieties are linked by the relation(lo)

K,,C = a*(Km) + Lso,

(10)

where a* (KM) is the pullback of KM under the transformation a: M --+ M and Lse is the bundle of the divisor of stationarity So of this transformation. 4 Since M is an algebraic variety, then there is a global meromorphic form

w on it of bidegree (n, 0); for instance, the restriction to M of such a form

on pN i M. Let this form be given in local coordinates z, z(p) = 0, in a neighborhood of the point p in the form

w(z) = f(z)dzl A...Adz,,, where f E 0(U), since without loss of generality it can be assumed that the polar singularities of w lie outside U. Using expression (4) for the transformation a, we write down the pullback of w in a neighborhood U C M: a'w(S') = f o a((")d(S, Si) A ... A d(SY) A ... A d(Sl a*f(S`,)(S,)n-'dci A ... A dsn. (lo)in contrast with Chapter I, we will here and further on use additive notation, i.e., we will write L, + L2 instead of Ll - L2.

§12. DIVISORS WITH SINGULARITIES

157

After writing this formula f o r all v = 1, ... , n, we see that within the neighborhood U of the divisors [a*w] = a* [w] + (n - 1)E

(11)

(we have used the fact that k n U,, = [!;,v]; here brackets denote the divisor of the function or form inside the brackets). Since outside t the divisor [a'w] = a` [w] by the biholomorphy of a, (11) is true on all of Al. We also observe that, as we can see from (4), the Jacobian of the transformation a in U,. equals a(z1i ... , zn)

jo ( S v)

'

a(s ,... ,c )

S 0 l

0

sv

Si

12

0

0

...

...

0

...

0

1

...

Sn

Si

(the line without zeroes is in the with place). Therefore, the divisor (n-1)E = So is the divisor of stationarity of a, and (11) can be rewritten in the form

[a`w] = a*[w] + S. It remains to recall that the canonical bundle on an n-dimensional manifold is the bundle of the divisor of a meromorphic (n, 0)form on the manifold (see subsection 6 of Chapter II). Therefore, the bundle of the divisor [a"w] is the same as KM and the bundle of the divisor [w] is KM. To the sum of these divisors there corresponds (in additive notation) the sum of their bundles.

The quadratic transformation just described, with center at p E M, can be generalized. First let B be a ball in C' and let E _ { z E B : z17,+1 = = zn = 0} be the intersection of B with an m-dimensional plane; let w be a point of pn-7n-1 with homogeneous coordinates [W.+1, ... , wn]. We consider the manifold

B={zxwEB

xpn-m-1:

zjwk =zkwj,m+1 < j,k
(12)

B the restriction to b of the projection B xpn-'"-1 -+ B. The transformation a, and also the manifold h, is called the quadratic and denote by a: B

transformation with center E, and the set E = a-1(E) is called the exceptional divisor.

The manifold b can be covered by neighborhoods (J zxwEB to,, 54 0), v = m + 1, ... , n, which have local coordinates S" = (Si , , Sn ),

158

IV. GENERALIZATION OF THE MAIN THEOREMS

where

for j = 1, ... , m.

z3

S=

u?

_ °j

W,

ZL)

z

for j=m+1.... A...,n, V forj=v.

(13)

In these coordinates the set t n U is described by the equation SL" = 0. As B is, up to biholoin the case of a point center, the transformation a: B morphic isomorphism, independent of the choice of holomorphic coordinates z` = g(z) in B such that E is described by the equations z1.+1 = = z'n = 0. Pn-m-1 : z'.w/ = z w', }, then the biholoActually, if B' _ { z' x w' E Is x k k morphic equivalence g: B \ E -> B' \ E' defined by the mapping g can be extended to k by setting g(z, w) = g(z) x w', where z E E and r Wr = [wk+1, ... ,

i

w >k=m+1 azl (z)uk. k

The last remark allows us to apply this construction to an arbitrary ndimensional complex manifold M with a given m-dimensional submanifold E. Let { B,, }, a E A, be a family of coordinate balls on M covering all of E and such that in every ball the set E n B« is given in local coordinates by the equations z,,,,+1 = = zn = 0. If aa: BQ - B,, is the quadratic transformation with center E n B0, then in the intersections U,,,,a = B,, n B3 biholomorphic isomorphisms are induced: a,,;j: as 1(Ua;s) - o,' (U0a). By using these, one can glue the manifolds B0, a e A, into one manifold B and construct a single mapping a: B U B0. Setting the mapping a-1 to be the identity on M \ U B0, we extend b to a manifold M and call this manifold

along with the mapping a: M M the quadratic transformation of M with center E; the set E = a-1(E) is called the exceptional divisor. As in the case where the center is a point, one can check that the transformation a: M --+ M is biholomorphic on the set k \ E and the restriction a: E -+ E is the projectivization of the normal bundle over E. The latter can be explained in this way: When E = p is a point, then the whole tangent space Tp(M) Cn is normal to it and its projectivization is the space Pn-1. In the general case, for a point p e E the tangent space Tp(E) = { (zl, ... , z,,,,, 0, ... , 0) } _ Cm, the normal space Np(E) c Cn-m and its projectivization = a-1(p) is the fiber of bundle U: k -+ E. P"-ii-1

Further, for any analytic subset N C M containing E, the set N is defined as the closure in k of the inverse image a-1(N \ E). The intersection N n t is a fibration over E whose fibers are the projectivizations of the tangent cones Cp(N) at points p E E (compare this with the analogous property in the case

of a point center). In particular, if N = D is a divisor on M and at every point p E E the degree po of the tangent cone Cp(D) is the same, then as in

§12. DIVISORS WITH SINGULARITIES

159

the case of a point center the following formula is true:

o*(D)=15+poE=D+(µo-1)E,

(14)

where b = o-1(D). There is a generalization of Lemma 1 which is also true: for an n-dimensional algebraic variety M, the canonical bundles of the variety and its quadratic transformation M with m.-dimensional center E are related by

K,u = o*(KM) + (n - m - 1)LE = o*(KM) + Ls,,

(15)

where LE and Ls, are the bundles of the divisor k and the divisor of stationarity SQ. To prove this we choose on M a meromorphic (n, 0)-form which in a neighborhood U of a point p E E can be written in local coordinates z, z(p) = 0, in the form w(z) = f (z) dz1 A

A dz,,,

f E 0(u).

If U, is the neighborhood on o-1(U) defined above, with the local coordinates S" of (13) acting in it, then a*(dzj) = do for j = 1,...,m,v and o*(dzj) =

d(c S?)=S,vdq- +q-dS for j=(m+1),...,(m+v-1),(m+v+1),...,n. Therefore,

*w

= o*f (s")n m-1dSi n ... A

From this follows a relation between the divisors: [o*w] = o*[w]+(n-m-1)E. The remainder of the proof proceeds as in the case where the center is a point. EXAMPLE. A quadratic transformation with a manifold as a center can be used to eliminate singularities in the following situation. Let a divisor D on an n-dimensional manifold M be the union of q divisors Dj of holomorphic sections of a positive line bundle L - M, where the divisors are manifolds. Suppose further that the following conditions are satisfied: a) D1, ... , D. intersect in general position on M \ E, where E C M is a submanifold of codimension k.

b) D1 n...nDk = D1 n...nDk+1 = E.

c) Dj, , ... , Djk, Dk+1+1...... D. for any choice 1 < j1 < - < jk < k + l intersect in general position on all of M. The singularity of the divisor D thus consists in the presence on E of l extra divisors Dj which violate the condition of intersection in general position. The quadratic transformation a4 f --+ M with center E attaches at every point p E E a projective space Pk-I (we have m = dim E = n - k; therefore, -

n - m - 1 = k - 1), and thus the exceptional divisor E = E X Pk-1. In our situation, the transformation of M can be described globally as the manifold M = {(p, w) E M X

Pk-1.

sji (p)wj2 =

8,2

(p)w3,,

1 < jl, i2 < k},

(16)

where the sj are the holomorphic sections of the bundle L - M whose divisors are the manifolds D j a n d [w, ... Wk] are the homogeneous coordinates on

160

IV. GENERALIZATION OF THE MAIN THEOREMS

pk-1. From condition c) it follows that all the D3, j = 1, ... , k + 1, have distinct tangent planes TT(D;) at the points p E E. Therefore, the intersections with k of the various A = a-1(D.j \ E) do not intersect each other. From this and from condition a) it follows that the divisor b = a-1(D) has selfintersection in general position; the quadratic transformation has eliminated the singularity of D caused by intersections not in general position. We observe further that in our example, in accordance with a generalization of (12), the divisor of stationarity of the transformation a is

Sa = (k - 1)E,

(17)

and since the degree of the tangent cone Tp(D) at the points p E E is equal to lro = k + 1 (k + 1 manifolds intersect on E), by (14)

a'(D)=D+(k+l)E=D+(k+1-1)E. 1

(18)

6. Singularities of intersection. We will determine the changes in the second main theorem for mappings f : A - M, with dim A = dim M, when the divisor D as before is composed of manifolds but the condition of intersection in general position is not fulfilled. In §6 examples were presented which show that without this condition the theorem itself is not valid in the given formulation, nor is the defect relation which is a consequence of it. To begin with, we will limit ourselves to the situation described in the last example: the condition of intersection in general position only fails on some submanifold E C M of codimension k on which, besides the divisors Dl,..., Dk, another 1 extra divisors Dk+1, ... , Dk+1 intersect (a precise description of the situation is given at the end of the previous subsection). THEOREM 1. Let A be an afne manifold and M a projective manifold, both of dimension n, and let f : A -r M be a nondegenerate holomorphic mapping. Let D be a divisor which is the union of q manifolds D; , each of which

is the divisor of a holomorphic section of the positive line bundle L -+ M and which together satisfy conditions a), b), and c) of the previous subsection. Then

gTf(L,r) +Tf(KM,r) +N(Sf - S, r) _

Nf(Df,r) +lmf(E,r) + R(r), j=1 (19)

where

m f (E .r )= 2Js In>

aq

=(

,

lIs°Ill a is the proximity function(11) of the manifold E = {p E M : si(p) _ F II3f°1112

s ln

(20)

= sk(p) = 0) on which the condition of intersection in general position fails, (11 )The symbol 11

11 denotes the Hermitian metric on the bundle L.

§12. DIVISORS WITH SINGULARITIES

161

and the remainder admits the estimate R(r) < c In r + O(ln Tf(L, r)) outside a set of finite 6-measure.

4 We perform a quadratic transformation o: M --a M with center E and consider the lifting f : A -# M of the mapping f , i.e., the mapping such that o o f (p) = f (p) for all p E A. Since the divisor D = u-1(D) is made up of manifolds intersecting in general position, Theorem 2 of §11 can be applied (see the remark just after the theorem); this yields T f(LD + KM, r) + N(S1 - S.,, r) = N f(D, r) + R(r),

(21)

where R(r) 5 a In r + O(InTf (Lb, r)) outside a set of finite 6-measure.(12) Now we must turn to the manifold M and the mapping f . Using the formula KM = O' (KM) + (k -1)LE (see (15), where n - m is set equal to k) and also the obvious corollary

Q*(LD) = Lb + (k + l - 1)LE

(22)

of formula (18), we get that Lb + KM = a*(LD + KM) - IL p. From this follows the analogous relation with the Chern forms of the bundles (see §2, where the multiplicative notation is used for the operation on bundles) and thus for their characteristic functions: T f (L b + KM, r) = Tf (o * (LD + Km), r) - lT f (LE, r)

(23)

= Tf(LD + KM, r) - IT f (LE, r) (we have also used the equality T fp* (L)r = Tf (L, r), which follows from the obvious relation f * (CL) = f * (ce. (L)) between the Chem forms of the bundle L and its pullback.). Further, from the definition of the counting function we get

N1(D,r) = Nf(D,r),

N1(E,r) = Nf(E,r) = 0,

(24)

In the last equality we take into account that here we consider the counting functions of sets of dimension n - 1, but the dimension of f 1(E) is less than that. For the same reason

N(Sf, r) = N(Sf, r),

(25)

since the Jacobian Jf(p) differs from J f(p) by a factor of Jv(f (p)) that is zero on the same set f -1(E). Now applying the first main theorem to the mapping f and the bundle LE and taking (24) into account, we get Tf(LE, r) = N1(E, r) + mf(E, r) + 0(1)

= mf(E, r) + 0(1).

(26)

(12)In the theorem we replaced L by the bundle of the divisor D = E Dj and used some obvious relations.

162

IV. GENERALIZATION OF THE MAIN THEOREMS

Then if we choose the metric on LE which is the lift of the metric on the bundle L M, we find that

ml (k, r) = f *1190f11 In QA

f

aA = m f(E, r),

In

(27)

1180A

,

where s = (s1, ... sk) and the sj are holomorphic sections of L whose divisors are the Dj. Substituting (23)-(27) into (21), we rewrite it in the form

Tf(LD + KM, r) + N(Sf - Sam, r) = Nf(D, r) + Im f(E, r) + R(r),

which is the same as (19). It remains only to prove that in the estimate

R(r) < elnr + O(InTf(LD,r)) the quantity Tf(LD,r) can be replaced by Tf(L, r). But from (22), based on the relation previously used between the Chern forms of the bundle and its pullback, we get that f(cLD) =

f.(cLD)+(k+I - 1)1`(cLk),

From this and (26) follows the relation

Tf(LD, r) = Tf(LD, r) + (k + I - 1)m fE, r) + 0(1). Since m f(E, r) can be assumed to be nonnegative (see §4), the inequality Tf(LD, r) > ATj(Lb, r) + 0(1) is true with some constant A; from this it follows that the quantity O(InTf(LD, r)) is also O(lnTf(L, r)). Thus in this situation the second main theorem differs from the one proved

in §11 only in the additional term lmf(E,r) determined by the set E where the condition of intersection in general position fails and by the number I of divisors which violate this condition. The same additional term also occurs in the defect relation, which is obtained from Theorem 1 in the usual manner: THEOREM 2. Under the hypotheses of Theorem 1, the sum of the defects of the divisors Dj and the index of stationarity k

Ebf(D,)+Of < inf{ACR: Ac(L)+c(KM)>0} j=1

(28)

++c+ linl where

= rlim

Imf(E,r) Tf(L,r)

N(S, . r)

Tf (L, r) if A = C' or if f is a transcendental mapping.

0

In particular, for holomorphic mappings f : C" P" and hyperplane divisors the lower bound in the preceding formula is equal to n + 1 (see §6), and

§12. DIVISORS WITH SINGULARITIES

thus

163

q

ri+1+

bf(Dj)+Of

lim T

j=1

00

1

mTf(r)r)

(29)

EXAMPLE (B. SHIFFMAN [3]). Let f : C2 _ P2 be the mapping from subsection 11 of §6 defined in homogeneous coordinates by the formula f (z) = , ez'], and let D be a divisor made up of four lines: D1 w1 = 0), D2 = { w2 = 0 }, D3 = { wl + awe = 0, a # 0 }, and D4 = { wo = 0 }.

[1, ezl

The condition of intersection in general position does not hold at the point E = [1, 0, 01, through which pass the three divisors D1, D2, and D3. Since E = D1 n D2 = D1 n D2 n D3, we have the situation considered above with

k=2and1=1. Since sl = w1 and 32 = w2, by (20) and (17) of §8 we get 2kinIfi12+

mf(E,r)=

If212

For the restriction f,, of the mapping f to the line z = .1S (A E C2, Al I= 1

2mfa

and S E C) we have, clearly,

f7,

ln(1 + Iea1SI2 + Ie- 2112) do

(E, r) =

f27, 47r

ln(IeA.,12 + le>'

l2) dB,

moreover, according to the result of Ahlfors proved in Chapter I (see (19) in §2), these integrals are equal (up to the addition of a bounded term) to, respectively, Pr/27r and Plr/27r, where P = IA1I + IA21 + IA2 - Al I is the perimeter of the convex hull of the points 0, A and A2, and P1 = 21A2 - Al is the perimeter of the convex hull of the points Al and )'2. Therefore, m f, (E, r) = 27r(IAII + 1A2I - IA2 - A1I) + 0(1),

and, using the computation in subsection 12 of §6 (see there the derivation of (21)), we find that

mf(E,r) =

3

r+O(1).

Since by (21) of §6 the characteristic function T f(r) = (2+ f)r/31r+0(1), it follows that the additional term in the defect relation (29) is

mf(E,r) _ 2-vf2

HE r-.ao T f (r) and this relation takes the form

2+ v'2-

=3-2f

4

Eb1(Dj) <6-2f j=1

(we have the divisor of stationarity S f = 0).

(30)

IV. GENERALIZATION OF THE MAIN THEOREMS

164

v2

1 a

b

C

FIGURE 10

We observe that, since these divisors are not taken on by the mapping, and by (23) of §6, b f(D3) 3 - 202. Thus the sum on the left side of (30) is also

equal to 6 - 2 f , and this example shows the sharpness of the estimate in Theorem 2.

1

7. Arbitrary singularities. Some singularities are resolved by consecutively performing several a-processes. EXAMPLE 1. Let us consider in C2 the curve D = { x2 = y3 } with singularity at the origin (Figure 10a, here x and y are complex coordinates).

The quadratic transformation a with center at 0 for y # 0, according to (4), reduces to the substitution x = uv, y = v in the equation of D. Therefore D has the equation v = U2; since the exceptional divisor k = { v = 0) also appears here, a-1(D) = { v = u2 } U { v = 0 } (Figure 10b). The next transformation a1i which for u # 0 reduces to the substitution u = u1i v = u1v1, turns a` (D) into the divisor { u1 = vi }U{ v1 = 0 }U{ u1 = 0 }, where { u1 = 0 } is the exceptional divisor E1 (Figure 10c). At this stage we have gotten rid of the tangency of the divisor components, but there still remains an intersection not in general position. The latter can be avoided by still another transformation; to perform this transformation, it is convenient to first rotate the u1 and v1 axes by, say, 45° (so that as above it is possible to do with one chart). With the new axes, the divisor al 1 o a-1(D) is described by the equation (u2 - v2) u2 = 0, and the last transformation 0-2: u2 = u3 V3 V2 =V3 (v2 # 0) reduces it to the divisor f u3 = 1 } U { u3 = 0 } U { v3 =01, where { v3 = 0 } = E2 is the exceptional divisor (Figure 10d). I EXAMPLE 2. The singularity of the surface D = {x2 = yz2 } in C3 (Figure I la; x, y, z are complex coordinates) can be resolved by four a-processes.

The first of these is a: x = uw, y = vw, z = w (z # 0); it transforms D into the divisor { u2 = vw } U { w = 0 }. The second, a1: u = u1 v1, v = v1, w = v1w1 (v1 # 0), gives { w1 = u2 } U { viw1 = 0 }; the third, a2: u1 = U2, v1 = u2v2, w1 = u2w2 (u1 # 0), results in { w2 = u2 } U { u2v2w2 = 0). Further, one performs the transformation u2 - w2 = U3, u2 + W2 = W3, V2 = V3,

§12. DIVISORS WITH SINGULARITIES

165

a

FIGURE 11

and then the last or-process a3: u3 = u4W4, V3 = v4w4, w3 = W4 (w3 j4 0) leads to the divisor {U4 = 0 } U {(U4 - 1)v4w4 = 01 (see Figure 11b, where the exceptional divisor is set off by shading). I In the general case singularities can be resolved using the theorem of Hironaka, which says that given a projective manifold M and a hypersurface D on M having on a set E singularities that are different from self-intersections in general position, there exist a projective manifold M and a holomorphic mapping a: M -> M with the following properties: 1) D = a-1(D) has only self-intersections in general position.

2) The mapping a maps M \ E biholomorphically onto M \ E, where E = 9-1(E) is the exceptional divisor. The mapping a can be a a-process, or a superposition of a-processes or a transformation of a more general type. In particular, it is proved in algebraic geometry that singularities of algebraic curves can always be resolved by finitely many o--processes.

Already in the very simple examples which have been presented, it can be seen that the exceptional divisor t = a-1(E) can consist of several irreducible components: we will assume that k consists of J such components E; . If as

before we denote by b the closure on k of the inverse image a (M \ D), then instead of (14) we get the relation J

J

a'(D) = D+EpA = D+E(p; - 1)E." j=1

(31)

j-1

where the p, are positive integers (we took into account that every compo-

nent Ei occurs in D = or-'(D) once). The divisor of stationarity of the

166

IV. GENERALIZATION OF THE MAIN THEOREMS

transformation a also is composed of the Ej with positive integer coefficients: J

SQ =

(32)

q., Ej.

The second main theorem in the general case is proved just as in subsection

6. Instead of (15) we have the relation KM = a*(KM) + >i gjEj; from this and from the equality J

- 1)k,

a'(LD) = Lb + Dpi 1

deduced from (31) it follows that J

LD+KM =a*(LD+ KM)+(qj -pj+1)Ej. This leads to a relation between the characteristic functions: J

Tf(LD +KM,r) =Tf(LD +KM,r) +J(qj -pj + i

generalizing (23).

The rest of the reasoning proceeds justs as in subsection 6 and leads to the appearance on the right-hand side of the second main theorem of the additional term J

E(pj - qj - 1)mI(LE; , r),

(33)

1

which in the situation considered in subsection 6 is the same as dmf(E, r). This same term appears also in the defect relation. It reflects the singularities of the divisor D under consideration. To simplify the estimate it is convenient to replace the set E of the singularities of D by an arbitrary hypersurface H on M containing E. Let J

o,' (H) = FI +Er3Ej, j_i

where H is the closure on M of the set a-1(H \ E) and the rj are positive integers. If we set

r = max(pj - qj - 1, 0) and 'I = max(rj+ /rj ),

(34)

§12. DIVISORS WITH SINGULARITIES

167

then the additional term of (33) can be estimated thus: J J

E(pj - qj - 1)mf(Ej,r) <ErJ mf(Ej,r) 1

1

J < y E rjm f(Ej, r)

(35)

j=1

< ymf(a*(H),r) = ym f(H,r). This estimate leads to the defect relation in the following formulation (for simplicity we leave out the terms containing the divisors of stationarity): THEOREM 3. Let L -p M be a positive line bundle and let D1,. .. , D. be

divisors of holomorphic sections of it which pairwise have no common com-

ponents. If E is the set of singularities of the divisor D = Ei Dj which are not self-intersections in general position and if H is a hypersurface on M containing E, then for any nondegenerate holomorphic mapping f :A - M, bf(Dj) < inf{A E R: Ac(L)+c(KM) > 0} (36)

j=1

+ y inf{It E R: lcc(L) - c(LH) > 0}.

4 The first term on the right is the usual one for the defect relations considered earlier, so we only need to discuss the second term responsible for the singularities which are not in general position. If we denote the infimum that occurs in this term by /zo, then for any p > go we have c(LH) < pc(L),

and hence Tf(LH, r) < pTf(L, r). From this by the first main theorem for the bundle LH we conclude that

mf(H,r) < ltTf(L,r) + 0(1). Substituting this in (35), we obtain an estimate for the additional term in the second main theorem,

j E(pj -qj -1)mf(Ej,r)
j=1

which leads in the defect relation to the additional term yµ. It remains to take the lower bound of all such p and we obtain the second term in (36). From this an assertion of Picard type follows: COROLLARY. Let D be a divisor on M with a positive bundle LD and let

H be a hypersurface on M containing the singularities of D which are not self-intersections in general position. If inf{A: Ac(LD) +c(KM) > 0} + yinf{µ: Ecc(LD) - c(LH) > 01<1, then any holomorphic mapping f : A -p M \ D is degenerate.

(37)

168

IV. GENERALIZATION OF THE MAIN THEOREMS

In conclusion we present an example, also due to Shiffman, which shows that the estimate obtained is sharp. EXAMPLE. Let us consider the curve D C P2 defined in homogeneous coordinates by the equation k-1 k-1 k +w2 ); W1 = wo(w1 for k > 2 this has a singularity as the point [1, 0, 0]. In local coordinates x = wl/wo, y = w2/wo, the equation of the curve has the form xk = xk-1 +yk-1, so the tangent cone To(D): xk-1 +yk-1 = 0 consists of k - 1 distinct complex lines y = For k > 3 we have a singularity not in general position (-1)1/k-lx

which is resolved by a single a-process: x = u, y = uv (u # 0). We have pl = k - 1; since the Jacobian a(x, y)/a(u, v) = u, we obtain q1 = 1 and thus

ri =k-3.

Choosing as H any complex line in P2 passing through the point [1, 0, 0], we get that a` (H) = H + E. Therefore, r1 = 1, and thus -y = k - 3. We have c(LD) = kw, e(KM) _ -3w, and c(LH) = w, where w is the Fubini-Study form on p2. Therefore, the first term on the left side of (37) equals 3/k, the second is (k - 3)/k, and their sum is exactly equal to I. Inequality (37) does not hold, and one can produce a nondegenerate holomorphic mapping

f(zi,z2)= carrying C2 to p2 \ D.

(1 - ex2(l+zi 1)) (1+zk-1)

,1,z1

CHAPTER V

Further Results In this chapter we will set out a number of results supplementing the main theorems of multidimensional value distribution theory and will also consider some special classes of holomorphic functions.

§13. Results using capacity Various concepts of capacity are widely used in complex analysis and in particular in value distribution theory. We begin with an exposition of the concept of plurisuperharmonic capacity and its application to the study of sets of defective values discovered recently by A. Sadullaev.

1. P-measure. The plurisuperharmonic measure (abbreviated Pmeasure) of sets in a complex manifold is a natural generalization of the harmonic measure of sets on the complex line. It was introduced in several different forms by several authors; we will follow the exposition of Sadullaev [2] and restrict our attention to subsets of the complex projective space PN. Let G be a domain in pN and let E be an arbitrary subset of it. An admissible function for the pair E, G is defined to be an arbitrary plurisuperharmonic(1) function in G which is nonnegative everywhere in G and which takes on values no less than 1 on E . The class of such functions will be denoted by P(E, G). We will consider the function w(z, E, G) =

inf

uEP (E,G)

u(z)

(1)

and define the P-measure of the set E with respect to G to be the regularization of this function, i.e., w. (z, E, G) = lim w(z', E, G).

(2)

(')We recall that a function is called phaisrgierhannonic in a domain G C PN if it is a lower semicontinuous function u: G -. (-oo, ool whose restriction to any complex line I is superharmonic on the open set G fl l_ One can learn about the properties of these functions in Vladimirov [11 or Shabat H. 169

V. FURTHER RESULTS

170

We observe that the function w is not in general plurisuperharmonic, since it is not in general even lower semicontinuous; but the function w. is always plurisuperharmonic in G. In fact, w. is lower semicontinuous in C by definition; one only has to prove that its restriction to any complex line i satisfies the inequality characterizing superharmonic functions (see Shabat I, p. 310).

But by this inequality, for functions u E P(E,G) at points z' E G fl l for sufficiently small r,

f27(z' u(z')

+ rest) dt

27r

(we keep the same notation for functions and their restrictions to 1). Since the functions u are bounded from below (they are nonnegative), by Fatou's lemma one can pass to the lower bound over the functions u E P(E,G). For the same reason one can also pass to the lower limit as z' -> z, which gives the needed inequality 2n

1

w. (z. E, G) >

27f

fo

w4 (z + re", E, C) dt.

REMARK. Let the domain G C C1`' be strongly pseudoconvex in the sense of Zakharyuta* (z-pseudoconvex); that is, it is defined by an inequality p(z) < 0, where