What is Several Complex Variables?people.aub.edu.lb/~fb31/304/krantz.pdf1987] WHAT IS SEVERAL COMPLEX VARIABLES? 237 quite the opposite: to limit oneself to the study of one complex

What is Several Complex Variables?Author(s): Steven G. KrantzSource: The American Mathematical Monthly, Vol. 94, No. 3 (Mar., 1987), pp. 236-256Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2323391 .

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236 STEVEN G. KRANTZ [March

expositor. Gail's excellence as a teacher at all levels is a corollary to a remarkable intellectual breadth and depth coupled with a rare sensitivity to the needs of his students. He has had nineteen Ph.D. students.

The simple truth is that everything Gail has done-and he has done enormously much-has been well-done, useful, and important.

Our final word should be this: Gail, though now retired, is still an active member of our community and our profession-long may it be so, for we need him for his wisdom and we enjoy him for himself.

What Is Several Complex Variables?

STEVEN G. KRANTZ* Department of Mathematics, Washington University, St. Louis, MO 63130

When I am asked what sort of mathematics I study, my stock response is "several complex variables." But the reaction this usually elicits makes me feel as though I have said "generalized theory of fluxions." Whereas "algebraic geometry" and "partial differential equations" arouse a glimmer of recognition (if not genuine understanding) in most mathematicians, several complex variables usually draws a blank. It is natural for the listener to suppose that we who work in the subject ran out of things to do in one complex variable, so now we are busy juggling multi-indices. The thesis of this article, formulated rather aggressively, is in fact

Steven G. Krantz was born in San Francisco and raised in Redwood City, California, in the San Francisco Bay area. He was an undergraduate at the University of California at Santa Cruz and a graduate student at Princeton University where he wrote his dissertation under the direction of E. M. Stein.

*Author supported in part by a grant from the National Science Foundation. He would like to thank Steven R. Bell for several inspiring conversations on the topic of this paper.



1987] WHAT IS SEVERAL COMPLEX VARIABLES? 237

quite the opposite: to limit oneself to the study of one complex variable is to do complex analysis with one eye closed.

Let me briefly elaborate on this last point. More than any other subject that I know, several complex variables (SCV for short) is proof that many different parts of mathematics can interact fruitfully. The symbiotic relationships that algebra, differential geometry, partial differential equations, algebraic geometry, and Banach algebras enjoy with SCV have led to major developments in all of these subjects. Does one need to be expert in all of these diverse fields in order to begin to appreciate what SCV is about? Fortunately the answer is no, and I intend to prove this point in the present article.

No article nor any book could introduce the reader to all the aspects of this subject. What I hope to do here is to provide some simple yet striking examples of how SCV differs from nineteenth century complex function theory. Many of these examples were known seventy years ago, but I intend to shed some modern perspectives on them. I will try to weave these examples into a cohesive essay, and to provide some views of current concerns in the subject. Along the way, we will see some new ways to think about one complex variable, which is of course yet one more reason for considering the broader perspective of SCV.

Judging from the number of items in the Math Reviews, one finds that SCV has grown nine-fold in the last twenty years or so. A particular high point occurred in the academic year 1980-1981 when there were five different complex analysis seminars every week in Princeton. I was lucky enough to have participated in this burst of activity and would like to share some of the excitement with the mathematical community at large. An expository article in the MONTHLY seems like an ideal place in which to do so.

1. Some Preliminaries. Complex analysis of several variables is done on the space consisting of the n-fold product of C with itself:

Cx ... xC.

This space is denoted by the symbol C n. A typical element has the form

Z = (Z1, ** z- Z) E Cn.

Since C is simply R2 with some additional algebraic structure, we realize that C n is (topologically) R 2n with some additional algebraic properties. We have a natural way to identify points in C n with points in R 2n. This is described by the scheme

Cn ? (z1) ... Zn) *- (Xi ? 1Yl,..., Xn + iYn) *- (xl, Yl, x*** Xny,) E 2n

In particular, we measure distance in Cn in the customary Euclidean fashion: if z = (Z1 ..*, zn) and w = (wl, wn) are points in C?n, then

- l (z1- W112 ? * ? z - W 2 1/2




By the same token, it is sometimes useful to think of C ' as a vector space over C. If z = (z1,..., Zn) and (w1,..., w") are elements of Cn, then we can define an inner product by

<,W 1, 1 + Zn Wn -

With this notion of inner product, the familiar notion of orthogonality can be used to aid in the study of function theoretic questions. Indeed, we shall use such geometric insight to guide our thoughts in Section 2.

Rather than use a/axj and a/ayj for doing calculus on C n, it is more convenient to consider

d 1| d d

and

d |I d d

ai- 2\ adxj ayj

Check for yourself that

d az1 Zk = jk'y dzj

d az.Zk = k0

d

ad Zk = 0.

Here ajk is the Kronecker delta. The real significance of a/azj and d/ld- will become apparent after we define analytic (or holomorphic) functions.

Recall that there are several different ways to think about analyticity/holomorphy in one complex variable: two very important ones are the Cauchy-Riemann equations and local power series expansions. These points of view will be useful in SCV as well, as we shall now see.

What is an analytic function of several complex variables? A simple working definition is that if S2 c C is an open connected set (a domain) and if f: S2 -? C, then we call f analytic if for each fixed P = (Pl,..., Pn) E Q2 and each fixed




j E { 1, .. ., n }, the single variable function

C E= Z -*f (pl,** pj_-1 pj + Z, Pj+1l***. Pn) is analytic for z small. In other words, f is analytic if it is analytic (in the classical one variable sense) in each variable separately. As in one complex variable, the word "holomorphic" is used interchangeably with "analytic."

It is reassuring to know that no aberrations can occur: an f which is analytic according to our definition must (by a non-trivial theorem of F. Hartogs) be continuously differentiable to arbitrarily high order as a function of the 2n real variables xl, Y1,..., Xn, yn. Like Goursat's version of the Cauchy Integral Theorem, this result of Hartogs is more an aesthetic than a useful one: the holomorphic functions which arise in practice are usually smooth by inspection. But Hartogs's result makes the theory cohesive.

If f is holomorphic on Q c C n' then f satisfies the Cauchy-Riemann equations in each variable separately: if f = u + iv then

Au av

dxj dyj and

Au av

dyj dxj'

for j = 1,..., n. Our new calculus notation makes the Cauchy-Riemann equations particularly easy to write down. For if f = u + iv satisfies

d d_f =O0 j = 15... n ,

IZ

then we may calculate as follows.

d d 1 a d d 0= f= a(u?+iv) a ia (u+iv)

1{ d d 1{ d d =~ 2t - a V

+ i t v+ -u. 2 }ai d :yj)?2(dij aayj>

Taking real and imaginary parts we see that f satisfies the Cauchy-Riemann equations in each variable. Since the calculation runs backwards too (try it!), we see that a continuously differentiable function of several complex variables is holomorphic if and only if

d da

f =, j= 1,...,n.




Typical examples of holomorphic functions are polynomials such as

(Z1, Z2, Z3) = Z1Z2- (Z3),

rational functions such as

z1

r(z1, Z2) = (z22

and convergent power series such as

00

q(Z) E (ZlZ2)z k=O

While the first example is a well-defined holomorphic function for all z = (z1, Z2), the second is only defined when z2 7 + i and the third only when Iz z21 < 1. To see that the power series truly defines a holomorphic function, one needs to check that a power series may be differentiated termwise on its domain of convergence. Such matters are best left to the reader, or see [12].

In general it is quite hard to see what is the (largest) domain of definition of a given holomorphic function. It is also very difficult in several complex variables to construct holomorphic functions with specified properties. Indeed, many of the most powerful tools of one complex variable (Mittag-Leffler and Weierstrass theorems, Blaschke products, inner-outer factorizations, the Riemann mapping theorem, conformal mapping, Rouche's theorem) are unavailable in SCV while others (the Cauchy Integral Formula, residues, harmonic functions) are considerably less useful. These obstructions have become major themes in the subject of SCV and have led to the development of powerful new tools.

2. A Fundamental Discovery of Hartogs. In 1906 Hartogs made a startling discovery which helped to establish SCV as a subject in its own right. The result concerns "natural boundaries" for holomorphic functions. Most students encounter the notion of natural boundary for the first time in the context of the Hadamard gap theorem: certain lacunary power series produce functions which are holomorphic on the open disc, continuous on the closed disc, yet cannot be analytically continued to any larger open set. The circle is said to be the natural boundary for such a function. Proceeding informally, we might ask whether similar functions exist on any open set in the complex plane. If they do, then they surely cannot be constructed using power series. Instead, the Mittag-Leffler theorem enables us to prove the following result:

THEOREM 1. Let sa c C be an open set bounded by a simple closed curve. Then there exists a holomorphic f on Q with the following property: if sa is any open set which strictly contains Q, then there is no holomorphic F on sa such that F restricted to sa equals f.




Proof. Let 9 = {q1} C sa be a countable set which (i) has no accumulation points in Q2,

and (ii) accumulates at every boundary point of da . (It is a good exercise in plane geometry to construct such a set-see Fig. 1.) By

Weierstrass' theorem, there is a holomorphic function f on Q2 whose zero set is precisely {qj}.

FIG. 1

Now if there were a P Eca d and r > 0 such that f continued analytically to Q- U {z: Iz - PI < r}, then P, being an accumulation point of .9, would be

an interior accumulation point in Q2' of f-'({0}). See Fig. 2. Thus f 0, yielding a contradiction. U

FIG. 2




Does a result like Theorem 1 hold true for holomorphic functions of several complex variables? Hartogs's discovery is that sometimes the answer is yes and sometimes no. In a moment we shall see why. First we introduce some notation:

If P E C and r > 0, then let

D(P, r) {z e C: z - Pi < r},

D(P, r) {z E C: Iz - PI < r},

D D(0,1),

D D(0,1),

and if P= (p1,...,p)eGCand r>0, then

D n(P, r)--z E C: lz - pjl < r,j =l,...n),

-Dn(P. r)-{En 'II J Ar, j = 1,...,n},

B(P, r) = {z E Cn: IZ,12 + + lz 12 < 1)

B(P, r) = {z E Cn: jz112 + + jz,,2 < 1}2

Now we return to our discussion. Let p be the nonextendable holomorphic function on D whose existence is guaranteed by Theorem 1. Let 52 = D2(0, 1). Define a holomorphic function f on ?2 by

f(Z1, Z2) = T(Z1) * ((Z2)

It is immediate that f is a non-continuable holomorphic function of two complex variables on the domain l: there is no larger open set containing 1 to which f can be analytically continued.

Notice that the very same argument shows that every product domain in C2 exhibits this non-extension property. So we have found without much effort a large family of domains which, like domains in one complex variable, are the "natural domain" for some holomorphic function. Such a domain is called a domain of holomornhy.

If SCV were as trite as this last construction suggests, then the subject would bask in well-deserved obscurity. We begin to see some texture in the subject as we now turn to a domain which is not a domain of holomorphy.

THEOREM 2 (Hartogs [9]). Let r > 0 and define

9 = D2(0, r) \D2(0, r/2).

(See Fig. 3.) If f is holomorphic on 0, then there is a holomorphic function F on the domain

S2-D2(0, r)

such that F = f. (See Fig. 4.)




Z2

'fz- Z1 -y

FIG. 3

Z2

FIG. 4

Proof. For any fixed 1z1 < r we may expand f in a Laurent series about zero in the Z2 variable:

00

f(Zl, Z2) E [ak(Zl)] (Z2)k. k= =-oo

Referring to Fig. 3, we see that the series surely converges for r/2 < I Z21 < r. Notice also that the formula

ak(ZI) = k!if( ,k+1) d'

guarantees that ak is a holomorphic function of z1 (use Morera's theorem).




Referring again to Fig. 3, we now make the crucial observation that when r/2 < Iz1 I < r, then f(zl, *) is holomorphic on D(O, r). Thus

ak(Zl) = 0 when k < 0 and r/2 < IzIl < r.

By analytic continuation (the one variable result), ak- when k < 0.

Thus we may define 00

F(Z1, Z2) = E [ak(Zl)I (Z2), k=O

and F will have all the desired properties. U

Several important phenomena came to the surface in this proof: first, the fact that our function f is holomorphic in each variable separately plays a crucial role; second, that there are enough dimensions for us to be able to "move around" and find an open set on which ak = 0 for k negative.

We now see that something definitely new is going on in SCV, but we have not a clue as to how to identify which domains exhibit the phenomenon of Theorem 2, and which, like the product domains, support non-extendable holomorphic functions (and are thus domains of holomorphy). Consider the ball for example. It is not a product domain (it could conceivably be "equivalent" to one, but we shall see in Section 5 that such is not the case). Is it still a domain of holomorphy? Just to illustrate that we are dealing here with a fairly subtle problem, we now present the following result:

THEOREM 3. There is a function f holomorphic on the unit ball B = B(O, 1) E C 2 such that f cannot be analytically continued to any larger open set.

Proof. If P = (Pl, P2) is in the boundary of the ball, IpLo2 + IP212 = 1, then define

cpp(z) = -[(Z1 + P) 1 + (Z2 + P2) *P2I. 2

It is instructive to notice that the formula defining rgp is an instance of the inner product discussed at the beginning of Section 1:

1 TPp() - . (z + F, F). 2

Linear algebra now tells us that Tp will be largest when z (considered as a vector) points in the same direction as P and is as far from the origin as possible. Thus cp(P) = 1 and IgpI is strictly smaller elsewhere in B. A more quantitative way of saying this is that if 0 < r < s < 1, then there is a constant k = k(r, s) such that for any z E B(O, r) it holds that




Let a = { I1 } be a sequence of elements of B which accumulates at every point of dB. Let a be the sequence

al, al, a2, a1, a2, a3, a1, a2, a3, a4,..

Then a contains all the elements of a and each element is repeated infinitely often. We will write ac= {)a. For each a1 choose rj as large as_possible so that Dj1=D2(aj, rj) c B. For each j E {1, 2, 3,...} define Kj = B(O, 1-1/j) and choose Zj e Dj \ KJ. Let p1 = zJjl /zj and observe that the function sj(z) = qp, (z) has larger modulus at z1 than it does on Kj. If we define

1

C1 = si(z1)

then

hi(z) = c1 S

satisfies

hj(zj) = 1 and lhjl1K < tj< 1

for some constant tj. If positive integers Nj are chosen sufficiently large, then the functions

mj(z) [hj(z)]Nj

satisfy

mj(zj) = 1 and imjIiK < 21.

Define 00

h(z) = r7 (1 -m(Z))j- j=1

Then the product converges uniformly on each Kj since

Ej- 2- < x.

Thus h is holomorphic and not identically zero, and h has a zero of order at least j at zj. Since each ad (and hence each D ) is repeated infinitely often in the sequence a, each Dj contains points at which h vanishes to arbitrarily high order. Any analytic continuation of h to a neighborhood of a point P E8 d 2 is a continuation to a neighborhood of some Dj. Hence the domain of h would contain an accumulation point zo of a sequence of zeroes of increasing order of h. At zo, h would vanish to infinite order. Hence h would be identically zero (by the one variable identity theorem). That contradicts the fact that h comes from a convergent infinite product. .




It should be noticed that the special nature of the ball plays no role in the proof of Theorem 3. What is crucial is the existence, for each boundary point P of S, of functions gp which are "big" near P and "small" away from P. In fact domains of holomorphy are characterized by the fact that they possess such functions Tp for each boundary point P.

It was a problem of long standing, called the Levi problem, to give a purely geometric description of domains of holomorphy. This problem was solved in the early 1950's by Oka, Bremermann, and others. The geometric notion which char- acterizes domains of holomorphy is called pseudoconvexity. The precise definition of pseudoconvexity is rather technical, but we can give an informal description as follows.

Consider the collection 9 of all domains which can be obtained from a convex domain by applying to it a holomorphic mapping:

(Z1,-.- Zn) =Z -4 >(z) =(q91(Z),., pn(Z)),

each pj holomorphic. Let Y be the collection of domains 2 whose boundary can be paved (see in Fig. 5) by finitely many elements of S. Here we say that a 2 is "paved" by the boundaries of domains 2j, j = 1, . . ., k, if d . = u(dd n d Qj).

FIG. 5

Finally, let C be the collection of domains which can be obtained by increasing unions of elements of Y. The collection 9 coincides precisely with the collection of pseudoconvex domains, which in turn coincides with the collection of domains of holomorphy.




If 0 is pseudoconvex and P Ea d 2, then it is possible (using deep techniques such as sheaf theory or partial differential equations) to construct a function pp which is large near P and small elsewhere. With these functions in hand, the proof of Theorem 3 can be mimicked to prove that 0 is a domain of holomorphy. This is the hard half of the Levi problem (see [12, p. 130] for a discussion of the Levi problem). As an exercise, the reader may wish to construct functions pp for a convex domain 2 and P E d8 and then verify that the proof of Theorem 3 may be successfully

imitated to see that 2 is a domain of holomorphy. This is much easier than the general case.

3. Some Consequences of Hartogs's Theorem and Related Results. A fundamental fact about a non-constant holomorphic function f of one complex variable is that its zero set is discrete: if the zeroes of f accumulate in the interior of the domain of f, then f 0. Moreover, by Weierstrass' theorem, any discrete set can be the zero set of a non-constant holomorphic function. This is startlingly false in dimensions two and higher, as we shall now see.

THEOREM 4. Iff is holomorphic on a domain s c C n, n > 2, then f has no isolated zeroes.

Proof. Suppose that P E 0 is an isolated zero of f. Let Dn(P, r) c s be a polydisc such that {z e Cn: f(z) = 0} n Dn(P, r) = P. See Fig. 6. Then g(z) _ 1/f(z) is holomorphic on Dn(P, r) \ Dn(P, r/2). By Theorem 2, g continues analytically to Dn(P, r). In particular, g is well defined at P, so f cannot vanish at P.E

FIG. 6

Here is another result about zero sets which is of a similar flavor, but its proof is a bit different and shows once again how the existence of several complex dimensions can be exploited to obtain new results.




THEOREM 5. Let f be holomorphic on 1, a bounded domain in C n, n > 2. Define "= {z E S1: f (z) = 01. Then S is either empty or non-compact in U.

Proof. Suppose that .9 is non-empty and is compact in U. Fix a point X outside S2. Choose a point P in .9 which is as far as possible from X. Let v be a unit vector in the direction XPP and let wi be a unit vector which is orthogonal to the vector v) (in the sense of the inner product (, )). See Fig. 7. For r > 0, small and fixed, and for j sufficiently large, we define holomorphic functions of one complex variable by the formula

T1(j) =f (P + (l/j) + rw), E D.

Then each function j is zero-free since the complex disc

P?+ (l/j)v+ rtw: E D} lies in S2 but outside f9 (provided that r is small enough). But the functions pj have the limit

To) =f (P + rw)

ex

FIvG./

FIG. 7




x

FIG. 8

as j oo. And the function Po vanishes at D = 0. By Hurwitz's theorem, we must conclude that Po 0, and that is false (since P is as far as possible from X- see Fig. 8). Thus 9 cannot be compact. U

COROLLARY. If f is holomorphic on a bounded domain 0 in C n, n > 2, then every level set of f escapes to 8 U.

Proof. Let a be in the range of f and set g(z) = f(z) - a. Then the result follows by applying the theorem to g. a

Recall now the following form of the maximum modulus principle: if 2 is a smoothly bounded domain in C and f is non-constant, continuously differentiable on 02, and holomorphic on 0, then the maximum modulus of f occurs on the boundary and only on the boundary of 0 (we are assuming here a bit more than is necessary). This result holds in fact in any complex dimension, and the proof is the same as the classical one-variable proof. The maximum modulus principle suggests that functions like f(z) = zn, Z E C C C, are typical: the modulus of f on the boundary is greater than that in the interior. But notice that, in one variable, the open mapping theorem shows that it cannot be that the image of f is always




contained in the image of f | da (for f continuously differentiable on C2 implies that f( ad ) has no interior). The situation is just the opposite in dimensions two and higher:

THEOREM 6. If C C ? n is a bounded domain, n > 2, f is continuous on the closure 1 of SA, and f is holomorphic on SA, then the image off | contains the full image off

on ii.

Proof. Let P E U. If f(P) 4 f(d A), set w = f(P). Then f l(w), being a closed, bounded set disjoint from da , is compact in U. This contradicts the corollary to Theorem 5.-

We have seen now that zero sets of holomorphic functions of several complex variables are never discrete. Is there some way to describe zero sets of holomorphic functions in both C1 and C n simultaneously and in the same language, so that the two theories do not seem so disparate? The answer is yes, and is best formulated in terms of complex dimension. Unfortunately, a rigorous treatment of this topic would take us far afield. Therefore we will content ourselves with the detailed discussion of a simple example, together with a few cultural remarks.

EXAMPLE. Consider the holomorphic function C2 defined by

f (Zl Z2) = Zl *Z2.

Then

( (f= {(Z1, Z2) E C2: Z * Z2 = 0)

= {(z1, Z2) E C2: Z1 = 0) U {(Z1, Z2) E C2: Z2 = 0).

In short, T (f ) is the union of two complex hyperplanes. In particular, -T (f ) is a 1-dimensional complex surface (the number 1 being notable because it is one less than the dimension of the ambient space), except at the point (0, 0) of intersection of the two hyperplanes.

Compare the discussion of the preceding paragraph with the fact that the zero set of a holomorphic function in C1 is discrete. A discrete set in C1 is a set of complex dimension zero, which dimension is one lower than that of the ambient space.

In general one can prove, using a deep algebraic fact known as the Weierstrass Preparation Theorem, that the zero set of a non-constant analytic function in C n is a complex hypersurface of dimension (n - 1), except on a singular set of lower dimension. The discussion in the two preceding paragraphs illustrate this assertion. .

Let us consider yet another interpretation of Theorem 2. It says that a holomorphic f of at least two variables never has isolated singularities. In particular, if f is holomorphic on a domain in C2 and if the set of singularities of f has complex dimension zero then the singularities of f form a discrete set; hence the singularities are isolated. Therefore they are removable. Inductively, if we have shown that a



19871 WHAT IS SEVERAL COMPLEX VARIABLES? 251

singular set of complex dimension (n - 2) in C '1 is removable, then let g be holomorphic on a domain ? in C n+1, except on a singular set of dimension (n - 1) = (n + 1) - 2. Let 9' be a complex hyperplane and consider g restricted to A. Then, generically, the restricted function is a holomorphic function on 9Y with a singular set of dimension (n - 2). By induction, this singularity is removable. Since this last assertion is true for generic hyperplanes 9, it can be argued that the full singularity of f is removable.

4. Inner Functions and Related Topics. An inner function f on the unit disc D C C is a bounded holomorphic function with almost everywhere radial boundary values on the unit circle having modulus 1. There are a great many inner functions on D. Any function of the form

p(Z) = Zk

is certainly inner, any Blaschke factor z - a

Ba(Z) =1-z

for a e D fixed and z E D is inner, and any Blaschke product

V1_ B,, }(z)

is inner. Other inner functions may be obtained as exponentials of Cauchy integrals of singular measures. Indeed there are so many inner functions on the disc that the closed lnear span of the inner functions in the uniform topology gives all bounded analytic functions (this is a deep theorem of D. Marshall [14]).

In the early 1960's Walter Rudin and A. G. Vitushkin posed the problem of determining whether there are non-constant inner functions on the ball in C 2. It soon became apparent that if there are such functions, they must be highly pathological. Let us see why.

LEMMA. A non-constant inner function on D c C cannot be bounded from zero.

Proof. Let T be non-constant and inner on D and suppose that

p(z)j > y > 0 on D.

Then the function

p(z)

is bounded and holomorphic. Since (p is the Poisson integral of its boundary values (which have unit modulus), it follows that I p(z) I < 1 for all z E D. But the same reasoning applies to at(z). Thus Ip(z)j I on all of D. By the maximum modulus principle, qp must be constant. That is a contradiction. U




LEMMA. If p is a non-constant inner function on D, then the range of (p is dense in D.

Proof. Let a E D be fixed and suppose that the range of q omits an open disc centered at a. Then the function

Ba O 9

(where Ba is a Blaschke factor as defined above) is bounded from zero and is still inner. By the first lemma, that is impossible. a

THEOREM 7. Suppose that f is a non-constant inner function on the unit ball in C 2.

Then for any P E dB the cluster set

W( P){ w E C: there is a sequence B wI P such that f (wj) -+w}

equals the entire closed disc D.

Proof. For almost every complex hyperplane .2 which intersects B it holds that f restricted to 92 n B has unimodular boundary values. But then, by the second Lemma, this restriction has range which is dense in D. Fix P E dB. Now let gj be a sequence of these complex hyperplanes which approach P as shown in Fig. 9. Then the restriction of f to each j has range which is dense in D. But this simply means that W(P) = D. U

FIG. 9




We know from the Corollary to Theorem 5 that the level sets of a holomorphic function of several complex variables all escape to the boundary of the domain. But for a non-constant inner function in ? n, n > 2, we see that the behavior is genuinely pathological: a dense collection of level sets escapes to every point in the boundary of the domain B. It thus came as a shock when in 1981 it was discovered that non-constant inner functions exist. (Relevant papers are [1], [8], [13]. See [17] for a discussion of the rather delicate priority question.) Indeed there are enough inner functions so that their closed linear span, in a suitably weak topology, generates all the bounded holomorphic functions on the ball.

While the construction of inner functions on the ball is elementary, it is extremely technical and ingenious and we cannot reproduce it here. But we have described these results to bring out the fact that relatively basic phenomena are still being discovered in the subject of SCV.

We conclude this section by considering a factorization problem. If S is a bounded domain in ? n, p 0, f is holomorphic on S, and f(P) = 0, then can we "factor out" the zero?

In one complex dimension, the answer to this question is easily "yes": If f is not identically zero, then there is a unique integer k > 0 such that

f(z) = (z -P) * g(z)

for some holomorphic g on S such that g does not vanish at P. In more than one dimension, matters are more complicated. For one thing, the

zero cannot be isolated. In addition, as we have seen in previous considerations, our ability to answer this question depends on the shape of the domain. In case

2= B c C 2 then it is elementary to see that f (z) is given by a convergent power series:

00

E ajk(Zl)j(Z2 ) k.

j, k=O

Suppose for simplicity that P = 0. Then f(P) = 0 means that aoo = 0. Hence 00 00 00

f (Z1 Z2) =Z E E ajk(Zj(Z2) + Z2 E aOk(Z2k1)

j=1 k=O k=1

= *i ,g1(zl Z2) + Z2 *g2(Z1, Z2).

This is a natural generalization for the ball in two dimensions of the familiar result in one variable. But what of more general domains? If S is any domain (for simplicity assume that it is in C 2), P E 0, f is holomorphic on 0 and vanishes at P, then we can restrict attention to a small ball centered at P and decompose f as

f(ZlI Z2) = Z1g1(ZlI Z2) + Z2g2(Zl, Z2)

on this small ball. But what about a decomposition on the entire domain U? It turns out that the decomposition can be achieved whenever S is a domain of

holomorphy, but the proof is essentially equivalent to the solution of the Levi




problem. In fact this is one of the unifying features of SCV: the construction of functions (in this case the gj's) is very hard, can usually only be performed on domains of holomorphy, and involves (or in many cases is equivalent to) the solution of the celebrated Levi problem. One of the important features of the recent solution of the inner functions problem is that it gives a very powerful technique for constructing non-trivial holomorphic functions which does not use the solution of the Levi problem. It should prove to be an important tool in years to come.

5. Holomorphic Mappings. A holomorphic mapping of a domain g2 C Cn to a domain S ' c Cm is a function

(Zl * * * Zn) =

((P1(Z1, * Zn) ...... Pm(Z1, ***Zn)),

where each pj is a complex-valued holomorphic function in the usual sense. In this section we will be concerned with biholomorphic mappings: mappings which are holomorphic, one-to-one, onto, and have a holomorphic inverse (this last require- ment is redundant, but we include it for simplicity and clarity). Merely for topological reasons it will be the case for biholomorphic maps that m = n.

When n = 1, biholomorphic mappings are just conformal mappings. And the geometry of conformality is a considerable aid in studying these maps. As soon as n > 2 then a mapping is never conformal (in the sense of preserving angle and length infinitesimally) unless it is rational. Thus our mappings have less classical geometry in them, but they are clearly the right "functors" to use in complex function theory: if there is a biholomorphic map D: 2 -S Q' then, under composi- tion, holomorphic functions on S' may be pulled back to S and vice versa.

The most striking result about conformal mapping in the classical one variable theory is the Riemann mapping theorem: a simply connected proper subdomain of the plane is biholomorphic to the disc. One of the earliest results in several complex variables, due to Poincare, is that the analogous result in C2 is false.

THEOREM 8. The ball B and the bidisc D2(0, 1) in C2 are not biholomorphic to each other.

Proof. We introduce a biholomorphic invariant which distinguishes B from D2. If 92 c C2 is a bounded domain containing a point P, then consider

X( P) G Ec 2

I there is a holomorphic mapping qp: D -S satisfying \ x(~2, P) e c2Iff(p (j) P and (p'(O) =

Here p: D -S is an ordered pair of functions, )p(T) = (qp(lT), p2(0)), and (p'(T) denotes (q4(p'), 2(p))

If D: 92 - Q`2' is a holomorphic map, then the Jacobian matrix

Jac D (Zk

maps X(S, P) to X(S', D(P)) (this is just an exercise with the chain rule). If (D is




biholomorphic, then the map of X(S, P) to X(S', ?D(P)) must be linear, one-to-one and onto (just examine D-1).

Now suppose that 4D is a biholomorphic map of B to D2. By composing 4D with suitable M6bius transformations of the disc, we may suppose that 4D maps 0 to 0. Thus Jac 4) is a linear bijection of X(B, 0) to X(D2, O). We claim that

X(B,O) = B

and

X(D 2,O) .

This would certainly complete the proof. There could be no linear, one-to-one, onto mapping of B to D2 since the ball has smooth boundary while 8D2 has corners.

Verification of the claim amounts to judicious application of the one variable Schwarz's Lemma. For the first assertion, notice that if t E B, then the map

maps D into B and satisfies p(0) = 0 and p'(0) = t. Therefore B c X(B, 0). Conversely, if t e X(B, 0), then let p: D -* B satisfy p(0) = 0 and P'(0) = (. If

p is the complex linear projection of B to {(z1, 0): Iz1 I < 1) and a is any unitary rotation, then p o a o (p maps D to D and takes 0 to 0. Schwarz's Lemma then gives that I (p o a o p)(O) I < 1. Using the chain rule and writing this out gives Ip o a(t)J < 1. Since a was arbitrary, it follows that J<J < 1. Thus X(B,O) C B. So X(B,O) = B.

The proof that X(D2, 0) = D2 is left as an exercise for the reader. U

Poincare's remarkable discovery (obtained, incidentally, by entirely different methods), led to the general question of determining when two domains are biholomorphically equivalent. The theorem shows that even in the topologically trivial case there are subtle obstructions. Poincare initiated a program, in the case of smoothly bounded domains, to calculate differential geometric invariants in the boundaries of domains which would behave canonically under biholomorphic mappings. These could then be used, in principle, to classify domains in C '.

In order for such a program to be successful, one would have to know that biholomorphic mappings extend smoothly to the boundaries of the relevant domains. That such is the case in one complex variable is an old, but still rather difficult, result (see [2]). In several complex variables, theorems of this nature (for several special classes of domains) have only come about in the last fifteen years (see [3], [4]). The proofs of these results use an enormous amount of machinery from geometry, algebra, differential equations, and analysis. It is still an important open question whether biholomorphic maps of arbitrary smoothly bounded domains in C ' extend smoothly to the boundary.

6. Concluding Remarks. In this article I have touched on several important and central themes in the subject of several complex variables: domains of holomorphy,




pseudoconvexity, analytic varieties, removable singularities, inner functions, factorization of holomorphic functions, and biholomorphic mappings. The list of important topics which I have omitted is considerably longer. But my aim has been to give the reader just a taste of the subject.

While I have been careful in this article to attribute most of the theorems presented, I have not attempted to give a complete history of the subject. Many important names (H. Cartan, Grauert, Oka, etc.) have not even been mentioned. The reader who wishes to have more detailed bibliographical information should consult [5], [6], [7], [10], [11], [12], and [15].

By the same token, many significant aspects of SCV, including differential geometry, sheaf theory, commutative ring theory, partial differential equations, and probability, have been ignored in this article. The reader who has been tantalized will find that the references in the last paragraph will provide further information on these topics.

REFERENCES

1. A. B. Aleksandrov, The existence of inner functions in the ball, Math. USSR-Sb., 46 (1983) 143-159.

2. S. Bell and S. Krantz, Smoothness to the boundary of conformal maps, Rocky Mountain J. Math., to appear.

3. S. Bell and E. Likocka, A simplification and extension of Fefferman's theorem on biholomorphic mappings, Invent. Math., 57 (1980) 283-289.

4. C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math., 26 (1974) 1-65.

5. B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables, Translations of Mathematical Monographs, American Mathematical Society, RI, 1963.

6. H. Grauert and K. Fritzsche, Several Complex Variables, Springer-Verlag, Berlin, 1976. 7. R. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall,

Englewood Cliffs, NJ, 1965. 8. M. Hakim and N. Sibony, Fonctions holomorphes bornees sur la boule unite de C'n, Invent. Math.,

67 (1982) 213-222. 9. F. Hartogs, Zur Theorie der analytischen Functionen mehrener unabhangiger Veranderlichen

insbesondere Uber die Darstellung derselben durch Reihen, welche nach Potenzen einer Verander- lichen fortschreiten, Math. Ann., 62 (1906) 1-88.

10. L. Hormander, Introduction to Complex Analysis in Several Variables, North-Holland, Amsterdam, 1973.

11. S. Kobayashi and H. Wu, Complex Differential Geometry, Birkhiiuser, Boston, MA, 1983. 12. S. Krantz, Function Theory of Several Complex Variables, Wiley, New York, 1982. 13. E. Low, A construction of inner functions on the unit ball of CP, Invent. Math., 67 (1982) 223-229. 14. D. E. Marshall, Blaschke products generate H', Bull. Amer. Math. Soc., 82 (1976) 494-496. 15. R. Narasimhan, Several Complex Variables, University of Chicago Press, Chicago, IL, 1971. 16. H. Poincare, Les fonctions analytiques de deux variables et la representation conforme, Rend. Circ.

Mat. Palermo, 23 (1907) 185-220. 17. W. Rudin, Inner functions in the unit ball of C', J. Funct. Anal., 50 (1983) 100-126.



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What is Several Complex Variables?people.aub.edu.lb/~fb31/304/krantz.pdf1987] WHAT IS SEVERAL COMPLEX VARIABLES? 237 quite the opposite: to limit oneself to the study of one complex