Analytic cycles and vector bundles on non-compact algebraic varietiespublications.ias.edu/sites/default/files/analyticcycles.pdf · 2010-12-16 · Analytic Cycles and Vector Bundles

Inventiones math. 28, 1 - t06 (1975) �9 by Springer-Verlag 1975

Analytic Cycles and Vector Bundles on Non-Compact Algebraic Varieties*

Maurizio Cornalba (Pisa) and Phillip Griffiths (Cambridge, Mass.)

Table of Contents w o. Introduction

A. Orders of Growth on Algebraic Varieties . . . . . . . . . . . 4 w 1. Review of the Classical Theory . . . . . . . . . . . . . 4 w 2. Generalization to Algebraic Varieties . . . . . . . . . . . 6 w 3. Exhaustion Functions and K~ihler Metrics on Special Affine

Varieties . . . . . . . . . . . . . . . . . . . . . . . 9 84. Order of Growth of Analytic Sets . . . . . . . . . . . . 13 w 5. Order of Growth of Holomorphic Mappings; the First Main

Theorem of Nevanlinna Theory . . . . . . . . . . . . . 15 w 6. Orders of Growth of Holomorphic Vector Bundles . . . . . 17

B. Holomorphic Vector Bundles with Growth Conditions . . . . . 22 w 7. Statement of Main Results about Algebraicity of Vector

Bundles . . . . . . . . . . . . . . . . . . . . . . . 22 w 8. Statement of Main Results about Transcendental Vector

Bundles . . . . . . . . . . . . . . . . . . . . . . . 26 w 9. The Kodaira Identity and Consequences . . . . . . . . . 28 w 10. Applications of Nevantinna Theory . . . . . . . . . . . 31 w l 1. Proof of Theorems I and III(ii) for Affine Varieties . . . . . 35 w 12. Proof of Theorem [I . . . . . . . . . . . . . . . . . . 37 w 13, Proof of Theorem ltI . . . . . . . . . . . . . . . . . 42 w 14. Proof of Theorem IV . . . . . . . . . . . . . . . . . 47 w 15. Proof of Theorem V . . . . . . . . . . . . . . . . . . 50

C. Applications to Algebraic Geometry . . . . . . . . . . . . . 52 w 16. G.A.G,A. and Serre's Theorems A and B . . . . . . . . . 52 w 17. An ApproximationTheorem . . . . . . . . . . . . . . 53 w 18. The Algebraic de Rham Theorem . . . . . . . . . . . . 56

D. Grauert 's Theorem and the Oka Principle . . . . . . . . . . 61 w 19, Complex Structures on Vector Bundles and Holomorphic

Homotopies of Complex Vector Bundles . . . . . . . . . 61 w 20. Statement and Proof of Grauert 's Theorem . . . . . . . . 64

E. A Lower Bound on the Growth of Analytic Subvarieties Represent- ing a Given Homology Class . . . . . . . . . . . . . . . . 69 w 21. Statement of Theorem VII . . . . . . . . . . . . . . . 69 822. Proof of Theorem VII . . . . . . . . . . . . . . . . . 73

" This research was partially supported by NSF grant GP 38886. I Inventiones math,, Vol. 28

2 M. Cornalba and P. Griffiths

F. Discussion of Some Open Problems . . . . . . . . . . . . . 75 w 23. Runge Problem for Analytic Sets . . . . . . . . . . . . 75 w 24. Extension Problem for Analytic Sets . . . . . . . . . . . 78 w 25. A Characterization of Algebraic Varieties . . . . . . . . 79 w 26. On the Growth of Holomorphic Vector Bundles and Related

Matters . . . . . . . . . . . . . . . . . . . . . . . 80 w 27. Order of Growth and Hodge Theory . . . . . . . . . . . 82

Appendix 1 : Chern Classes of Holomorphic Bundles and the Funda- mental Class of an Analytic Subvariety . . . . . . . . . . . . 86

Appendix 2: Vector Bundles and Chern Classes/ll) . . . . . . . . 9l

Appendix 3: Solving the c5-Equation with Dependence on Parameters . . . . . . . . . . . . . . . . . . . 99

w 0. Introduction

a) In this pap6r we shall use funct ion theory with growth cond i t ions to s tudy some ques t ions on a lgebra ic variet ies which have analyt ic , but not necessar i ly a lgebraic , solut ions. The ma in such ques t ion arises from the obse rva t ion that , on a s m o o t h affine var ie ty A, the c o h o m o l o g y H2k(A, Q) is genera ted by fundamen ta l classes of ana ly t ic subvar ie t ies 1. These subvar ie t ies canno t in general be a lgebraic , and one of ou r main discover ies is that the t r anscenden ta l level of the subvar ie ty may be related to the H o d g e type of its fundamen ta l class. We are able to p rove a lower b o u n d for this (Theorem VII), and to show that the lower b o u n d is sharp for divisors (Theorem V). However , we could not find good upper b o u n d s for h igher cod imens ion cycles, and so must be content to s ta te some conjec tures whose so lu t ion would comple te our p r o g r a m (w167 26 and 27).

Once we began the s tudy of cycles f rom a " transcendental po in t of view, we were soon led to the conclus ion tha t an ex t remely large number of na tu ra l ly posed geomet r i c p rob l ems on affine a lgebra ic variet ies admi t ana ly t ic but not necessar i ly a lgebra ic solut ions. To p rope r ly unde r s t and the na tu re of these solut ions, it seems necessary to p rove the Oka principle with growth conditions on such a var ie ty z. The p r imary example of the O k a pr inc ip le is the t heo rem of G r a u e r t [4], which states that on a Stein mani fo ld M the na tu ra l m a p p i n g

(0.1) Vectho ~ (M) ~ Vect,o p (M)

is a b i jec t ion of sets. One of our main p r o b l e m s was to prove (0.1) with growth cond i t ions in case M is an affine variety. A l t h o u g h unab le to do this with wha t p r e s u m a b l y are the correct g rowth condi t ions (except

A complete proof of this fact is given in the appendices to this paper. 2 Roughly speaking, the Oka principle states that, on a Stein manifold M, an analytic problem has an analytic solution if, and only if, it has a topological solution.

Analytic Cycles and Vector Bundles on Non-Compact Algebraic Varieties 3

in the case of line bundles), we are able to give a new proof of Grauert's theorem based on the LZ-methods for the 0-operator and a certain lineari- zation trick (w167 19 and 20), a proof which does give some growth conditions and which may suggest a better method for treating the problem (cf. w 26).

In addition to an estimate on the growth of cycles in a given cohomology class and proof of Grauert's theorem, a third main part of this paper is a systematic discussion of the order of growth of holomorphic vector bundles on algebraic varieties. Included are i) a characterization of algebraic vector bundles and their cohomology (Theorems I and II), and ii) a proof of the equivalence of several definitions of growth for line bundles (Theorem III; we do not know the analogous result for vector bundles of rank > 1). Numerous examples ([11]) together with our work here and further heuristic reasoning, also given below, indicate that the finite order category is the correct one in which to prove the Oka principle for affine varieties. Moreover, the precise order of growth should, in many cases, be an integer which can be calculated by Hodge theory. It is this principle which has been our working hypothesis throughout.

As a fourth part of this paper (Section C) we have in w167 16-18 used the general techniques to prove some linear results in algebraic geometry. These include Serre's G.A.G.A and special cases of his Theorems A and B for locally free algebraic sheaves, Grothendieck's algebraic de Rham theorem, and an approximation theorem for sections of an algebraic vector bundle over an affine variety.

A final part of our work (Section F) is devoted to formulating a few of the many open questions which arose in our study. We are able to prove special cases of all these problems and to give additional heuristic arguments for most, but the general questions eluded us, for reasons due primarily to a lack of techniques for handling global non-linear geometric problems in codimension > 1.

b) We want to discuss briefly the two main techniques, the Nevanlinna theory and the LZ-estimates for the O-operator, which we have used in this paper. The latter gives a good method for handling linear problems with growth conditions. All that is really necessary beyond the techniques in Chapter IV of [20] is to put the basic estimate in intrinsic geometric form, which is accomplished by the Kodaira identity (w 9), and then to carefully construct suitable plurisubharmonic exhaustion functions and complete Kiihler metrics with "bounded curvature" on a non-compact algebraic variety (w 3), and find a suitable "partition of unity" argument for the Zariski topology (w 13).

The Nevanlinna theory is used to relate the order of growth of analytic sets (w 4) to the growth of the defining functions, and is reviewed in w 5. Due to the Cornalba-Shiffman example (cf. [5]), this theory works 1"


well only in codimension one, and this is perhaps the basic reason why we are unable to do very much in the general case. When combined, these two techniques give, as far as they go, a very nice method for studying non-compact algebraic varieties. However, because of the failure of the analytic Bezout theorem, more subtle techniques seem necessary for higher codimension questions.

c) To conclude this introduction, we wish to give a few remarks concerning the organization and contents of this paper, i) In w167 1 and 2, we have recorded some historical and informal remarks concerning growth conditions on algebraic varieties. This area does not seem to be widely known among algebraic geometers, and it therefore seemed a good idea to establish a frame of reference for our later work. ii) In the appendices we have given a proof of the fact that, on a Stein manifold M, the even cohomology H2k(M, (~) is generated by analytic subvarieties. This result follows" formally from

a) Grauert 's theorem (a proof of which is given in w 19),

b) standard results concerning the fundamental class of an analytic subvariety and Chern classes of holomorphic vector bundles (Appendix I), and,

c) the Atiyah-Hirzebruch spectral sequence in K-theory (our reference for this is [8]).

The latter is based on Bott periodicity, a deep result on the 2g-homotopy of the unitary groups. However, all we really need is a Q-version of Bott periodicity, and it seemed a good idea to give a complete proof of this; one intended for non-topologists. Based on ideas of John Morgan and Dennis Sullivan, we have given such a proof in Appendix II.

A. Orders of Growth on Algebraic Varieties

w 1. Review of the Classical Theory

In classical complex function theory, the concept of the order of growth is of fundamental importance. In order to establish a frame of reference for this paper, it is useful to review what for our purposes are the essential aspects of this theory:

a) Associated to an entire meromorphic function f(~), there is an order function [28] T O (f, r), a convex function of log r which measures the growth o f f In particular,

(1.t) f(~) is rational <=~ To( f, r)= O(log r).

b) In case f(~) is holomorphic and we set

M ( f r) = max log Jf(()t [(l =< r,


one has To~, r)< M(f,,r)+O(1)

(1.2) M(f,r)<CoTo(f, Or)+O(1 ) (0> 1).

c) If n(a, r) is the number of solutions to the equation f (O=a in the disc t~l =<r and

r N(a, r)= ~ {n(a, t ) - n(a, 0)} dt + n(a, 0)log r

o t

is the logarithmically averaged counting function, then

(1.3) N(a, r) <= To(f, r) 4- O(1, a).

This implies the less symmetric estimate

n(a,r)<CoTo(f, Or)+O(1, a) (0> 1).

d) Given a divisor O = ~ n ray (n~>0) on C, we may define the 10

counting function N(D, r) as above. Then there exists an entire holomorphic function f ( 0 with f - 1 (0) = D and where T O (f, r) may be estimated by N(D, r). In case D hasfinite order in the sense that

N(D, r )= O(r ~) for some 2, we set [lo..,O,r:]

ord (D)= ,~o~lim log r '

and then f may be chosen to have order equal to ord (D).

The one variable theory has been extended to divisors and meromorphic functions on 112" by Lelong [26], Stoll [33], and Skoda [32]. Again there is defined an order function T o ~ r) associated to an entire meromorphic function f (~l , - . - , ~,) such that (a) and (b) remain valid. To give the generalizations of (c) and (d), we let D be an effective divisor (i.e., all multiplicities are non-negative integers) on ll~" and D Jr] the intersection of D with the ball II?" [r] of radius r. Then we define

v(D[r]) n(D,r)=al r2,_2 ,

" dt N(D, r) = ~ {n(D, t ) - n(D, 0)} t + n(D, O) log r

0

where v(D [r]) is the Euclidean volume of D(r) and cq is a constant such that n(lI?" 1, r )= 1. With these conventions, (c) and (d) (in the finite order case) remain valid.


In the case of several entire holomorphic functions

A (0 . . . . . L(Oee(~"),

the analogous estimate to (1.3) on the size of the set fl . . . . = f k = 0 of common zeroes in terms of the growth of thef~'s is false [5]. This failure of the analytic Bezout theorem creates a major difficulty in the theory.

On the other hand, given a codimension k analytic subvariety Z ~ ~ ' , we set 3

v(Z[r]) n(Z,r)=~ r2,_2k ,

r dt N(Z, r )= ~ {n(Z, t ) -n(Z , 0)} t + n ( Z , 0)log r

0

as before. In case Z has finite order, it has been proved by Skoda [32] that Z is the set of common zeroes of n+ 1 functionsf~ecg(IE') ( i=0, ..., n) with o r d f = ord Z.

The effect of this positive result is perhaps somewhat muted by the lack of unity present in the codimension one case:

If D is a divisor of finite order 2 and f, g are two entire holomorphic functions of order 2 defining D, then

f =eP g

where P(0 is a polynomial of degree [2] (=greatest integer in 2). This follows from (1.2) and the relation [28]

To(f/g, r)<= To(f, r)+ To(g, r)+ 0(1).

In the higher codimension situation, the analogue of this result is unknown and may even be false for reasons similar to the failure of Bezout.

w 2. Generalization to Algebraic Varieties

There seems to be no essential difficulty in generalizing the above theory to an algebraic variety A (unless mentioned to the contrary, we shall always assume that A is smooth). One may first cover A by finitely many affine open subsets A~ and measure growth by doing so on each A~. In case A is an affine algebraic variety, and is thus an n-dimensional algebraic subvariety of ~ , we may find a large number of finitely sheeted branched algebraic coverings

re: A--+ (E"

3 It is a result of Lelong that n(Z, r) is a non-decreasing function of r and Thie proved that n(Z, O) =~im ~ n(Z, e) is the multiplicity of Z at the origin ([13]).


and essentially reduce questions on A to 112". There would seem to be little point in carrying out this generalization except for the facts that:

a) A may be topologically very complicated, and

b) it follows from K-theory and a result of Grauert that the cohomology H 2k (A, ~ ) is generated by fundamental classes u z of codimension k analytic subvarieties on A (cf. Appendix 2 below). In general we cannot take Z to be algebraic, and in fact there is a lower bound on the growth or transcendental level of Z in terms of the Hodge type of the cohomology class u z (cf. w 21).

What this suggests is that we study growth questions on A but in contrast to the classical case, we should perhaps concentrate on mini- mizing growth within the collection of analytic objects representing the same topological situation. Examples of this would be (a) analytic subvarieties representing a given homology class, and (b) holomorphic mappings f : A ~ G (r, m) (= Grassmannian of m - r planes in t12") representing a given topological vector bundle. Some examples of this are discussed in [11], and in this paper we shall present further results on the general question.

The affine varieties A we shall use are usually not of the most general sort, but will be special affine varieties defined as follows:

Definition. Let V be a smooth projective variety and D an ample divisor with simple normal crossings 4. Then the complement A = V - D is a special affine variety.

Remarks. a) Any algebraic variety B has a finite open covering (in the Zariski topology) by special affine varieties, and so most questions concerning growth can be dealt with by restricting to special affines.

b) We let A = {l~l < 1} be the unit disc and A* =A - {0} the punctured unit disc. Note that the boundary circles Iffl= 1 are included in both cases. A k-fold punctured polycylinder P* is a product

p* =(A*) k x A "-k.

A special affine variety has a finite open covering by punctured poty- cylinders. Those with no punctures (k=0) are polycylinders lying entirely in A, and those with k > 0 represent neighborhoods at infinity.

Using punctured polycylinders to localize at infinity is a useful tool in studying growth questions on general algebraic varieties.

c) The simplest special affine varieties are complements of smooth ample divisors D.

In this case the neighborhoods at infinity are punctured polycylinders P*= A* x A "-1 having a single puncture. As a general rule, for no-

4 This means that D = D1 + ' " + DN where the irreducible components D i are smooth and intersect transversely.


D 2

A = V - (D t + D2)

Fig. 1. Punctured polycylinders

tational simplicity we shall consider only such special affine varieties, although the statements and proofs will work in general (cf. I l l ] for discussion of the general case).

d) The rason that the outer boundaries are included in the punctured polycylinders is that we are interested in growth behaviors as one tends to the divisor at infinity D; i.e., towards the puncture in the neighborhoods at infinity.

Fig. 2

Thus, questions about growth at the boundary circles are irrelevant, and so we shall assume that all holomorphic functions, analytic sets, etc. are defined across the boundary circles ]~1 = 1 of the various P*.

e) Finally, the reason for using special affine varieties is that these admit especially favorable exhaustion functions and metrics, which will be constructed in the next section.


w 3. Exhaustion Functions and KShler Metrics on Special Affine Varieties i) On ~" one has the two natural plurisubharmonic exhaustion

functions r I1~11 ( ~ c n ) ,

Zo =log(1 + ll(ll 2) ( ( ~ ' ) .

The first has E.E. Levi form ddCr satisfying the identity

(ddC~)"=O,

and is thus a generalization of a harmonic exhaustion function in one variable. The Levi form

dd~zo=CO

of the second is 2n times the restriction to r of the standard K~ihler form o) on IP", and is thus everywhere positive definite. We shall give analogues of qJ and % for any special affine variety A. More precisely, instead of ro we will frequently use the slightly modified function

z = C log(1 + II(IL2)- log(log(1 + I[(112)) 2 ,

which is still strictly plurisubharmonic and has somewhat more favorable properties near infinity than %, and the analogue of z will be given.

Before doing this we need to briefly discourse on the ubiquitous Poincar6 metric ~Op, on a punctured polycylinder P*. On the punctured disc 0 < [(1 < 1, the metric

d~ dr ~ % , = c - , fl 2 (/~> 1)

(3.1) 1(12 (l~ ~-~ - )

=c' dd~log (log 1@-) 2

has, for suitable c and c', constant Gaussian curvature - 1 . By the Poincar6 metric o)p, on a punctured polycylinder P* =(A*)k x A "-k, we shall mean the product of the o~d,'s on the punctured discs and the standard Euclidean metric on the remaining discs. The second equation in (3.1) suggests the use of potential functions of the form log(logu) 2 where u is a positive plurisubharmonic function.

Given a (1, 1) form r/on a special affine variety A, the notation

1/= O (%,)

shall mean that, relative to a suitable finite covering of A by punctured polycylinders Po*, the conditions

[~1[ ~ C ~p,


are satisfied where they make sense and for a suitable c~)nstant c>0. In case q is positive, the notation

r / ~ top.

shall mean that both r/=O(~op,) and m e , = O ( q ) where these estimates make sense.

(3.2) Proposition. On a special affine variety A there exists an exhaustion funct ion z whose Levi f o rm satisfies 5

dd~z ..~e)e,.

P r o o f Write A = V - D where D is an ample divisor with simple normal crossings on the smooth projective variety V. Let D = D 1 +- . . + D N where the D i are smooth, and choose metrics in the line bundles [Di] such that the induced metric in

N

[o] ~ | [Di] i=l

has positive Chern class (cf. w 0 of [13] for notation and terminology). Finally let 0.ie(9(V, [Di]) be a section which defines D i and has length ]ail< 1, a = a I | | a s , and define

1 (3.3) z = C log ~ - - l o g { ( l o g [0.112) 2 ... (log [aN[2)2}.

We will show that ddCz ~ cop, in the case where D = D 1 is smooth, referring to w 6 of [13] for the calculations in the general case.

To begin with, the C~(1, 1) form

1 1 1 2n ~ ddCl~ [0.]2'

is the Chern class of [D]-+`4, which by assumption is positive on `4. Locally, D is given by z = 0 for a suitable coordinate z on .4, and la[2= z. ge" where u is C ~176 By multiplying a by a constant, we may assume that I0.t < 1 on `4, and then

t ,, / log [a[~- T

= - - ~ { ~ ~ l o g l 0 . [ 2}

8 log [a[2 ^ t~log [0.12 9•log [0"12 = Jr

(log [0"[2) 2 log [0.12

5 The formula (3.3) for z will also be important for later use.


Now ~log10"[ 2 = d z + 0 u , and so Z

1 - d d ~ loglog [a]2

_ 1 / - 1 s dz/',d2 dz/xcg~ 3u^d~ (3.4) - 2 [[zl2(loglal2) 2 + z(loglal2) 2 + ~(10gl0-[2) 2

Ou^& } (log lal2) 2

6O + (log I0-[z) "

Using the inequality

2 ~ _ _ _ ~ 2 + 1 - - ~ ~ ,

the cross terms satisfy estimates

e l ~ - l d z A d g 1 6o<_l / - - ldzAd~<el /~- ldzAd~ [zlZ(log [0"12) 2 /3 -- [Zl(1og 10"12) 2 = IZl2(1og [aI2) 2

Moreover, the terms

1 - - + - - 6 o .

~U A (~U

and (log I 0"12) 2

6o

log la] 2

may be absorbed into C6o for sufficiently large C. Combining this with (3.4) gives the proposition. Q.E.D.

The analogue of log II { H on ~" may be found by considering a generic projection

relative to a suitable embedding A c C u, and then setting

~b (~) = log II zt (0 H (~ ~ A).

This gives a special exhaustion function in the sense of w 2 in [13], to which we refer for properties of r

It will be proved in Lemma (4.4) below that the essential properties of r vis fl vis growth questions are independent of the particular projection A --* ~".

ii) We now come to the construction of suitable K~ihler metrics on a special affine variety A. Given a Hermitian metric ds2=~gi~dzid~i with associated (1, 1) form i,i

g

i , j

where


on A, we consider the volume form

~0 = ~0" = det(gij) 1- [ dz iAd , i = 1

and its corresponding Ricci form

Ric q~ = dd c log(det gij)

(cf. w 0 of [13] fora general discussion of volume forms and their associated Ricci forms).

(3.5) Proposit ion. There exists a complete K~hler metric ~o on A whose associated Ricci form satisfies

Ric q) = O (~oe,).

Proof For q0 we take the positive (1, 1) form ddCz constructed above. Since the Poincar6 metric is complete, it follows that (p is complete and it is obviously K~ihler. Referring to (3.4) we see that

n i = 1

( l o g 1 0 [ 2 ) 2 c~ + log 10-[ 2 fl+T hl h - z

Iz[Z(log Io'[2) 4 ]z[Z(log I0"t2) 4

with ~, fl, 7 C ~ functions and ~ > 0. Now d d c log (log I0"[z)z is 0 (oge,) by the proof of Proposition 3.2, and so we must verify that

c~h 1/x ~h 1 h 2 ,

0c~h 1

hi

are each O(~e, ). Using the calculations in the proof of Proposition 3.2, this is straightforward to check. Q.E.D.

Remark. The three conditions that

a) ~p be K~ihler,

b) ~0 be complete, and

c) Ricq~ be bounded by ddCr

are all essential. Property (a) gives us the Kodaira identity (cf. w 9 below), thus avoiding messy and hard-to-estimate torsion terms in the L2-esti -


mates. The second condition allows us to conclude that the compactly supported forms are dense in suitable L2-spaces, thus avoiding the appearance of boundary integrals in the basic estimates. Finally, the third property gives a bound on the curvature arising from the tangent bundle in the Kodaira identity.

w 4. Order of Growth of Analytic Sets Let A be a special affine variety. In this section we fix an embedding

A ~ C N such that the closure of A in IP N is a smooth projective variety V, and where the hyperplane at infinity in II? N meets V in a divisor D having simple normal crossings. On A we let 0 ( x ) = l o g [prc(x)[J be a special exhaustion function corresponding to a generic projection

re: A - * C " (n =d ime A),

and denote by o) a Kiihler form on V. We also set

A [r] = {xeA" 0 ( x ) < l o g r}.

Given a k-dimensional analytic set Z cA, we denote by Z [ r ] = Z c~ A[r ] the part of Z in the ball of radius r. We shall measure the growth of Z by the area in IP N of the Z[r] 's . Thus far the use of the area to measure growth of analytic sets in C rr has been the only really profitable method, although the curvature and torsion have appeared in at least one geometric problem (cf. [t2]). We define then the counting function

(4.1) U(Z,r)= i{ ~ (22k} d t o z[t] t

The growth of N(Z, r) is essentially independent of choices. So far, the main result is the theorem of Bishop-Stoll (cf. [34]):

(4.2) Z is algebraic ~- Scok< Go ~=~ N(Z, r)=O(logr). A

For technical reasons arising from the relation (dd c r 0, it is useful to consider the second order function

(4.3)

where

N(Z,r)=~{n(Z,t)-n(Z,O)} -n(Z,O)logr 0

n(Z,t)= ~ (dd~) k, Z[t]

n (Z, 0) = lira n (Z, r). r ~ O


When Z does not meet n - ~ (0), (4.3) reduces to

i{ I (ddC#} d• o z[t] t

In the cases which most concern us, N and ]V have essentially the same growth. More precisely, we have the

(4.4) Lemma. i) N(Z, r)=O(logr) o iV(Z, r)=O(logr); ii) N(Z, r) has finite order p r N-(Z, r) has finite order p 6.

Proof. Since ddC~ < Co~ outside a compact set, it will suffice to bound N(Z,r) in terms of N(Z, r). We do this in three steps:

Step one. Let n: V-~ IP" be a projection, 0 = n* (K~ihler form on IP"), and

No(Z ,r )= i{ I Ok} dt o z[t] t

Obviously, N o (Z, r) is bounded by O (N(Z, r)). Conversely, if N o (Z, r)=< X (r) for all such projections, then N(Z, r )5 x(r).

Step two. For a given projection, we may bound Ne(Z, r) by ]V(Z, 70 for 7 > 1 by the same argument used in the proof of Proposition 2.7 in [3] - cf. the discussion just below Eq. 2.3 in that paper.

Step three. Finally, given two projections n: A - ~ " and n': A - - ~ " , the corresponding order functions N(Z, r) and fi ' (Z, r) may be estimated in terms of each other by applying again the same argument used in Proposition (2.1) in the above paper. For completeness, we will sketch the proof here. Set z = ( o n and z' = (' o n' where (, (' are the coordinates on ~". Then ~ = l o g IlzlI and ~ ' = l o g l]z'[I are the corresponding special exhaustion functions. The standard integration by parts twice argument (cf. the proof of (1.2) in [3]) gives

i { I (dd~b')" A (ddCr k-p} d t o zm t

<log I]z']I S (ddetP') p-~ /x(ddCr k-p+1 z[t]

~ r

< C , S { 5 (ddr A(ddC~) k-p+1} dt = t 0 Z[tl

for a > 1. Applying this recursively in p bounds/q(Z, a r) for a > 1. Q.E.D.

Remark. Because of this lemma, we shall make the (somewhat imprecise) agreement to use N(Z, r) and h~(Z, r) interchangeably. In particular, this will be done in w 21 below.

6 Recall that this means that !ira [(log N(Z, r))/(log r)] =p and similarly for IV(Z, r).


w 5. Order of Growth of Holomorphic Mappings; the First Main Theorem of Nevanlinna Theory

a) Let .4 be an affine variety, W a projective variety with K~hler form ~0, and

f: A - ~ W

a holomorphic mapping. In practice, W will frequently be the Grass- mannian G (r, N) of (N - r)-planes in C N and f the mapping given by a set of N holomorphic sections spanning the fibres of a rank r holomorphic vector bundle over A. To define the growth of f, we introduce order functions

n q, dt (5.1) Tq(f ,r)= i ( ~ (f*q~)qJco-)T" 0 A[t]

These order functions are extensively discussed in [3], and we shall only recall a few of their properties here. Before doing this, we remark that, for reasons similar to those in w 4 above, it is technically convenient to also consider the related order functions

(5.2) Tq(f,r)=i{ ~ (f*q))q^(ddC~) "-q} dt 0 A[t] t

Just as before, the growth of Tq(f, r) and Tq(f, r) is essentially the same (cf. the proof of Proposition (2.1) in [3]), and we shall use the two interchangeably.

A first basic property of the order functions is

(5.3) f is rational ~ T 1 (f, r)=O(logr) => Tq(f, r)=O(logr)

for q = 1, c .... n. Because of this it seems at least plausible that T 1 (f, r) should be the dominant one of the n order functions Tq (f,, r). This is true in some cases, but is unknown in general (cf. [3] for further discussion). In any case we shall call

T(f, r)= T 1 (f, r)

the order function for the holomorphic mapping f : ,4 --~ W.

b) Next we come to the First Main Theorem (FMT) of Nevanlinna theory: Let E - , A be a rank r holomorphic vector bundle having Hermitian metric h with curvature matrix O. From O one constructs a (q, q) form cq(E, O) representing the q-th Chern class of E -~ A in the de Rham cohomology of A (cf. [7]). Suppose next that 0-1 . . . . . G are holomorphic sections of E -* A such that the Schubert cycles

Zq = {~i A... A ~,-q+1 =0}

16 M. C o r n a [ b a a n d P. Gr i f f i ths

have codimension q. The homology class of Zq also represents the q-th Chern class of the bundle (cf. Appendix 1 below), and the FMT gives a relation between the area of Zq and the growth of the q-th bundle order function

(5.4) Tq(E, O, r)= i { ~ Cq(E, O)/x (ddCO) "-q} d t 0 A[tl t

To state this, we recall that there is defined on A an integrable (q - 1, q - 1) form A(Zq) having the following properties: (i) A(Zq) is I/~ ~ on A-Zq and A(Zq)>O in case O > 0 and the sections a i have bounded length (cf. [7] for explanations); (ii) the equation of currents

(5.5) d d c A (Zq) = Zq - cq (E, 6))

is valid.

Using A(Zq) we define

m(Zq, r)= ~ A(Zq)AdCt~/x(ddCO)"-q (proximity form), OA [r l

S(Zq, r)= ~ A(Zq)/x (ddC~) "-q+l (remainder term). A [rl

By twice integrating the equation of currents (5.5), one arrives at the FMT (cf. [73)

(5.6) N(Zq, r)+m(Zq, r)= Tq(E, 6), r)+S(Zq, r)+ O(l).

In particular, if O > 0, and the lengths jail are < 1, then both the proximity form and remainder term are non-negative and

(5.7) N (Zq, r)< Tq(E, 6), r)+ S(Zq, r)+ O(1).

Unfortunately, this Nevanlinna inequality does not in general seem to have a great deal of analytical use, since the remainder term may dominate the order function (5.4). Indeed, the Cornalba-Shiffman example shows that N(Zq, r) may grow arbitrarily faster than Tq(E, 6), r) for q> 1. In case q = 1, however, S(Zq, r)=-0 because of (ddCO) "= 0, and (5.7) reduces to the beautiful estimate 7

(5.8) N(Z1, r)< T~ (E, O, r)+ O(1).

It is this inequality which is our main reason for the use of special exhaustion functions.

c) Now suppose that f : A ~ G (r, N) is a holomorphic mapping of A into a Grassmannian. Let U-~ G(r, N) be the universal rank r vector bundle over G(r, N) and E = f * U. We may think of./" as being given by

7 Reca l l t ha t ZI is the zero set o f a sec t ion as ^ "" ^ G e ( 9 ( A , A r E ) where la t ^ --- ^ G [ < l.


holomorphic sections tr 1 . . . . . a N ~ (9 (A, E) which span the fibres of E -+ A, and where

f ( x ) = (~i, . . . , ~NIeCN: ~ i ~ i (x)=O . i = 1

The bundle U ~ G(r, N) has a standard invariant metric, and we denote by Oq the corresponding universal q-th Chern form and set

r

(5.9) Tq(fE, r)= ~{ ~ f*OqA(ddCO) "-q} d t o a[t] t

The various codimension-q Schubert cycles. ZqcG(r,N) are parametrized by an irreducible algebraic variety ,~q. The action of the unitary group on G(r,N) permutes the various Z~'s and induces a transitive action on Eq. Letting

N (Zq, r)= N( f - l (Zq), r)

the FMT (5.6) may be averaged over the unitary group, with the result being the Crofton formula (cf. [7])

(5.10) TO(f E, r)= ~ N(Zq, r) dZq. Zq~,~q

Since obviously To( f, E, r )~ CTq(f r), we find the average estimate

(5.11) ~ N (Zq, r)dZq<= C TO (f, r).

If one were able to estimate TO(f, r) in terms of T 1 ( f r), then there would be at least one Schubert cycle Zq for E-~ A whose growth could be bounded by the growth of the mapping f In any case, this suggests that TO(f, r) has to do with the behaviour of f vis ~t vis the codimension q subvarieties of G(r, N).

w 6. Order of Growth of Holomorphic Vector Bundles a) Having defined the notion of growth for analytic sets and holo-

morphic mappings on algebraic varieties, we come to the problem of defining a good notion of growth for holomorphic vector bundles. It is not immediately clear how this should be done, since a vector bundle is an equivalence class of analytic objects rather than a single holomorphic mapping or subvariety. We shall give a principal definition of the order of growth of a holomorphic bundle, and in addition three alternate definitions 8. For line bundles these are all equivalent, but for general bundles we do not know very much about the relation between the various definitions.

8 For extendible vector bundles, there will be a fourth definition of what it means to have finite order.

2 lnventiones math.,Vol. 28


Let A be a special affine variety and E ~ A a ho lomorph i c vector bundle. To define the growth of E we shall use the auxil iary quanti t ies

z = the exhaus t ion function (3.3),

h = Hermi t i an metr ic in the fibres of E ~ A.

We set q~=ddCz, so that in the no ta t ion o fw 3

q) ~ (Dp,

in the punctured polycyl inders at infinity on A (cf. w167 2 and 3 above). Associa ted to the metr ic in E is the curvature matrix

O= ~ Ouvijeu@e*@dziAdzj, .Iz,v,i,j

written here relative to a uni tary f rame {e,} for E and local ho lomorph i c coordina tes z 1, . . . , z. on A, and curvature form

Z dz, d~j,

which is a real (1, 1) form depending on the direction given by a nonzero vector a in E. F o r line bundles, O ( ~ ) = O is an ord inary real (1, 1) form on A.

N o w suppose that x(r) is a posit ive increasing function of r which is regular in the sense that

)~(Ar)<= C(A) z(r) (A>O) for large r.

We have in mind such funct ions as

z (r) = C ,

){ (r) = log = r,

x ( r ) = r e

which m a y be used to measure growth.

Definition I. E -+ A has order <__ Z if there exists a fibre metric for this bundle whose curvature form satisfies

(6.1) [ O (a)[ ~ Cz(er (a~Ex).

The bundle has finite order if it has order < Cr ~ for some 2 < ~ , and in this case we define the order by

ord E - - inf2.


Remarks. We shall abbreviate (6.1) by

O = O (X),

which should read as saying that, locally near infinity "the curvature form O(a) grows at most like Z times the Poincar6 metric". Using standard properties of the curvature, we observe the functoriality properties

i) E has order <X ~ E*, @PE, APE have order < L

ii) E, E' have order <Z ~ E| E| have order <X. b) There are several other possibilities for defining the order of growth

of a holomorphic vector bundle, and we want to discuss these now. i) Given a holomorphic vector bundle E-+ A over an affine variety

(hence a Stein manifold), an easy Baire category argument allows us to find finite dimensional subspaces of (9 (A, E) = H ~ (A, (9 (E)) which generate every fibre, and consequently there are holomorphic maps

f : A ---, G (r, N)

which induce E from the universal bundle over the Grassmannian. More precisely, as an easy consequence of Grauert's theorem (cf. w 20 below), for large enough N, there is a bijection of sets

(6.2) [A, G (r, N)]ho , ~ Vect~,ol(A),

where the left hand side denotes the holomorphic homotopy classes of maps to G(r, N) and the right side denotes the equivalence classes of rank r holomorphic vector bundles over A. Given such a map f as above, the order function

T(f, r): T 1 (f, r) is defined as in w 5 above.

Definition If. E has order ~ Z if there exists f : A ---, G (r, N) inducing E and satisfying

T(f, r)=< Cz(r ) logr.

Remark, The reason for the log r is that T ( f r) has been logarithmically averaged.

ii) Let P* = {(~1 . . . . . ~ , ) ~ " : I~i] =< l, ~1 ... ~k# 0} be the k-fold punctured polycylinder and

P*(r)={~P*: l ~ i l > l for i= 1 . . . . . k}.

Given a holomorphic function ~ (0 on P*, set

M(O, r)= max log [0(01. ~eP*(r)

2*


(6.3) Lemma. If ~k 4=O, then

(~---, c tr)+ O(logr) (~> 1). M(t~,r)<C~M

Sketch of Proof. We shall do the case n = k = 1. Set

m(O,r)=~-~ S l~ d0 ~1 =1

Since log [@1 is harmonic,

log I~I dO=O(logr)

and consequently (cf.- Proposition 1.4 in [6])

(6.4) m (~, r) = m ( 1/~, r) + 0 (log r).

It is obviously the case that

m(~, r)__< M(~, r),

and using the Poisson-Jensen formula [28] the opposite inequality

M(tp, r)< C~m(~9,~r)+O(1) (~>1)

can be proved. Combining this with (6.4) gives the lemma. Q.E.D.

Now we consider the group GL(r, (9 (P*)) of holomorphic mappings

g: P* -~ GL(r, ~).

By considering the growth of the individual matrix entries in g(~), we may define what it means for g to have order =< g 9. Using Lemma 6.3 and Cramer's rule, the set of geGL(r, (9(P*)) having order =<Z forms a subgroup GL(r, (gz(P*)).

We now give another definition of growth for a vector bundle E -~ A. For a general A the definition is somewhat complicated because the topological (or analytic, which is the same) K - ring of a punctured polycylinder P* is non-trivial unless P* has only one puncture [11]. Therefore we limit ourselves to the case when A has a smooth completion A such that D - - , 4 - A is smooth. The general case can be done using the techniques developed in [11].

Definition 111. The holomorphic vector bundle E -~ A has order < X if, relative to a suitable finite open covering { Uv} of A the restrictions El Uv are holomorphically trivial and the transition functions lie in GL(r, (gx (P*))

9 We assume that x(r)> log r.


for every polycylinder P e A such that P * = P n A is a once-punctured polycylinder contained in U v n U,.

iii) Finally, we have the

Definition I V. E ~ A has order < )~ if there exist sections

a 1 . . . . . G~(9(A, E)

in general position such that the Schubert cycles

Z ~ = {al A .-. ^ ar-~+1 = 0 }

satisfy estimates N (Zq, r) = C X (r) log r.

Remark. When E is a line bundle, Z 1 uniquely determines E, but in general the analogue of this statement is false. However, the rational cohomology classes of the Zq's depend only on the class of E in

Kho~(A)|

so one might attempt to define the growth of elements in Kho~(A)| Although this may be a desirable end to look towards, at the moment this seems too speculative to be discussed further.

c) For a special class of vector bundles on A, namely those which extend Coo to vector bundles on a smooth completion A, we may give stitt another definition of what it means to have finite order. Bundles with the above property will be called extendible. Let E ~ A be one such bundle and let F - , A be a C ~ bundle on a smooth completion of A such that E = F[A as C ~ vector bundles. F ~ d will rarely admit a holomorphic structure. In fact, as a consequence of the theorem of Grauert proved in w 20 below, every C ~ vector bundle on i[ restricts to a bundle on A which admits a complex structure.

E -~, A may be viewed as being given by the data of its underlying C ~ bundle and of its c~-operator. Thus it makes sense to measure the growth of E by seeing how much singularity ~ has along the divisor H = i [ - A . Let a be a holomorphic section of [H] which defines H.

Definition V. The extendible vector bundle E-- , A has finite order if there exist a smooth completion d of A, a C ~ vector bundle F ~ d and an integer k > 0 such that F] A = E as C ~ vector bundles and

I~lkJ extends to a smooth operator on all of A.

Remarks. Here ]a[ is the length of a relative to a fibre metric for [HI -~ V. Intuitively, E -* A has finite order according to this definition


in case the O-operator of E has only a finite pole along H. In some sense, this is the simplest possible singularity which ~ may have, and this is the appeal of this definition.

(6.5) Proposition. For extendible vector bundles Definition V implies Definition I.

Proof According to Definition I, we must find a metric in E ~ A whose curvature satisfies certain growth conditions. A natural metric to try is a Coo metric in F ~ A, and this will be seen to work.

Choose a point x~H and a smooth frame e I . . . . . e r for F over a neighbourhood U of x. If q = ~ r/i e~ is a section of F, we may write

(6.6) cSr/= ~ e~ | (~r/i + ~ ~/, ej | 0j'~

where 0"=(01j) is a rr)atrix of (0, l)-forms. If f = 0 is a local equation for H n U, according to Definition V we may write

q, (6.7) 0" = - -

I f l k

where q/is a C ~ matrix of (0, 1)-forms. If we choose the frame e 1 . . . . . e r to be unitary, then the connection matrix 0 for the metric connection 17= 17'+ ~ satisfies the Hermit ian symmetry condition (cf. [-7])

(6.8) 0 + ' 0 = 0 .

Compar ing (6.7) and (6.8), we find that

(6.9) 0 = ~ - ' ~ ]f[k �9

According to the Cartan structure equation the curvature matrix O is given by

(6.10) O=dO+O A O.

Assuming that k > 1, it follows from (6.9) and (6.10) that

where ~o is any smooth metric on U, which implies our proposition. Q.E.D.

B. Holomorphie Vector Bundles with Growth Conditions w 7. Statement of Main Results about Algebraicity of Vector Bundles

a) Let A be an algebraic variety. On A we may work in either the algebraic category or analytic category. The former means that we


consider the Zariski topology on A and the structure sheaf (9,Lg of rational, holomorphic functions. (In general, the subscript "alg" means that we are in the algebraic category.) The latter means that we consider A as a complex manifold in the usual topology and with the structure sheaf (9 of holomorphic functions. When A is complete (--compact), the two points of view are essentially equivalent (cf. [30]), but not in general otherwise (cf. [11]) 1~

An algebraic vector bundle Ealg ---* A is given by rational holomorphic transition functions relative to a suitable Zariski open covering. In this case there is a smooth compactif ication/[ of A and algebraic (= analytic) vector bundle E ,1g~A whose restriction to A is the given bundle. Associated t o Ealg ~ A is an analytic vector bundle E--, A given by the same transition functions relative to the same open covering. A holomorphic vector bundle which arises in this way is said to have an algebraic structure. Not all holomorphic vector bundles have algebraic structures, and isomorphic analytic bundles may arise from distinct algebraic ones (cf. [11]).

Theorem I. A holomorphic vector bundle E--, A has an algebraic structure if and only if, there exists a Hermitian structure for E whose curvature form satisfies

(7.1) 0 (~) = O (o)e,)

in the punctured polycylinders at infinity on A.

Remarks. i) In other words, when A is a special affine, E---, A arises from an algebraic vector bundle if, and only if, E has order 0(1) in the sense of Definition I. The Hermitian structure determines the algebraic structure in a manner to be made precise by Theorem I! below.

ii) The algebraic structure given by Theorem I will be seen to be functorial in the following sense. Let E be a holomorphic vector bundle with a Hermitian structure such that (7.1) is satisfied. Let B be an algebraic variety and let f : B ~ A be a rational holomorphic mapping. Then f * E inherits from E a Hermitian structure satisfying (7.1). This Hermitian structure induces an algebraic structure on f * E . Func- toriality means that this structure is the pullback of the algebraic structure of E.

iii) Let E,t g ~ A be the restriction of a holomorphic bundle E ~ on a smooth completion /{ of A. Taking any C ~ metric for E gives a metric for E such that (7.1) is satisfied. Thus, assuming (7.1) for E - * A , we must produce an algebraic structure. We shall do this when A is a special affine variety. Taking into account Theorem II below, we may cover a general A by finitely many special affines and the resulting

10 This part of G.A.G.A. will be proved in w 16 below.


algebraic structures will patch together on intersections of two such open sets.

b) Let E---, A be a Hermitian vector bundle whose curvature form satisfies (7.1). We shall tell how to compute the algebraic sheaf cohomology H*(A, (.Oalg(Ealg)) of the algebraic structure Ealg--~ A given by Theorem I.

We assume, for the moment, that A is a special anne. In addition to the given Hermitian structure in A, we consider the K/ihler metric q~ in the tangent bundle of A given in w 3 and denote by 4~ = q~" the resulting volume form. We let Aq(E, O) be the vector space of C ~ E-valued (0, q) forms q on A which satisfy the U-conditions

A

(7.2) ~ IcSr/l 2 e -N~ q~ < oo A

where z is the plurisubharmonic exhaustion function given in w 3 and N is some real number. Intuitively speaking, (7.2) means that both r/and gr/ should have at most poles at infinity in the L 2 sense.

When A is a general, smooth algebraic variety we choose a finite covering { U~} of A by special affines and define A* (E, O) to be the vector space of all smooth E-valued forms (p such that

(pl U~e A* (El U~, O)

for every index i. It is easily seen that this is independent of the covering {U/}. By definition, {A*(E, O),(3} forms a complex and we denote by H~' (E, O) the resulting cohomology.

Theorem If. Let E--~ A be a Hermitian vector bundle whose curvature form satisfies (7.1). 7hen for the algebraic structure given by Theorem I,

u* (A, ealg (Gg))'~ H? (A, o).

Corollary. When A is a special affine, the global algebraic sections (galg (A, Ealg ) are the holomorphic sections a ~ C (A, E) which satisfy

(7.3) ~ Irrt z e -u~ 4~< oo A

for some N >> O.

Remark. The corollary will actually be proved before the full Theorem.

c) We begin by proving an elementary result, which is in fact a consequence of Theorem I.

(7.4) Proposition. Let E - * A be a Hermitian vector bundle whose curvature form satisfies (7.1). Then the Chern classes Cq(E)e H2q(A, t~) are the restrictions of classes ?qeH2q(.4, (12) which have Hodge type (q, q).


Proof It is well known that, via de Rham's theorem, Cq (E) is represented by a polynomial cq(E, O) in the curvature matrix O (cf. [7]). Because of (7.1), we have an estimate n

(7.5) eq(E, O) = 0 (o)%)

in the punctured polycylinders at infinity on A. Let 0 be any Coo(q, q) form on A which satisfies (7.5). Since by (3.1)

• O )p * < O O , zl*

we may define a current TO of type (q, q) on A by

T0(r/)=lim ~ 0A~ r ~ oo A [r l

where q is a Coo (n q, n - q) form on ,4. If dO = 0, then by Stokes' theorem

dToO1)- To(dtl)= lim ~ 0A~/=0, r ~ o o OA[r]

the last step being again derived from the explicit expression (3.1) for the Poincar~ metric. It follows that T o is a closed current of type (q, q) on A, and since the cohomology computed from currents is the same as that computed from C ~ forms ([29]), T o defines a class O~Hq'q(,~) whose restriction to A is the class of 0. Q.E.D.

Remark. Both Theorem I and Proposition (7.4) are false if the curvature form O (a) is allowed to have a singularity of the type

~ 1 (7.6) (7) = ~- +-~- + ~ 5- (fl/x d~ +/~/x d~)

with ~, /3 bounded in the whole polycylinder. In fact, given a smooth projective variety V and sufficiently ample smooth hyperplane section D with complement A = V - D , every class u e H2(V, 7/) restricts on A to the Chern class of a hotomorphic line bundle L-~ A having a metric whose curvature form satisfies (7.6) (cf. w 15 below). But generally L can not be taken to be algebraic.

It is easily seen that d) satisfies estimates

G~ = O ( ~ ; T oP*)

i t cf. The proof of Proposition 9.7 below.


for every positive e. However, it is a consequence of Theorem VII that for extendible line bundles an estimate on O of the type

1 IO1__< C ~ - ~ , (u<l)

implies algebraicity.

w 8. Statement of Main Results Concerning Transcendental Vector Bundles

a) Let A be a special affine variety, and we refer to w 7 for the various definitions of order of growth of holomorphic vector bundles over A.

Theorem Ill. i) For a holomorphic line bundle L--~ A, the Defini- tions I - IV of L having finite order <= 2 are all equivalent.

ii) In general, Definition I implies Definition II.

We do not know to what extent the four definitions of growth are equivalent for general vector bundles.

b) To state our next result, we let E - * A be a holomorphic vector bundle of finite order (according to Definition I). Using the given metric in E, complete K~ihler metric q~ and exhaustion function p = e ~/2 on A, we define the finite order sections of E ~ A by

(gf.o.(A, E ) - {a~(9(A, E): S]a] 2 e-PN ~ < ~ for some N>0} . A

Thinking of ~ as being approximately log ]tz[i 2 for a suitable embedding A c ~ " , e pN is approximately e -IIzlIN, and so (gf.o. (A, E) are the holomorphic sections having exponential growth in the L2-sense.

Given two finite order vector bundles E, E' over A, Hom(E, E') has finite order, and consequently the notion of a finite" order isomorphism between E and E' is defined. Thus the set of equivalence classes

Vect~.o.(A)

of finite order vector bundles of rank r over A has a meaning. In particular, for r = 1 we define the finite order Picard variety by

Picf.o. (A) = Vect~.o (A).

Theorem IV. Let A be an affine algebraic variety. Then the Chern class map

c 1 : Picf.o.(A) --* H 2 (A, Z) is an isomorphism.

Remarks. This gives us the "Oka principle with growth conditions" in the simple case of line bundles. We do not know to what extent Theorem IV holds for general vector bundles. Further discussion of this problem is given in w 26 below.


c) Let V be a smooth projective variety. On the rational cohomology H 2 k (V) are defined the Hodge filtration F p H 2 k (V) and Lefschetz filtration UH2k(v) as follows (our indexing and terminology are not standard)

FP HZt(v)= { { e H2k(V): { GHk+ p'k- P(V) + ... + Hk- p,k+ P(V)},

LPH2k(V)= {images of H2p(Z) s, , H2k(V)}

where Z runs over all smooth varieties having a non-degenerate holomorphic mapping f : Z--+ V onto a codimension ( k - p) subvariety f (Z) of V. Intuitively, FPH2k(v) are the classes whose Hodge decomposition is at most p steps off from being of type (k, k)

FPHZk(v)

(2 k, 0) (k + p, k - p) (k, k) ( k - p, k + p) (0, 2 k)

and LPH2k(V) are the classes in H2k(V)'~H2,_2a(V,q)) which are supported in a subvariety of dimension n - k + p. In particular, LkH 2 k(v) are the algebraic classes. It is standard that

LP H2k(V)~ FP H2a(v),

and the famous Hodge conjecture is that these are equal (for p > 0, a slight refinement is necessary-cf . [15]).

We now define a third filtration, the filtration by order of growth, G~p2a(v) in the primitive part p2k(V)cH2k(v) of the rational cohomology. (The primitive cohomology is discussed in w b e l o w - i t forms the "building blocks" for the whole cohomology.) To do this we let H ~ V be a sufficiently ample hyperplane section and A = V - H the resulting special affine variety. For an analytic subvariety Z cA , the order ord(Z) is defined by

- - l o g N ( Z , r ) ord (Z) = lim

�9 ~ ~ l o g r

Thus 0 < o r d ( Z ) < + c~, and Z has finite order if o rd (Z)< oo. For each such Z of codimension k, uz~HZk(A) denotes the fundamental class (Appendix 1). It is a consequence of Grauert 's theorem (w 19) and standard K-theory (Appendix 2) that every class in HZk(A) is of the form ~u z for some a ~ Q and codimension k subvariety Z. We set

G;~p2k(v)={~EP2k(V): ~[A=au z where ord(Z)<2} .

Intuitively, GZp2k(V) are the classes in p2k(v) whose restriction to A is represented by an analytic cycle with area growing like O(r ~+~) for any e > 0. A priori, 2 runs over all real numbers in the filtration by growth. It is proved in w below that

Fv p2R(v)~ Gp p2R(V),

28 M. C o r n a l b a a n d P. Gr i f f i ths

which is a lower bound on the growth of analytic cycles in a given homology class. Taken together with what was said previously, we have the inclusions

(8.1) Lp p2k(V) c Fp p2k(V)~ Gp Pzk(V).

Theorem V. For k = 1, we have

Lpp2(V)=Gpp2(v).

Thus, for k = 1, the inclusions in (8.1) are all equalities.

Remarks. The equality L ~ = F ~ is the classical Lefschetz theorem (with Q-coefficients), and a byproduct of Theorem V is a slightly new proof of it. More interesting perhaps is the conclusion that filtration G ~ p2(v ) occurs non-tr!vially only when 2=0, 1, whereas any real 2 is a priori possible. It seems unlikely that this should be an isolated phenomenon.

w 9. The Kodaira Identi ty and Consequences

a) Let A be a complex manifold having a Hermitian metric ds 2, E ~ A a Hermitian vector bundle, and denote by A~ the C% compactly-supported, E-valued forms on A. Using the given metrics on A and E, there is a pointwise norm ]q]2 for r/~A~ and from this an L z norm

A

where �9 is the volume form associated to the ds a on A. Letting c5" be the adjoint operator to ~, the Kodaira identity is a formula for the Dirichlet norm

I@~ll/+ I@* ~tll 2 (q6A~

To explain this formula, we choose locally a unitary frame {e,} for E and unitary coframe {q~i} for A (thus ds 2 =~0~Cpi ). Using multi-

i index notation I = ( i 1 . . . . . iq), opt= ~oil /x ... ^ ~oi, , etc., we write

1

Denoting the curvature matrix for E by {O,~ij} and Ricci form associated to ds 2 by {R0. }, we define the pointwise operators

(9.1) < Rtl, q>= ~ R6quI iqu i j


and then the integrated versions

A

(R~,.)= i (R,1, ~),P. A

Finally, we let Vtl= Z VjtluIeu @ ~oj@UPl

where ~ signifies covariant differentiation.

Proposition ([23]). I f the metric on A is Kiihler, then

(9.2) II0r/II 2 + I/~* r/tl 2 = II~r/[I 2 + ( O rl, r/) + ( e r/, r/) (qA~

Remarks. i) Conversely, the Kodaira identity implies that the metric on A is K~ihlerian. If this is not the case, then nasty terms involving the torsion appear on the right hand side of (9.2).

ii) The Kodaira identity is most useful when the right hand side is positive. To do this, we try to select metrics for A and E such that (O 17, q ) + (R t/, t / )> 0. Having chosen metrics to begin with, the metric for E may be multiplied by weight functions e -z (geometrically, these weight functions may be thought of as metrics in the trivial line bundle). The new curvature form Oz(a), (acE) is then

(9.3) Ox (a ) = Ot a) + d dr z.

This suggests the type of functions Z which should be used to modify our given metrics.

Let L~ be the completion of A~ using the norm [[ql[ 2. De- noting by S, T the maximal closed extensions of 8, we obtain a "complex" of Hilbert spaces

LO,~-,(e) S,LO, q(e) ~,LO, q+~(E)

where S and T are closed, densely defined operators. Letting @R denote the domain of definition of the (unbounded) operator R, the intersection

is a Hilbert space using the graph norm

ll~ll~ + liTr/l[~ + IIS*~ll2 (~e~s. n~O.

(9.4) Proposition ([1]). I f the metric on A is complete, then A~ is dense in ~s* c~ ~T relative to the graph norm.


Corollary (Basic estimate). I f the metric on A is complete and Ki~hler, then

(9.5) II Trtlt 2 + IIS* rtll 2 >_-(Or/, q)+(Rq, q)

for all ~ l~s ,c~@T.

c) Suppose now that A is a special affine variety with exhaustion function z and complete K~hler metric q~ given as in w 3. Let E -~ A be a Hermitian vector bundle whose curvature form satisfies

(9.6) [O(a)] < CddCz.

According to Proposition (3.2), this is equivalent to the basic growth condition (7.1). In the notations of w b, we shall prove the

(9.7) Proposition. H~(E, O)=Ofor q>0.

Proof. We begin with an observation concerning the condition (9.6). If {eu} is a unitary frame for E and {~oi} a unitary co-frame for A, then (9.6) is equivalent to

I O..jl <= c2

for all sets of indices. Indeed, taking a = e u in (9.6) gives IO..~jl<Cx. Next, setting

1 a = ~ (e u + e~)

and using the estimate just proved gives ]O,~0. L < C z .

Next, we may multiply the given metric in E ~ A by e -N~162 without changing the complex {A*(E, 0) ; c5}. Referring to (3.5), (9.1), and (9.3), we may choose N o such that, for the new metric,

(9.8) ( O r/, n) + (R q, r/) > Ir/l 2.

Finally, given a complete K~ihter manifold A and Hermitian vector bundle E--*A whose curvature satisfies (9.8), the basic estimate (9.5) allows us to use the arguments in 1-20] Chapter IV, to prove the vanishing Theorem (9.7).

d) Keeping E ~ A as just above, we denote by

O(E, O)= H~(E, O)

the holomorphic sections a~O(A, E) satisfying the L z growth conditions

A

for some N > 0.

Analytic Cycles and Vector Bundles on Non-Compact Algebraic Varieties 3 l

(9.9) Proposition, Given finitely many distinct points x~6A and e,~Ex,, there exists cre(9(E, O) with cr(x,)=e~.

Proof. This may be done by the method of [24], by first blowing up A at the {x~} in order to convert the ideal sheaf of S ,x , into a line bundle so that &methods apply, and then combining the curvature estimates from the proof of (9.7) with those used in [-24]. Q.E.D.

Corollary. Given finitely many points x~6 A, there exist

a 1 . . . . . ~,ECP(E, 0 )

such that crl(x,),..., ar(x ~) give a basis for Ex .

Remark. It is not evident that the analytic set where a~ A.. . /x o-~=0 is an algebraic subvariety of A. Questions of this sort seem to require Nevanlinna theory, and will be discussed in the next section.

w I O. Applications of Nevanlinna Theory

a) Let A be a special affine variety and E---, A a Hermitian vector bundle whose curvature satisfies the basic estimate (7.1). Recall that

(10.1) (9(E, O)= {crE(9(A, E): ~ [a[z e - ~ 4 ) < cc for some N>0}. A

We want to characterize C(E, O) by a pointwise estimate, and for this we use the following analogue

(10.2) M(a, r) = max log ]~r (x)] x~A[r]

of the maximum modulus growth indicator of a holomorphic function.

(10.3) Proposition. We have the characterization

(9(E, O)= {ar E): M(a, r) = O(logr)}.

Proof. If M(a, r)< C log r, then (cf. (3.3))

Io-[z e- N~--_ O(1)

for N > C , and consequently the L 2 estimate (10.1) holds for N > C since ~ ~ < oc.

a Thus, we shall assume (10.1) and attempt to bound the growth of

M(g, r). The idea is: i) to make loglcrl plurisubharmonic by suitably modifying the

metric in E ~ A; and

ii) to use a variant of the Poisson-Jensen formula to estimate M(a, r) by an integral.

32 M. C o r n a l b a a n d P. Gr i f f i ths

Replacing the given metric h in E by eN~ does not change (10.1) or the growth of (10.2), but, by (9.3) and (9.8), does allow us to assume that the curvature form 0(o9 is non-positive. Then a straightforward calculation gives

ddC log la,2 = _ O(a) + ~ _ l { la'2 (D' a' D' a) - (D' a' a) A (D' a' a) } �9 lal4 >_-0

so that log Ial 2 and subsequently

{al=er

are plurisubharmonic on A. Next we recall the special exhaustion function ff (x)= log IT n(x)ll where

n: A ~ I E " is a generic projection. The ( 2 n - 1) form

A=dC~ A(ddC~b) "-1

is closed since (ddC~9)"=O, and may be described as n*(S) where Z is the standard unitarily invariant form on C"-{0} having constant integral over all spheres Ilzll = r.

(10.4) Lemma. The function

v(a , r )= ~ [~IA OA [r]

is increasing, and satisfies the estimate

v (o-, r) = O (rX).

Proof of Lemma. We shall apply the standard "integration twice" calculation of Nevanlinna theory to the plurisubharmonic function

[ a[ = e �89 log I~12.

This function is not smooth, but Stokes' theorem may still be applied. Thus, for r < R, and setting A [r, R] = A JR] - A Jr]

r dt 0<~ { ~ dd~lalA(ddC~)"-'} -

r n o ] t

r dt =~ { S dClol^IddCq')"-l} Z - (Stokes')

r OA[t]

= ~ df f^dCla[^(ddC~) "-1 (Fubini) A It, R]

= ~ d[alAdCr "-1 (bytypeconsiderations) A[r ,R]

= ~ d(lald~kA(ddCO)"-a) ((dace)"=0) A [r, R]

= v (a, R) - v (a, r) (Stokes').

Ana ly t i c Cycles and Vector Bundles on N o n - C o m p a c t Algebra ic Var ie t ies 33

Next, by the Cauchy-Schwartz inequality

laI '/' = ~ I~rl e(-NO/2 e (N~)/2 A Jr] A [r]

=<( f [O'[2e-N~))�89 f e N ~ ) ~ A [rl A [r]

~< C f )J

since ae(9(E, O) and e m q) = O(rZ').

A [r]

Thus, since v(a, r) is increasing,

1 2~ d t v(~, r )<==-= ~ v(o, t ) - -

log z r t

1 - - l o g 2 ~ [aIAAd~9

A[r, 2r]

=< C 1 r ~

since A A de < (cf. w 3). Q.E.D. i]/2 = C 2 ~

Recalling the interpretation A = re* (X) mentioned above and letting p ---, 0 in

v(a ,p)<v(a ,r ) (p<=r) gives the estimate

(10.5) ~ la(x)l_-< C3 v(a, r). ~(x)= 0

This inequality is, in fact, just the sub-mean-value principle for the plurisubharmonic function

xe~ - l(z)

on IIY. We want to generalize (10.5) by removing the special role played by the origin in IlY. This may be done by finding a linear fractional transformation T of the ball t r"[R] into itself sending z to the origin. When this is done, we find

~(z )<C S a T * A . Or

Now for r < R, an explicit computation gives

T* A 1c~r [R] < C (r, R) A[ 112" [R] 3 Inventiones math.,VoL 28


where C(r, R) is homogeneous of degree 0 in (r, R). Choosing R= 2 r , we obtain for Ilzlt < r

[a(x)[ <= C' v(a, 2r)=O (ra), xerc- l(z)

which obviously gives M(a, r)=O(logr). Q.E.D.

b) Suppose now that A is a special affine variety and L--~A is a

Hermitian line bundle whose curvature satisfies the basic estimate (7.1).

(10.6) Proposition. The divisor (or--0) of a section a~(9(L, 19) (cf (10.1)) is an algebraic hypersurface on A.

Proof Letting D be the divisor of a, the First Main Theorem of Nevanlinna theory gives

N(D, r) q- 1 l o g ~ - A = T~(L, O,r)+O(1) OA [r]

since the remainder term S(D, r )=0 due to (ddC~b)" =0. Since T(L, (9, r)= O(log r) because of 69 = O(q~) and

log [crl 2 A = O(log r) OA[r]

by (i0.3), we obtain N(D, r) = O(log r),

which by (4.2) implies that D is algebraic. Q.E.D.

(10.7) Corollary. Let E--~A be a Hermitian vector bundle of rank r whose curvature satisfies (7.1), and %, ..., a,E(,9(E, O) sections such that a 1 ̂ . . . ^ a,~O. 7"hen the divisor a 1 ̂ ... ^ a , = 0 is algebraic, and moreover the degree may be estimated.

Proof Proposition (10.3) gives us that

e l , . . . , g,e(_9(E, O) ~ t r I A ... ^ a,e(9(A" E, O)

(this is not evident from the LZ-definition). Now (10.6) may be applied. Q.E.D.

Remarks. i) We are unable to prove (10.6) directly for a bundle of rank r > 1, since in this case the nasty remainder term appears non- trivially in the FMT.

ii) The same proof as given above shows that, if L-+ A is a line bundle of finite order and a~tgf.o.(A, E) has divisor D, then D is an analytic set of finite order on A. More precisely, if L has order < r a and

f 10"12 e- '~ ~ < + o 0


where p = e */2, then the order of D does not exceed sup(2, p). Similarly, if E - * A is a vector bundle of order _-<r x and a t . . . . . a,e(gf.o.(A,E) satisfy estimates

~ [0"/12 e - P " # < + oo i=1 . . . . . r

then the order of D = {a t A-.. ^ a~=0} is less than sup(2,/0-

w II . Proofs of Theorems I and I I I ( i i ) for Affine Varieties

a) We shall give two proofs of Theorem I, the first being under the additional assumption that A is a special affine variety.

Applying (9.10), we may choose 0-1 . . . . . a~e(9(E,O) such that a 1 A--. A a,~g0. By (10.7), the divisor D = {a 1 ̂ - - . ^ a ,=0} is algebraic. Choose a point x~ on each of the finitely many components D~ of D. Applying (9.10) to the x~, we may find

at+l, ..., 0"2,e(9(E, O)

such that ar+l(X~)^...Aa2r(X~)~:O. Thus a 1 . . . . . aa, span the fibres of E--*A outside a codimension two algebraic set.

Continuing in this way, we may find a I . . . . . aNe(9(E, O) ( N = r(n + 1)) which span the fibres of E---, A.

N

Set ( = ~ (ray for (~v)etE N. There is a holomorphic mapping (eval- V=I

uation mapping) f : A - , G(r, N)

given by f ( x ) = {(~o)elI~N: ~(x)=0}, and then E--*A is induced from the universal bundle. It will suffice to show that f is a rational map.

For this we consider the composition h = p o f where

p: G (r, N) ~ IP (~)-1

is the Pliicker embedding. The inverse images h- l (H) of hyperplanes are linear combinations of divisors given by a h ^ . - - ^ a ~ =0, where thus the h- 1 (H) are algebraic hypersurfaces of uniformly bounded degree on A. It follows, either directly or by combining (5.3) and (5.10), that h is rational 12. Q.E.D.

b) For the second proof, we assume for a moment that A is a special affine

(11.1)

i)

ii)

f~ are

12 Cf. L e m m a (6.19) in [13].

3*

variety and shall prove the following.

Lemma. Suppose that a, a t . . . . ,ar~(9(E, O) satisfy

a 1 A-.. ix a, ~ 0, and

a= ~ f i a i where fiE(9(A) are holomorphic functions. 7"hen the i = t

rational functions.

36 M. C o r n a l b a and P. Grif t i ths

Proof. We consider f l , and for this function will prove that the level sets D , = { f l = a } are algebraic hypersurfaces of uniformly bounded degree. Replacing a by a - a a l , it will suffice to consider Do= {fl =0}. N O W O 0 c D = { o A (72 A - - . A O r = 0} , while D is itself an algebraic hypersurface of uniformly bounded degree by (10.7). Q.E.D.

To prove Theorem I for a general variety A, we may cover A by finitely many open sets A, on which we have a frame a,, 1 . . . . . %, r e (9(E, O) with a,. 1 A--./~o-,,r4=0. On the intersections A c~A~, the transition functions passing from one frame to the other are algebraic by (11.1). Q.E.D.

c) We now prove (ii) in Theorem III. The idea is to construct finitely many sections al . . . . , a~(Of.o.(A, E) which span every fibre E,~ (xeA). The resulting mapping f : A --+ G(r, N) will induce E --+ A and will have finite order by the same argument used in w 11 (a) above (cf. the second remark following (10.7)).

Suppose then that E--+A is a Hermitian vector bundle whose curvature form satisfies

(11.2) [O(a)[ < C pa . (p

where p = e ~/2. Multiplying the given metric in E by e -kp~ gives a new metric whose curvature form O(a) is (cf. (9.3))

O(a) = O(a) + kdd~ p ~

k2 a k22 (11.3) = O ( c r ) + ~ - p d d ~ z + ~ p ~ d z / x d ~ z

> - I O( a ) l+~- pXdd~z .

Comparing (11.2) and (11.3) we see that, with the notations ofw for large enough k

(11.4) (Or/, t/> + (R r/, t/> > [r/l 2

for every compactly supported form t/.

Applying the arguments in w167 9 (c), (d), we may find a I . . . . . are (_0f.o.(A, E) which satisfy a: A... A a, 4=0 and

~[ail2e-kp*o< +oo ( i=1, .,.,r).

More precisely, the following holds:

(11.5) Proposition. Given finitely many distinct points x~EA and e~eE~., there exists ae(fico.(A , E) with a(x~)=e~ and

5 [o'[Z e-kP~ O < +oO.


The difficulty now is that the set D = {a~/x---/x a ,=0} is not algebraic and thus may have infinitely many components. This can be overcome by using.

(11.6) Proposition. Given a diverging sequence of distinct points x~eA and subspaces F~ ~ E~, there exists a~ (.Of.o. (A, E) such that

51r~12 e-kP~ r +oo.

Proof Choose e ~ e E x - F ~ and sections % such that

~,,(x e) = 0 (~ < ~),

a=(x=)=e=,

~la,12 e-kP~ ~b< +oo.

Inductively on ~, choose c==#0 such that

Z ceae(x,)r ~<=~

1 f [ C a o ' a l 2 e - t P ' ~ 2-- 7-

Then a=Y, G% satisfies the requirements of the Proposition. Q.E.D. ~t

w 12. Proof of Theorem II

a) Let E--+A be a Hermitian vector bundle whose curvature form satisfies (7.1). It has been proved (w 1 i) that E--+A comes from an algebraic vector bundle, and we now want to characterize the algebraic sheaf cohomology according to the statement of Theorem II.

To begin with, it follows from Proposition (10.3) and Lemma (11.1) that

(12.1) H~ (_galg (Ea,g)) = (P(E, 0),

thus proving Theorem II for H ~ (cf. (7.3)). A useful consequence of (t2.1) is the following: L e t / 7 ~ 4 be a_holomorphic (=algebraic) vector bundle over a smooth completion A of A which restricts to E-+A. (More precisely, there is a holomorphic bundle isomorphism E IA g E; cf. the remark below.) Choosing a C ~ metric in E--+A gives a new curvature O' for E--* A, and we claim that

(12.2) (9(E, O)= (.0(E, O').

38 M. C o r n a t b a a nd P. Griffi ths

Proof Recall that the algebraic sections of E---~A are those a~(9(A, E) which satisfy

(12.3) max log I o-(x)[= O(log r) x ~ A Iv]

(cf. Proposition (10.3)), where [al is the length using the given metric in E--,A. However, (12.3) is obviously true for the new metric obtained from E---,A, which then proves (12.2).

Remark. Suppose that E---~A is the trivial bundle A • r with metric given by

h ( x ) = e Re S~x~

where f~(9(A) is an arbitrary entire function. The curvature ddClogh of this metric is zero, so that (7.1) is trivially satisfied. Using (12.3), (_Oalg(A, E) are the holomorphic functions ~ on A which satisfy

max (log (~k (x)) + �89 Re f(x)) = O(log r). xeA[r]

Thus the mapping _ Re y Ip --, tp e 2

takes (.gang(A, E) isomorphically onto (gang(A).

Turning now to the higher Hq's, the sheaf cohomology groups Hq(A, (9~g(E~lg)) are defined using the (~ech procedure relative to coverings of A by Zariski open sets and sections of the sheaf (gang (Ea~g) discussed above. We define sheaves ~r176 in the Zariski topology as follows: s~~ is the sheaf associated to the presheaf

V ~ A~ O)

where U ~ A is a Zariski open set and A~ O) are the C *, E-valued (0, q) forms ~p on U satisfying the growth conditions (7.2), which we shall henceforth refer to as: ~p and Oq9 having L2-poles at infinity. Making the identification (12.1), we consider the complex of sheaves

(12.4) 0--,dglg(E, lg)--osC~176 ~ ,~r176 ~ ,--..

The vanishing Theorem (9.7) gives the "Poincar6 lemma" for the complex (12.4). Thus Theorem II will follow in the standard way from the following:

(12.5) Proposition. H q(A, ~r v (Ea~g)) = 0 for q > O.

Since we are working in the Zariski topology, the usual partition of unity argument does not apply to prove (12.5). In fact, our proof of (12.3) is somewhat involved, but does have an advantage in that the


relationship between cohomology with "L2-poles at infinity" and with "Coo-poles at infinity" will be clarified.

b) We shall give some preliminaries to the proof of Proposition (12.5).

(12.6) Lemma. Let X be a smooth algebraic variety and D 1 . . . . . D h effective divisors such that 0 Di = ~" Then there exist Coo functions Pi on X such that i

i) Pi vanishes to infinite order along Di;

ii) pi>=O and ~ p i = l . i

Proof. Choose metrics in each of the line bundles [D~] and let aie(9(X, [DI]) define D i. Set

1

21= e - l ~ -

Then 2~ vanishes to infinite order along D i and is positive on X - D I . The functions

2i Pi = ~ 2i

i

satisfy the requirements of the lemma. Q.E.D.

The pi's may be considered as an analogue, in the Zariski topology, of a partition of unity subordinated to the covering {X-Di} of X. How- ever, this partition of unity can not be used directly for C | sheaves. For example, given X and D1, D2 and a C ~ function f~ 2 on

( X - D1)c~(X-DE)= X - ( D 1 uD2)

having L2-poles along D~ u D 2 the function px f~2 is generally not Coo on X - D 2. Thus, we shall first prove an L2-version of (12.5)(where the partition of unity will work), and then give a smoothing argument to obtain the Coo case.

Letting E --, A be as above, we denote by

K*(E, O)=(~K~ O) q

the complex of E-valued, locally L 2, (0, q) forms r which have L2-poles at infinity and are such that t~cp (in the sense of distributions) is locally L 2 and has L2-poles at infinity. Evidently, K*(E, O) is the L 2 analogue of A*(E, O). The proof of Proposition (7.7) can be repeated verbatim to prove the following:

(12.7) Lemma. I f A is a special affine, then {K*(E, O); 3} has no higher cohomology.


The next result is an L2-analogue of Theorem II.

(12.8) Proposition. There is a canonical isomorphism between thecohomol- ogy of {K* (E, O); c5} and H*(A, (galg(Ea,g)).

Proof Define a complex of sheaves 3((*(E,ig)=@5(~ in the Zariski topology by the same procedure as q

~1" (Ealg)

was defined at the end of w 12 (a). Note that if gEK~176 O) satisfies ~q0 =0, then q~ is holomorphic and is in (_galg(A , Ealg ). Thus the complex of sheaves 0 __~ (Qalg (E) __+ ff{~o, o (Ealg) 3 ) ~s~/~O,1 (Ealg) 3 )

is exact in the Zariski topology, by Lemma (12.7). Moreover, the partition of unitary argument s apply to the sheaves ,.~~ and thus HP(A,,~~ for p > 0 . N o w (12.8) follows in the usual way.

Q.E.D. Our next result is a local regularity proposition. Let O be an open

subset of IR n, f2' a relatively compact open subset of f2, f an analytic function on f2. We set D = { f = 0 } , O * = O - D . If u is a function, or an r-tupte of functions, on f2*, which is locally L 1 on f2, then u determines a distribution (or an r-tuple of distributions) on ~2, which we shall denote by ft.

(12.9) Proposition. Let A be a linear elliptic partial differential operator of order k on (2. Suppose that u is C ~ on Q* and that, for N >= No, fN u, fN A u

are locally L 2 on f2 and f"ff'Au has L 2 derivatives up to order I 2 - k. Then,

for large enough N, fNu has locally L 2 derivatives up.to order # on f2'.

Proof For simplicity, we limit ourselves to the case k = 1, as this is the only case we shall need in the sequel. It is clear that, for any v

A(fMv)= fM Av+ BM(fM-lv)

where B M is an operator of order zero. Now choose N > N 0. Then

A ( f '~u)= f '~A u + B u ( ~ u ) + T

where T is concentrated on D. Therefore, for large enough M, fM T = 0 on g2' and

(12.10) A(fN+Mu)= fN+M Au+ BN+M(f N+M-1 U)

on f2'. It follows from (12.10) that, i f fU+M-lu has locally L z derivatives

up to order 2 (2 <p), then ~ has locally L 2 derivatives up to order 2+1. Q.E.D.


Now let U be an open set in 112" and leO(U) a holomorphic function. Set D = { f = 0 } and U * = U - D . A smooth function u on U* has Coo poles along D if, for each # > 0 , fNu has a CU-extension to U for N>0 . We may also speak of forms having a C ~ pole along D.

Let E ---, A be the restriction of E --* A as above, and as metric in E we take any C ~ metric for/~---,/1. Denote by B~ the space of smooth, E-valued, (0, q) forms on A which have C ~ poles at infinity (i.e., along D = A - A ) in the sense explained above. Clearly B~176 and we set B* (E)=@B~

(12.1t) Proposition. I f A is a special affine, then the complex {B*(E); ~} has no higher cohomology.

(12.12) Proposition. There is a canonical isomorphism between the cohomology of {B* (E); c?} and the cohomology H* (A, Calg(E)).

Since the "partition of unity" constructed in Lemma (12.6) applies to the sheaves M~ ) constructed by localizing B~ in the Zariski topology, (12.12) follows from (12.11 ) by the usual argument.

Proof of (12.11). Let eeB~ satisfy ~c~=0. As a consequence of Proposition (9.7), we may write c~=c5/3 for some ~eA~ 0). Let a be a section of [D] which defines D = A - A and choose smooth metrics for/[ and [D]. Relative to these metrics, for N >0

S I~12 I~ N q ' < + ~ , A

~1/~12 Icrl N ~e< + ~ A

where 7 j is the volume form of the metric of A. Setting

(12.13) (~o, @)-- ~ (~o, q,> [al N 7 ~ A

we may write ~q =/~1 +/?2, where

ffl/~ilz lain ~ < + ~ ( i= 1, 2), A

~/~ = 0

and where/~t is othogonal (in the sense of (12.13)) to the space of the locally L z forms y on A such that

~'[yl 2 Io-I N ~ < + ~ , A

~ = 0 .

Then cSfl~ =~ and fll is in A~ O) by the regularity theory for elliptic P.D.E.'s. Now let c~* denote the formal adjoint of~ relative to the metrics


of ,4 and E. Clearly

(12.14) ~* fl~ =Fill

where F is an order zero operator which has C ~ poles at infinity (up to sign, F is inner product with c3 laiN). We will prove, inductively on #, the following statement, which, in view of Sobolev's theorem, is equivalent to fll having C ~ poles at infinity:

(12.15) For every # and M >>0, aMfll has locally L 2 derivatives up to order/t.

Here, as in Proposition (12.9), aMfl~ stands for the unique E| valued current which restricts to anti1 on A. (12.15) is obviously true for # =0. If anti1 has locally L 2 derivatives up to order p - 1, then, by (12.14), a M c~* fl~ has locally L z derivatives up to order p - 1 (to have this it may be necessary to take a larger M). Applying Proposition (12.9) gives our assertion. Q.E.D.

c) We now prove Theorem II for the higher Hq's. Using (12.8) and (12.3), we have inclusions

B*(E)~A*(E, O)cK*(e, O)

where B* (E)c K* (E, O) induces an isomorphism on c~-cohomology, and where either may be used to compute H*(A, (galg(Ea~g)). Thus H~(E, O) maps surjectively onto the cohomology of K*(E, 0), and Theorem II will follow if we show that this mapping is injective. Suppose that ~ A ~ (E, O) satisfies

for some fl~K~ 0). Using the same notations 'as in the proof of (12.11), we may suppose that

S I/~1 z I~rl N ~ < ~ A

and that fl is orthogonal to the space of the locally L 2 forms ~ on A such

that ~y = 0,

ITI 2 la[ ~ ~ e < 00. A

Then ]~ is in A~ O) by the regularity theory for elliptic P.D.E.'s. Q.E.D.

w 13. Proof of Theorem III Letting I, II, III, IV refer to the various definitions of growth for a

line bundle over a special affine variety A, we shall show that these are


all the same by p rov ing the impl icat ions

I ~ II ~ I V ~ I I I ~ I .

The first step has been done in w 11 (c).

(II =:- IV) Given a h o l o m o r p h i c m a p p i n g

f : A --. IP m

whose order funct ion satisfies

(13.1) T(f, r ) = O(r~),

we set D = - f - I O P " -1 ) so tha t [ D ] = L

where L---, A is the pul l -back of the s tandard posit ive line bundle over IP". Since T(f , r ) = T t (f, r) in the no ta t ion of w 5, the First Main Theo- rem (5.6) and subsequent Neva l inna inequali ty (5.8) give

N(D, r)< T t (f, r)+ O(1).

C o m b i n i n g this with (13.1) gives the impl ica t ion II =*, IV.

(IV =~ II I ) Let P* = {(z, w)er x ~3"- l: 0 < tz l < 1, Iwl < 1} be a once- punc tured polycyl inder and P* [ r ] = {(z, w)~P*: ]z] > 1/r} 13. Deno te by

o9 = (dz A d~) + ~ dwv A dwv V=I

the Eucl idean K~ihler fo rm on (12 • t~ "-1. Fo r a divisor D = P * , set D Jr] = D ~ P* [r] and

N ( O , r ) = i { ~ O)n-1} d t o PIt] t

The ma in step in our a rgumen t is the following:

(13.2) L e m m a . Set P* - { O < [ z l < 1 - 6 , [wl<e}. I f N(D, r)=O(r~), then F. ,~ - -

for sufficiently small e and 3, D c~ P~*~ is the divisor o f f 6(_9(P~*o) with

M ( f , r ) = max log [f(~)l = O(ra+~) ~i%*,6[r]

for every q > O.

Proo f Choose an integer q > [2] and let 1 + l + + 1

E(z, q)=(1 - i /z) e ~ -2r ' ~

be the Weierstrass primary factor (for l/z). In the case where P * = {0 < l zl < 1} is a punc tured disc in one variable, D = zl + z z + . . . and the

~3 As mentioned before, we shall treat the case A =,4- H with H being smooth; the general situation may be done by combining these methods wih the techniques of [ 11] dealing with multiply-punctured polycylinders.


function

k = l

is ho lomorph ic on P*, has zero divisor D, and satisfies the required growth estimate (cf. [28]).

In general, we consider the punctured discs

A~'(w0)=P~; ~ {w=w0}.

w o A~( Wo )

Z - Q X I S

W - C IX IS

Fig. 3. The punctured discs d~(w 0)

In our applications we may assume that the given divisor D contains no disc A*(w), and it is convenient to make this assumpt ion here. Then, for ~, 5 sufficiently small, D will not meet any of the boundary circles Izl=l-6, tw]<a.

Thus we may write

D~A~(w)=Dw=z_l(w)+ z2(w)+...

where all fzk(w)L < 1. The expression

i i E z k=X Zk(W q =f(z,w)

will, if shown to converge uniformly on compact sets, be our required function. To do this, it will suffice to prove that, for each w,

(13.3) N(Dw, r)= O(r ~)

with a uni form " O ' .

N o w since HI(P*,(9)=O=H2(p*,7Z), we may find ge(fi(P*) with (g) = D 14. By Jensen's theorem for an annulus (cf. Proposi t ion (1.4) in [6])

(13.4) v(w, 1 ) + N ( D w, r)--v(w, r)+O(logr)

~4 This is where we-are using that P* has only one puncture.


where

1 S log e i~ w dO. v(w, r )=57 o

Since v(w, r) is plurisubharmonic in w,

v(w,r )<C o ~...~ v(u,r)du ld~ 1 . . . du ._ i d u . _ l . luv- wd <=v

On the other hand, setting n - - l ~

qo u = ~ du v/x d~v, u=l

by (13.4)

~ ... ~ v(u, r) du i d~ i ... du._ ~ dfi._ 1 Iu~-wvl=<o

= ~...~ N(D , , r )du ldfi i . . . d u . _ l d~ ._ l+O( logr ) [uv- ~,v[ No

= i ( ~ ' " Q n ' D , , t ) d u l d " i " ' d u , - l d u , - 1 ) ~-dt +O,logr) 1 l u v - w d = o

, 1 , dt < i ( ~ ~ . - ) ~ - + O ( l o g r )

1 D[t]

< N(D, r) + 0 (log r)

= 0 fix).

Combining this with (13.4) gives

N(Dw, r)<= CrY+ C',

which then proves (13.3). Q.E.D.

Now let f g be holomorphic functions on the punctured disc A*= {0< ]zl < 1} and assume that:

M(f, r) and m(g,r) are O(r ~)

h = j / g is holomorphic.

(13.5) Lemma. Under the above assumptions,

M(h, r)= O(rX).

Proof Following the notations and argument in the Lemma (6.3), it will suffice to show that

(13.6) m (h, r)-- O (ri).

Now clearly

(13.7) M(h, r)<-_m(f, r)+m(1/g, r).

proof of


On the other hand, Jensen's theorem (cf. [6]) for A* gives

N ( D, r) + m (1/g, r) = m (g, r) + O (log r)

where D = {g(z)=0} is the divisor of zeroes of g. Thus

re(l/g, r)< m(g, r)+ O(log r),

and both re(f, r) and m(g, r) are O(r~). Plugging this into (13.7) gives (13.6). Q.E.D.

Proof of I V ~ III. Cover A by finitely many punctured polycylinders {P~*} such that

D c~ P~* = (f~) where, in the above notation,

M ( L , r)= O(r") for a suitable #. Setting

f~# = f~/f#,

the {f,#} give transition functions for the line bundle [D]--~A and, by Lemma 13.5,

M(f,#, r)= O(r"). This proves our assertion.

(III ~ I) We begin with the following:

(13.8) Lemma. On the punctured polycylinder

P * = { ( z , w ) O E x ~ " - l z 0 < l z l < 1 and twl<l} ,

let f (z, w) be a non-vanishing holomorphic function with

M(f , r)= O(rZ). :l-hen we may write

f (z, w)= e ~ . z l

where g is holomorphic in the whole polycylinder P.

Proof Suppose 1 d f

Then f / z z is non-vanishing and has a global logarithm on P*. Thus

f = e h z l

where h e(9 (P*). The growth estimate on f gives

max Re h(z, w)= O(rl), l~l>-a/r Iwl=<l

from which it follows that h(z, w) has at most a pole of order k = [2] on the puncture z=0. Q.E.D.


Suppose now that L---~ A is given by finite order transition functions {f,~} relative to a covering {P~} of A by punctured polycylinders. Applying Lemma (13.8), we may write

L~ = e (~-") - - - z'

where z = 0 defines the puncture in P~ and g,ge(,0(P~ n P~). The function

log IL~l" Iz, l"

is, for m>>0, of class C ~ on P~* n_P~*, and class C t2) on P~n P~. Writing A = A - H , we may choose ~ ( 9 ( A , [HI) which defines H and set

p=l~l"

for any metric in [H] ~ A. Then

p log If~l =0,r

is of class C ~2) in P~ n Pp, and

0 ~ + 0 ~ = 0 ~

where defined. Using a partition of unity, we may write

2 0~ = 7~ - 7~

where 7, is C ~~ on P~* and class C ~2) on P~. The functions

a~ = e ~'/~ satisfy

a,=tLat 2 a a

in P~ n P~, and thus define a metric for L ~ A. The curvature of this metric is

O = dd c log a,

and thus satisfies the correct growth condition. Q.E.D.

w 14. Proof of Theorem I V

We shall prove Theorem IV only when A has a smooth completion/1 such that H = A - A is a smooth divisor. In the general case, the proof is somewhat complicated by the fact that analytic Picard group of a polycylinder with more than one puncture is non-trivial (cf. [11]). Alter- natively, a proof of Theorem IV can be given along the lines of the proof of Grauert 's theorem in w167 19-20: in this case Eq. (20.7) reduces to the usual inhomogeneous Cauchy-Riemann equation and can be solved globally with good growth estimates.


a) Before proving Theorem IV, we need some preliminary remarks. L e t / [ be a smooth compactification of A such that A = A - H where H is a smooth divisor. Let j: A ---, A be the inclusion.

(t4.1) Lemma, Hq(A, (9*)~-Hq(A, j.(9*) for q>O.

Proof By Leray's theorem there is a spectral sequence {El 'q} abutting to H*(A, (9*) and with is

E~,~ = I4p(i, I~ I (9*).

If we show that the direct image sheaves R~.(9*=0 for q>0 , then the result will follow. This is a local question in a punctured polycylinder P*. On P* we have

0---, :g ~ (9 ~ (9" ~ 0

and Hq(P* ,O)=0 for q>0 . Thus H~(P*,(9*)=O for q > 0 and we are done. Q,E.D.

Next we let (9/!o. c j , (9* be the subsheaf o n / 1 of functions of finite order.

(14.2) Lemma. Hq(,4, (9~'.o.)~Hq(~{,j. (9*) for q>0 .

Proof We observe first that a non-vanishing holomorphic function uE(9*(P*) has finite order if, and only if, d logu has only finite poles on the puncture P - P * (cf. Lemma (13.8)). It follows that the sheaf sequence o n / 1

0 C*--~ (9~o ' ~ 1~ f2~z(poles) 1 1 --~ ~ Rj. ~/Ri. ~ ~ 0

is exact, where (2~z(poles) is the sheaf of closed meromorphic one-forms having finite poles along H = A - A .

Now consider the sheaf j , f2 q of closed q-forins on A which are holomorphic on A and have arbitrary singularities along/4. We claim that

(14.3) HP(A, (2~:(poles)) - , HP(/I, j . (2 q)

for p, q > 0. The proof is by descending induction on q, the case q = n being easy since g2~= (2" and thus both sides are zero, For q > 0 we have exact sequences

0 , ~2~:(poles) ,f2 q (poles) ~ ~ (2~+ 1 (poles) , 0

1 O- , j . O ~ + j , Oq a , j . ( 2 ~ + 1 - - , 0 .

~s Whal we are observing is that a suitable covering of A by punctured po]ycylinders is acyclic for the sheaf @*.


H ~ (A, f2 q (poles))

1 H ~ (A, j . ~ )

By the holomorphic and algebraic de Rham (cf. w 18 below)

From the five lemma it follows that (14.3) for q + l and p> 1 implies (14.3) for q and p>2. To get p = 1, we consider

, H ~ (A, s + 1 (poles)) ~ H 1 (m, (2~:(poles)) , 0

1 1 HO(~,j , oq+ l, H 1 (A, j , t2~) ,0.

theorems respectively

0 --,.j, (9" ,0 .

Q.E.D.

and this gives (14.3) for p= 1.

Now consider the exact sequences

0 ~ll~*

------~ tl~*

-~ (9;'.0. -~ 0 dlog , ~2~:(poles)- , R~j. 112/RJ. Z

, R L ~ / R L 2~

Applying (14.3) and the five lemma once more gives the lemma.

b) Combining (14.1), (14.2) and the isomorphism

H 1 (A, (9") ~ H 2 (A, 7Z) ~ Vectlo p (A),

which holds because Hq(A, (9)--0 for q>0, we obtain

(14.4) H1 (~[, (gf*.o.) ---- Vect~op(A)-

Taking into account Theorem III, we see that every C ~ line bundle on A has at least one complex structure of finite order.

To prove uniqueness, suppose that L--, A is an Hermitian line bundle whose curvature is O(r z. q)). By the same arguments as in w 10, the finite order sections (gf.o. (A, L) are those holomorphic sections a which satisfy

(14.5) M(a, r)= O(rU).

(14.6) Lemma. Suppose that ae(9~.o.(A, L) and f~_(gf.o.(A) is a holomorphic function of f inite order such that p = a / f is holomorphic. Then p e (gf.o. (A, L).

Proof Using the notations of w 10, Lemma (10.4), the argument given these shows that it will suffice to show that

O(p,r)= S l~ + IPlA OA[r]

4 Inventiones math., Vol. 28

H ~ (A, j . (2~: + X)/dH ~ (,4, j . oq) ~ H q + 1 (A, r

H ~ (A, Q ~ + I (poles))/dH o (,~, f2q (poles)) ~ H q +~ (A, r

50 M. Cornalba and P, Griffiths

is O(rZ). This is because, by suitably modifying the metric in L---~A, tog ]Pl will be a plurisubharmonic function on A, and thus log [p(x)] for xeA [r] may be estimated by O(p, R) for R>p. Now

O(p,r)=O(a,r)+ ~ log+-~-I A 0A[r] Ifl

and 0(a, r)<M(a, r) is O(r") for suitable /~>0. On the other hand, by Jensen's theorem ([6]) and setting D = {f=0},

N(D,r)+ ~ log + 1 TTi - A = ~ log + If l A+O(I ) O A [r] I J l OA[r]

<M(fr )+O(1) , so that

log + 1 oaf,] ~- f~A<M(fr)+O(1)

and we are done. Q.E.D.

Now suppose that L--, A is topologically trivial. We must find a non- vanishing section p of finite order. Choose a section ae(gf.o.(A, L). The divisor D = {a= 0} is a divisor of finite order by the First Main Theo- rem (5.6). Following the proof of Theorem III in w 13, we may find a finite covering {P~*} of A by punctured polycylinders and f~e(gf.o.(P~* ) such that Oc~P~*=(f~). The cocycle L~=L/A is trivial in H1(/[,(9~.o.)~ //2(A, 7/), and thus f~o=gp/g, where g,e(9~.o.(P~* ). Then f={ f~g ,} is global finite order holomorphic function on A with ( f ) = D. Applying Lemma (14.6) p =a/fe(Pf.o.(A, L) is a non-vanishing finite order section.

Q.E.D.

w 15. Proof of Theorem V a) Let V be a smooth projective variety and H c V a smooth ample

divisor with

(15.1) H2(V, (9 [HI )=0 .

Given a C ~ line bundle L v - , V, we choose a metric in Lv and ~-connection Vo'. Then (Vd') z =~ is a C ~~ c~-closed, (0, 2) form on V. Using (15.1), we may write

where t/is an [//]-valued C ~ (0, 1) form on Vand ee(9 (V, [//]) defines H. Then


gives a complex structure in L--~A where A = V - H and L=LvlA. Moreover, a V"

is a C oo operator on all of V. Let P be a polycylinder around a point on H, P* -- P n A, and e a Coo section of unit length for LvtP. Set Vo'e = 0 o. Then

is the connection matrix for the metric connection in LIP* relative to the unitary frame e. The curvature O = dO is then of the form

where e, fl, 7 are Coo forms on P and z = 0 defines Hc~P. It follows from (3.1) and (15.2) that, in the punctured polycylinder P*,

so that L-+ A has finite order _<_ 1.

If the order were less than one, then by the arguments in w 13 we would have L = [D] where D is a divisor of order < 1 on A. Choosing a finite covering {P=*} of A by punctured polycylinders, we could then write

D ~ P~* = (f~)

where f~(O(P~*) has order < 1. The ratios

f=e = f=/f~ e O* (P~* c~ P~*)

are non-vanishing holomorphic functions of order < l, and then by Lemma (13.8) the f=a have only finite poles on H c~ P~* n Pp*. But if this happens, L ~ A carries an algebraic structure. Thus

ord L < 1 r L--+ A is algebraic (15.3)

ord L = 1 otherwise.

If the Chern class c 1 (Lv) is not of type (1, 1), then

(15.4) ord L > 1

by Theorem VII. Combining (15.3) and (15.4) gives Theorem V. Q.E.D.

Remark. This proof is different from the usual proof of the Lefschetz theorem in that the lower bound given by Theorem VII has been played off against an upper bound which can be explicitly determined, essentially because GL(1, C) is abelian and the exponential map C ~ C* ties in so close with the logarithmic potential methods of Nevanlinna theory. 4*

52 M, Cornalba and P. Griffiths

C. Applications to Algebraic Geometry

w 16. G.A.G.A. and Serre's Theorems A and B

a) G.A.G.A. [30], and its subsequent refinement by Grothendieck, states that:

On a complete, complex algebraic variety, the analytic and algebraic theories coincide.

More precisely, given a coherent algebraic sheaf ~a~g on a variety V, there is an associated analytic sheaf ~ n and induced mapping

H* (V, ~ g ) --) H* (V, ~n) .

Serre's results state that, in case V is complete.

(16.1) The assignmont ~ i , ~ ~ n induces a bijection

abelian category of] (abelian category of] coherent algebraic [ ~ ~ coherent analytic ~ ; sheaves on V J (sheaves on V J

and moreover the induced maps

H*(V, ~lg) --~ H*(V, ~ ) are isomorphisms.

We wish to observe here that, in the case when V is smooth and f f is locally free, (16.1) follows from Theorems I and II applied when A = V is compact so that the growth conditions are vacuous.

A consequence of(16.1) if Chow's theorem, which is deduced as follows: Let Z c V be an analytic subvariety and (9 z the structure sheaf in the

usual (Hausdorff) topology. Then there is a unique coherent algebraic sheaf ~ g whose associated analytic sheaf is O z. An easy argument then shows that, in either the Zariski or usual topology,

supp ~'~l, = supp •z = Z.

It follows that Z is an algebraic subvariety.

Conversely, our proof of Theorems I and II were ultimately based on Chow's theorem, as proved in w 2 of [13-], and the subsequent growth characterization (4.2) of the algebraic subvarieties among the analytic ones, this being proved in w 4 of [13].

b) Theorems A and B for an affine variety A state that (cf. [31]):

(16.2) I f ~x , is a coherent algebraic sheaf on A, then

Hq(A,~1, )=O for q > 0 ,

and H~ (A, ~lg) generates every stalk (~lg)x as an ((~alg)x-module.


In case A is a smooth special affine variety and ~ g is locally free, these results follow from (9.7) and (9.9). The difficulty in using our methods to prove (16.2) in the case when A is a general smooth affine variety and ,~,~g is locally free is that there may be no smooth compactification ,4 of A such that D = / [ - A is ample and has simple normal crossings, and thus the basic estimate (9.5) will not hold. In fact i r a is special affine and ~lg ~ ~ restricts to ~ g ---, A, then the proof of (9.7) gives that

(16.3) H%4,~ lg [kD] )=O for q>0, k > k o.

On the other, Theorem A is equivalent to the weaker assertion

(16.4) H q (/l, ~ g [k D]) ---, H" (,4, o~g [(k + l) D])

is zero for q > 0 and t> lo(k ). It is almost certainly the case that (16.4) may be proved by suitably refining the methods used to prove (16.3), but we will not pursue this here.

The full statement of Theorem A for smooth affine varieties will be used in our discussion of Grothendieck's algebraic de Rham theorem in w 18 below.

c) A final remark is concerning the famous Kodaira vanishing theorem ([23]), which states that if L ~ V is a positive line bundle over a smooth, complete variety V with canonical bundle K, then

(16.5) Hq(K (9(L | for q>0 .

We wish to observe that (16.5) may be proved directly from the Kodaira identity (9.2), without using harmonic forms, by the argument given in w 9 (c).

It is perhaps interesting to note that the variant of(16.5) given by [27] seems to require the use of harmonic forms.

,~ 17. An Approximation Theorem

The Laurent series of an analytic function in It* shows that the rational, hotomorphic functions on ~2" are dense in (9(tF*) with the topology of uniform convergence on compact sets. The following generalization of this should have been noticed before, but we are unable to find it.

(17. l) Proposition. Let E --* A be an algebraic vector bundle over a special affine variety. Then the space (galg(A, E) of rational, holomorphic sections qJ" E ~ A is dense in the space (9 (A, E) of all holomorphic sections, the topology being uniform convergence on compact sets.

We shall deduce this result from another, more differential geometric statement, which we now explain. Let M be a complex manifold having


a complete K~ihler metric with corresponding (1, 1) form q~, volume form

q~=g0 A --- A q , n t ime~

and Ricci form "Ric". Suppose that E ---, M is a Hermitian vector bundle with curvature form 0 . Finally, assume that we are given an exhaustion function z: M ~ IR + and that all of this data satisfies the following inequalities (c being a generic constant):

IRicl~cq~

(17.2) IO(e)[<ccp (e~E)

ddCr >_ctp.

(17.3) Proposition. ,Under these conditions, the space of holomorphic sections t r~O(M,E) which satisfy ~ lal a e - N ~ < ~ for some N is dense

M

in the space ~o (M, E) of all holomorphic sections.

Proof. We use the almost magical methods of [20], pp. 95-96. The topology on (P(M, E) is equivalently given by uniform convergence on compact sets or by La-convergence on compact sets. For every continuous linear form 2 on (9(M, E), by the Hahn-Banach theorem we can find a compactly supported L2-section ~ of E --* M such that

M

Suppose that we have ~(~,~)~:o

M

for all a~(9(M, E) which satisfy the estimate

M

for some N. We want to show that

(~, ~) ~ - - o M

for all a~ (9 (M, E). For this it will suffice to solve the distributional equation

(17.4) (=c~* 7

for a compactly-supported, square-integrable, E-valued (0, 1) form 7- This is because (17.4) implies that

~(,~, ~)~= ~o^ ,a*~--+ ~ ^ ,~=0. M M M

Suppose now that ~(x)=0 if r(x)>R, and choose an increasing convex function Z such that Z( t )=0 for t<R and Z(t)=t for large t.


Choose N o large enough so that

(17.5) ( N ~ ddC~>{lRic, ,IO(e)l ,q)},

and define ~N (x) = No �9 (x) + N Z (~ (x)).

Let L~ be the E-valued (0, q) forms ~ which are locally L 2 and satisfy the global estimate

~ t~12 e-~'N cb< oo. M

Here, the point norm t~[ 2 is computed using the K~ihler metric on M and the given metric in E.

Consider the sequence of Hilbert spaces

(9(E, N)--~ L ~ o (N) ~ L~ _ L ~ LO2 �9 2 (N)

where C(E, N) are the holomorphic sections a of E satisfying

~ lot2 e-(N+No)~ ~b < oo, M

and SN, T N are the closures of 0. Considering the metric (,)N on E which is the given metric times the weight function e -e'N, the curvature for this metric is given by

ON(e)=O(e)+ No ddC~ + N ddC z(r)

> O ( e ) + N o ddCr.

For a C ~~ compactly supported, E-valued (0, 1) form r/, the Kodaira identity (9.2) gives

II Tev~lP[ 2 + JIS~ll 2 = IlVnll 2 +(Ricr/, ~l)N+(ONrl, ~I)N

_>-cltr/[l~ by our choice of N 0. Since the K~ihler metric is complete, we have the basic estimate

(17.6) pi ZNrlll~v+ llS*nll~ >__cllrll[~

for all n e ~ r ~ c~ ~s~, (cf. (9.5)).

By assumption, ~e ~N is orthogonal to the null-space (9(E, N) of SN, and therefore belongs to the range of S~,, i.e. we have

e *'~ = S~, o>N

for some o ~ e L ~ (N), which may be chosen to be in the null-space of TN. Then (17.6) gives

IIoJNII~__ < ~ - IjCe~"llg. r


Explicitly, this amounts to 1

5 I~ 2 e-~ q } < - - 5 [(I 2 eON q~, M C M

S* {e- q*N o-}N) =

where S is the closure of ~ using the original metric in E and S* is its adjoint. By our choice of Z, 5 [il e e~ q} is independent of N as N--+ oo. Writing M

O)N = eON )'N,

we thus have j" I~NI 2 e *N {/} < c ,

M

Let 7 be a weak limit of the ?N- Since Ou(x)-*oo if r ( x ) > R , it follows that supp y c M JR] and S*y = ~ in the sense of distributions, and this is what we wanted. Q.E.D.

Proposition(17.1) follows from Theorem II in w taking into account the construction of z and q~ given in w 3.

~" 18. The Algebraic de Rham Theorem

We shall give a proof of the algebraic de Rham theorem due to [16]. The argument will use the results in w 12, and will thus be ultimately based on the c?-operator. Although we shall not do so here, the method may be pushed a little further to yield a proof which does not assume resolution of singularities. Finally the result may be viewed as a special case of a de Rham theorem with growth conditions on any Stein manifold having a plurisubharmonic exhaustion function with a finite number of critical values.

a) De Rham's theorem with Growth Conditions. Let M be C ~ manifold and r: M - * Ill + an exhaustion function having finitely many critical values, all of which are less than t 0. Setting M r = z -1 (t), the M, are all smooth real hypersurfaces with normal dr for t > t o. Let g be a Rie- mannian metric on M. Using g we may identify dr with a vector field ~?/dr normal to the M t, and integration of the vector field

~ / ~ 2

~r ~T gives diffeomorphisms

f : M,o - -~* M,.


Definition. We shall say that g has polynomial growth (relative to z) if

i) [dr] 2 and Idr]~ 2 are bounded by a polynomial in t;

ii) the norms of d f and dr,-t are bounded by polynomials in t.

For a metric g which has polynomial growth, let Aq(M, r, g) be the vector space of Coo q-forms r /on M such that

M

(18.1) S Id.l %< + oo M

for some N > 0, where 4~g is the volume form of g, It is clear that if h is another metric with polynomial growth such that g < c r u h, h < c ' z u ' g, then Aq(M, z, g)= Aq(M, ~, h). By definition we have a complex

�9 ." ~ A q - 1 (M, z, g) - ~ Aq(M, r, g) ~ ""

and we let H~R(M, ~, g) be the resulting cohomology. If H~R(M) is the usual de Rham cohomology of M, there is a natural map:

(18.2) H~R(M, z, g) ~ H~a(m).

(18.3) Proposition (C ~ de Rham with growth conditions). The mapping (18.2) is an isomorphism.

Proq[~ Set : M[t l , t2] ~- {xEMl t l ~z(X)~ t2},

M [ q , ~ ] = { x ~ M l q ' < r ( x ) } .

Using f, there is a diffeomorphism:

(18.4) M It o , ~ ) ~ M , o x [to, oo).

On M [t o, co) take a metric which is the product of a fixed metric on M,o and dt 2 on [to, oO); then extend this to a smooth metric h on M. By the remark below (18.1 ) we may replace g with h without affecting A* (M, z, g). In what follows, if 0r, is a family of forms on M,, t > t o , depending smoothly on t, we shall denote by 6 the unique form on M [ t o, oo) such that ~'[M, = a, and (c3/~? t, 6 ) = 0, where ( , ) stands for contraction.

Let (p be a closed q-form on M. Using the isomorphism (18.4), the restriction of q0 to M[t , oo) is cohomologous to ~O,o =(plM, o, viewed as a form on the product M,0x[ to , oo). Thus we have q~-dO=q~,o on M[to, oo). Choose to < t l <t2 and let p be a C oo bump function which is one on M [ t 2 , oo) and zero on M[0 , fi]. Then

cp - d ( p 0 ) = q~o


where

M

for any N > 1. Thus the mapping (18.2) is onto.

It is a little harder to prove injectivity. For this, suppose that tpeAq(M, z, h) is closed and q~ =dr / for some C ~ form t /on M. Then each restriction

is exact, and using Hodge theory with smooth dependence on parameters ([25]),

r

Mt Mt

where ~, is C ~ in t; and where h, is the induced metric on Mr using (I 8.4). Then

M [ t o , oo) M[to. ~ )

Letting p be a bump function as above, the form p2 is in Aq-I(M, z,h)

and ~ =q~ - d ( p 2 ) has the property that r =dr 1 for some C ~ form t/on M, and q/ ,=~,[M,=0 for t>t2.

Thus on M[t2, ~) we may write ~=~Adt . From O=dql=d~Adt, it follows that d~, = 0. Using [25] again we may write

~ , - d y , = ~ , where ~, is harmonic and y,, ~, are C | in t,

and where

Mt Mt Mt

Thus the forms ~ A dt and ~ A dt are in A* (M [t2, co), ~, h) (using obvious notation) and { A d t -d (~ A dt)=~ A dt. Using the bump function p as before we may replace ~p by ~ A dt where ~, is harmonic. Set

t

t2

Then d(~=~ Adt, and an easy integration by parts shows that

(~r oo), ~, h).

Using again the bump function p, ~ A dt-d(p6~)=O on M[tz, c~). Our conclusion is this:

If tpeAa(M, ~, h) is a closed form such that tptM~=0 in H~(M,) for large t, then we may find taaAa-~(M,z,h) such that ~ p - d a = 0 on M[t~, oo) for large t z.


To complete the proof, we may suppose now that ~o is a closed C ~ form on M such that q~=0 in M[to, ~), and q~=dr/for some C ~~ form r/. Then dq=O in MEt o, ~), and we may repeat the above surjectivity argument to find a C ~ form ct such that r / -&t~Aq-l(M[to, co), z,h). Then q~ =d( r / -d~) , and we are done. Q.E,D.

b) The Case of Affine Varieties. Let A be a smooth affine variety,

(18.5) ...__,oq t ~ ~oq+l(A)---~.. . ~alg ~za] -- a~alg

the complex of rational holomorphic forms on A, and

Hqg (A, alg)

the cohomology of the complex (18.5). There is a natural map

(i 8.6) H*R (A, alg) ~ H3R (A),

and Grothendieck's comparison theorem states that (18.6) is an isomorphism.

To prove this, we let A be a smooth completion of A, D = , 4 - A the divisor at infinity for A, aEC(A, [D]) be a section defining D, and

1

an exhaustion function for A. An easy computation shows that z has only finitely many critical values (cf. w 2 (b) of [13]), and any metric g o n / [ has polynomial growth with respect to r. In what follows we shall also assume that g is K/ihler.

Let q) be a closed form representing a class in H~R (A, z, g). Then

q9 = ~pq, o + --- +tpo,q

where ~q00, q =0. Also ~Oo. ~ has an L a pole at infinity and thus by Serre's Theorem B for affine varieties

~Po,q = ~r/o,q-1

for some r/0,q_ I with an L 2 pole at infinity. Assume for a moment that we may choose r/in such a way that 0r/has L 2 pole at infinity. Replacing ~p by r -d r /e l imina tes the (0, q) component of q~, and repeating the argument q - 1 times we see that (18.6) is surjective. Similar reasoning shows that it is injective. It remains to show that, if ~b,.~ has an L z pole at infinity and ~ , =0, then we may write

(18.7) ~9 =c~q . . . . 1


where r/and 0r/have an L z pole at infinity. To do this, let ( , ) be the inner product on forms induced by the metric g. Set:

Ilull-- (~l/~,u~, Ilullu= IIr-NJ2ull.

Denote by ~* and 0* the formal adjoints oft? and 6 (relative to g). Arguing as in w 12 we may solve (18.7) with an t/such that c~*r/has an L 2 pole at infinity. If N is large enough:

IIt/llu-2 < +~v,

H~II~v_ 2 < +vO,

H~*r/IIN_2 < +oo.

Moreover, by taking N even larger, if necessary, we may suppose that there is a sequence {t/z} of compactly supported smooth forms such that:

Jilnoo(tlr/z- r/HN_ 2 ~-ll(~r/z- 01IN_ 2 + H~* r/z--(~* qHN_ 2)=0.

Then:

II0qill~ + II0* ,,11~ =(0t/,, 0(~-Nr/,))+(0* r/z, 0* (r- '%))

-(0qz, & - u A r/i) + (t/i, Or-u A 0* qz)

_-< [(~t/,, 0(~-N r/z))+ (~* r/i, ~* (~-%))[ (18.8) + k (tl0r/ilIN + II0* t/;tlu)

=< t[c~r/z[l~ + 1[~* r/,l[~ + k(ll0t/zHN + I[0* r/zllN) t - +k (It0t/~IIN + II~* t/zll~)

< C + k(ll0r/zllu + I[0* r/;llN)

where C, k, k' are suitable constants. To derive (18.8) we have used the fundamental K~ihler identity:

&~* + ~ * ~ = o 0 * + o * o

plus the remark that ~z -u is equal to z -N+~ times a bounded form. Using the inequality

f-

we infer from (18.8) that II0r/;llN + II0* r/zIJN is bounded. Therefore Or/is the weak limit of a subsequence of {0r/i} and

[Icgr/IlN< +co . Q.E.D.


D. Grauert's Theorem and the Oka Principle

w 19. Complex Structures on Vector Bundles and Holomorphic Homotopies of Complex Vector Bundles

This section, the following one and Appendix 3 are devoted to a proof, in the spirit of P.D.E.'s, of the theorem of Grauer t cited in the introduction, namely that the natural mapping (0.1) is a bijection of sets. The core of the proof is in w 20, while Appendix 3 contains some results of a rather technical nature, which are used in w 20.

a) Let M be a complex space, E a differentiable vector bundle over M with fibre tlY. A complex structure on E is a locally free sub-(~M-module of the sheaf ~f (E) of all differentiable sections of E which generates Cs as a module over the sheaf of all differentiable functions on M. Two complex structures are said to be equivalent if there is an au tomorphism r/ of E that carries one into the other, homotopic (differentiably homotopic) if there is a continuous (smooth) family ~/, of au tomorphism of E depending on a real parameter t such that r/0 is the identity and r/1 transforms one complex structure into the other.

Suppose that M is a manifold. A ~-connection on E is a linear differential operator V that carries differentiable sections of E into differentiable E-valued (0, 1)-forms and satisfies the relation:

V (fs) = s | ~ f+ fVs

for every section s of E and every differentiable function f. We may give a meaning to the expression V(a), where a is an E-valued (0, q)-form by

setting: V (s | q)) = s | ~q) + Vs /x ~o

for every section s of E and every (0, q)-form ~0. Clearly, i fa is an E-valued (0, q)-form and p is a (0, p)-form:

V(p Aa)=~p A a + ( - - 1)Pp A Va.

It is easily seen that, i f f is a function and s is a section of E, then

V2( f s )= fV2s

in other terms V 2 is a Hom(E, E)-valued (0, 2)-form. V is said to be flat (or integrable) if V 2 =0.

Let E, F be vector bundles with ~-connections V E, VF. Then there are naturally defined ~-connections on E*andE @ F (and hence on H o m (E, F), etc.). Explicitely:

(VE, ~0)(e)= ~q0 (e)-~o(Vee),

VE|174 V E e Q f +e | VFf

where e is a section of E, f a section of F, ~o is a section of E*.


If V is a 0-connect ion on E, we shall usually denote by the same symbol the connect ions that V induces on E*, Horn(E, E), etc. Not ice that these connect ions are flat if V is.

(19.1) Proposition. Let n: E---~M be a differentiable vector bundle of rank r and let V be a flat ~-connection on E. Then the sheaf Y of all the sections of E that satisfy the equation Vs = 0 is a complex structure on E.

Proof This is an easy appl ica t ion of the Newlande r -Ni renbe rg theorem [20]. Tha t 5 p is an (gM-module is clear, therefore all we have to do is finding enough local solutions of Vs = 0. It will be nota t iona l ly more convenient to do this for E* ra ther than for E. We may suppose that M is a ball in ~" with coordina tes z~ . . . . . z r and that E is differentiably trivial: let ~1 . . . . ,4r be fibre coordinates on E. We shall view ~i, . . . , ~, as sections of E* or', al ternatively, as complex-va lued functions on E. Similarly we may also think of V~ as a form on E. With these identi- fications we may give E, considered as a manifold fibered over M, an a lmos t complex structure compa t ib le with the complex s t ructure on M, _by letting_t_he (0, 1)-forms on E be the linear combina t ions of dS~ . . . . . dS,, ~341 . . . . ,04~, where

It is easily seen that the condi t ion that V be fiat t ranslates into the condi t ion that this a lmost complex s t ructure be integrable. Let r . . . . , ~kr be h o l o m o r p h i c functions on E such that ~k~ . . . . , if,, Zl . . . . . z, form a system of local coord ina tes in a ne ighborhood of the point z = 0 , 4 = 0 . We m a y suppose that ~bi = 0 on the zero section of E. Not ice that 0~; = V~i, hence the restriction of ~i to any fibre of n is ho lomorphic . Therefore we m a y write:

~i(z, 4) = ~ aij(z) ~ + a te rm of higher order in r J

It is immedia te that ~ e u 4 j is ho lomorphic , i.e., that V ( ~ e i j 4 j ) = 0 for every i. On the other hand the condi t ion that ~kl . . . . , 0 r , za . . . . , z, be a system of local coord ina tes means tha t the system of sections of E* :

Z O~ij(Z) CJ i= 1, . . . , r J

is a local frame. This concludes the proof. Q.E.D.

b ) L e t M be a complex space, D a disk (the disk { z l ] z ] < l } to fix ideas). Set X = M x D, p = project ion of X on to M, q = project ion of X onto D, and let re: E ~ X be a ho lomorph ic vector bundle. The locally free d)x-module o~ is, by definition, the sheaf whose sections are the relative vector fields on E ~ X that can be written, locally, as

0 ~..,z a,jr i ~ + b Oz


where 41 . . . . . 4, are fibre coordinates on E and alj, b are holomorphic functions on X. o~ fits into an exact sequence:

(19.2) 0 ---~ Horn(E, E ) - , J TM > O--~0

where Hom(E, E) is the sheaf of germs of endomorphism of E and O is the sheaf whose sections are the relative holomorphic vector field along the fibres of p.

Suppose that c?/~?z lifts to a global section of o~, 2.2 generates a holomorphic flow az with complex parameter z. The differential equations that define a are linear in the fibre coordinates: it follows that a~(e) is defined whenever z + q (~ (e)) belongs to D. Another consequence is that a~ maps fibres of rc into fibres of rc linearly.

When M is Stein O/Oz can always be lifted to a section of o~ By the above remark as(e ) is defined for every z~D when q(rc(e))=0. Denote by E 0 the restriction of E to M x {0}. Then:

(e, z) - - , ~ (e)

is an isomorphism of E 0 x D onto E. We have thus proved the following.

(19.3) Proposition. Let M be a Stein space, D a disk, p the projection of M x D onto M. Then every holomorphic vector bundle E on M x D is of the form E =p* (F) for some holomorphic vector bundle F on M.

(19.4) Corollary. Let E be a differentiable vector bundle on M and let 5P be a complex structure on p* (E). Let t o, t~ be two points of D. ~9 ~ induces complex structures 5Po, ~ on E via the inclusions:

M ~ - ~ M x { t i } - * M x D i=0, 1.

Then 5Po and ~ are differentiably homotopic.

(19.5) Corollary (H. Cartan's lemma on holomorphic matrices). Every holomorphk vector bundle over a polycylinder is trivial.

(19.6) Remark. Proposition (19.3) also holds (with the same proot) if D is replaced with IIL It follows in particular, that every holomorphic vector bundle on ~" is trivial.

(19.7) Remark. A relative version of Proposition(19.3) also holds. Namely, let M be a Stein space, N an analytic subspace of M, D a disk (or I12), to a point of D, p the projection of M x D onto M, E a holomorphic vector bundle over M x D. Let F be the vector bundle over M induced by E via the inclusion:

M ~ , M x { t o } - - - ~ M x D .

Then every isomorphism between p* (F) tN x D and E /N x D extends to an isomorphism between p* (F) and E. The proof is essentially the same


as the proof of Proposition (19.3): the only difference is that the vector field O/~?z must be lifted to a section of ~ that takes on prescribed values on N x D, which is clearly possible since M is Stein.

(19.8) Remark. It should be noticed that if in Proposition (19.3) M is supposed to be affine algebraic, D is replaced with 112 and E is supposed to be algebraic, our argument does not show that E is algebraically isomorphic to a bundle of the form p*(F): in fact, this is false. (19.2) is an algebraic sheaf sequence, O/Oz generates an algebraic flow and can be lifted algebraically, but in general the flow generated by 2 will not be algebraic. It is an unsolved problem of Serre whether there exist non- trivial algebraic vector bundles on IIY.

w 20. Statement and Proof of Grauert's Theorem

Our goal in this section is to prove the following result.

Theorem Vl. Let M be a Stein complex space, N a closed subspace of M, E a differentiable vector bundle over M, 5 P a complex structure on E IN. Then"

(i) ~9 ~ extends to a complex structure on E.

(ii) Any two extensions of ~ "~ to a complex structure on E are differentiably homotopic through automorphisms of E that reduce to the identity on N.

Theorem VI is a special case of Grauert 's Oka principle [4]. One of its obvious consequences, in the light of (19.3), is that the natural mapping (0.1) is a bijection of sets, as announced in the introduction. It should also be noticed that, in view of Remark (19.7), part (ii) of Theorem V1 is a consequence of part (i) applied to M x ~, (M ~ {0, 1 }) w (N x ~), E x C.

(20.1) Lemma. Let M, N, E, 5" be as in Theorem VI. Suppose that M is smooth. Then 5P extends to a complex structure on E[ U, where U is a suitable neighborhood o f N.

Proof Induction on the maximum dimension of the components of N. When this is zero the lemma is trivial; here is the inductive step. Choose a holomorphic function f that does not vanish identically on any component of N but vanishes on its singular set. Let H be the subset of N where f vanishes. By inductive hypothesis there are an open neighborhood V of H in M and a complex structure ~ on EI V that induces 5 P on H. We may suppose that 5~ agrees with 5 ~ on Vm N. In fact if this is not the case we may find an automorphism ~/ of E over a neighborhood of H that reduces to the identity on H and carries ,~JN into 5 p (this is a consequence of Cartan's Theorem B): we may then replace the original

with q (~) .


By a theorem of Docquier-Grauert [18] an open neighborhood W of N - H holomorphically retracts onto N - H, hence 5~ [ N - H extends to a complex structure 5e2 on EI W. We may suppose that V and W are Stein. It follows, again by Cartan 's Theorem B, that the identity automorphism of EI Vc~ N extends to an au tomorphism r /of E over a neighborhood 1/i of Vc~ N which carries ~ onto 5P2 off a closed subset K of V. Therefore ~ and q ( ~ ) are complex structures which agree in I/1 c~ ( W - K) and induce ,9 ~ on N. Since N is contained in I/1 w ( W - K ) this concludes the proof of the lemma. Q.E.D.

Let K be a closed subset of M. Consider the following statements:

(1, K) 5~ extends to a complex structure on E over a neighborhood of K.

(2, K) Let ~ , 5P2 be two extensions of 5 P to complex structures on E over a neighborhood U of K. Then there is a neighborhood V of K such that V= U and ~ ] V, 5PzlV are homotopic through automorphisms of E that induce the identity on E IN.

(20.2) Lemma. Let M 1, m 2 . . . . be a sequence of open subsets of M such that U Mi = M and M i c Mi + 1. I f (1, Mi) and (2, Mi) hold for every i, then

i

(1, M) is true.

Proof By hypothesis there are complex structures ~ on E IMi that extend ~9 ~ and smooth one-parameter families rh, , of automorphisms of E] Mi such that:

qi,0 ~-- identity,

Set:

rh, , IN r~ Mi = identity,

qi, l(~)=~+11M~.

P i j , t = r l i _ l , t . . . . . r l j , , if j<i, --1

[)ij, t : D j i , t if i<j.

The compatibili ty conditions:

P i j , t o Djh , t = P ih , t

are obviously satisfied, hence gluing by means of the pu.,'s is possible and yields a smooth family E, of differentiable vector bundles over M such that E o = E and E 1 is naturally endowed with a complex structure ~ ' . Moreover there is a smooth family a, of isomorphisms between E IN and E, [N such that a O is the identity and al is compatible with the complex structures on E] N and EI t N. Then al extends to an isomorphism between E and E1 and a~ 1 (~,) is the desired extension of ~ to all of E.

Q.E.D. 5 Inventiones math.,Vol. 28


(20.3) Remark. Lemma(20.2) reduces the proof of Theorem VI to showing that (1, U), (2, U) hold for (say) every relatively compact open subset U of M such that U has a fundamental system of Stein neighborhood, for M can be exhausted by open sets of this sort. In effect it will be sufficient to prove (1, U), in view of Remark (19.7).

(20.4) Remark. Another consequence of Lemma (20.2) is that we may limit ourselves to the case when M is an open Stein subset of ~". In fact, suppose we want t o p r o v e (I, U), where U is a relatively compact open set in M such that U has a fundamental system of Stein neighborhoods. A neighborhood V of U can be embedded as a closed analytic subset is some big It". El Vextends differentiably to a vector bundle E' over a Stein neighborhood W of M. If we know that Theorem VI holds for W, 5 ~ extends to a complex structure on E', hence on E I V.

(20.5) Remark. In proving Theorem VI we may limit ourselves to the case when M is an open Stein subset of tlY and N is defined by a single equation. This can be proved by induction on the number of analytic equations that define N (as a point set) as follows. Choose nonzero analytic functions f~ . . . . . fh on M such that

N = {m6M If1 (m) . . . . . fh (m) = 0}

as a point set. Set

N'= {me M I fz (m) . . . . . fh(m) =0},

H = { m 6 M ] f l ( m ) = O } .

Then 5 p extends to a complex structure on E,f U, where U is a suitable neighborhood of N. Choose a Stein neighborhood V of H such that Vn N' c U. 5O extends to a complex structure 5~ on E I N ' c~ V. By induction hypothesis ~ extends to a complex structure on E] V, hence 5* extends to a complex structure 5~z on Et H. Then ,~ extends to a complex structure on all of E.

We have thus shown that Theorem VI is a consequence of the following.

(20.6) Proposition. Let M be an open Stein subset of ~", K a compact subset of M, f a nonzero holomorphic function on M. Let N be the subset of M where f vanishes. Let E be a differentiable vector bundle over M and suppose that E IN has a complex structure 5~. Then 5 ~ extends to a complex structure on E f V, where V is a suitable neighborhood of K.

Before actually proving Proposition (20.6) and Theorem VI with it, we wish to describe the principle of the proof which makes sense for M


a general complex manifold, not only for open subsets of ~". The idea is to find a flat ~-connection on E by solving suitable differential equations. Suppose for simplicity that N is empty. E certainly has 0-connections: let V be one. Any other cS-connection on E is obtained by adding a Horn (E, E)-valued (0, 1)-form to V. Therefore all we have to do is find a Hom(E, E)-valued (0, 1)-form r/such that (V-q) 2 =0. Set O---V 2. �9 is a Horn(E, E)-valued (0, 2)-form. The condition that q must satisfy can also be written as follows:

(20.7) Vq - ~l A q = O.

Eq. (20.7) is nonlinear unless E is a line bundle and we are unable to deal with it directly, although this would be desirable. Therefore we are forced to solve (20.7) by a sort of induction on the dimension of M or, in other words, to do things "one variable at a time". To illustrate what this means notice first that, if M has dimension one, then every ~-connection on E is flat. Suppose next that there is a smooth fibering:

n: M - - * U

where U is an open subset of C. Suppose moreover that we can find a ~-connection V on E such that for every t~ U, the connection V, induced by V on E[n -~ (t) is flat. This means that �9 is divisible by dL i.e., that we can write:

(20.8) �9 = O ^ n* (dl)

where O is a Hom(E, E)-valued relative (0, 1)-form along the fibres of n. Let 8 , be the restriction of O to n-1 (t): it is easily checked that F t Ot =0. If we can find a section q~ of Horn(E, E) such that:

(20.9) Vt ~o, = O,

for every t e U , then [7-~pg*(di) isa flat c~-connection on E. In view of Proposition (19.1) solving (20.9) amounts to solving a linear 3-equation with smooth dependence on parameters. We may also notice that the problem of solving (20.9) is a local on U: we can make up a global solution out of local ones by means of a partition of unity on U.

Proof o f Proposition (20.6). Let zl . . . . . z, be linear holomorphic coordinates in ~" and set:

~k(z l . . . . . z . ) = ( z ~ + l . . . . . z . ) .

By Lemma (20.1) we may choose a ~-connection F on E which is flat in a neighborhood of N and induces ow on E tN. We shall prove, induc- 5*


tively on k, the statement:

A(k): There are a Stein neighborhood U o f K and a Horn(E, E)-valued (0, 1)-form qk on U such that V - - f Ilk has f l a t restriction to the f ibres o f Tr k.

If k = 1 the statement is obviously true (every ~-connection is flat when the base is a curve). Suppose now that we have proved A (k - 1 ). Set V = V- - f qk_ 1 and let 17k be the relative ~-connection along the fibres of 7: k induced by V. Set @= VkZ: 4~ is in Horn(E, E)-valued (0, 2)-form involving only the differentials d ~ l , . . . , d~k, Moreover, by induction hypothesis, it is divisible by dSk and by f, i.e., we may write"

45 = f O /x dZ k

for some Horn (E, E)-valued (0, 1)-form O involving only d~ 1 . . . . . d~k_ 1. It is easily checked that V k_l O =0. Choose a relatively compact Stein open neighborhood of K in U: let it be U'. Cover U' with finitely many relatively compact open subsets of U, V~ x 14/1, . . . , V h x W h, where each Vi is a Stein open set in the coordinates zl . . . . . Zk_ ~ and W~ is a ball in the coordinates Zk, . . . , Z,, such that, for any ( = (Zk . . . . . Z,) and any i, either lr/21 (()c~ U ' c V/,. x W/or 7rU1 (~)c~ U' does not intersect V/• Wi. It can be shown that we may find sections ~'i of Horn (E, E) over V~ • W~ such that Vk_ 1 ~, i=O (a proof of this will be sketched in Appendix 3). Choose differentiable functions Z~ (z, , . . . , z,) wi th compact support in ~ such that ~ Zi= l on a neighborhood of 7rk_ ~ (U'). Set ~, =~Xi~'~- Then

i v~_, r

on U'. This implies that F - J ' O d~k is flat along the fibres Of~k and we may set:

t lk~-tlk_ 1 + ~ d z k. .

We have thus proved that there is an Horn(E, E)-valued (0, 1)-form r/ such that [~-fr/ is flat in a Stein neighbourhood V of K. To prove Proposition (20.7) it remains to show that [~-f t / induces 5 ~ on El Vc~ N, i.e., that every section of 5~ Vc~ N extends to a section s' of E over a neighbourhood of Vc~ N that satisfies the differential equation:

~ s ' - f r / s ' = 0.

We can certainly find a section s" of E over a neighbourhood of Vc~ N that extends s and satisfies the equa t ion

l?s" =0 .

Then r/s" is (l~-fq)-closed, therefore we may find a section u o r e defined over a neighbourhood of Vc~ N, such that:

[~u - f r / u = q s".


Then we may set s '=s"+fu . This concludes the proof of Propo- sition (20.6) and, consequently, of Theorem VI. Q.E.D.

Some final remarks are in order. We have seen that proving Grauert 's theorem is essentially equivalent to solving (20.7). We are unable to solve (20.7) globally on M. What we do, instead, is solving (20.7) by the "one variable at a time" trick, on relatively compact subsets of M: a global solution is then obtained by a sort of limiting process. In doing this all information about the growth of the solution is lost. One might hope that solving (20.7) directly on M by potential theoretic methods would yield estimates on the behavior of the solution at infinity.

When M is an affine algebraic variety one might try to proceed as follows. Fibre M over ~ by means of a generic hyperplane pencil. Let

be this fibering. Assuming that we can solve (20.7) - with suitable dependence on pa rame te r s - fo r varieties of dimension one less than the dimension of M, solve (20.7) on M by the "one variable at a time" argument. One trouble here is that 7: will usually have singular fibres, but, with some effort, this can be overcome. A far more serious trouble is that this method does indeed give some estimate on the growth of a solution of (20.7) but not the estimates that would seem geometrically reasonable (see w 26 for more details on this).

E. A Lower Bound on the Growth of Analytic Subvarieties Representing a Given Homology Class

w 21. Statement of Theorem VII

Let V be a smooth n-dimensional projective algebraic variety and H c Va smooth hyperplane section relative to some projective embedding of V. The complement A = V - H is an affine variety, and hence a Stein manifold. Combining Grauert 's theorem in Section D above with standard results in topology, it follows that:

Every class ueH2k(A, t~) is a rational multiple u=)LUz(XeQ) of the fundamental class of an analytic subvariety Z ~ A.

(A complete proof of this result is given in two of the appendices to this paper.) For reasons arising from Hodge theory, Z can not generally be taken to be an algebraic subvariety (cf. Lemma (21.4) below), and we will find a lower bound on how far away from being algebraic Z must be.

To explain this, we let L-* V be the hyperplane line bundle and ae(9(V,L) a holomorphic section which defines H. Choosing a metric


with positive curvature in L--, V, the function

1 t o = log la12

gives a plurisubharmonic exhaustion function on A whose E.E.Levi form

dd c ro = ~

is a K~ihler form representing the Chern class of L---, V. The level sets A [ r ] = { x E A : % ( x ) < 2 1 o g r } give an exhaustion of A by pseudo- convex domains. (Note: The projective embedding V c I ~ and metric in L ~ Vmay be chosen so that A is an algebraic subvariety in CNcIPN,

Zo.= log (1 + JIzlJ 2) (z = (zl . . . . , zN)~u) .

b)=dd c log (1 + IJzt{ 2)

is the standard K~ihler form on IP N, and

h [r] = {zEA: lfll-+/Iztl2__< r}

is essentially that part of A in the ball of radius r in Cu.)

We set Z[r] = Z c~A Jr] for any subvariety Z of A and define

n(Z, r)= ~ J Z[rl

= volume of Z [r]

(d = dimension of Z)

by the Wirtinger theorem ([34]). It is a theorem of Stoll, recalled in w 4 above, that

Z is algebraic,~n (Z, r)= O(1).

If Z is not algebraic, then n (Z, r) may grow like any increasing function of r, such as (log r)', r a, e r" etc., as may be seen by taking the case when Z consists of points. We shall use n(Z, r) to measure the growth or transcendental level of Z, and Z is said to be of finite order if

n (Z, r) = 0 (r ~)

for some 2 (cf. the discussion in w 4).

Concerning the cohomology of V, the basic tools for understanding it are (i) the hard Lefschetz theorem, (ii) the primitive decomposition, and (iii) the Hodge decomposition. The hard Lefschetz theorem is the isomorphism

(21.1) e)k: H,-k(V, ffj,)~,_~H,+k(V, Q).


Using it, to understand the cohomology Hq(V,Q) we may generally restrict our attention to q<n. (Note: Geometrically the hard Lefschetz theorem says that all cycles 7eH._k(V, Q) are of the form H k. 2 for some 2eH,+k(V, Q), so that, for instance, ~ is algebraic in case 2 is, etc.) The primitive cohomology Pn-k(V, ~ ) c H ' - k ( V , ~ ) is defined to be the kernel of

o ) k + l : H"- k (V, (I~)--~ H" + k + 2 (V, ~1~),

and the primitive decomposition is

(21.2) H ' - k (V, Q ) = ( ~ (o' p , - k - :'(V,, Q). 1

Using it, to understand Hn-k(V, I1~) it suffices to understand the primitive pieces P'-k(v , Q), which may be viewed as the fundamental "building blocks" of the cohomology of a variety. Finally, the Hodge decomposition

(21.3) H"(V,~)= (~ Hv, q(V) p + q ~ m

decomposes the complex cohomology of V into (p, q) type ([-35]). It is compatible with the primitive decomposition (21.2), and thus

pm(v,r | p~,.(~) p+q=m

where Pv, q(v) = P"(V, ~) c~ H v" q( V).

A class ueH2k(V, Q) is said to be algebraic in case u = 2 u z (2eQ) is a rational multiple of the fundamental class u z ofa codimension-k algebraic subvariety Z in V. A necessary condition for this is that the Hodge decomposition

1.1---- E 13 p , q p + q = 2 k

of u satisfy Up, q = 0 for all (p, q) 4= (k, k),

i.e. u should be of type (k, k). The famous conjecture of Hodge is that this condition is sufficient.

According to what was said earlier, the restriction u(A)eH2k(A, ~ ) of u to A is always of the form

u(A)=2 Uz (2~)

for some analytic subvariety Z of A.

(21.4) Lemma. I f u is primitive and u k + t, k- ~ 4= 0 for some 14= O, then Z is not algebraic.


Proof If Z were algebraic, then the closure z~ of Z in V is algebraic of codimension k and the difference

u - uzEker {H2k(V, Q)---~H2k(A, Q)}.

Using the isomorphisms (Q-coefficients)

H~(V, V - H ) ~ H ~ ( U , 8U)

,~ H l- 2 (H)

(excision),

(Thorn),

where U is a tubular neighborhood of H in V, in the exact cohomology sequence of the pair (V, V - H ) gives the commutative diagram

H 2 k - 2 ( H , Q ) ,H2k(V, Q) -

/-/2k- 2 (V, Q)

, HZk(A, Q)

Since 2 k - 2 < n - 2, p is an isomorphism by the usual Lefschetz theorem. Thus

ker {HZk(V, Q ) ~ H Z k ( A , Q)} ~o~-H2k- 2 (V, Q),

and by (21.2) the intersection

Pz~(V, Q) c~ ker {HZ~(V, Q)~H2k(A , Q)} =0 .

In particular, u is the primitive part of u 2, which then contradicts uk+z, ~_ ~+0 for some k 4= 0. Q.E.D.

The lemma remains true, with the same proof, provided that the primitive component u ~ ofu has 0 uk+ z, k- ~ + 0 for some 14= 0. An equivalent condition is

S U A U /k O),2n- 2k jt= 0

(21.5) v v~ p2 k(V, Q) and is of type (k + 1, k - I) for some l=4= 0

as follows from the non-degeneracy of the Hodge bilinear forms. In case u satisfies this condition and u(A)--2Uz as above, it makes sense to seek a lower bound on the growth of Z.

Theorem VII. Let u~H2k(V, Q) be a class which satisfies (21.5) and suppose that the restriction u (A) = 2 u z for some analytic subvariety Z c A. Then

n(Z, r ) ~ c r l for some constant c > O.


w 22. Proof of Theorem VII

The proof proceeds in several steps. For simplicity of notation we will assume that dim c V=2n and k=n, so that we are working in the middle dimension. The general argument is the same.

Step one. Let U be a tubular neighborhood of H in V and suppose that v~H2"(V,~) is primitive and of type (n+l, n - l ) .

(22.1) Lemma. In U, we may write v=dw where

+~ W= Z [~7[P+Wn+t-p-I,n--I+p

with w.+t_ p 1..-t+p a continuous form of the indicated type and p + = max (p, 0).

Proof In the commutative diagram

H2 n ( V, (I~) " ,H2n(H,(I~)

H2"+2(KII?),

where the vertical arrow is obtained by Poincar6 duality from H2,_ 2 (H)--, H2,_2(V) and is thus as isomorphism by the usual Lefschetz theorem, we have co- v = 0 since v is primitive. Thus the restriction p(v) ofv to H is zero in cohomology.

Suppose first that U looks holomorphically like a neighborhood of /4 in its normal bundle. Then there is a holomorphic fibration ~: U - ~ H . Since v = 0 in HZ"(H, t12), we may write

vIH =d~l

where q has type (n+l, n - l - 1 ) (cf. Lemma 2.13 in [14]). Consider the homotopy

u x[0,1]

U

where f(z , t) = t z, and look at f * v. Writing

f * v = a + b A d t

where a, b do not involve dt (this may be done globally on U • [0, 1]), we have

O=df* v = d a + d b ndt .


This implies the relations

d U a = O ,

Ca at + d e b=O

where "du" means "exterior derivative with respect to the coordinates in U". From this we obtain

1 dt), where b has type (n + 1, n - 1 - 1)+ (n + 1 - 1, n - l), and

a(z, O) =dzt* r/.

Taken together these give

1

v=a(z, 1)=d (n*q- ~b(z, t )dt):dw ~, 0

where w has type (n + 1, n - 1 - 1) + (n + 1 - 1, n - 1).

In general there is a diffeomorphism g: U-+ W of U to a tubular neighborhood W of H in its normal bundle such that

g* (d(i) =- Z ~ij d~(la]) J

where (~, ~j are local holomorphic coordinates in W, U respectively and the ~is are continuous. By composing with g we may make a C ~ fibration re: U--,.H and homotopy f : U• 1 ] ~ U as before, and then

where

Moreover,

v[H =&?

7~* ~-~-E1r p§ ~n+l--p,n--l+p--l" P

b=~ [tTl p+ bn+l_ 1 --p,n--l+p P

using the previous notations. Now the proof is completed as before.

Step two. We now complete the proof of the theorem. Using Lemma (22.1) we may find a C ~ form w on V such that v + dw has compact support in A, and where w has the type decomposit ion indicated in the statement of the lemma. Multiplying v+dw by a constant and using

Analytic Cycles and Vector Bundles on Non-Compac t Algebraic Varieties 75

(21.5), we have by Stokes' theorem that

0< f . ^ v = S u ^ ( v + d w ) V V

= S(v+dw) Z

= l i m ~ (v + dw) t~oo Z[rl

= lim f w r ~ ao OZ[rl

since S v=O

Z[r]

because v has type (n + l, n - I) for some l=t= 0. Thus

(22.2) S w>c'>O OZ[rl

for sufficiently large r.

Type considerations imply that

w l Z = ( I c r [ Z w . - x , n + [~rl ' - l w . . . . x)l/ therefore

1 ~ w = / - a S f

OZ[rl OZlrl

where f is a continuous form on V Combining this with (22.2) yields

c'r'-l-- -< S f OZ[rl

for large enough r. Integrating this equation from r o to r gives

c'/+O(1)< I f ^dlal<c'' I a)n Zlro, rl Z[ro, rl

since f^dlal is bounded. Q.E.D.

F. Discussion of Some Open Problems

w 23. Runge Problem for Analytic Sets One of the most important properties of a Stein manifold are the

various approximation theorems or Runge theorems, such as Proposition(17.1). Traditionally, these approximation theorems deal with linear data such as sections of vector bundles, but they make sense in geometry and lead to an interesting problem.

Let M be a Stein manifold with strictly plurisubharmonic exhaustion function ~. Setting M[r]={xEM: "c (x) =< log r}, we consider the space


A~(M[r]) of C ~ compactly supported 2 n - q forms on M[r] , and let ~q(M[r]) be the dual space of currents of degree q on M[r]. Two currents are close if their values on each form are close (we do not make this precise here), and thus ~q(M[r] ) is a topological vector space ([29]).

A codimension k analytic subvariety Z of M[r] defines a current Tz~2k(M[r]) by integration:

Tz(~)= ~ a (a~A2k(M[r])). Z~eg

Runge Problem (first form). Given Z~M[r], does there exist an analytic subvariety W c M such that W[r]=Wc~M[r] is close to Z in the sense that Tw~,~ is close to Tz?

Contrary to the linear case, there is a non trivial necessary condition.

(23.1) Lemma. If Z can be approximated by analytic subvarieties IV,, then the fundam'ental class Uz is in the image of the restriction map

H 2 k (M, ~)--~ H 2 k (M [r], ~) .

Proof. The compactly supported de Rham cohomology of M [r] is finite dimensional, and we may then select forms qh . . . . . ~o,,~A2k(M[r]) giving a basis for H~-RZk(M[r]). Next given e > 0 we may find an analytic subvariety W c M such that

S ( i=1 . . . . ,m). Z W[r]

Since u z and Uw~rl are both integral classes, it follows that u z=uw~rJ in H2k(M[r], ~). Q.E.D.

Remark. This lemma is almost certainly still valid with :~-coefficients.

Runge Problem (second form). Let ZcM[r] be an analytic subvariety such that

(23.2) Uz e image {H 2 k (M, ~ ) -+ H2 k (M [r], Q)}.

Then for some integer 2 >0, can 2Z be approximated by analytic subvarieties W < M ?

Proof for k = 1 (cf. [19]). In this case Z is a positive divisor, and consequently there is a holomorphic line bundle LeHI(M[r],r H2(M[r],2~) and holomorphic section aE(O(M[r],L) such that Z = {a=0}. Replacing Z by 2Z if necessary, we may find a holomorphic line bundle K~HI(M,O*)~-HZ(M,Z) such that KIM[r]=L. By the Runge theorem for holomorphic sections of a vector bundle, we may approximate a uniformly on compact subsets of M[r] by sections pe(9(M, K). Thus we are reduced to proving the following:


(23.3) Lemma. Let fk, fe(9(A") be such that f k - ~ f uniformly on A", and let ct be a C ~ form with compact support. Then

lim ~ ct= ~ ~. k ~ cx~ { f k = O} { f = O}

Proof For any h~(9(A"), the Poincark equation (cf. (1.2) in [13])

ct=--I S l~ hlddcc~ h = O 7[ A n

is valid. Since f~-- , f uniformly on A", log Ifkl---' log If] in L 1 (loc, A"), and consequently

loglfklddC~ ---* ~ l og[ f l dd ~ . A n gl n

Combining this with the Poincar6 equation gives the lemma.

Remarks. a) The reason for our use of Q-coefficients is that, in addition to (23.2), there are further topological restrictions of a torsion nature in order that Z can be approximated by a W. Namely, by resolution of singularities, the fundamental class uzEHZk(M[r], Z) is the image of a smooth manifold mapped in smoothly, and thus Uz satisfies torsion conditions (e.g. S q3 Uz=0 ) which arise in Thorn's work. Similar conditions must also be satisfied by Uw, and this is the reason for stating the problem with Q coefficients.

b) In case M = A is an affine algebraic variety, we claim that we may take the divisors W (in the codimension one situation) to have finite order, but in general can not take them to be algebraic.

Proof Following the notations in the proof for k = 1 above, we may assume that K ~ A is a finite order line bundle (cf. Theorem IV in w In general, however, we can not take K to be algebraic. Next, we may approximate a by finite order sections p~(gr.o.(A, K). Finally, the zero locus p = 0 of such a finite order section is a divisor of finite order on A (cf. remark (ii) at the end of w 10). Q.E.D.

This suggests that in the algebraic case we refine the Runge problem to the following:

Runge Problem in the Algebraic Case. Given Z and W as above, can we take W to be an analytic set of finite order on A ?

c) The special case M=II~" deserves comment. Here we are given an analytic subvariety Z c ~" [r], the ball of radius r, and we wish to approximate Z by an entire analytic set W. Since there is no Hodge theory involved, it may be that W can be taken to be algebraic (O.K. for codimension one). Even the special case when Z is an analytic curve in ~3 [r] seems to be unknown.


Remark Added in Proof Recent work by A. Cassa seems to give the Runge theorem for curves in ~3.

w 24. Extension Problem for Analytic Sets

Similar to the Runge problem for analytic sets is the extension problem, to be discussed now. Let M be a Stein manifold and S ~ M a complex submanifold; hence S is again a Stein manifold. Given an analytic subvariety Z c S, we may ask if there is an analytic subvariety W c M such that

W.S=Z.

An obvious necessary condition is that the fundamental class u z come by restriction from a. cohomology class on M. This leads to the following:

Extension Problem. Given S c N as above and a codimension-k analytic subvariety Z ~ S such that the fundamental class

(24.1) Uz~Image {H2k(m, (I~)---+ Hzk(s, (I~)},

is there an analytic subvariety W c M such that

W.S=2Z for some integer 2 > 0 ?

Remarks. a) The reason for using Q-coefficients is to get rid of topological obstructions of a torsion character, similar to the Runge problem discussed above.

b) Our particular motivation for the extension problem is the following:

On a Stein manifold M, we consider the "effective analytic cycles Z. We say that Z and Z' are Q-analytically equivalent, written

Z~,Z',

if on M • A there is an effective analytic cycle 3 such that

3 " M • { t l } = 2 Z ,

3" M • {t2} = 2 Z '

for a positive integer 2. If Z,,~Z', then Uz=Uz, in HZk(M, ~), and the extension problem would imply the converse.

(Proof Replace M by M x A, S by (M x {tl} ) u (M x {tz}), and Uz by Uz+Uz,. An easy exact sequence argument shows that: "Uz=Uz , in H2k(M, ~) implies the condition(23.1)", and we may then take 3= W in the solution to the extension problem.)


Thus, assuming the extension problem, we find the following holomorphic cobordism-type description of the even cohomology of a Stein manifold:

Let M be a Stein manifold. Then the cohomology H2k(M,~) may be described as cycles a. Z where a~Q and Z is a codimension k subvariety on M, with the equivalence relation Z ~ Z' described above.

w 25. A Characterization of Algebraic Varieties

A suitable theory of complex analysis with growth conditions should allow one, at some point, to give an intrinsic characterization of which complex manifolds are quasi-projective algebraic varieties, thus generalizing Kodaira's famous theorem about Hodge manifolds ([24]). As a tentative start on this, we mention the following:

Problem on Characterizing Affine Varieties. Let M be an n-dimensional complex manifold with an exhaustion function ~b which satisfies the conditions

~k has finitely many critical values;

(25.1) ddC~ >O,

(ddC~t)"-l~0 but (ddC0)"=0.

Then is M an affine algebraic variety ?

Remarks. a) In case M is an affine variety, the special exhaustion function constructed in w 2 of [-13] has the properties (25.1).

b) For n= 1, this problem has an affirmative answer as follows: Since ~ has only finitely many critical values, the level sets M[r] =

{x~M: ~9 (x)<log r} are, for r~0, Riemann surfaces with finitely many smooth boundary curves.

Thus the complement M - M [ r ] is differentiably a finite number C1 . . . . . CN of punctured discs. Let C be one of the C~. The restriction of ~ to C is a harmonic function, and, after multiplying by a suitable positive constant, we may assume that S doff = 1. Letting n: C ~ C

oc be the universal covering of C and f/the harmonic conjugate to ~ = n* ~, the function f = ~ - l ( ~ + I / Z 1 f/) on C is holomorphic and

(25.2) f(~" 7)=f(~)-+ 1

where ~ generates the group covering transformations of C-~C. Moreover, the imaginary part of f is positive, and thus we obtain a


conformal mapping f C - f---* H

C - J - ~ A*

of C to the upper-half-plane H which induces an injective mapping f of C to the punctured disc A*. Since 0 is proper, the image f(C) covers a neighborhood of the puncture in A*. Thus each C v is biholomorphic to a punctured disc, and M is an affine algebraic curve. Q.E.D.

c) Using ~ we may define the counting function of an effective divisor D on M by

(25.3) U(D,r)= ~( S (ddC~/)"-t) dt ro M[r] t

The order function To( f r) of a meromorphic function f on M is then given by

(25.4) To(fr)=N(D~,r)+ y log+[fldC~A(ddC~z) "-~, ,~M[r]

and with this definition the First Main Theorem in the original form of R. Nevanlinna is valid (cf. w 3 (b) of [13]). If M is to be an affine algebraic variety, then the field of rational functions on M should be those meromorphic functions f which satisfy

(25.5) To(f, r)= O(log r).

Conversely, one might hope to prove that. M is algebraic by using L2-c 5 methods to construct sufficiently many holomorphic functions satisfying (25.5).

w 26. On the Growth of Holomorphic Vector Bundles and Related Matters a) Perhaps the most interesting problem to arise from our work is the

need to understand the Oka principle with growth conditions on an affine variety A. The naturally posed linear problems, such as the algebraic de Rham theorem (w 18) or approximation by rational sections (w 17), can be solved in the algebraic category. However, the naturally posed non-linear geometric problems cannot, for reasons involving the Hodge theory on A, in general be solved algebraically. Indeed, the Hodge theory gives in some cases (cf. Theorems V and VII) a lower bound on the transcendental level of holomorphic solutions to various geometric problems.

Analy t ic Cycles and Vector Bundles on N o n - C o m p a c t Algebra ic Variet ies 81

Specifically, it seems to us necessary to first prove Grauert 's theorem VI with correct growth conditions. The one-variable-at-a-time method used in w 19 to linearize the ~-equation for complex structures in a C ~~ bundle E ~ A leads to a complex structure which grows roughly like a function Z (r) with

(26.1) log(,., log(log z(O)=O(logr ) (n=dimcA). (n -- 1 )-t imes

For n = 2 we have log Z (r)= O(log r),

i.e. E o A has finite order. In view of this and of Theorem IV, we pose the following:

Problem on Order of Growth of Vector Bundles. Does every C ~ bundle E ~ A admit a complex structure of finite order? More precisely, is the natural map (w 8)

Vectf.o. (A) ~ Vecttop (A) a bijection of sets?

As further heurestic evidence for this, we offer the following: Given a complex structure of finite order in E--~ A, this means that we may find a Hermitian metric whose curvature form satisfies

(26.2) O(~r) = O(r ~ q~).

The Chern classes cq(E, 0), as computed from the curvature matrix for E, will then be C ~176 (q, q) forms satisfying

tcq(E, 0)1<= C (r~ q~q),

which clearly implies the L2-estimate

(26.3) ~ ]cq(E, 61)[ 2 e -Nl~ ~ < c t 3 A

for N ~ 0 . As was seen in w 18, the complex of C ~ forms r on A satisfying the L2-growth conditions

j'l~12 e - N l o g ~ < ~ A

(26.4) f [dlpl2 e-N1~ < oo

A

is the "correct one" for studying the C ~ de Rham cohomology on A. Consequently, according to (26.3), the condition of finite order is the 6 Inventiones math., Vol. 28


proper one to insure that the naturally defined Chern classes land in the complex (26.4).

b) Related to the question of growth conditions on holomorphic vector bundles are two long standing problems in algebraic geometry which we wish to mention.

Problem (Serre). Is every algebraic vector bundle Ealg ~ t~ n algebraically trivial ?

This question has been answered in the affirmative (cf. [2]) when E is stabilized; i.e., he proves that

Kalg (tl~") ~ Kalg (point).

One interpretation of this result is that the problem on finite order bundles should perhaps also be stabilized; i.e., is the natural map

(26.5) Kf.o. (A) ~ Kto p (A)

an isomorphism? Since passing to the stable situation has in some sense the effect of making the general linear group commutative 16, it is suggestive to think that proving (26.5) might be easier in that the non-linear bracket term in the cS-equation of w 20 might be made to somehow disappear from the analysis.

The second problem is when a smooth algebraic curve Cct~" is a complete intersection? If this is the case, then the algebraic normal bundle of C is algebraically trivial, and hence the canonical bundle K(C) is algebraically trivial, which need not always occur. However, the canonical bundle is analytically trivial, even analytically in the finite order category (cf. Theorem V). Now Forster and Ramspott (cf. [9]) have shown that any smooth analytic curve in a Stein manifold is complex analytically a complete intersection.

Problem. Is a smooth algebraic curve in t12 a the intersection of two analytic surfaces of finite order meeting transversely?

w 27. Order of Growth and Hodge Theory

The basic proble m of this paper was an attempt to understand, in terms of growth conditions, the analytic cycles which may be used to generate the even-dimensional cohomology (over II~) of an affine algebraic variety. Except in special cases we have been unable to do this, but on the other hand, perhaps enough evidence has been collected to allow us to formulate two problems which would be profitable to think about. The first one is the following:

16 Note that [A, GL(oo, IE)]ho ~ is an abelian group, because of Grauert 's theorem and since this is true topologically.


Problem on Growth of Analytic Cycles in a Homology Class. Let A be an affine algebraic variety. (i) Is the homology H2k(A, Q) generated by fundamental classes of analytic sets of finite order? (ii) Given two analytic sets Z, Z' of finite order whose fundamental classes are the same, is Z Q-analytically equivalent to Z' (cf. remark (a) in w 24 above) through cycles of finite order?

Remarks. Informally, the problem asks whether H2k(A, I1)) may be computed from the analytic cycles of finite order, the equivalence relation being analytic homotopy through finite order cycles? Finite order is the least growth condition which may be used (Theorem VII), and the problem is true for k = 1 (Theorem V).

An heuristic argument for this problem may be given as follows: Recall that the currents are the continuous linear forms on the space

Ac(A ) of compactly supported C ~~ forms on A. Among these currents are those linear forms Tsuch that both Tand dT(in the sense of currents) are represented by measures; i.e., may be extended to the space ~*(A) of continuous compactly supported forms. Intuitively, these currents do not involve derivatives of the forms on which they operate. Examples of such currents are (i)

Tr S0 A q) A

where q~eA*(A) is a C ~~ form on A, and (ii)

Tz(0) = ~ O Z

where Z is an analytic subvariety of A.

Given a current T such that T and dT are represented by measures, we say that T has finite order in case

IT(O)l < CrZ [lOlt,

[dT(~)[=< Cr z I[0]l

for all ~ e A* (A) having support in A Jr], and where ]l... It is a sup-norm. Referring to examples (i) and (ii) above, the condition of finite order means respectively

(27.1) sup [q~ (x)], [d~o (x)[ = O (r") xeA[r]

for the current T~o, and Z has finite order

for T z. Let A*o.(A ) and ~*o.(A) be respectively the C ~~ forms satisfying (27.1) and the currents of finite order. Thus A*o.(A)=N*o.(A)c~A*(A). 6*


It seems reasonable that the natural map

(27.2) H* ({A~.o.(A), d}) - , H ~ ( A )

from the cohomology of the complex {A~o.(A),d } to usual de Rham cohomology should be an i somorphism- in any case, the U-version of (27.2) was proved in w 18 (el. w 13 (b)). Assuming this, the general philos- ophy of computing cohomology wither from smooth forms or from currents (cf. [29]) suggests that the isomorphism

H*((@~.o.(A), d})~ H*(A, if;)

may be valid. If one accepts this, and if we recall that the currents were introduced to provide a means for interpolating between geometric cycles on the one .hand and smooth forms on the other, then the analytic cycles of finite .order appear naturally in the problem of representing H . . . . (A, Q) by analytic subvarieties.

Coming to our second problem, we let V be a smoo~h projective variety and H c V a smooth sufficiently ample divisor with complement A = V - H . We may view the primitive cohomology P2~(V)=p2k(V, Q) as a subspace ofH2k(A, ~) (cf. the proof of Lemma (21.4)), and on P2k(V) the filtration G~p2k(V) by order of growth is defined (w 8 (c)). Assuming the above problem about generating H . . . . (A, II~) by analytic cycles of finite order, this filtration has a good meaning. We recall (Theorem VII) that

(27.3) F v p2~(V)c G v P2k(V)

where F p is the Hodge Filtration.

Problem Relating Order of Growth to Hodge type: Do We Have Equality in (27.3)?

In other words, is the minimal growth of analytic cycles in a given primitive cohomology class ~ computable from the Hodge decomposition of ~? This would imply in particular that the filtration G x by order of growth occurs only where ,~ is an integer, a fact which is by no means evident. An affirmative answer to this problem would not formally imply the Hodge conjecture, since an analytic cycle may have order zero and not be algebraic. However, the proof of this problem would almost surely lead to the construction of algebraic cycles by suitably refining the growth indicator functions. We note that the problem is true for k = 1 (Theorem V), and this together with (27.3) constitute all the evidence we have for the question.

To conclude this section we should like to give a few comments concerning extremal problems and existence and uniqueness theorems


in several complex variables. The existence of an analytic subvariety of minimal growth in a given homology class is relatively easy due to the normal families result of [34] :

Given a sequence {Z,~} of codimension-k subvarieties Z v of an affine variety A such that the 2k-dimensional volumes of the Z v are locally bounded, there exists a subsequence Z,Iv) converging (in the sense of closed sets) to a codimension-k analytic subvariety Z c A . However, finding out anything specific about Z seems quite difficult, perhaps due to the tack of uniqueness. As a general comment, in complex analysis the question of existence of an extremal is frequently trivial, due to some sort of normal families argument. The question of uniqueness is generally more difficult: For example, if M is a complete hyperbolic manifold in the sense of [22], then for each point (x, ~) ( ~ Tx(M) ) in the complex tangent bundle of M, there is a disc of maximal radius R for which there is a holomorphic mapping

f: {tz[<RI---~M

satisfying f(O) = x and f , (O/Ot) = ~. The Kobayashi length of ~ is then

1 F(x, ~)=~-.

It is not known (even in concrete examples such a s I P 2 - {5 lines in general position}) whether f is unique, even in the variational sense. Moreover, even when existence and uniqueness are known, it may be quite difficult to understand the extremal. For example: On a simply-connected bounded domain f2 c IE containing the origin, we consider holomorphic functions f(z) with

[f'(z)l 2 dz d2 = 1,

f (0) = 1,

and seek to maximize Re(f'(0)). It is essentially trivial that a unique extremal f exists, but much less easy to show that fo is the Riemann mapping function.

As a general remark, the lack of effective variational techniques, such as Euler equations of some sort, could account for some of the difficulty we have faced in finding some reasonable upper bounds for the minimal growth of analytic cycles in a given homology class. Even more so, the complete lack of any good uniqueness results may be causing trouble, as witnessed by the lack of a good "syzygy with growth condi- t ions" for entire analytic sets Z c ~" of codimension > 1 (even the case of points in tEz is not understood).


Appendix 1 Chern Classes of Holomorphic Bundles

and the Fundamental Class of an Analytic Subvariety

a) Throughout this appendix, homology and cohomology will be singular with Z-coefficients. Assuming Poincar6-Lefschetz duality for manifolds, we shall define the fundamental class of an analytic subvariety on a complex manifold.

Let A" be the unit polycylinder in C" and A " - k c A " the sub-polycylinder given by z 1 . . . . . z k = O.

(A.I.1) Lemma. H q ( A " , A " - A " - k ) ~ 0 q+ 2k q = 2 k .

Proo f Using,the homotopy equivalence

(a n, A"-An-k)--(A k, Ak- {0})

and excision, we obtain

Hq(A ", A n - A " - k ) ~ Hq(A k, A k - {0})~ Hq(A ~, 0Ak),

from which the lemma follows. Q.E.D.

(A.1.2) Lemma. Let M be a complex manifold and Z c M an analytic subvariety o f codimension k. Then H q (M, M - Z) = 0 for q < 2 k.

Proo f Inductively we may assume the result for singular set Z~ of Z. From the piece

(A.1.3) Hq(m, M - Z s ) - , H q ( m , m - z ) - - , H q ( m - z s , M - Z )

of the long exact cohomology sequence of

M - Z c M - Z ~ c M ,

we see that it will suffice to prove the lemma when Z is smooth. Let U be a closed tubular neighborhood of Z in M which topologically looks like the unit neighborhood of Z in its normal bundle relative to a suitable metric, z

Fig. 4


By excision and the Thom isomorphism for the normal bundle of Z

Hq(M, M - Z ) ~ H q ( U , U - Z )

~Hq(U, OU)

H q - 2 k (Z)

=0 when 2 n - q > 2 n - 2 k

where d i m c M = n so that d i m ~ Z = 2 n - 2 k . Q.E.D.

(A.1.4) Proposition. Let M be a complex manifold and Z an analytic subvariety of codimension k. Then there exists a unique class

uz~H2k(M,M-Z)

with the property that u z localizes to a generator of H2k(A n, A n - d n-k) around the simple points of Z.

Proof Lemma (A.1.2) applied to the singular set of Z and the exact sequence (A.1.3) give the isomorphism

H2R(M, M-Z),~HEk(M-Zs, M-Z).

Thus it will suffice to prove the result when Z is smooth, and in the notation of the proof of Lemma (A.1.3)

H2k(M, M - Z ) ~ HER(U, U - Z)

H2k(U, OU)

,-~ H ~ (Z) (Thorn).

In case Z is irreducible and hence connected, H~ is canonically isomorphic to Z, and there is a unique class uz~H2k(M, M - Z) corresponding to 1 ~ H ~ In the general case, Z = ~ Zi is a union of complex

submanifolds, and there is a unique u z which localizes to I~H~ for each i. Q.E.D.

Remarks. (i) The class u z is called the fundamental class of Z in M. In case Z = ~ n~Zi is an analytic cycle with irreducible components Z~

having multiplicities n~, the fundamental class is Uz= ~ ni Uz. i

(ii) If we are using differential forms and de Rham cohomology, then an analytic subvariety Z defines a current T z on the compactly supported C ~ forms A*(M) by integration

Tz(q,)= Zr


where Z , = Z - Z s is the set of smooth points on Z (cf. [34]). Stokes' theorem in the form

Tz(d0)=0 (OeA*(M))

is valid ([34]), so that T z defines a linear function on the compactly supported de Rham cohomology

H*DR(M)~ H*(M , 112).

On the other hand, the usual de Rham group

(A. 1.5) H~R (M) g H* (M, ~)

also defines linear functionals on H* oR (M) by

(r/,~o)= j'r/A~o (tI~H~R(M) and ~oeH*(M)), M

and under the isomorphism (A.1.5)

Tz= u z.

Note that both the definition of u z and of T z use that the singular set of Z is removable for cohomological purposes.

(iii) Suppose that M is projective algebraic or that M is Stein and Z is smooth. Then the sheaf (9 z has a global resolution (syzygy) by locally free sheaves

(A.I.6) 0-- ' gm ~ d~- 1 ~ '---" N0 ~ (9z ~ 0 -

Each ~ corresponds to a vector bundle E i--~ M, and we may consider the Grothendieck element (cf. Appendix 2 for a discussion of K-theory)

7Z "= ~ ( -- 1 )i Ei �9 Ktop (M, M - Z ) i=o

(cf. [8]; ~z defines a relative class since (A.1.6) is exact on M - Z ) . It is a result of Atiyah-Hirzebruch that the Chern character (cf. Appendix 2 below)

(A. 1.7) c h (Tz) = (k - 1)! u z + (higher order terms).

Sketch of Proof First, one shows by an Euler characteristic argument that 7z is independent of the resolution (A.1.6) (this essential step is implicit in the definition). Next by a similar argument as in the proof of Lemma (A.1.2), we may assume that Z is smooth. Finally, to show that c h ( v z ) = ( k - 1)! u z modulo terms Hq(M, M - Z ) for q>2k , it will suffice to do this locally where M=A" , Z = A "-~, and the resolution (A.1.6) is the standard Koszul complex. This is done by an explicit computation (cf. [8] for details).


Using (A.1.7), Atiyah and Hirzebruch (toc. cit.) showed that certain cohomology operations, such as Sq 3, applied to u z are zero. On the basis of this they concluded that not all torsion classes in HZk(M, Z) are fundamental classes of analytic subvarieties for k > 1.

Note. In case M is Stein but Z may not be smooth, a resolution (A.1.6) exists over relatively compact domains on M. It exists globally, provided that the gl appearing in all local resolutions may be chosen to have bounded rank.

b) Let M be a complex manifold and E ~ M a hotomorphic vector bundle of rank r. Assume given holomorphic sections a~ . . . . . a re (9(M, E) such that the following general position requirement is satisfied:

(A.1.8) Z k = {a 1 A ... A at_k+ 1 =0} has codimension k.

By definition, Z k is the k-th Schubert cycle associated to the sections ~r i, and we will prove the following result of Chern.

(A.1.9) Proposition. The fundamental class Uz~ is equal to the k-th Chern class Ck(E ) in HEk(M).

Remarks. In Proposit ion (A.1.4) we defined u k as a relative class in H2k(M, M - Z), and here we are considering its image in

HZk(M, M - Z ) - - ~ H 2 k ( M ) .

The definition of Chern classes may be taken to be via the axioms or that of Grothendieck (cf. [17]). In either case the most important properties are (i) the duality formula (in particular, the relation c~(E)=0 in case rank E = l and E has a non-zero section) and (ii) the relation

(A.I.10) c~ ( [D])= u D

for a divisor D with associated line bundle [D] -~ M. The latter is easily verified by an explicit computat ion using the local defining equations for D relative to a suitable covering of M.

Proo f Because of our conventions, the multiplicities take care of themselves, and so we wilt assume that Z k as defined in (A.1.8) has all components occurring with multiplicity one. Let W c Z k be an analytic subvariety of codimension one in Z k. Using Lemma(A.1.2) we may replace M by M - Wand Z k by Z k - W. Letting W be the union of Zk+ 1 with the singular set of Z k, we may assume that (i) Z k is smooth and (ii) there is an exact sequence of holomorphic vector bundles

O _ , tFr- k __. E ---~ F --~ O


over M, where ~E r- k is the trivial sub-bundle of E generated by a t . . . . . at_ k- The projection a of a,_,+ 1 in F gives a holomorphic section a~(9(M, F) such that Z ,= {a=0} , thus we may assume that Z is smooth and is the zero set of a holomorphic section a e (9(M, E).

On M - Z , a is non-zero and thus splits off a trivial line sub-bundle. By the duality formula, CR(E ) = 0 in HZk(M - Z ) and consequently

Ck(E)Eimage { H2k(M, M - Z ) - * H2k(M)}.

Suppose for a moment that Z is irreducible and let U be a tubular neighborhood of Z in M as in the proof of (A.1.2). If dk(E ) is any lift Of Ck(E) to H2k(M, M - Z ) , then since by the proof of (A.1.4)

H2k(M, M--Z) ,~H2k(u, ~U)m 2g,

rig(E)=2, u z in.H2k(M, M - Z ) for some constant 2. In case k = 1 so that Z is a divisor, 2= 1 by (A.I.10).

In the general case we may prove that 2 = 1 (and somewhat more) as follows: Let n: M--~M be the monoidal transformation with center Z, 2 = n-l(Z), and L -1 = [Z]. Now L | n*F has the non-vanishing section n* a, and consequently c k (L | n* F) = O. But

Ck(L@7~, F ) = E ( _ 1)k-iTz, c i (F ) k - i U~ , i

and applying the Gysin homomorphism n , gives

0 = ~ ( - 1)k-ici(F) n ~u k--i~ , ~ Z /" i

But n , u ~ - J = 0 for 0 < j < k and n, uk=(--1)R-lUz . Hence Ck(E)= Ck(F )=u z. Q.E.D.

Remark. We recall that the holomorphic vector bundle E-~ M is said to be sufficiently ample in case there are global sections zl . . . . . zn~ (9(M, E) such that (i) the zi(x ) span all fibres E~ (x6M), and (ii) the differentials dz(x) of the sections z = ~ aiz ~ (aide) which vanish at x span E x | T*.

i In this case, suitable linear combinations a~,=~a~.~z i (v= 1 . . . . . r) give

sections a~ . . . . . a, which satisfy the general position requirement (A. 1.8). Moreover, the structure sheaf (gz~ will have standard local resolutions (A.1.6) by locally free sheaves of bounded rank. In particular, if M is Stein, then there will be a global resolution (A.1.6) of (gz~ (cf. [21] for a treatment of the singularities of Schubert varieties).

c) To give an application of Proposition (A.1.9), we shall use the following lemma, which may be proved by a standard Baire category argument.


(A.I.I1) Lemma. Let E~*M be a holomorphic vector bundle of rank r over a Stein manifold M. Then the set of

(a 1 . . . . . ar)eX'(9(M, E)

such that (A.1.8) is satisfied is of second category.

Combining this with Grauert's Theorem VI above and proposition (A. 1.9) gives the following:

(A.l.12) Proposition. Let E - ~ M be a topological vector bundle over a Stein manifold M. Then the Chern classes ck(E)~ HER(M) are fundamental classes of analytic subvarieties of M.

Appendix 2 Vector Bundles and Chern Classes/Q

a) If we combine Proposition (A.l.12) with Grauert's Theorem VI, then we see that on a Stein manifold M the Chern classes of any topological vector bundle may be represented by analytic subvarieties. The question of how much of the cohomology

H . . . . (M, 7/) = ( ~ H2k(M, 7I) k

is representable as fundamental classes of analytic subvarieties then becomes the purely topological problem of seeing how much of H . . . . (M, 2[) is generated by Chern classes of continuous bundles. With integer coefficients this problem involves thorny questions of torsion, but over the rationals it has a very nice answer, due originally to Atiyah- Hirzebruch [8] and making essential use of Bott periodicity. In this appendix we shall sketch a self-contained treatment, based on rational homotopy theory and intended for analysts, of their result. When applied to Stein manifolds, one obtains the following:

Theorem VIII. On a Stein manifold M, every class u~H2k(M, if)) is a rational multiple of the fundamental class u z of an analytic subvariety Z of M.

Remarks. Although not proved here, it seems likely that we may write

(A.2.1) u = 2 . u z

where 2EII~ and Z is irreducible. It is much less clear whether of not we may take Z to be smooth (cf. [21]), although by resolution of singularities there will exist a complex manifold Z o and proper holomorphic mapping f: Zo-- ,M with f (Zo)=Z.


(Note. Schubert varieties admit canonical desingularizations; cf. [21].)

The question of uniqueness of Z in (A.2.1) seems quite interesting and was discussed in w 24.

b) Let X be a finite dimensional CW complex and K(X) the Grothendieck group constructed from the complex vector bundles on X. Since X has finite dimension, every vector bundle E - , X has an additive inverse in the sense that there exists a vector bundle F---, X such that

E|

where I" is the trivial bundle of rank r. Thus elements of K(X) have the form

(A.2,2) ~ = E - I"

where the vector bundle E--*X is determined up to stable equivalence. (Recall that E and E' are stably equivalent if

E O I ' ~ E ' O I ' "

for suitable r, r'.) The additive structure in K(X) is induced by " | of vector bundles, and the ring structure by " | ".

The fundamental invariants of a vector bundle E---,X are its Chern classes cq (E)e H 2 q(X, ~g) (q = 0 . . . . . r = rank (E)). Letting c (E) = c o (E) +--- + c,(E) be the total Chern class, then the Whitney formula

c(E @ F) = c(E). c(F) gives a map

c : K(X)---~H . . . . (X , ~ )

taking sums into products. Since the Chern classes are the basic quantities measuring the non-triviality of vector bundles, it is desirable on purely formal ground to use them to construct a ring homomorphism from K(X) to cohomology. Such a mapping is given by the Chern character

(A.2.3) ch: K(X)---~ H .... (X, r

(Q-coefficients are necessary here essentially because of the divisibility part of periodicity; cir. (A.2.16).) The Chern character is uniquely determined by the requirements that (i) it is a functorial ring homomorphism, and (ii)

ch (L) = e c' ~L)

for a line bundle L. If ~eK(X) satisfies Co(i) . . . . . cq_l(~)=0, then

(A.2.4) ch(~)- cq(~) ~-(higher order terms). ( q - l ) !

(The (q - 1)! appearing here is the same coefficient as appears in (A.1.7).)


The result we wish to discuss is due originally to Atiyah-Hirzebruch.

(A.2.5 Proposition. The mapping

ch: K ( X ) | . . . . (X, Q)

is a ring isomorphism.

This result states informally that:

Giving a stable vector bundle/Q is the same as giving its Chern classes/Q.

This is the generalization to vector bundles of the standard identification of the group of line bundles with H 2 (X, 7/) via their first Chern class.

Proposition (A.2.5) follows fairly easily from Bott peridocity, and this was the original proof. On the other hand, periodicity is one of the deeper theorems in topology, and is a Z-statement whereas our result is over Q. Thus it seems reasonable to prove Proposition (A.2.5) by proving a Q version of periodicity, which should be more elementary than the Z-statement. The purpose of this appendix is to outline such a proof using elementary homotopy theory '7. Specifically, Proposition (A.2.5) will come out quite naturally by introducing the rational homotopy category and then observing a few basic properties such as the Hurewicz and Whitehead theorems.

c) In this section we shall work with the class of countable but generelly infinite-dimensional C W complexes X which are simple in the sense that n 1 (X) is abelian and acts trivially on the homotopy and homology of the universal covering of X. We want to discuss the localization X(o ) of X, which is a Q-space (definition below) X(o ~ together with a map

X ~ X(o~

such that the induced map on homotopy is given by

~, (x) ~ % ( x ) | Q ( , > 0).

Passing to X(o ) has the effect of removing all torsion and divisibility phenomena from the homotopy type of X, thus focussing on the Q- information there. A reference for this section is lectures 9-10 of [10].

Recall that an abelian group A is a Q-vector space if A --, A | is an isomorphism. This is equivalent to being able to uniquely solve the equation a x = b for ae2g-{0}, beA.

Definition. X is a Q-space if the homotopy groups ni(X ) are Q-vector spaces for i > 0.

17 From homotopy theory we shall assume the definition of the Eilenberg-MacLane spaces K(n, n) together with elementary properties such as [X, K(n, n)] ~ H"(X, n), basic obstruction theory, and some familiarity with Postnikov towers, which are iterated fibrations of K(n,n)'s built up via obstruction theory. All of this discussed in lectures 1-9 of [10].


Examples. The infinite mapping telesope T obtained by successively forming for each n > 1 the mapping cylinders for the mapping S 1 ~ S 1 of degree n, has n1(T)~ Q and rc i (T)=0 for i > 1 ~8. Thus T i s a Q-space, and in fact T is a K (Q, 1).

f)) /(" \ x

\

Fig. 5. The infinite mapping telescope

In general, the Eilenberg-MacLane spaces K(Q, n)are Q-spaces.

(A.2.6) Lemma. 7ti(X ) ( i>0) are Q-vector spaces,~the Hi(X, Z) ( i>0) are Q-vector spaces.

Sketch of Proof (i) One first proves inductively on n that the homology groups Hi(K(Q, n), ~) are Q-vector spaces. Fo r n = 1, this was observed above. In general, the result follows from the homology spectral sequence of the path fibration 19

(A.2.7) K(Q,n)--~P--~K(Q,n+ I).

An easy inductive consequence of the cohomology spectral sequence of the fibration (A.2.7) is the computa t ion of the cohomology

(A.2.8) H*(K(Q, 2n), Q)= Q [x2.]

U*(K(Q, 2n- 1), Q)= A [Y2.-,]

where I~['X2n ] is the Q-po lynomia l algebra on one 2n-dimensional generator x2, and /~ [-Yzn-1] is the Q-exter ior algebra on one ( 2 n - 1 ) - dimensional generator Y2,- a 2~

~a The divisibility property ofn~ (T) is clearly illustrated by this example. 19 Recall that the loop space ofa K(n, n+ 1) is given by OK(n, n+ 1)=K(n, n), as follows from the homotopy sequence of the fibration f2K(n, n+ 1)~ P ~ K(Tr, n+ 1), P being the (contractible) path space of K(Tt, n + 1). 20 Since K(•, n) is a Q-space, the reduced cohomology/4*(K(Q, n), 7I)~/~* (K(~, n), ~). Thus the Z-homology and cohomology are much simpler for K(Q, n)'s than for K(JE, n)'s, a first indication that ~-homotopy theory is less complicated than 7Z-homotopy theory.


(ii) To prove the result in general, we shall use the Postnikov tower associated to a space X. This is a sequence of spaces

x n + 1

X , X n

where X" is obtained from X by killing all the homotopy in dimension > n using the obvious process of attaching cells. The map X"-- ,X"-I is a fibration with fibre K(rc,(X), n), and is obtained from the universal fibration

K(~.(X), n)-~ P--~ K (Tr.(X), n+ 1)

by a cohomology class k .6H"+l(X"-X, Tr.(X)), the so-called n-th k- invariant. This picture of a simple space as an iterated fibration of K(~, n)'s is a basic theoretical tool in homotopy, although it is perhaps less useful for specific computations.

Assuming our lemma for X "-~, and using that it is true for K(Tr.(X), n) by (i) above, we may deduce the statement for X" using the fibration K(rc,(X), n)-- - ,X"-~X "-1. Q.E.D.

Remark. IfX is a Q-space whose homotopy is finitely generated over Q in each dimension, then the Postnikov tower of X is an iterated fibration o f K ( Q r", n)'s with k-invariants k,6 {H "+ l(Xn-1, Q)}r..

(A.2.9) Lemma. Let X be a space, X(o ) a Q-space, and f'. X---~X(o ) a continuous map. Then the following are equivalent:

i) f , : rc i (X) | Q --~ rc i (X~o)) is an isomorphism for i > 0;

ii) f , : Hi(X)| is an isomorphism for i>0 ; and

iii) given any Q-space Y(o) and map g: X -* Y(o), there is a unique up to homotopy factorization h

) (0)

~0)"

Sketch of Proof. Assuming (ii), we shall prove (iii) by obstruction theory. Using the mapping cylinder, we may think of f as an inclusion, and then


the obstructions to extending h over the successive skeleta of X~o ) are classes in H"+I(X(o), X); n.(Y~o))=0 since Y~o) is a Q-space. Similarly, h is unique up to homotopy.

Given (iii), we take Y~o)=K(H,(X)| n) so that g: X--, Y~o)induces an isomorphism g*: H"(Y~o))~ H"(X) | Q to obtain (ii).

The implication (i)---,(ii) is proved first for X=K(n , n) (then X(o~= K (n | Q, n)) by induction on n using the spectral sequence of the fibration K(n, n)~P---,K(n, n+ 1). Knowing the result for K(n, n)'s, the general case follows by induction on n in the Postnikov tower for X. The converse is also proved by a Postnikov tower argument (cf. [10] for details).

(A.2.11) Lemma. Given a simple space X, there exists a unique up to homotopy simple Q-space X(o ) and map f: X---~Xto ) such that (i)-(iii) of Lemma A.2.10 are satisfied.

Proof When X = K (n, n), X(o ) = K (n | Q, n). In general, X(o ) may be obtained by "tensoring the Postnikov tower of X with | one forms the new Postnikov tower whose homotopy groups are n , (X) | and whose k-invariants are obtained from k, ~ H "+ 1 (X"-1, n,(X)) by | Q. Uniqueness follows from (iii) in (A.2.9).

Definition. The space X(0 ) just constructed is the localization of X at zero.

The map f: X--* X(o ) realizes geometrically the algebraic operator "@ Q" on the homotopy and homology groups of X. Thus X~0 ) contains the Q-information from X.

Example 1. As mentioned before, K(~, n)~0)=K(gQQ, n).

For odd spheres, the map S(2o")-I--*K(Q, 2n-1) given by the generator of H2"-1(S~o]-I, Z ) ~ Q induces an isomorphism by (A.2.8). Applying (iii) in Lemma (A.2.9) we see that ~2 , - 1 ~0) ~ K ( Q , 2 n - 1).

f For even spheres, we consider the analogous map S(Zo] , K(Q, 2 n). The homotopy-theoretic fibre F o f f is a Q-space, and has cohomology H*(F, Q)~-/~ (Y4,-1). Both of these easy consequences of the spectral sequence o f f It follows that F is a K (Q, 4n - 1), and thus the even dimensional Q-sphere is the total space in the fibration

K ( Q , 4 n - 1)--* S{0]--, K (Q , 2n)

whose characteristic class is given by the transgression relation d 2. y = x 2, y being a generator of H4"-I(K(Q, 4n-1)) and x a generator of H2"(K (Q, 2 n)). In particular

7[ zo2n_l , ( Q i = 2 n - - 1 7~i(s2n-1)| i [•(o) ) = ~ 0 o t h e r w i s e

(A.2.12) = f Q i=2n, 4 n - 1 7~i(s2n) | Q ~ 7~i(S~o;)

otherwise,


a relation first due to Serre, and one which perhaps best explains the simplicity of Q-homotopy theory as opposed to Z-homotopy theory.

Example2. The inclusion C N c C N§ induces inclusions of Grass- mannians G(n, N)c G(n, N+ 1), and we let

BU, = lim G(n, N) N ~ c~

(recall that the homotopy of G(n, N) stabilizes). We claim that the localization

(A.2.13) (BU,)~o)~- f i K(Q, 2q). q = l

Proof. By (ii) in Lemma (A.2.9) and the usual computation of the cohomology of the Grassmannian,

/~* ((n U~)co ,, ~)~-Q [x~, x~, ..., x~~

Each x2q gives a map (BU~)~o)~K(Q, 2q), and by (A.2.8) the product map

(BU,)~o)-~ f i K (Q, 2q) q = l

induces an isomorphism on cohomology.

It follows from (A.2.13) that

t0 (A.2.l 4)

To explain this result, we let

BU = lim G(n, 2n) n~co

Q.E.D.

i = 2 , 4 . . . . . 2n otherwise.

using the inclusions G(n, 2n)cG(n+ 1,2n+2). Then Bott periodicity gives that

(A.2.15) ni(BU)={ ~ i=2q otherwise.

Thus (A.2.14) is an unstable Q-version of Bott periodicity. Roughly speaking, (A.2.15) seems to be equivalent to the Q-statement (A.2.14) and the divisibility statement (cf. (A.2.9))

(A.2.16) - - c h ( 0 EH2n(san ,•) (n- 1)!

for ~ /~(S 2") (cf. (A.1.7)).

Example3. Let H be a homotopy commutative H-space whose homotopy groups ni(H ) are finitely generated in each dimension. Then 7 Inventiones math.,Vol. 28


it follows formally from (iii) in Lemma (A.2.9) that the localization Hto ) is again a homotopy commutative H-space whose homotopy groups ~i(H(0)) are finite-dimensional Q-vector spaces. Moreover, the homotopy groups of the homotopy-theoretic fibre F o f f : H ---, Hto ) are finite groups in each dimension.

Now we denote by ~(f (X) and ~0 (X) respectively the functors

x , I x , H]

(A.2.17) I f

x , IX, H~0d

from C W complexes to abelian groups.

(A.2.18) Lemma. ~Yg0 (X)~ ~ (X) | and the mapping f: ~f~(X) ---, ~o(X) in (A.2.17) is " |

Proof This is a formal consequence of the following observations: (a) Given g: X--~ H~o ), the obstructions to lifting g in

H~ F

X g ~ m(0 )

are classes (~.(g) in H"(X, n._ I(F)); (b) given two maps 21 gi: X--~ H~o ) (i = 1, 2)

(9, (gl + g2) = (9,(ga) + C,(g2);

and (c) the aforementioned comment that the'n,_ 1 (F) are finite groups.

e) Classifying spaces and the proof of Theorem VIII. Let X be a finite dimensional CW complex and Vect"(X) the isomorphism classes of rank n vector bundles over X. Denoting by G(n, N) the Grassmannian of (N - n)-planes in 112 u, there is a universal n-plane bundle U, ~ G (n, N). The classification theorem for vector bundles says that any EeVect"(X) is of the form f * E for a unique-up-to-homotopy map f : X--* G(n, N). Here N is large relative to n and dim X. Thus, using the notations of Example 2 above, there is a bijection of sets

(A.2.19) IX, BU,] ~ Vect"(X).

There is a similar bijection of sets

(A.2.20) [X, BU] ~ [((X)

=l The sum gl +g2 is defined using the H-space structure on H~0).


where/((X) = ker {K(X) ~ K(xo)} is the reduced K-group (for connected X, / ( (X) is by (A.2.2)just the equivalence classes of stable vector bundles over X). Note that B U is a homotopy-commutative H-space, where the group structure on [X, B U ] is induced by the direct sum of vector bundles.

Consider now the cohomology class c h e H . . . . (BU, Q) given by the Chern character of U, ~ G(n, 2n) for each n. From c h we obtain the map c h in the diagram

B U ~h , [ i K(Q, 2q) q / / ~

/ /

/ / oh(o) (A.2.21) / /

8U~o~

where the completion of the diagram by the dotted arrow c h(o ) follows from (iii) in Lemma (A.2.9). The map ch~o ) in (A.2.21) iiaduces an isomorphism on cohomology by (A.2.13), and is thus a homotopy equivalence 22. Moreover, the induced map

(A.2.22)

EX, BU] ~h , [X , l-[ K ( Q , 2q)]

K ( X ) ch , H . . . . (X, Q)

renders the diagram commutative, and shows that ch in (A.2.21) is a map of H-spaces.

Proposition (A.2.5) now follows from Example 3 above, which gives the commutativity of

(A.2.23)

IX, By] , IX, B~o~]

~(x) | ,R(x)|

Appendix 3 Solving the r with Dependence on Parameters

We are given: (i) Open Stein subsets of C N, U c c V.

(ii) A differentiable vector bundle E over V with fibre C'.

22 This is a different, hut somewhat more na tu ra l homotopy equivalence from that given in Example 2.

7*


(iii) A family V t of flat c~-connections on E, depending smooth ly on t = ( t 1 . . . . . t,)eIR".

( iv )A family q)t of V,-closed E-valued (0, q + l ) - f o r m s depending smoo th ly on t~IR".

We shall sketch a p roof of the following.

(A.3.1) Proposition. There is a smoothly family tl, o f E-valued (0, q)-forms defined over U for It] < 1, such that

V, tl , = q~ ,

in U, for every t such that I t [< l.

Proof Choose a comple te K~ihler metr ic on U. Let dv be the volume element associated to it. Also choose a Hermi t i an metric on E. For every E-valued form .a set:

tidal = [ I t~l: dv]~ U

where [a] is the pointwise length of a compu ted with respect to the metrics we have chosen. Let ~o .p ) be the no rmed space of all E-valued (0, p)-forms a such that i]aH < + oo. Let Vt* be the formal adjoint of V~. It is possible to choose the metr ic on E so that, for every p > O, every t such that [tl = 1, and every form ct belonging to ~to,p)

(A.3.2) lie[ 12 < Gi l l V, all 2 + IF v,* ~ In 2]

where C is a constant independent of t (see w 9).

F o r m u l a (A.3.2) implies that, when a 1, a2 are forms of degrees (0, p + 1), (0, p - 1) such that lea1 II < + oo, liaR11 < + oo, V~ a = 0 and 0- 2 is o r thogona l to the null space of V t, then there is a unique (0, p)-form u such that Ilull < + oo and:

V t u z 0 - 1 ,

~t* U =O" 2

(see w 9). Therefore, when q > 0, we m a y find a family r/t of (0, q)-forms defined over U for [ t [< 1, such that :

V, t h = q~,,

V,* r t ,=0 .

In this case we shall p rove Propos i t ion (A.3.1) by showing that t/, depends smooth ly on t. This can be done by a slight var iant of a me thod due to Nirenberg [25]. T o simplify nota t ions we will assume n = 1.

Let ar be a family of E-valued (0, p)-forms on U. We will say that a, is C" in t if and only if all the U-derivat ives of a, are of class C v in the U-variables and t. We will say that a t depends continuously on t in the L 2

Analyt ic Cycles and Vector Bundles on Non-Compac t Algebraic Varieties 10I

sense if a t is C ~ in t, llat]] < + ov for every t and a t depends cont inuously on t as an element of ~(o,p). We will say that a t is C 1 in t in the L 2 sense if a t is C 1 in t and moreover :

(i) a~, Oat/& depend cont inuously on t in the L 2 sense

(ii) for every t: at+h--at Oa t ___>

h c~t 0 as h - * 0 .

We will say that a t is C" in t in the L 2 sense (v> 1) if a t is C ~-1 in t in the L 2 sense and (~v- i at)/(Ot v- 1) is C 1 in t in the L 2 sense.

We shall prove inductively on v, the statement:

A(v): Let q)t, tPt be families of E-valued forms of degrees (O,q+ 1), (0, q - l ) on U such that I[~otll<+oo, [ ]Ot l l<+oo, Vt(pt=0 and Ot is orthogonal to the null space of V~ for every t. Let rh(]tl< 1) be the unique family of (0, q)-forms such that Ilr/tt] < + oo, V t t h = ~o t ~* ~lt = Ot for every t such that It I < 1. I f qot, tPt are C ~ in t in the L 2 sense, so is tit.

Let S be a relatively compact open subset of U. In what follows 1t llS,k will denote Sobolev k-norm on S, whereas I ]S,k will denote the supremum in S of all derivatives of order up to k. Set ~ t = Vt V,* + Vt* V t. Let S c c T be relatively compac t open subsets of U. It follows from (A.3.2) and Sobolev 's and Friedricks ' inequalities that there are constants C~.k independent of t such that for every form ueCY(0,p }, p > 0 , defined on U and every t such that It[ < 1:

(A.3.3) lull,,__< Cl,k[]l~tUHr,k" ~ ]](V t "-]- Vt*)Ul] ]

if k>> l.

Proof of A(O). F r o m (A.3.3) we get

Ith-tl~l,3 <= Cl,k[l[ Vt* q)t-- Vt* qGllr,k 4-1] Vt 6t - Vt@~llT,k

~- I] ( ~ t - - ff@~) ?/~ II T, k "Jr- I[ q)t - - (/Or II "1- II 0 t - - 0~ II

+ II(g- v~) ~tll + I IG*- v~*) ~ttl].

The right hand side clearly goes to zero when t tends to z; this shows that t h is C o in t. On the other hand (A.3.3) implies that r h depends cont inuously on t in the L 2 sense.

Proof of A(1). While proving A(1) we shall show that

(A.3.4) Vt \ ~t ] Ot 8t (t/t)=~'

Vt* \ ~t ] ~t ~t (tlt)=flt"


It is easi ly seen that s t (resp. fit) is Vcclosed (o r thogona l to the null space of ~) tha t ct t be longs to ~to,~+l) (~(0,q-1)) and depends con t inuous ly on t in the L 2 sense. There is a un ique family 2 t of(0, q)-forms such tha t :

IGII < + oo.

We k n o w that 2 t depends con t inuous ly on t in the L 2 sense. Set:

\(" r/t + hh-- t/t (h) Zt ! - 2 t. I

We then have :

�9 1 Oct t t?V t 1 Vt (,'(,(h)) = T le t+h- q~t] - ~ - + ~ ) - ( r / t ) - T (Vt+h-- Ft) r/t+h

hence tl v, z, (h)ll --, 0 as h -~ 0 (recall that V t + h - ~ is an o p e r a t o r of degree zero def ined on V and tha t U is re la t ively compa c t in V). S imi lar ly llV,*xt(h)II and II~tzt(h)llr,~ go to zero as h - - , 0 . C o m b i n i n g this with (A.3.2) and (A.3.3) gives tha t :

as h - - , 0

Ilz,(h)l] - - ,0

hence &l]Ot exists and is equal to 2 t. M o r e o v e r r/t is C 1 in t in the L 2 sense.

Proof of A(v); v > 1. I nduc t i on on v. R e p e a t e d app l i ca t ions of (A.3.4) show tha t

and

v,*

are C 1 in t in the L 2 sense, therefore

c3v- 1 ~/t

c3 t ~- 1

is C 1 in t in the L 2 sense. This conc ludes the p r o o f of A(v).

W e now turn to the p r o o f of P r o p o s i t i o n (A.3.1) when q = 0. If c~ is an e lement of ~to,o~ we will deno te by Pt(e) the o r t hog ona l p ro jec t ion of


onto the null space of V~. It follows from (A.3.2) that:

(A.3.5) llel[ < IIP,~II + C It v,~ll

for every eeY)(o,o ) and every t such that I t l< l . This implies that, if 0"1e,~(0,1), O'2 e;~(O,O), Vto" 1 = ~tO'2 = 0 , there is a unique ue~(o,o ) such that Vtu=a 1, P , u = a 2. An analogue of formula (A.3.3) also holds, namely if S c o T are relatively compact open subsets of U, there are constants CI. k independent of t such that, for every ~e~(o,o):

(A.3.6) Ic(]s.t ~ C~,k [H~t(x[I T,k "j- l[ V,~lt + IIP,.H]

when k >> l.

We will prove, inductively on v, the statement:

B (v): Let (Pt be a family o f elements o f 9r ,1), Ot a fami ly o f elements o f ~to,o)- Suppose that Vt tP t = Vt ~k t = O for every t. Let r/t(]t]__< 1) be the unique family of elements o f 9~(0,0 ) such that Vtt/t=(pt, P,th=Ot for every t such that ]t]< 1. I f (Pt, Ot are C v in t in the L 2 sense, so is tit.

The proof is along the lines of the proof of A(v): as in that case, we will assume, for simplicity, that n = 1.

Step I. Proo f o f B(O). We may suppose that Or=0 for every t. B(0) follows from (A.3.5) and (A.3.6) as A (0) follows from (A.3.2) and (A.2.3). The only point that might not be entirely obvious is the proof that []p,~bl[ = I[(P,-P,)r/~][ goes to zero as t tends to t. This can be seen by writing ~/~ = ~*u (this is possible since ~/~ is orthogonal to the null space of Vt), and by noticing that:

liP• ~11 = liP, v~* ull = Ilp,(v~*- v,*)ul[ _-_ II(v,*- v~*)ull.

Step 2. Suppose that cq depends continuously on t in the L z sense. Then P, et depends continuously on t in the L 2 sense.

To prove this we will use formulas (A.3.5) and (A.3.6). We obtain:

IP, ~ , -P~ ~&,_-__ C~,k [II~, P, ~llr, k + liP, ~ t -P , P~ ~ll + l] gP,~,ll]

liP, c~,-P, cgll < lIP, ~,-P, ~ ~11 + C II g P, ~lI.

We may write:

II V, P~ ~11 = II(V,- ~) P, ~11.

Both terms clearly go to zero as t ---, t. As for the remaining term, write:

~,=p , ~ + ~* ~ .


Then

_-< II~t-~lt + H(v,*- v~*)s~ll

which goes to zero as t ---, r. This concludes Step 2.

Step 3. Proof of B(1). As in the p roof of A(1) we also obtain, as a byproduct , that:

V, \ Ot ! Ot Ot ,t (A.3.7)

P~\ ot ! \ at

where St is such ihat IIs, ll < + ~ , e,s,=0, r t t=0,+ V,*s,. Except for the obvious adaptat ions, the a rgument is the same we used to prove A(1).

Step 4. Let c~ t be a family of sections of E that is C 1 in t in the L 2 sense. We want to show that P, c~ t is C 1 in t in the L 2 sense, too.

Let St be the unique element of ~(o,o) such that :

g S t=0 ,

V,* S, = at - P~ c~t. We will also prove that :

(A.3.8)

e VV-I 0t! It is clear that

at

is g-closed, therefore we may find a family o- t of elements of N'(0,0 ) such that :

v, a, = - -g~- (~ c~,),

P, ~, = ~ - N - - P,-bT- s,.

By Step 1 a t is cont inuous in t in the L 2 sense. Step 4 can be completed by the same method we used to prove A (1) and B(1). In other terms, write:

z t (h)= Pt+n C~t+h-- Pt at h ~


W h a t is to be p r o v e d is tha t Ii~tMt(h)l[T,k, [IPtzt(h)l I a n d II~z~(h)ll go to ze ro as h---,O. Le t us e x a m i n e IIPtz~(h)l I. T h e o t h e r t e r m s can be dea l t w i th in a s imi la r way. W e can wr i t e :

= Pt( h ~_~t~t] ~- Pt~t~rh~t+hh P'Ti II IIP~ z,(h) ll <

+ v,+h g 0v,*~, = h (?t h 7t+h 6~ t

B o t h of these two t e r m s c lear ly go to ze ro as h -* 0. T h i s c o n c l u d e s S tep 4.

Step 5. Proof orB(v), v > 1. F i r s t no t i ce that , ifct t is C v-1 in t in the L 2

sense, so is Pt c(t. If v = 2 this is S t ep 4. If v > 2, the r igh t h a n d sides o f f o r m u l a s (A.3.8) c lear ly a r e C v-2 in t in the L z sense, t he re fo re B ( v - 2 )

impl ies tha t (~Pt ~,)/Ot is C ~- 2 in t in the L z sense.

N e x t no t i ce tha t the r igh t h a n d sides o f f o r m u l a s (A.3.7) a re C " - 1 in t

in the L 2 sense, h e n c e so is (Oqt)/Ot. This c o n c l u d e s S t e p 5 and the p r o o f

o f P r o p o s i t i o n ( A . 3 . 1 ) . Q .E .D .

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Maurizio Cornalba Istituto Matematico "L. Tonelli" Universit~ di Pisa 1-56100 Pisa, Italy

Phillip A. Griffiths Department of Mathematics Harvard University 1, Oxford Street Cambridge, Mass. 02138, USA

(Received April 16, 1974)

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