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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
}
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 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) 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) 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 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) 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)
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 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
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 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 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 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` 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
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)
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) 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) 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 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°drk=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) 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) 01 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
0, there exists a function u e P(E,G) such that u(z°)
- w(z°, E, G) < E.
(3)
We denote by C, a domain containing E U {z° } which is relatively compact
in G, and write M = SUPZErl ,p(z). Since M < 0, the function .p/M is plurisuperharmonic in D, as is the function V(Z) (z)
min(u(z), p(z)/M) p(z)/:M
for z E C,
for z E D\ G
(one must take into account that on 8G both methods of defining v coincide,
since there min(u, p/M) = -p/M). Clearly, G = { z E D : v(z) > 0): and since v is continuous in D \ G, there exists a domain G2
v(z) > -E. We also introduce the set Go = { z c G2
:
G in which
v(z) > 1 - E },
which is open by the lower semicontinuity of v and which contains E. Finally, *Translator's note. See V. P. Zakharyuta, &b rno1 plurisubhannonic fww ons, Hr7Gert scales and isomorphism of spaces of analytic functions of several variables. (Teor. Funktsii Funktsional. Anal. i Prilozhen_ vyp. 19 (1974), 133-157, Definition 3.4.]
§13. RESULTS USING CAPACITY
171
we denote by b the smaller of the distances p(E, 8G0) and p(8G, 8G2) and consider the average
v(z) =
J
v(z + bc)K(S) dVS
with a smooth kernel K concentrated in the unit ball. The function w =v + E is non-negative, plurisuperharmonic and smooth in G, and on E it takes values no smaller than 1, i.e., it belongs to P3. But w3(z°) < v(z°) + E < w(z°, E, G) + 2E (we have used (3)); and since E is arbitrary, we have w3(z°) < wz°. Taking into account the obvious reverse inequality and the fact that z° is an arbitrary point of G, we obtain the desired identity w3 (z) - w(z, E, G).
The value of the regularization of the function w at an arbitrary point, if it differs at all from the value of w itself, can clearly he only smaller. As H. Cartan proved, the set N = { z E G : w. (z, E, G) < w(z, E, G) }, where the regularization does not coincide with the function w, has zero capacity and consequently also has Lebesgue measure 0.(2) From this another property of the set N is deduced: for any superharmonic function v in G, at any point
zEG lim
v(z') = v(z).
(4)
-*z
z'NU{z}
Since clearly the P-measure w. (z, E, G) _> 0, by the minimum principle for plurisuperharmonic functions it is either everywhere positive in G or else identically equal to 0. In the last instance, as was proved by Lelong [21, the set E is P-polar in G; that is, there is some plurisuperharmonic function in G, not identically oo, which equals +oc on the set (or what is the same thing, a plurisubharmonic function not identically -co equals -oc). Conversely, if E is P-polar in some neighborhood of G, then w. (z, E, G) - 0. Using these rather refined results we prove, following Sadullaev, the properties of P-measure that we need. 1°.
Boundedness: 0 < w. (z, E, G) < 1
2°. Monotonicity:
G1 C G2, E1 C E2 = w.(z, E1. G1)
1 on E since uk, E P(E, G)); and since uk0 (z) + e/2 E P(U, G), then uko (z) + e/2 > w. (z, U, G). From this it follows that c(U)
µ(G) Jc w`
(z, U, G) dy
p(G)
fc
\uk0 (z) + 21 d#
< c(E) + e. Since k C U, (11) now follows from property 3°. 6°. For any decreasing sequence of subsets Ek C G c
I
I
Ek = kim c(Ek).
(13)
k=1
4 This property follows from the preceding one. We start by choosing a sequence Ek \ 0 and for each Ek choose a neighborhood Uk of the set E = n Ek such that c(Uk) - c(E) < Ek. Since the E, are shrinking to E, for any k there is a number jk such that E; C Uk for j > ik; thus for j > jk we also have 0 < c(E,,) - c(E) < Ek. From this follows (13). From property 3° it follows that for an arbitrary set E C G the P-capacity c(E) is the lower bound of the P-capacities of the open sets U C G containing E; or in other words, it is the outer capacity ce (E). One can as usual introduce the inner capacity ci(E), setting it equal to the upper bound of the c(K) over all compact sets K C E. It is clear that for any compact set K C G the inner capacity ci(K) = c(K). The following property shows that ci(U) = c(U) also for open sets U C G. 7°.
For any increasing sequence of open subsets Uk C G
c U Uk = kym°c(Uk)
(14)
k=1
,4 It is clear that limk_ w. (z, Uk, G) E P(U, G), where U = U Uk. Thus this limit is no larger than w. (z, U, G). On the other hand, since w. (z, Uk, G) _< w. (z, U, G) by the monotonicity of P-measures, this limit does not exceed w. (z, U, G). Thus, limk. w. (z, Uk, G) = w. (z, U, G); integrating this relation, we obtain (14). In conclusion we observe that in Sadullaev's paper [31 another concept of capacity is introduced which is in a certain sense equivalent to the one described above. Let E be a subset of a domain G on an n-dimensional complex manifold, which for simplicity of formulation we will assume to be zpseudoconvex. We set w1(z, E, G) = sup u(z) over the class of all nonpositive
plurisubharmonic functions on G which are in C2 (G) such that U E < -1;
§13. RESULTS USING CAPACITY
177
then we set
wi(z,E,G) = lim wl(z',E,G). z'-+Z
It is clear that wi (z, E, G) = -w* (z, E, G), where w* is the P-measure introduced above. If wi E C2(G), then dd`wi > 0 because wi is plurisubharmonic, and we set
cl(E, G) =
f(dd'wfl'.
Bedford and Taylor [1] proved the following maximum principle for plurisub-
harmonic functions of class C2(G) fl C(G): If two such functions u and v coincide on 8G, then u < v in G
fd'u)n >_
J(dd'vY.
Having this principle in mind, for an arbitrary set E C G we can set by definition
cl(E, G) = inf
JG
(dd`u)",
(15)
where the lower bound is taken over the class of all plurisubharmonic functions in C2(G) f1 C(G) for which u1 E < -1 and ulaG > 0. An extremal function for this problem, if it belongs to the class C2 (G), satisfies the so-called MongeAmpere equation in G \ E:
(dd`u)" = 0, (16) In recent years this equation has been encountered in many problems in complex analysis.
Developing the methods proposed by Bedford and Taylor, Sadullaev established for c1(E) = c1(E,G) the properties 1°-6° of the capacity c(E). In particular, cl (E), like c(E), is equal to zero if and only if E is a P-polar subset of G. In this sense these capacities are equivalent.
3. Polarity of the set of defective divisors. We already observed in Chapter II that, in contrast to the one-dimensional case where the number of defective values is at most countable, in the multidimensional case the set of divisors with positive defect can be uncountable. In subsection 12 of Chapter II we presented Shiffman's example of of a nondegenerate holomorphic mapping f : C2 -+ P2 for which every complex line passing through the point (1, 0, 0] is defective; this is the simple mapping f (z) = [1, ez', ez'] Nonetheless, for nondegenerate holomorphic mappings f :A -, M, where A is an m-dimensional affine manifold and M is an n-dimensional projective manifold on which is defined a positive line bundle L -, M, the set of defective divisors cannot be very large. Namely, Sadullaev proved in [2] that the set of such divisors is P-polar in the projective space PN of all divisors of holomorphic sections of the bundle L (see subsection 6 of Chapter I). From this it follows in particular that the Hausdorff (2N - 2 + c)-measure of this
V_ FURTHER RESULTS
178
set (see below, subsection 4) equals zero for any c > 0; this answers one of the questions posed by Griffiths and King [1]. In fact the result of Sadullaev is stronger; it is not concerned with the set of divisors with positive defect
bf (D) = 1 - rlim (Nf (D, r)/Tf (L, r)),
(17)
which was considered in Chapter II and which is naturally called the defect in the sense of Nevanlinna, but with divisors with positive defect
Af(D) = 1- lim (Nf(D,r)/Tf(L,r))
(18)
Vf = {D E pN: A f(D) > 0},
(19)
r-oo This quantity is called the defect in the sense of Valiron, and the set
clearly contains all the divisors which are defective in the sense of Nevanlinna. We begin with the case of a nondegenerate holomorphic mapping f : Cm Pn with the hyperplane bundle on Pn. The hyperplanes { [w] E P" : >o 0 } will be viewed as points of the projective space (Pl)* with homogeneous coordinates [ao,... , an]. For simplicity of notation we will omit the asterisk in designating (Pn)the hyperplane with equation E 0 will be denoted
by a. As always, Ua will be the domain { a E Pn : as # 0 } of the standard covering of Pn, Sr will denote the sphere { z E Cm : z = r }, and or is the Poincare form.
LEMMA 1. For hyperplanes a E U0, the counting function is representable in the form
Nf(a,r) = u,, (a, r) +hQ(a), where
ua(a,r) = J
In Sr
Ef
v=0
(20)
a
as
(21)
is a plurisubharmonic function in the local coordinates
(ao/a...... a0-i/a«,a«+i/aa..... a./a.), and n
h0(a) = -In
E
V=0
aQ
Y(o)
(22)
is a function summable with respect to the standard measure on P. 4 By Jensen's formula (9) in §4 we have n
Nf(a, r) _
j In E a., f or -In E a,, ,
0
'0'
I
§13. RESULTS USING CAPACITY
179
from which (20) follows immediately with the expressions (21) and (22) for ua and ha. The plurisubharmonicity of the function ua in local coordinates Si = ao/aa, , Sn = an /a,, follows from the plurisubharmonicity in C' (S) of
InIfa+f0Sl+...+fa-1Sa+fa+ica+1+...+fncnl for any fixed parameters fo,... , fn, and from the fact that integration with respect to z does not destroy plurisubharmonicity. That ha is summable can be seen by observing that this function on Pn has only logarithmic singularities
on the hyperplanes as = 0 and >2 a
0.
The next theorem is proved as the analogous theorem is proved in R. Nevanlinna [1], 1st ed., Paragraph 225, 2nd ed., Paragraph 233.
THEOREM 1. For any nondegenerate holomorphic mapping f : Cm
Pn
and for all hyperplanes a E Pn, except for a P-polar set E of them, the counting function N f(a, r) satisfies the inequality
Nf(a,r) > Tf(r) =
Tf(r) InTf(r),
(23)
for r > ro (a), where Tf (r) is the characteristic function.
t We set )t(r) = 1 Tf(r)1nTf(r); this function is clearly increasing, as is the function Tf(r) - .1(r). Using this, we can construct step by step an increasing sequence of numbers rk -+ oo such that (24) k = 0,1..... Tf(ro) > 1, Tf(rk+1) - A(rk+l) = Tf(rk), We set E, _ { a E Pn : N f (a, r) < T f(r) - A(r) } and prove that for a V E,k and r c Irk, rk+1]
Nf(a,r) > Tf(r) - 2A(r).
(25)
In fact, since Nf is increasing, for r > rk and a V E,k we have Nf(a,r) ? Nf(a,rk) > Tf(rk) - a(rk) t
1
Tf(r) - 2A(r) + [Tf(rk) - (Tf(r) - A(r))] + IA(r) - A(rk)],
and since Tf - A and A are increasing, we have Tf(r) - A(r) < Tf(rk) for
r < rk+1 and A(r) > A(rk) for r > rk. Thus under our conditions, both bracketed expressions in the last relation are positive. Discarding these, we obtain (25).
Let 000
00
E=(I
UE,k
I
(26)
j=1 k=j
and the hyperplane a ¢ E. Then a ko can be found such that a V Ukko Ell , i.e., a V E,k for all k > ko. According to what was proved above, it follows from this that (25) and thus the equivalent inequality (23) are true for all r > rk0.
V. FURTHER RESULTS
180
It remains to prove that the set E defined by (26) is P-polar in P". From (20), (21) and the Nevanlinna inequality (subsection 2 of §4) it follows that f o r points a = [ao, ... , a,,] E Ua
ua(a,r) = Nf(a,r) - ha(a) < Tf(r) +In
IaI Iaf(0)I
Therefore, if we fix an arbitrary bounded domain G C UQ and denote by c1 the greater of the two numbers 0 and maxaEC ln(Ial If(0)I / IaQI), then for fixed r the plurisuperharmonic function vQ (a, r) = (T f (r) - ua (a, r) +cl)/A(r) is nonnegative on G. On the set Er, on which by definition Nf (a, r) < Tf (r) A(r), we have va(a,r) =
Tf(r) + h,,(r) - Nf(a,r) + cl > 1
h«(a) + cl
-
A (r)
.1(r)
Since h0(a) + cl > 0 for a E G, the inequality va(a,r) > 1 is true for
aEE nG.
Thus the function va is admissible for P-measure, and consequently w, (a, E, n G, G) < va (a, r), while the P-capacity
c(E, n G) = u(G) Jc w. (a, E, n G, G) dp < k(GI fi(r) +
J{2,f(r) + In
1
I
W(G)a(r) c
{c1 - in
1
f
va (a, r) dp
u(G) JaIIIaf (I )I - U,, (a, r) l dp
lal if (0)I IaQI
dt. 1
Using the fact that the second integral is nonnegative and that the first integrand is nonnegative on all of Ua, we get
c(Er n G)
M the set E of divisors of holomorphic sections of L which are defective in the sense of Valiron is a P-polar set in the projective space of all such divisors. We observe that independently of A. Sadullaev an analogous result has been
obtained by Ronkin [2]. The latter result is weaker, since Ronkin obtained it in terms of the I'-capacity which he had introduced, but it applies to a somewhat larger class of mappings. 4. On the Bdxot&t problem. As we observed in Chapter II, for sets of codi-
mension greater than 1 the so-called transcendental Bezout theorem is in general not true, that is, there does not exist a general estimate from above of the counting function in terms of the characteristic function (see the example of Cornalba and Shiffman in subsection 4 of Chapter II). However, such an estimate becomes possible if we neglect a set which is in some sense thin. We present here a result of Carlson [3] which gives an estimate in terms of the
V. FURTHER RESULTS
182
characteristic function for the counting function of the preimages of points C'n. The thinness of the exceptional under holomorphic mappings f : C" set is formulated in terms of what is called the a-capacity. One can find out about this, for example, in Landkof's book [1]; we will limit ourselves only to the definitions and formulations of the properties that we will need. We consider in C" a kernel of the form & (z) = 1/ IzI', where 0 < a < 2n, and a measure µ concentrated on the set E; the potential and the energy of this measure are defined to be, respectively, VA (z) = fc K. (S - z) dµ(S),
'
'(µ) = f n V, (z) dµ(z)
(29)
For a given compact set E C C' there exists what is called an equilibrium distribution of the measure; this is a measure d1 with support E with µ(C") =
p(E) = Q such that the energy is minimal. It is unique for fixed Q (see Landkof [1], Chapter II, §1.3). We denote by V = maxVV(z) the potential of this measure, and call the a-capacity of the set E the quantity ca (E) = Q/V. (30) This capacity has a number of properties common to capacities; in par-
ticular, it is monotone and countably subadditive, and property 6° of the preceding subsection is true for it. It is related to the Hausdorff measure of sets, which is defined in the following manner. Cover the set E C C" with a finite or countably infinite set of balls with centers zj and radii rj, and set HQ,b (E) = inf >f rJ , where the lower bound is taken over all coverings with radii rj < 6. The Hausdorff measure of order a (briefly, a-measure) of the set E is defined to be HQ (E) = limbo Ha,b (E). If E is a real k-dimensional manifold, then the k-measure is proportional to
its volume. Moreover, H(E) = oc for a < k and H,,(E) = 0 for a > k. The link between a-capacity and Hausdorff measure is expressed, in particular, by the following fact: the Borel sets of zero a- capacity also have zero (a + E)measure for any E > 0 (for a proof see Landkof [1], Chapter III, §4). We pass now to the presentation of Carlson's result. The Nevanlinna inequality for holomorphic mappings f : C" -p C" and the preimages of points a E C" has the form
Nf(a,r) 0. If the function Nf(a, r) is continuous in r (for example, if f -1(a) is discrete), then the sets Ek defined in (36) are open and thus E is a countable intersection of open sets (a set of type G,5). Such sets are locally polar, i.e., for every point of E there is a neighborhood U and a superharmonic function on it, not identically equal to oo, but equal to oo on E n U (see Landkof [1], Chapter III, §1.1). In the same paper of Carlson [3), this theorem is generalized to preimages of points under holomorphic mappings of m-dimensional Stein spaces to C" and also to inverse images of planes A C P' of codimension k under mappings
in Pn, where in the last instance Nf(A,r) is estimated in terms of Tj(k)(r).
§14. Mappings of finite order For simplicity we will limit ourselves to holomorphic mappings f : C'" C". If as in Chapter II we denote
Mj(r)
EB
1 + ff(z)I2,
(1)
then the order pf and the type a f of the mapping f can be defined by the standard formulas
pf = li 00
hzlnlMj(r)I nr
of = rli hzM,(r) -00
(2)
This section is devoted to mappings of finite order and some special classes of such mappings.
V. FURTHER RESULTS
186
As we know, the kth
5. Estimates of characteristic functions from above. characteristic function T(k)(r)
-k,
dt
J
JBt(f*W)k AWE
(3)
is responsible for the distribution of inverse images of planes A C C` of complex codimension k under holomorphic mappings f: Cm -+ Cn. Here f *(w) = ddc ln(1 + If 12) and wo = ddc In Izl2 (see subsection 3 of §4). Our immediate task is to describe an estimate for Tf(k) in the case of mappings of finite order obtained recently by Degtyar' [4]. His proof is based on a lemma; to formulate the lemma, we set Br(a) = { z E C" : Iz - al < r } and SS(a) = aBr(a) and introduce the form wa = ddc In Iz - a12. LEMMA 1. Let u be a positive plurisubharmonic function of class C2 in
Ct, and let 4) be a closed (m - k, m - k) form such that for any integer I, 0 < I < k, the form (ddcu)k-1 A wa A it is positive and integrable. Then for any integer 1, 1 < l < k, and any 0 > 1, t
f
(ddcu)k A-0 0, then If (rz)l grows like er°h!(z) as r increases. If this limit were to be attained uniformly on the whole sphere Si and were everywhere positive, then we would have a simple estimate from below for If (rz) I and thus for T f (r). However,
such an assumption is too restrictive; for example, if there is a sequence of points in f -1(0) converging to infinity, then the limit of (9) over the sequence equals 0. Therefore, it is natural to relax the requirement of a uniform limit, dropping it on some small set. Following Degtyar', we will say that a set E C C'
has relative q-measure 0 if it can be covered by a system of balls B(J) = { B,, (aj) }jEJ with centers aj and radii rj < 1, satisfying the following condition: if JR C J is a set of indices such that the system B(JR) covers the intersection of E with the sphere SR, then
RXD R9 3EJR rig = 0.
(13)
.
As q grows, the quantity inside the limit sign clearly decreases, so if a set has zero q-measure, then it also has zero q'-measure if q' > q. We will say further that a holomorphic mapping f : Cm - C" of finite order p and finite type with regularized indicator hf is a mapping with q-regular growth if there exists a set E C C' of zero relative q-measure such that the following limit exists uniformly on the sphere S1: rlim00 rzoE
In
1+If(rz)12 p rp
= hf(z).
V. FURTHER RESULTS
190
We note that if f is a mapping with q-regular growth, then so is f - b for any b E C". In fact
rpIn
11+flf()-b12
1+If(rz)-b12=rpIn l+If(rz)I2+2r P
and the limit of the second term as r -f oo equals 0; this can be seen from the following elementary inequalities:(") 1/2(1 + IbI2) < (1 + If
-
b12)/(1 + IfI2) 5 2(1 + IbI2).
REMARK. Usually the indicator is defined as lim In If(rz)I = hf(z) rP r- oo
(15)
If for z E S1 in (15) there exists an ordinary rather than an upper limit, then it can be seen from what was said at the beginning of this subsection that the indicator introduced by us is hf(z) = I hf(z), 0,
if hf(z) > 0, if hf(z) < 0.
Levin and Pfluger (see Levin [1]) introduced and studied a class of entire functions f : C -+ C of completely regular growth, for which there exists an ordinary limit in (15) except for a certain exceptional set. Azarin [1] introduced
exceptional sets in R- (our definition (13) is a generalization of his) definition and on the basis of this defined subharmonic functions with competely regular growth. Using this definition, Agranovich and Ronkin [1] generalized the concept of functions of completely regular growth to functions of several complex variables and proved in particular that their class includes the class of functions f : Cm C introduced by Gruman [1]. These latter functions are those for which f (Az), for almost all z E C"`, is a function with completely regular growth in the variable A E C. Degtyar' [3] proved that all the functions investigated by these authors are functions of (tin - 1)-regular growth. I To obtain estimates from below for the characteristic functions of mappings of q-regular growth, we need two lemmas. The first of these is a slight modification of a well-known theorem of Hartogs (see Shabat I, p. 313). (4)The right inequality follows from the fact that
1+If+b12 0 and K C G, there is a to such that
forallt>to andzEK.
ut(z) 0, points zi E K and functions uj = ut, such that
Without loss of generality we can assume that zi --i z° E K. Let the ball B2,.(z°) with center z° and radius 2r be relatively compact in G; then, by the subharmonicity of the uj, the mean value v(zJ) +'F.
uJ
m rm
If I zi - z° I < r, then, by the nonnegativity of uj and the monotonicity of the mean value,
v(z') + E
C" be a holomorphic mapping with q-regular
growth. Then fork < m + 1 - q/2, Tjk)(r) > Hfk)rk" + o(r' ),
(21)
where p is the order of the mapping.
4 By Lemma 2 in §1, Tfk)(r) = 1 f ln(1+If12)wf-1 A k-1 2 ,
-If 2
B
ln(1 + If12 )wk-1 A wo -k+1
f
(22)
where w f = dd° ln(1 + I f I2). Let E be the exceptional set in the definition of q-regularity. Since f has finite type, we have In Mf(r) < ArP with some constant A, and by Lemma 3 (in which u(z) = ln(1+If(Z)12 ) and k is replaced by k - 1) we get that ln(1+If12)wf-1
TAP f
AQk-1
1 we get
.
Or
J
InOtk-1(r)
0, then
or converge to zero as r from the identity
2
1 + If I2 = 1 + e2rm(s) E F,
cAeir Im(S,A)
L=1
where b > 0, it can be seen that in (34) there exists an (ordinary) limit equal to m(s), provided that the following conditions are not satisfied simultaneously:
F,
cAe(s,A) = 0,
v = 1,...,n,
(35)
AEA, (c)
where z = rS. But all these conditions cannot be satisfied simultaneously with respect to z; consequently, the exceptional set E described by them is an analytic set of codimension at least 1. In particular, for every S E S1, the intersection of E with the real line { z = tS } consists of at. most countably many points. Thus the upper limit in (34) always exists and the indicator hf(S) = m+ (S) = max(m(S),0). As can be seen from its definition, m+ (S) is the support function for the convex hull of the set A U {0 }, so this convex hull coincides with the body G f introduced above.
Since the conditions in (35) have a special character, in the situation of general position the exceptional set E is empty. In this situation there exists an ordinary limit in (34), and it is attained uniformly on the sphere S1, so the mapping (33) has 1-regular growth. However, even in the case where the set E is nonempty, the mapping (33) can have 1-regular growth. Let us consider, for example, the mapping f : C2 C2 given by the functions f1 (z) = e" + e",
.
f2 (z) = eiz. - eZ' ;
It is nondegenerate, since Jf (z) = -ez' (ez' + ietzl) 0 0, and its indicator hf (S) = max(Re c1, Re c2, -Im S1, 0).
The exceptional set E consists of points (z1, z2), where
z1 = (7r/2 + kiir)(1 - i) and z2 = (7r/2 + kiir)(1 + i) + 2k2?ri, with k1 and k2 integers. Only on one real line passing through z = 0, namely, on the line l° _ { z = tS° }, where S° = ((1-i)/2, (1+i)/2), are there infinitely
many points of E; moreover, f = 0 at these points. Since hf(S°) = 1/2, for S = 0 ° in (34) only the upper limit exists. But it is not difficult to verify that the points of E fl 1° can be covered by balls whose union has zero relative 1-measure. Thus, outside these balls, the limit in (34) is attained uniformly on the sphere S1. Thus (33) is a mapping with 1-regular growth.
§14. MAPPINGS OF FINITE ORDER
203
EXAMPLE 2. The mapping f : C' -f C' realized by sums of exponentials with polynomial exponents, ff(z)
ePk-lzl,
=
v = 1,...,n,
(36)
kEJ where J, is a finite set of natural numbers and the Pk, are polynomials, was considered by Degtyar' [3]. He proved that in the situation of general position such mappings also have 1-regular growth. I
SUPPLEMENT
A Brief Survey of Other Work Here we wish to describe briefly some results in multidimensional value distribution theory which did not appear in the main text of the book. We begin with holomorphic curves. This is the part of the theory which is most closely connected to the one-dimensional theory and the part which is best developed.
The theory of entire curves f : C -i Cn is the object of the work of V. P. Petrenko [1] - [7] and his students (Krutin' [1], Krytov [1], Babets [1] et al.). One aspect of this work is that instead of the proximity function m f(D, r),
which measures the deviation of f from the hyperplane D in the integral metric, the function
Lf(D,r) = maxln If (z)Ilal IZI =T
(1)
I(f (z), a) I'
is introduced; it evaluates the deviation in a stronger uniform metric (a E C" is the vector defining D as a hyperplane in Pn-1 in homogeneous coordinates). Instead of the defect in the sense of Nevanlinna or Valiron, the quantity Of (D) = lim
Lf(D,r)
T-X Tf(r)
'
(2)
appears; it is called by Petrenko the deviation of f from D. It is clear that the Nevanlinna defect bf (D) < Of (D), while for curves
f of finite lower order it is proved that if the Valiron defect t f (D) = 0, then Of (D) = 0 also. Thus the research on the deviation carried out by Petrenko and his students gives information also about the defects of holomorphic curves.
Holomorphic curves are related to algebroid functions-that is, multivalued analytic functions of one variable z which are defined by polynomial equations in w + An(z) = 0 (3) Ao(z)wn + with entire coefficients A3 . To every such function is associated the holomorphic curve [A0,.. , An]: C - Pn. On the basis of this relation, Petrenko has .
205
206
SUPPLEMENT
established a member of properties of algebroid functions. He has also applied his results to the study of the asymptotic behavior of solutions of linear differential equations of ntli order with entire coefficients and to the study of algebroid solutions of algebraic differential equations. One can become acquainted with this research in Petrenko's book [6] and his later paper [7]. In the work of E. I. Nochka a defect relation for holornorphic curves f : C P" is presented, which takes into account multiplicity and degeneracy. One says that f intersects a hyperplane D = { [wj E P" : a0w0 + + a"w,1 = 0 } with multiplicity v if all the zeros of the functions fD = (f, a) have orders at least v and if at least one zero is of order v (if f (C) C D or f (C) fl D = 0, then v is considered to be oo). The curve f is called k-nondegenerate if f (C) is contained in some k-dimensional subspace of Pn but is not contained in any subspaces of lower dimension. Then this is true: THEOREM. Let a k-nondegenerate curve f: C ---> P" be given and let the Dj C P" be q hyperplanes in general position. If f intersects every Dj with multiplicity vj, then
/
E+1q1
k1 0, except for r in an at most countable union of intervals of finite total length,
f s,
ln+
I a f/azi l. Ill
Q
=fsr ln+ I foafi /azi - fi afo/az; I Ifofil
(8)
n + 1 hyperplanes
D3 C P" in general position and any r > 0, except for those in an at most countable union of intervals of finite total length, q
ENf(Df,r) > (q - n - 1)Tf (r) + N(S, r) - blln r -b2,
(9)
j=1
where S is the divisor of stationarity and b1 and b2 are constants (cf. (33) in §5). From this, for arbitrary dimensions, he proves in the standard manner the usual defect relation: the sum of the defects b f (Di) of hyperplanes in general position does not exceed n + 1. We observe also that Griffiths and King [11 give a generalization of the concept of the logarithmic derivative to holomorphic mappings. Let M be an n-dimensional complex manifold and let 0 be a meromorphic (n, 0)-form on it whose polar divisor D has self-intersections in general position. (In the case of
SUPPLEMENT
210
M = P' with homogeneous coordinates [W0,. .. , (w1, .... wn), one can take
and affine coordinates
n
fI
E(_l)vW,dWo n ... A dli;,_1 n dWv+1 n ... A dWn
wo...wii a
dw1 A ... A dwn
w1...wn For a holomorphic mapping f : Cn -> M the pullback f * (f2) = A f (z) dz1 A A dzn and the quantity
vf(r)
= f In+IAfla
(10)
replaces the integral which appears in the lemma on the logarithmic derivative
(for mappings f: C -> PI we clearly have )t f(z) = f'(z)/f (z)). For this quantity, in Griffiths and King [1], p. 211, an estimate is obtained, which is however weaker than (9). For the case of curvilinear divisors, the defect relation for mappings which raise dimension is still insufficiently worked out. One of the first results in this direction is due to Shiffman [4]. Let q distinct irreducible hypersurfaces Dj be given in Pn which are defined by homogeneous polynomials of degree p;
suppose further that the D; intersect in general position, in particular that no more than n of them pass through any given point.. The Veronese mapping (see Shaferevich [1], Chapter I,§4), which is realized by monomials of degree p,
imbeds PI in the space P", where N = (nY P) - 1, so that the images of the D3 lie on hyperplanes_ Therefore, regarding the nondegenerate holomorphic mapping f : C"` Pn as a mapping to P'^', and using the defect relation for hyperplane divisors, we obtain the trivial estimate a
Ebf(D;) - n+p \ P
a=1
(11)
This estimate is clearly not optimal. The optimal estimate for the sum of the defects for a nonconstant holomorphic mapping f : C'n _ Pn is clearly 2n (see the right-hand side of (4) for k = 1). However, in Shiffman [4] this is proved only for an extremely special class of mappings. In conclusion, we briefly indicate some work where holomorphy is replaced by some metric-topological conditions. The first research in this direction was done by Ahlfors [1], who built up a theory of covering surfaces-a geometrical analog of Nevanlinna theory-- which is true not only for conformal mappings but also for the more general quasiconformal mappings of Riemann surfaces. Schwartz [1], [2], extended some of the results of this theory to the case of multidimensional real manifolds.
SUPPLEMENT
211
I. M. Dektyarev [1] [4] considered mappings from open orientable manifolds
to compact Riemannian manifolds of the same dimension, where conditions close to quasiconformality were imposed on the mappings. In particular, lie obtained sufficient conditions for quasi-surjectivity of such mappings; several of his results are related to complex manifolds and holomorphic mappings, Ronkin [2] considered manifolds with mixed structure--real-analytic fiber bundles over real-analytic manifolds whose fibers are complex lines. Such bundles are mapped real-analytically to compact complex manifolds, and the mappings are assumed to be holomorphic on each fiber. In this situation he obtains a generalization of the first main theorem in the form of Griffiths and proves a theorem about the set of defective divisors in the sense of Valiron (see subsection 3 of Chapter V).
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