Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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complex space

X and a proper, surjective, finite-

fibered holomorphic map

X ! X which is biholo-

morphic outside a nowher e-dense proper analytic

subset. Difficulties can be overcome by simply lifting

functions to this normalization and applying the

Hebbarkeitssatz.

Continuation theorems of Hartogs-type reflect the

fact that complex analysis in dimensions larger than

one is really quite different from the one-variable

version. The following is such a theorem. Let (z, w)be

the standard coordinates in C

and think of the z-axis

as a parameter space for geometric figures in the

w-plane. For example, let D

:= {(z, w):jwj < 1} be

a disk and A

:= {(z, w):1 "<jwj < 1} be an

annulus. An example of a Hartogs figure H in C

is the union of the family of disks D

for jzj < 1  

with the family A

of annuli for 1   jzj < 1.

One should visualize the moving disks which

suddenly change to moving annuli. One speaks of

filling in the Hartogs figure to obtain the polydisk

H := {(z, w); jzj, jwj < 1}. Hartogs’ continuation the-

orem states that a function which is holomorphic

on H extends holomorphically to

Cartan–Thullen Theorem

One of the major developments in complex analysis

in several variable s was the realization that certain

convexity concepts lie behind the strong continua-

tion properties. At the analytic level one such is

defined as follows by the full algebra of holo-

morphic functions O(D). If K is a compact subset of

D, then its holomorphic convex hull

K is defined as

the intersection of the sets P(f):= {p 2 D : jf (p)j

jf j

}asf runs through O(D). One says that D is

holomorphically convex if

K is compact for every

compact subset K of D.

The theorem of H. Cartan and Thullen relates this

concept to analytic continuation phenomena as

follows. A domain D is said to be a domain of

holomorphy if, given a divergent sequence {z

}  D,

there exists f 2O(D) which is unbounded along it.

In other words, the phenomenon of being able to

extend all holomorphic functions on D to a truly

larger domain

D does not occur. The Cartan–

Thullen theorem states that D is a domain of

holomorphy if and only if it is holomorphically

convex. In the next paragraph the relation between

this type of convexity and a certain complex

geometric convexity of the boundary bd(D)willbe

indicated.

Levi Theorem and the Levi Problem

Consider a smooth (local) real hypersurface 

containing 0 2 C

with n > 1. It is the zero-set

{ = 0} in some neighborhood U of 0 of a smooth

function with d 6¼ 0onU. This is viewed as a piece

of a boundary of a domain D, where U \ D = {<0}.

The real tangent space T

=Ker(d(0)) contains a

unique maximal (one-codimensional) complex sub-

space T

=Ker(@(0)) = H. The signature of the

restriction of the complex Hessian (or Levi form) i@



@

to H is a biholomorphic invariant of . In this

notation the Hessian is a real alternating 2-form

which is compatible with the complex structure, and

its signature is defined to be the signature of the

associated symmetric form.

If the restriction of this Levi form to the complex

tangent space has a negative eigenvalue, that is, if

the boundary bd(D) has a certain degree of

concavity, then there is a map F :  ! U of the

unit disk  which is biholomorphic onto its image

with F(0) = 0 and otherwise F(cl())  D. The

reader can imagine pushing the image of this map

into the domain to obtain a family of disks which

are in the domain, and pushing it in the outward

pointing direction to obtain annuli which are also in

the dom ain. Making this precise, one builds a

(higher-dimensional) Hartogs figure H at the base

point 0 so that

H is an open neighborhood of 0. In

particular this proves the theorem of E. E. Levi:

every function holomorphic on U \ D extends to a

neighborhood of 0. This can be globally formulated

as follows:

Theorem If D is a domain of holomorphy with

smooth boundary in C

, then bd(D) is Levi-

pseudoconvex.

Here the terminology Levi-pseudoconvex is used to

denote the condition that the restriction of the Levi

form to the complex tangent space of every

boundary point is positive semidefinite.

One of the guiding problems of complex analysis

in higher dimensions is the Levi problem. This is the

converse statement to that of the Levi’s theorem:

Levi Problem Is a domain D with smooth Levi-

pseudoconvex boundary in a complex manifold

necessarily a domain of holomorphy?

Stated in this form it is not true, but for domains

in C

it is true. As will be sketched below, under

stronger assumptions on the Levi form it is almost

true. However, there are still interesting open

problems in complex analysis which are related to

the Levi problem.

Bounded Domains and Their Automorphisms

The unit disk in the complex plane is particularly

important, because, with the exceptions of projective

542 Several Complex Variables: Basic Geometric Theory

space P

(C), the complex plane C,thepunctured

plane Cn{0}, and compact complex tori, it is the

universal cover of every (connected) one-dimensional

complex manifold.

In higher dimensions it should first be underlined

that, without some further condition, there is no

best bounded domain in C

. For example, two

randomly chosen small perturbations of the unit ball

:= {(z, w); jzj

þjwj

< 1}, with, for example, real

analytic boundary, are not biholomorphically

equivalent.

On the other hand, the follow ing theorem of

H. Cartan shows that bounded domains D are good

candidates for covering spaces :

Theorem Equipped with the compact open topol-

ogy, the group Aut( D) of holomorphic automorph-

isms of D is a Lie group acting properly on D.

The notion of a proper group action of a

topological group on a topological space is funda-

mental and should be underlined. It means that if

} is a convergent sequence in the space where the

group is acting, then a sequence of group elements

}, with the property that {g

)} is convergent,

itself possesses a convergent subsequence. As a

consequence, isotropy groups are compact and

orbits are closed.

In the con text of bounded domains D this implies

that if  is a discrete subgroup of Aut(D), then

X = D= carries a natural structure of a complex

space. If in addition  is acting freely, something

that, with minor modifications, can be arrange d,

then X is a complex manifold.

Many nontrivial compact complex manifolds arise

as quotients D= of bounded domains. Even very

concrete quotients, for example, where D = B

, are

extremely interesting. Conversely, if Aut(D) contains

a discrete subgroup  so that D= is compact, then

D is probably very special. For example, it is known

to be holomorphically convex!

Any compac t quotient X = D= of a bounded

domain is projective algebraic in the sense that it can

be realized as a complex (algebraic) submanifold of

some complex projective space. In fact the embed-

ding can be given by quite special -invariant

holomorphic tensors on D, and this in turn implies

that X is of general type (see below). For further

details, in particular on Cartan’s theorem on the

automorphism group of a bounded domain, the

reader is referred to Narasimhan (1971).

Stein Manifolds

The founding fathers of the first phase of ‘‘modern

complex analysis’’ (Cartan, Oka, and Thullen)

realized that domains of holomorphy form the

basic class of spaces where it would be possible to

solve the important problems of the subject con-

cerning the existence of holomorphic or mero-

morphic functions with reasonably prescribed

properties. In fact, Oka formulated a principle

which more or less states that if a complex analytic

problem which is well formulated on a domain of

holomorphy has a continuous solution, then it

should have a holomorphic solution. Given the

flexibility of continuous functions and the rigidity

of holomorphic functions, this would seem impos-

sible but in fact is true!

Beginning in the late 1930s, Stein worked on

problems related to this Oka principle, in particular

on those related to what we would now call the

algebraic topological aspects of the subject, and he

was led to formulate conditions on a general

complex manifold X which sho uld hold if problems

of the above type are to be solved. First, his axiom

of holomorphic convexity was simply that, given a

divergent sequence {x

}inX, there should be a

function f 2O(X) such that {f (x

)} is unbounded.

Secondly, holomorphic functions should separate

points in the sense that, given distinct points x

, x

X, there exists f 2O(X) with f (x

) 6¼ f (x

). Finally,

globally defined holomorphic functions should give

local coordinates. Assuming that X is n-dimensional,

this means that, given a point x 2 X, there exist

, ..., f

2O(X) such that df

(x) ^^df

(x) 6¼ 0.

Assuming Stein’s axioms, Cartan and Serre then

produced a powerful theory in the context of sheaf

cohomology which proved certain vanishing theo-

rems that led to the desired existence theorems. This

theory and typical applications are sketched below.

Before going into this, we would like to mention

that Grauert’s version of the Cartan–Serre theory

requires only very weak versions of Stein’s axioms:

(1) The connected component containing K of the

holomorphic convex hull

K of every compac t set

should be compact. (2) Given x 2 X, there are

functions f

, ..., f

2O(X) so that x is an isolated

point in the fiber of the map F := (f

, ..., f

):X !

. Of course the results also hold for complex

spaces.

Holomorphically convex dom ains in C

are Stein

manifolds, and since closed complex manifolds of

Stein manifolds are Stein, it follows that any

complex submanifold of C

is Stein. In particular,

affine varieties are Stein spaces. Remmert’s theorem

states the converse: an n-dimensional Stein manifold

can be embedded as a closed complex submanifold

of C

2nþ1

. A nontrivial result of Behnke and Stein

implies that every noncompact Riemann surface is

also Stein.

Several Complex Variables: Basic Geometric Theory 543

Basic Formalism

The following first Cousin problem is typical of those

which can be solved by Stein theory. Let X be a

complex manifold which is covered by open sets U

Suppose that on each such set a meromorphic function

is given so that on the overlap U

:= U

\ U

the

difference m

= m

 m

=: f

is holomorphic. This

means that the distribution of polar parts of these

functions is well defined. The question is whether or

not there exists a globally defined meromorphic

function m 2M(X) with these prescribed polar

parts, that is, with m  m

=: f

2O(U

If one applies the Oka principle, this problem can

be easily solved. For this one can assume that the

covering is locally finite and take 

to be a partition

of unity subordinate to the cover. Using standard

shrinking and cut-off arguments, one extends the f

to the full space X as smooth functions so that the

alternating cocycle relations f

þ f

= 0 and

= f

still hold. Then f



is a smooth

function on U

which satisfies f

 f

= f

on the

overlap U

. It follows that f := m

þ f

= m

þ f

is a

globally well-defined ‘‘smooth’’ function with the

prescribed polar parts. The Oka principle would

then imply that there is a globally defined mero-

morphic function with the same property.

The basic sheaf cohomological formalism for

Stein theory can be seen in the above argument.

Suppose that instead of applying extension and cut-

off techniques from the smooth category, we could

answer positively the question ‘‘given the holo-

morphic functions {f

} on the U

, do there exist

holomorphic functions {f

} on the U

such that f



= f

on the U

?’’ Then we would immediately have

the desired globally defined meromorphi c function

m := m

þ f

. This question is exactly the question of

whether or not the Cech coho mology class of the

alternating cocycle {f

} vanishes.

Let us quickly summarize the language of Cech

cohomology. A presheaf of abelian groups is a

mapping U ! S( U) which associates to every open

subset of X an abelian group. Typical examples are

U !O(U), U ! C

(U), U ! H



(U, Z), ... . The

last example which associates to U its topological

cohomology does not localize well in terms of

following the basic axioms for a sheaf: (1) Given a

covering {U

} of an open subset U of X and elements

2 S(U

) with s

 s

= 0onU

, there exists s 2 S(U)

with sjU

= s

. (2) If s, t 2 S(U) are such that

sjU

= tjU

for all i, then s = t. For this we have

assumed that the restriction mappings have been

built into the definition of a presheaf.

Associated to a sheaf S on X and a covering

U = { U

} is the space of alternating q-cocycles

(U, S), which is the set of alternating maps 

from the set of (q þ 1)-fold indices of the form

, ..., i

) 7!s

,..., i

2 S(U

,..., i

). Here U

,..., i

:= U

\\U

. The boundary mapping  : C

qþ1

defined by ()

,...i

qþ1

(1)

,..., i

k1

, i

kþ1

,..., i

qþ1

.It

follows that 

= 0, and H



(U, S) is defined to be the

cohomology of the associated complex.

In any consideration it is necessary to refine

coverings, shrink, etc., and therefore one goes to

the limit H



(X, S) over all refinements of the

coverings. The script notation S is used to denote

that we have then localized the sheaf to the germ

level. Due to a theorem of Leray one can, however,

always take a suitable covering so that

(U, S) = H

(X, S) for all q, where now S(U)

satisfies the above axioms.

One of the important facts in this cohomology

theory is that a short exact sequence of sheave s 0 !

!S!S

! 0 yields a long exact sequence

0 ! H

ðX; S

Þ!H

ðX; SÞ ! H

ðX; S

! H

ðX; S

Þ!H

ðX; SÞ ! H

ðX; S

Þ!

in cohomology.

A fundamental theorem of Stein theory, Theorem

B, states that for the basic analytic sheaves S of

complex analysis, the so-called coherent sheaves, all

cohomology spaces H

(X, S) vanish for all q  1. In

the above example of the first Cousin problem the

desired vanishing is that of H

(U, O).

Coherent Sheaves

Numerous important sheaves in co mplex analysis

are associated to vector bundles on complex mani-

folds. A holomorphic vector bundle  : E ! X ov er a

complex manifold is a holomorphic surjective

maximal rank fibration. Every fiber E

:= 

1

(x)is

a complex vector space, and the vector space

structure is defined holomorphically over X. For

example, addition is a holomorphic map E 

E ! E.

Such bundles are locally trivial, that is, there is a

covering {U

} of the base such that 

1

)is

isomorphic to U

 C

andontheoverlapthegluing

maps in the fibers are holomorphic maps ’

: U

(C). The number r is called the rank of the

bundle. Holomorphic bundles of rank 1 are referred to

as holomorphic line bundles. Of course all of these

definitions make sense in other categories, for exam-

ple, topological, smooth, real analytic, etc.

A holomorphic section of E over an open set U

is a holomorphic map s : U ! E with   s = Id

The space of these sections is denoted by E(U), and

the map U !E(U) defines a sheaf which is locally

just O

. It is therefore called a locally free sheaf of

544 Several Complex Variables: Basic Geometric Theory

O-modules. Conversely, by taking bases of a locally

free sheaf S on the open sets where it is isomorphic

to a direct sum O

, one builds an associated

holomorphic vector bundle E so that E = S.

It is not possible to restrict our attention to these

locally free sheaves or equivalently to holomorphic

vector bundles. One important reason is that images

of holomorphic vector bundle maps are not necessa-

rily vector bundl es. A related reason is that the sheaf

of ideals of holomorphic functions which vanish on

a given analytic set A is not always a vector bundle.

This is caused by the presence of singularities in A.

There are many other reasons, but these should

suffice for this sketch.

The sheaves S that arise naturall y in complex

analysis are almost vector bundles. If X is the base

complex manifold or complex space under consid-

eration, then S will come from a vector bundle on

some big open subset X

whose boundary is an

analytic set X

, and then on the irreducible

components of X

it will come from vect or bundles

on such big open sets, etc. These sheaves are called

coherent analytic sheaves of O

-modules. The

correct algebraic definition is that locally there

exists an exact sequence

0 !O

!!O

!S!0 ½1

of sheaves of O-modules. This implies in particular

that, although S might not be locally free, it is

locally finitely generated, and the relations among

the generators are also finitely generated.

Selected Theorems

The following efficiently formulated fundamental

theorem contains a great deal of information about

Stein manifolds.

Theorem B A complex space X is Stein if and only

if for every coherent sheaf S of O

-modules

(X, S) = 0 for all q  1.

Since S is a sheaf, it follows that H

(X, S) = S(X).

This is referred to as the space of sections of S over

X. As a result of Theorem B, we are able to

construct sections with prescribed properties. Let us

give two concrete applications (there are many

more!).

Example Let A be a closed analytic subset of a

Stein space X, and let I denote the subsheaf of O

which consists of those functions which vanish on A.

Note that this must be defined for every open subset

U of X . Then we have the short exact sequence 0 !

I!O

=I!0. The restriction of O

=I to

A is called the (reduced) structure sheaf O

of A.In

other words, for U open in A the space O

(U)

should be regarded as the space of holomorphic

functions on U.

Now, I is a coherent sheaf on X and therefore by

Theorem B the cohomology group H

(X, I)vanishes.

Consequently, the associated long exact sequence in

cohomology implies that the restriction mapping

(X) !O

(A) is surjective. This special case of

Theorem A means that every (global!) holomorphic

function on A is the restriction of a holomorphic

function on X. ^

Example Let us consider the multiplicative (second)

Cousin problem. In this case meromorphic functions

are given on the open subsets U

of a covering U

with the property that m

= f

,wheref

is holo-

morphic and nowhere vanishing on the overlap U

This is a distribution D of the zero and polar parts of

meromorphic functions, which in complex geometry is

called a divisor, and the interesting question is whether

or not there exists a globally defined meromorphic

function which has D as its divisor.

Now we note that GL



and thus

: U

! C



defines a line bundle L on X and we

regard it as an element of t he space H

(X, O



)of

equivalence classes of line bundles on X.HereO



is the sheaf of nowhere-vanishing holomorphic

functions on X. It is not even a sheaf of O-modules;

therefore coherence is not discussed in this case.

The long exact sequence in cohomology associated

to the short exact sequence 0 ! Z ! O !

exp



! 1

yields an element c

(L) 2 H

(X, Z), which is a purely

topological invariant. It is called the Chern class of L,

and one knows that L is topologically trivial if and

only if c

(L) = 0.

Coming back to the Cousin II pro blem, using the

same argument as in the Cousin I problem, we can

solve it if and only if we can find nowher e-vanishing

functions f



) with f

= f

. This is equivalent

to finding a nowh ere-vanishing section of L. But a

line bundle has a nowhere-vanishing section if and

only if it is isomorphic to the trivial bundle. In other

words, the Cousin II problem can be solved for a

given divisor D if and only if the associated line

bundle L(D) is trivial in H

(X, O



). For this, a

necessary condition is that the Chern class c

(L(D))

vanishes. But if X is Stein, this is also sufficient,

because the vanishing of H

(X, O) together with the

long exact sequence in cohomology shows that

(X, O



) !

(X, Z) is injective.

Hence, in this case we have the following precise

formulation of the Oka principle: ‘‘A given divisor

D on a Stein manifold is the divisor of a globally

defined meromorphic function if and only if the

associated line bundle is topologically trivial.’’ ^

Several Complex Variables: Basic Geometric Theory 545

A slightly refined statement from that above is the

fact that on a Stein manifold the space of topologi-

cal line bundles is the same as the space of

holomorphic line bundles. In the case of (higher

rank) vector bundles this is a deep and important

theorem of Grauert. It can be formulated as follows.

Grauert’s Oka principle On a Stein space the map

F : Vect

holo

(X) ! Vect

top

(X) from the space of holo-

morphic vector bundles to the space of topological

vector bundles which forgets the complex structure

is bijective.

In closing this section, a few words concerning the

proofs of the major theorems, for example, Theorem B,

should be mentioned. In all cases one must solve

something like an additive Cousin problem and one

first does this on special relatively compact subsets. For

this step there are at least two different ways to

proceed. One is to delicately piece together solutions

which are known to exist on very special polyhedral-

type domains or build up from lower-dimensional

pieces of such.

Another method is to solve certain systems of PDEs

on relatively compact domains where control at the

boundary is given by the positivity of the Levi-form.

An example of how such PDEs occur can already be

seen at the level of the above Cousin I problem. At the

point where we have solved it topologically, that is, the

holomorphic cocycle {f

} is a coboundary f

= f

 f

smooth functions, we observe that since



= 0, it

follows that  =



is a globally defined (0, 1)-form. It



@-closed, that is, the compatibility condition for

solving the system



@u =  is fulfilled. If this system can

be solved, then we use the solution u to adjust the

topological solutions of the Cousin problem by

replacing f

by f

 u.Westillhavef

= f

 f

,but

now the f

are holomorphic on U

To obtain the global solution to a Cousin-type

problem, one exhausts the Stein space by the special

relatively compact subsets U

where, by one method

or another, we have solved the problem with

solutions s

. One would like to say that the s

converge to a global solution s. However, there is no

way to a priori guarantee this without making some

sort of estimates. One main way of handling this

problem is to adjust the solutions as n !1by an

approximation procedure. For this one needs to

know that holomorphic objects, for example, func-

tions on U

, can be approximated on U

by objects

of the same type which are defined on the bigger set

nþ1

. This Runge-type theorem, which is a non-

trivial ingredient in the whole theory, requires the

introduction of an appropriate Fre´chet structure on

the spaces of sections of a coherent sheaf. This is in

itself a point that needs some attention.

Montel’s Theorem and Fredholm

Mappings

If U is an open subset of a complex space X, then

O(U) has the Fre´chet topology of convergence on

compact subsets K defined by the seminorms jj

Using resolutions of type (1) above, one shows that

the space of sections S(U) of every coherent sheaf S

also possesses a canon ical Fre´chet topology. This is

then extended to the spaces C

(U, S), and conse-

quently one is able to equip the cohomology spaces

(X, S) with (often non-Hausdorff) quotient

topology.

Elements of such cohomology groups can be

regarded as obstructions to solving complex analytic

problems. One ofte n expects such obstructions, and

is satisfied whenever it can be shown if there are

only finitely many, that is, a finiteness theorem of

the type dim H

(X, S) < 1 is desirable. Here we

sketch two finiteness theorems which hold in

seemingly different contexts, but their proofs are

based on one principle: use the compactness

guaranteed by Montel’s theorem as the necessary

input for the Fredh olm theorem in the context of

Fre´chet spaces.

Recall that a continuous linear map T : E ! F

between topological vector spaces is said to be

compact if there is an open neighborhood U of 0 2 E

such that T(U) is relatively compact in F.IfY is a

relatively compact open subset of a complex space

X, then Montel’s theorem states that the restriction

map r

: O(X) !O(Y) is compact. This can be

extended to coherent sheaves, and using the Fred-

holm theorem for certain natural restriction and

boundary maps, one proves the following funda-

mental fact.

Lemma 1 If the restriction map r

: H

(X, S) !

(Y, S) is surjective, then H

(Y, S) is finite

dimensional.

Since the methods for the proof are basic in complex

analysis, we outline it here. Take a covering

U of X

such that H

(U, S) = H

(X, S). Then intersect its

elements with Y to obtain a covering

U of Y. Finally,

refine that covering with refinement mapping  to a

covering V of Y such that H

(V, S) = H

(X, S)andso

that U

contains V

(i)

as a relatively compact subset

for all i.LetZ

(U, S) denote the kernel of

the boundary map  for the covering U, and consider

the map Z

(U, S) C

q1

(V, S) !C

(V, S)whichis

the direct sum    of the restriction and boundary

maps. By assumption it is surjective. Since  is the

difference of this map and the compact map ,

L Schwartz’s version of the Fredholm theorem for

Fre´chet spaces implies that its image is of finite

546 Several Complex Variables: Basic Geometric Theory

codimension, that is, H

(Y, S) = H

(V, S) is finite

dimensional.

Applying this Lemma in the case of compact

spaces where X = Y, one has the following theorem

of Cartan and Serre:

Theorem If X is a compact complex space and S is

a coherent sheaf on X, then dim H

(X, S) < 1 for

all q.

Grauert made use of this technique in solving the

Levi problem for a strong ly pseudoconvex relatively

compact domain D with smooth boundary in a

complex manifold X. Here strongly pseudoconvex

means that the restriction of the Levi form to the

complex tangent space of every boundary point is

positive definite. To do this he sequentially made

‘‘bumps’’ at boundary points to obtain a finite

sequence of domains D = D

 D

D

such a way that the restriction mappings at the

level of qth cohomology, q  1, are all surjective

and such that at the last step D is relatively

compact in D

. Applying the above Lemma,

dim H

(D, S) < 1. Using another bumping proce-

dure, it then follows that D is holomorphically

convex and, in fact, that D is almost Stein.

This last statement means that one can guarantee

that O(D) separates points outside of some compact

subset which could contain compact subvarietes on

which the global holomorphic functions are constant.

In this situation one can apply Remmert’s reduction

theorem which implies that there is a canonically

defined proper surjective holomorphic map  : D ! Z

to a Stein space which is biholomorphic outside of

finitely many fibers. One says that, in order to obtain

the Stein space Z, finitely many compact analytic

subsets must be blown down to points.

The above mentioned reduction theorem is a

general result which applies to any holomorphically

convex complex space X. For this one observes that

if X is holomorphically convex, then for x 2 X the

level set L(x):= {y 2 X; f(y) = f (x) for all f 2O(X)}

is a compact a nalytic subset of X. One then defines

an equivalence relation: x  y if and only if the

connected component of L(x) containing x and that

of L(y) which contains y are the same. One then

equips X= with the quotient topology and proves

that the canonical quotient  : X ! X=

=: Z is

proper. Finally, for U open in Z one defines

(U) = O

(

1

(U)) and proves that, equipped

with this structure, Z is a Stein space. This Remmert

reduction is universal with respect to holomorphic

maps to holomorphically separable complex spaces,

that is, if ’ : X ! Y and O

(Y) separates the points

of Y, then there exists a uniquely defined holo-

morphic map 

’

: Z ! Y so that 

’

  = ’ . It should

be noted that, even if the original space X is a

complex manifold, the associated Stein space Z may

be singular. This reflects the fact that it is difficult to

avoid singularities in complex geometry.

Mapping Theory

Above we have attempted to make it clear that

holomorphic maps play a central role in complex

geometry. It is even important to regard a holo-

morphic function as a map. Here we outline the

basic background necessary for dealing with maps

and then stat e three basic theorems which involve

proper holomorphic mappings.

Basic Facts

A holomorphic map F : X ! Y between (reduced)

complex spaces is a continuous map which can be

represented locally as a holomorphic map between

analytic subsets of the spaces in which X and Y are

locally embedded. In other words, F is the restriction

of a map F = (f

, ..., f

) which is defined by

holomorphic functions.

If X is irreducible and X and Y are one-

dimensional, then a nonconstant holomorphic map

F : X ! Y is an open mapping. This statement is far

from being true in the higher-dimensional setting.

The reader need only consider the example

F : C

! C

,(z, w) ! (zw, z).

Despite the fact that holomorphic maps can be

quite complicated, they have properties that in

certain respects render them tenable. Let us sketch

these in the case where X is irreducible. First, one

notes that every fiber F

1

(y) is a closed analytic

subset of X. One defines rank

F to be the codimen-

sion at x of the fiber F

1

(F (x)) at x. Then

rank F := max {rank

F; x 2 X}. It then can be

shown that {x 2 X; rank

F  k} is a closed analytic

subset of X for every k. Applying this for

k = rank F  1 we see that, outside a proper closed

analytic subset, F has constant maximal rank.

If F : X ! Y has constant rank k in a neighbor-

hood of some point x 2 X, then one can choose

neighborhoods U of x in X and V of F(x)inY so

that FjU maps U onto a closed analytic subset of Y.

By restricting F to the sets where it has lower rank

and applying this local-image theorem, it follows

that the local images of the set where F has lower

rank are at least two dimensions smaller than those

of top rank. Conversely, the fiber dimension

(x):= dim

1

(F(x)) is semicontinuous in the

sense that d

(x)  d

(z) for all z near x. Finally, we

note that if Y is m-dimensional, then F : X ! Y is an

open map if and only if it is of constant rank m.

Several Complex Variables: Basic Geometric Theory 547

Proper Mappings

By definition a mapping F : X ! Y between topolo-

gical spaces is proper if and only if the inverse image

1

(K) of an arbitrary compact subset in Y is

compact in X. This is a more delicate condition

than meets the eye. For example, if F : X ! Y is a

proper map and one removes one point from some

fiber, then it is normally no longer proper! On the

other hand, the restriction of a proper map to a

closed subset is still proper.

Remmert’s ‘‘Proper mapping theorem’’ is the first

basic theorem on proper holomo rphic maps:

Theorem The im age of a proper holomorphic map

F : X ! Y is a closed analytic subset of Y.

Given another basic theorem of complex analysis,

the reader can imagine how this might be proved.

This is the continuation theorem for analyt ic sets

due to Remmert and Stein:

If X is a complex space and Y is a closed analytic

subset with dim

Y  k for all y 2 Y and Z is a closed

analytic subset of the complement XnY with dim

Z 

k þ 1 at all z 2 Z, then the topological closure cl(Z) of

Z in X is a closed analytic subset of X with E = cl(Z)n

Z = cl(Z) \ Y a proper analytic subset of cl(Z).

Similar results hold for more general complex

analytic objects. For example, closed positive cur-

rents with (locally) finite volume can be continued

across any proper analytic subset (Skoda 1982). A

sketch of the proof of the proper mapping theorem

(for X irreducible) goes as follows. From the

assumption that F is proper, the image F(X)

is closed. If F has constant rank k, then, by the

local result stated above, its image is everywhere

locally a k-dimensional analytic set. Since the image

is closed, the desired result follows. If rank F = k

and E := {x 2 X; rank

F < k} 6¼;, then by induction

F(E ) is a closed analytic subset of dimension at

most k  2. Let A := F

1

(E) and apply the

previous discussion for constant rank maps to

Fj(XnA): XnA ! YnE. The image is a closed

k-dimensional analytic subset of YnE and its

Remmert–Stein extension is the full image F(X).

In this framework the Stein factorization theorem

is an important tool. Here F : X ! Y is again a

proper holomorphic map which we may now

assume to be surjective. Analogous to the construc-

tion of the reduction of a holomorphically convex

space, one says that two points in X are equivalent

if they are in the same connected component of an

F-fiber. This is indeed an equivalence relation, and

the quotient Z := X= is a complex space equipped

with the direct image sheaf. Thus one decomposes F

into two maps X ! Z ! Y, where X ! Z is a

canonically associated surjective map with con-

nected fiber, and Z ! Y is a finite map.

This geometric proper mapping theorem is a preview

of one of the deepest results in complex analysis:

Grauert’s direct image theorem. This concerns the

images of sheaves, not just the images of points. For this,

given a sheaf S on X one defines the qth direct image

sheaf on Y as the sheaf associated to the presheaf which

attaches to an open set U in Y the cohomology space

1

(U), S). Grauert’s ‘‘Bildgarbensatz’’ states the

following: ‘‘If F : X ! Y is a proper holomorphic map,

then all direct image sheaves of any coherent sheaf on X

are coherent on Y.’ ’

Complex Analysis and Algebraic

Geometry

The interplay between these sub jects has motivated

research and produced deep results on both sides.

Here we indicate just a few results of the type which

show that objects which are a priori of an analytic

nature are in fact algebraic geometric.

Projective Varieties

Let us begin with the algebraic geometric side of the

picture where we consider algebraic subvarieties X of

projective space P

(C). If [z

: z

:  : z

]arehomo-

geneous coordinates of P

, such a variety is the

simultaneous zero-set, X := V(P

, ..., P

), of finitely

many (holomorphic) homogeneous polynomials

= P

, ..., z

). Chow’s theorem states that in this

context there are no further analytic phenomena:

Theorem Closed complex analytic subsets of pro-

jective space P

This observation has numerous consequences. For

example, if F : X ! Y is a holomorphic map between

algebraic varieties, then, by applying Chow’s theorem

to its graph, it follows that F is algebraic.

Chow’s theorem can be proved via an application

of the Remmert–Stein theorem in a very simple

situation. For this, let  : C

nþ1

n{0} ! P

standard projection, and let Z := 

1

(X). Since Z is

positive dimensional, by the Remmert–Stein theorem it

can be extended to an analytic subset of C

nþ1

.The

resulting subvariety K(X) (the cone over X)isinvariant

by the C



-action which is defined by v ! v for  2



.Iff is a holomorphic function on C

nþ1

which

vanishes on K(X), then we develop it in homogeneous

polynomials f =

and note that

(f )(z) = f (z ) =



also vanishes for all .

Hence, all P

vanish identically and therefore the

ideal of holomorphic functions which vanish on K(X)

548 Several Complex Variables: Basic Geometric Theory

is generated by the homogeneous polynomials which

vanish on K(X) and consequently finitely many of

these define X as a subvariety of P

(C).

Complements of subvarieties in projective varieties

occur in numerous applications and are important

objects in complex geometry. Even complements P

of subvarieties Y in the full projective space are not

well understood. If Y is the intersection of a compact

projective variety X with a projective hyperplane, that

is, Y is a hyperplane section, then XnY is affine. If Y is

q-codimensional in X, then XnY possesses a certain

degree of Levi convexity and general theorems of

Andreotti and Grauert (1962) on the finiteness and

vanishing of cohomology indeed apply. However, not

nearly as much is understood in this case as in the case

of a hyperplane section.

Kodaira Embedding Theorem

Given that analytic subvarieties of projective space

are algebraic, one would like to understand whether

a given compact complex manifold or complex

space can be realized as such a subvariety. Kodaira’s

theorem is a prototype of such an embedding

theorem. Most often one formulates projective

embedding theorems in the language of bundles.

For this, observe that if L ! X is a holomorphic

line bundle over a compact complex manifold, then

its space (X, L) of holom orphic sections is a finite-

dimensional vector space V. The zero-set of a section

s 2 V is a one-codimensional subvariety of X.

Let us restrict our attention to bundles which are

generated by their sections which for line bundles

simply means that for every x 2 X there is some

section s 2 V with s(x) 6¼ 0. It then follows that for

every x 2 X the space H

:= {x 2 X; s(x) = 0} is a

one-codimensional vector subspace of V.ThusL

defines a holomorphic map ’

: X ! P(V



), x 7!H

Note that we must go to the projective space P(V



because a linear function defining such an H

is only

unique up to a complex multiple.

Projective embedding theorems state that under

certain conditions on L the map ’

is a holomorphic

embedding, that is, it is injective and is everywhere

of maximal rank in the analytic sense that its

differential has maximal rank. Here we outline a

complex analytic approach of Grauert for proving

embedding theorems. It makes strong use of the

complex geometry of bundle spaces.

Let L ! X be a holomorphic line bundle over a

compact complex manifold. A Hermitian bundle metric

is a smoothly varying metric h in the fibers of L.This

defines a norm function v 7!jvj

:= h(v, v)onthe

bundle space L. One says that L is positive if the tubular

neighborhood T := {v 2 L; jvj

< 1} is strongly

pseudoconcave, that is, when regarded from outside

T, its boundary is strongly pseudoconvex.

To prove an embedding theorem, one must

produce sections with prescribed properties. Sections

of powers L

are closely related to holomorphic

functions on the dual bundle space L



. This is due to

the fact that if  : L ! X is the bundle projection,



1



) ﬃ U



 C is a local trivialization, and z



a fiber coordinate, then a holomorphic function f on



has a Taylor series development

f ðvÞ=



ðnÞððvÞÞz



ðvÞ

The function f is well defined on L. Hence, the

transformation law for the z



must be canceled out

by a transformation law for the coefficient functions



(n). This implies that the s



(n) are sections of L

Hence, proving the existence of sections in the

powers of L with prescribed properties amounts to

the same thing as proving the existence of holo-

morphic funtions on L



with analogous properties.

The positivity assumption on L is equivalent to

assuming that the tubular neighborhoods of the zero-

section in L



defined by the norm function associated

to the dual metric are strongly pseudoconvex. The

solution to the Levi problem, which was sketched

above, then shows that L



is holomorphically convex,

and its Remmert reduction is achieved by simply

blowing down its zero-section. In other words, L



essentially a Stein manifold, and using Stein theory, it

is possible to produce enough holomorphic functions

on L to show that some power L

defines a

holomorphic embedding ’

: X ! P((X, L

)



Bundles with this property are said to be ample, and

thus we have outlined the following fact: ‘‘a line

bundle which is Grauert-positive is ample.’’

It should be underlined that we defined the Chern

class of L as the image in H

(X, Z) of its equivalence

class in H

(X, O



), that is, in this formulation the

Chern class is a Cech cohomology class. It is, however,

often more useful to consider it as a deRham class

where it lies in the (1, 1)-part of H

deR

(X, C). If h is a

bundle metric as above, then the Levi form of the norm

function is a representative c

(L, h) of the Chern

class of L



.Thusc

(L, h) is an integral (1, 1)-form

which represents c

(L). It is called the Chern form of L

associated to the metric h. The following is Kodaira’s

formulation of his embedding theorem:

Theorem A line bundle L is ample if and only if it

possesses a metric h so that c

(L, h) is positive definite.

Kodaira’s proof of this fact follows from his

vanishing theorem (see Several Complex Variables:

Compact Manifolds) in the same way the example

of Theorem A was derived from Theorem B in the

Several Complex Variables: Basic Geometric Theory 549

first example in the subsection ‘‘Selected theorems.’’

That an ample bundle is positive follows immedi-

ately from the fact that if ’

is an embedding, then

its pullback of the (positive) hyperplane bundle on

projective space agrees with L

Finally, one asks the question ‘‘under what natural

conditions can one construct a bundle L which is

positive?’’ The following is an example of an answer

which is related to geometric quantization.

Suppose that X is a compact complex manifold

equipped with a symplectic structure !, that is, ! is

a d-closed, nondegenerate 2-form. One says that ! is

Ka¨ hlerian if it is compat ible with the complex

structure J in the sense that !(Jv, Jw) = !(v, w ) and

!(Jv, v) > 0 for every v and w in every tangent space

of X. Note that if L is a positive line bundle, then it

possesses a Hermitian metric h such that ! = c

(L, h)

is a Ka¨ hlerian structure on X.

It should be underlined that there are Ka¨hler

manifolds without positive bundles, for example,

every compact complex torus T = C

= possesses

the Ka¨hlerian structure which comes from the

standard linear structure on C

. However, for n > 1

most such tori are not projective algebraic and

therefore do not have positive bundles.

If, on the other hand, the Ka¨hlerian structure is

integral, a condition that is automatic for the Chern

form c

(L, h) of a bundle, then there is indeed a line

bundle L ! X equipped with a Hermitian metric h

such that c

(L, h) = !. The condition of integrality can

be formulated in terms of the integrals of ! over

homology classes being integral or that its deRham

class is in the image of the deRham isomorphism from

the Cech cohomology H

(X, Z)  C to H

de R

(X, C).

Coupling this with the embedding theorem for positive

bundles, we have the following theorem of Kodaira:

Theorem If (X, !) is Ka¨hlerian and ! is integral,

then X is projective algebraic.

This result has been refined in the following

important way (a conjecture of Grauert and

Riemenschneider proved with different methods by

Siu (1984) and by Demailly (1985)): the same result

holds if ! is only assumed to be semipositive and

positive in at least one point.

For Grauert’s proof of the Kodaira embedding

theorem and a number of other important and

beautiful results, we recommend the original paper

(Grauert 1962).

Quotients of Bounded Domains

Let D be a bounded domain in C

and  be a discrete

subgroup of Aut(D) which is acting freely on D with a

compact quotient X := D=. For  2  let J(, z)bethe

determinant of the Jacobian d=dz and, given a

holomorphic function f, consider (at least formally)

the Poincare´series

f ((z))J(, z)

of weight k.Iff is

bounded and k  2, then this series converges to a

holomorphic function P(f )onD which satisfies the

transformation rule P(f )((z)) = J(, z)

k

P(f )(z).

Now the differential volume form  := dz

^^

transforms in the opposite way (for k = 1).

Therefore s(f ) = P(f )()

is a -invariant section of

the kth power of the determinant bundle

K :=



D of the holomorphic cotangent bundle

of D. In other words, s(f ) 2 (X, K

). Since the

choice of f may be varied to show that there are

sufficiently many sections to separate points and to

guarantee the maximal rank condition, it follows

that the canonical bundle K of X is ample. Compact

complex manifolds with ample canonical bundle are

examples of manifolds which are said to be of

general type (see Several Complex Variables: Compact

Manifolds). Thus, this construction with Poincare´

series proves the following: ‘‘Every compact quotient

D= of a bounded domain is of general type and is

in particular projective algebraic.’’

See also: Gauge Theoretic Invariants of 4-Manifolds;

Moduli Spaces: An Introduction; Riemann Surfaces;

Several Complex Variables: Compact Manifolds; Twistor

Theory: Some Applications [in Integrable Systems,

Complex Geometry and String Theory].