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

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A projective manifold is a compact manifold which

is a submanifold of some projective space P

.Of

course, a projective manifold can be embedded into

projective spaces in many ways. According to Chow’s

theorem (see Several Complex Variables: Basic

Geometric Theory), X  P

is automatically given

by polynomial equations and is therefore an algebraic

variety. This is part of Serre’s GAGA principle which

roughly says that all global analytic objects on a

projective manifold, for example, vector bundles or

coherent sheaves and their cohomology are auto-

matically algebraic. A compact manifold which is

bimeromorphically equivalent to a projective mani-

fold is called a Moishezon manifold. These arise

naturally, for example, as quotient of group actions,

compactifications, etc.

The most important birational invariant of com-

pact manifolds is certainly the Kodaira dimension

(X). It is defined in three steps:



(X) = 1 iff h

(mK

) = 0 for all m  1.



(X) = 0 iff h

(mK

)  1 for all m,and

(mK

) = 1 for some m.



In all other cases we can consider the meromorphic

map f

: X ! P

N ( m)

associated to H

(mK

) for all

those m for which h

(mK

)  2. Let V

denote

the (closure of the) image of f

. Then (X )is

defined to be the maximal possible dim V

Recall that f

is defined by [s

:  : s

] for a given

base s

of H

(mK

), cf. Several Complex Variables:

Basic Geometric Theory.

In the same way one defines the Kodaira (or

Iitaka) dimension (L) of a holomorphic line bundle

L (instead of L = K

We are now going to describe geometrically the

different birational equivalence classes and how to

single out nice models in each class. Using methods

in characteristic p, Miyaoka and Mori proved the

following theorem:

Theorem 1 Let X be a projective manifold and

suppose that through a general point x 2 X there is a

curve C such that K

 C < 0. Then X is uniruled, that

is, there is a family of rational curves covering X.

A rational curve is simply the image of noncon-

stant map f : P

!X. It is a simple matter to prove

that uniruled manifolds have (X) = 1, but the

converse is an important open problem. A step

towards this conjecture has recently been made by

Boucksom et al. (2004) if K

is not pseudoeffective,

that is, K

‘‘cannot be approximated by effective

divisors,’’ then X is uniruled. Here one also finds a

discussion of the case when K

is pseudoeffective.

Mori theory is central in birational geometry.

To state the main results in this theory, we recall the

notion of ampleness: a line bundle L is ample if L

carries a metric of positive curvature. Alternatively

some tensor power of L has enough global section to

separate points and tangents and there gives an

embedding into some projective space; see Several

Complex Variables: Basic Geometric Theory for

more details. The notion of nefness, which is in a

certain sense the deg enerate version of ampleness,

plays a central role in Mori theory : a line bundle or

divisor L is nef if

L  C ¼ degðLjCÞ0

for all curves C  X. Examples are those L carrying

a metric of semipositive curvature, but the converse

is not true. However, if L is nef, there exists for all

positive >0 a metric h



with curvature 



> !,

where ! is a fixed positive form. In this context

singular metrics on L are also important. Locally

they are given by e

’

with a locally integrab le

weight function ’ and they still have a curvature

current .IfL has a singular metric with 

bounded from below as current by a Ka¨hler form,

then L is big, that is, (L) = dim X, the birational

version of ampleness. If one simply has   0as

current, then L is pseudoeffective (and vice versa).

All these positivity notions only depend on the

Chern class c

(L)ofL and therefore one considers

the ample cone

amp

ðH

1;1

ðXÞ\H

ðX; ZÞÞ  R

and the cone of curves

NEðXÞðH

n1;n1

ðXÞ\H

2n2

ðX; ZÞÞ  R

The ample cone is by definition the closed cone of

nef divisors, the interior being the ample classes,

while the cone of curves is the closed cone generated

by the fundamental classes of irreducible curves.

A basic result says that these cones are dual to

each other. The structure of

NE(X) in the part

where K

is negative is very nice; one has the

following cone theorem:

Theorem 2

NE(X) is locally finite polyhedral in

the half-space {K

< 0}; the (geometrically) extremal

rays contain classes of rational curves.

A ray R = R

[a] is said to be extremal in a closed

cone K if the following holds: given b, c 2 K with

b þ c 2 R, then b, c 2 R. Given such an extremal ray

R 

NE(X), one can find an ample line bundle H

and a rational number t such that K

þ tH is nef

and K

þ tH  R = 0. Using the Kawamata–Viehweg

vanishing theorem, a generalization of Kodaira’s

vanishing theorem, which is one of the technical

corner stones of the theory, one proves the so-called

552 Several Complex Variables: Compact Manifolds

Base point free theorem Some multiple of K

þ tH

is spanned by global sections and therefore defines a

holomorphic map f : X !Y to some normal projec-

tive variety Y contracting exactly those curves whose

classes belong to R.

These maps are called ‘‘contractions of extremal

rays’’ or ‘‘Mori contractions.’’ In dimension 2 they

are classical: either X = P

and f is the constant

map, or f is a P

-bundle or f is birational and the

contraction of a P

with normal bundle O(1), that

is, f contracts a (1)-curve. In particular Y is again

smooth. In the first two cases X has a very precise

structure, but in the third birational case one

proceeds by asking whether or not K

is nef. If it

is not nef, we start again by choosing the contrac-

tion of an extremal ray; if K

is nef, then a

fundamental result says that a multiple of K

spanned. The class of manifolds with this property

will be discussed later.

The situation in higher dimensions is much more

complicated. For example, Y need no longer be

smooth. However the singularities which appear are

rather special.

Definition 1 A normal variety X is said to have

only terminal singularities if first some multiple of

the cano nical (Weil) divisor K

is a Cartier divisor,

that is, a line bundle (one says that X is

Q-Gorenstein) and second if for some (hence for

every) resolution of singularities  : X !X the

following holds:

¼ 



ðK

Þþ

where the E

run over the irreducible -exceptional

divisors and the a

are strictly positive.

A brief remark concerning Weil divisors is in

order: a Weil divisor is a finite linear combination

with Y

irreducible of codimension 1, but Y

is not necessarily locally defined by one equation.

Recall that if each Y

is given locally by one

equation, then the Weil divisor is Cartier. On a

smooth variety these notions coincide.

One important consequence is that (X) = (

X)in

case of terminal singularities, which is completely

false for arbitrary singularities. Also notice that

terminal singularities are rational: R





) = 0 for

q  1. Terminal singularities occur in codimension

at least 3. Thus they are not present on surfaces. In

dimension 3 terminal singularities are well under-

stood. The main point in this context is that for a

birational Mori contraction the image Y often has

terminal singularities.

Now the scheme of Mori theory is the following.

Start with a projective manifold X.IfK

is nef, we

stop; this class is discussed later. If K

is not nef,

then perform a Mori contraction f : X !Y. There

are two cases:



If dim Y < dim X, then the general fiber F is a

manifold with ample K

, that is, a Fano

manifold (discussed in the next section). Here we

stop and observe that (X) = 1. Of course one

can still investigate Y and try to say more on the

structure of the fibration f.



If dim Y = dim X,thenY has terminal singularities –

unless f is a small contraction which means that no

divisors are contracted. Thus if f is not small, we may

attempt to proceed by substituting X by Y.

As a result one must develop the entire theory for

varieties with terminal singularities. The big pro-

blem arises from small contractions f. In that case

cannot be Q-Cartier and the machinery stops. So

new methods are required. At this stage, other

aspects of the theory lead one to attempt a certain

surgery procedure which should improve the situa-

tion and allow one to continue as above. The

expected surgery Y * Y

, which takes place in

codimension at least 2, is a ‘‘flip.’’ The idea is that

we should substitute a small set, namely the

exceptional set of a small contraction, by some

other small set (on which the canonical bundle will

be positive) to improve the situation. Of course Y

should possess only terminal singularities. The

existence of flips is very deep and has been proved

by S Mori in dimension 3. Moreover, there cannot

be an infinite sequence of flips, at least in dimension

at most 4.

In summary, by performing contractions and flips

one constructs from X a birational model X

with

terminal singularities such that either



is nef in which case we call X

a minimal

model for X,or



admits a Fano fibration f

: X

(discussed

below), in which case (X) =  (X

) = 1.

Up to now, Mori theory (via the work of

Kawamata, Kolla´r, Mori, Reid, Shokurov, and

others) works well in dimension 3 (and possibly in

the near future in dimension 4) but in higher

dimensions there are big problems with the existence

of flips. Of course there might be completely

different and possibly less precise ways to construct

a minimal model. One way is to consider the

canonical ring R of a manifold of general type:

R ¼

ðmK

If R is finitely generated as C-algebra, then

Proj(R) would be at least a canonical model which

Several Complex Variables: Compact Manifolds 553

has slightly more complicated singularities than a

minimal model. However, it is known that this

‘‘finite generatedness problem’’ is equivalent to the

existence of minimal models. On the other hand, if

X is of general type with K

nef (hence essentially

ample) or more generally when some positive

multiple mK

is generated by global sections , then

R is finitely generated.

We now must discuss the case of a nef canonical

bundle. The behavior is predicted by the

Abundance conjectur e. If X has only terminal

singularities and K

is nef, then some multiple

is spanned.

Up to now this conjecture is known only in

dimension 3 (Kawamata, Kolla´r, Miyaoka). In

higher dimensions it is even unknown if there is a

single section in some multiple mK

.IfmK

spanned, one considers the Stein factorization

f : X !Y of the associated map, which is called the

Iitaka fibration (if not birational) and we have

dim Y = (X) by definition. The general fiber F is a

variety with K

 0, a class discussed in the next

section. If f is birational, then Y will be slightly

singular (so-called canonical singularities) and K

will be ample. Essentially we are in the case of

negative Ricci curvature.

Everything that was outlined above holds for

projective manifolds. In the Ka¨ hler case one would

expect the same picture, but the methods completely

fail, and new, analytic methods must be found. Only

very few results are known in this context.

We come back to the case of a Fano fibration

f : X !Y. By definition the a nticanoical bundle K

is relatively ample so that the general fiber is a Fano

variety. In this case there are no constraints on Y.

To see how much of the geometry of X is dictated

by the rational curves, one considers the so-called

rational quotient of X. Here we identify two very

general points on X if they can be joined by a chain

of rational curves. In that way we obtain the

rational quotient

f : X * Y

This map is merely meromorphic, but has the

remarkable property of being ‘‘almost holo-

morphic,’’ that is, the set of indeterminacies does

not project onto Y. In other words, one has nice

compact fibers not meeting the indeterminacy set. If

Y is just a point, then all points of X can be joined

by chains of rational curves and X is called

rationally connected. This notion is clearly biration-

ally invariant.

A deep theorem of Graber–Harris–Starr states

that, given a Fano fibration (or a fibration with

rationally connected fibers) f : X !Y, then X is

rationally connected if and only if Y is.

Manifolds X

which are birational to P

are

called rational. If there merely exists a surjective

(‘‘dominant’’) rational map P

* X, then X is said

to be unirational. Of course rational (resp. unira-

tional) manifolds are rational ly connected, but to

decide whether a given mani fold is rational/uni-

rational is often a very deep problem. Therefore,

rational connectedness is often viewed as a practical

substitute for (uni)rationality.

Often it is very important to compute the Kodaira

dimension of fiber spaces. Let us fix a holom orphic

surjective map f : X !Y between projective mani-

folds and we suppose f has connected fibers. Then

the so-called conjecture C

states that

ðXÞðFÞþðYÞ

where F is the general fiber of f. This conjecture is

known in many cases, for example, when the

general fiber is of general type, but it is wide open

in general. It is deeply related to the existence of

minimal models (Kawamata).

Biholomorphic Classification

In this section we discuss manifolds X with



ample anticanonical bundles K

(Fano manifolds),



trivial canonical bundles, and



ample canonical bundles K

Due to the solution of the Calabi conjecture by

Yau and Aubin, these classes are characterized by a

Ka¨ hler metric of positive (resp. zero, resp. negative)

Ricci curvature. In principle, in view of the results of

Mori theory, one should rather consider varieties

with terminal singularities, but we ignore this aspect

completely. Philosophically, up to birational equiva-

lence all mani folds are via fibrations somehow

composed of those classes via fibrations, possibly

also up to e´tale coverings.

Examples of Fano manifolds are hypersurfaces of

degree at most n þ 1inP

nþ1

, Grassmannians, or

more generally homogenenous varieties G/P with G

semisimple and P a parabolic subgroup. Fano

manifolds are simply connected. This can be seen

either by classical differe ntial geometric methods

using a Ka¨hler metric of positive curvature or via the

fundamental

Theorem 3 Fano manifolds are rationally

connected.

The only known proof of this fact uses, as in the

uniruled cri terion mentioned above, characteristic p

methods. By just using complex methods it is not

554 Several Complex Variables: Compact Manifolds

known how to construct a single rational curve (of

course, in concrete examples the rational curves are

seen immediately). One still has to obs erve that

rationally connected manifolds are simply con-

nected, which is not so surprising, since rational

curves lift to the universal cover.

At least in principle, Fano mani folds can be

classified:

Theorem 4 There are only finitely many families of

Fano manifolds in every dimension.

A family (of Fano manifolds) is a submersion

 : X!S (with S irreducible) such that all fibers are

Fano manifolds. The essential step is to bound (K

)

An actual classification has been carried out only

in dimension up to 3; in dimension 2 one finds

, P

 P

and the so-called del Pezzo surfaces (P

blown up in at most eight points in general position).

In dimension 3 there are already 17 families of Fano

3-folds with b

= 1 and 88 families with b

 2.

An extremely hard question is to decide whether a

given Fano manifold is rational or unirational. Even

in dimension 3 this is not completely decided.

The next class to be discussed are the manifolds

with trivial canonical class K

. This means that

there is a holomorphic n-form without zeros

(n = dim X). Important examples are tori and

hypersurface in P

nþ1

of degree n þ 2. Simply

connected manifolds with trivial canonical bundles

are further divided into irreducible Calabi–Yau

manifolds and irreducible symplectic manifolds.

The first class is defined by requiring that there are

no holomorphic p-forms for p < dim X whereas the

second is characterized by the existence of a

holomorphic 2-form of everywhere maximal rank.

A completely different characterization is by holonomy:

an irreducible Calabi–Yau manifold has SU-holonomy

whereas irreducible symplectic manifolds have

Sp-holonomy (with respect to a suitable Ka¨ hler metric).

The splitting theorem of Beauville–Bogomolov–

Kobayashi says

Theorem 5 Let X be a projective (or compact

Ka¨hler) manifold with trivial canonical bundle.

Then there exists a finite unbranched cover

X !X

such that

X ¼ A  X

 Y

with A a torus, X

irreducible Calabi–Yau, and Y

irreducible symplectic.

The key to the proof of this theorem is the

existence of a Ricci-flat Ka¨ hler metric on X,a

Ka¨ hler–Einstein metric with zero Ricci curvature.

Actually one has a stronger result: instead of

assuming K

to be trivial, just assume that

(X) = 0inH

(X, R). Then there exists a finite

unramified cover X !X such that K

is trivial. In

view of Mori theory, normal projective varieties X

with at most terminal singularities and K

 0 (i.e.,

 C = 0 for all curves) should also be investigated.

It is expected that similar structure theorems hold;

in particular 

(X) should be finite. The main

difficulty is that there are no differential methods

available; on the other hand an algebraic proof even

for the splitting theorem in the smooth case is

unknown.

Calabi–Yau manifolds play an important role in

string theory and mirror symmetry (see Mirror

Symmetry: A Geometric Survey). Here we mention

two basic problems. The first is the problem of

boundedness:

Are there only finitely many families of Calabi –

Yau manifolds in any dimension?

This prob lem is wide open; in particular one

might ask:

Is the Hodge number h

1, 2

bounded for Calabi–

Yau 3-folds?

The other problem asks for the existence of

rational curves. In all known examples there are

rational curves, but a general existence proof is not

known. The case where b

(X) = 1 seems to be

particularly difficult. If b

(X)  2, then in may

cases one can hope to find a fibration or a birational

map, at least for 3-folds. Given such a map, the

existence of rational curves is simple. For example,

if D  X is an irreducible hypersurface which is not

nef, choose H ample and consider the a priori

positive real number p such that D þ pH is on the

boundary of the ample cone. Then actually p is

rational and a suitable multiple m(D þ pH)is

spanned and defines a contraction on X. This

comes from ‘‘logarithmic Mori theory.’’

The above splitting theorem exhibits a torus

factor and all holomorphic 1-forms on X come

from this torus. This principle generalizes: given any

projective or compact Ka¨ hler manifold X, there

exists a ‘‘universal object,’’ the Albanese torus

AlbðXÞ¼H

ð



ðX; ZÞ

(which is algebraic if X is) together with a

holomorphic map

 : X !AlbðX Þ

the Albanese map. This Albanese map is given by

integrating 1-forms and is often far from being

surjective. The important property is now that,

given a holomorphic 1-form ! on X, there exists a

holomorphic 1-form  on the Albanese torus such

that ! = 



(). The universal prop erty reads as

Several Complex Variables: Compact Manifolds 555

follows: every map X !T to a torus factors via an

affine map Alb(X) !T.

There is a nonabelian analog, the so-called

Shafarevich map, but at the moment this map is

only known to be meromorphic. It is an important

tool to study the fundamental group 

(X). We refer

to Campana (1996) and Kolla´r (1995) .

In the following, Chern classes of holomorphic

vector bundles will be important. Let X be a

compact complex manifold and E a holomorphic

vector bundle on X. The jth Chern class of E is an

element

ðEÞ2H

ðX; QÞ\H

j;j

ðXÞ

It can be defined, for example, by putting a Hermitian

metric on E, computing the curvature of the canonical

connection compatible with both the metric and the

holomorphic structure and then by applying certain

linear operators coming from symmetric functions

such as determinant and trace. Actually Chern classes

can be attached to every complex topological vector

bundle on a topological manifold; then c

(E)will

simply live in H

(X, R). There is also a purely

algebraic construction by Grothendieck. We refer, for

example, to Fulton (1984) as well as for a discussion of

the elementary functorial properties of Chern classes.

Here we just recall that for a rank-r vector bundle E the

first Chern class

ðEÞ¼c



where the Chern class of the line bundle

E as

given in Several Complex Variables: Basic Geo-

metric Theory actually lives in H

(X, Z).

Finally we discuss manifolds with ample canonical

class K

. Here moduli question often plays a central

role. Moduli spaces of surfaces with fixed c

and c

are very intensively studied (by Catanese, Ciliberto,

and others). Here, without going into details, we

will concentrate on the very interesting topic of

Ka¨ hler–Einstein metrics.

AKa¨hler metric ! is said to be Ka¨ hler–Einstein, if

its Ricci curvature Ric(!) is proportional to !. The

proportionality factor  can be taken to be 1, 0, 1. In

case K

is ample or trivial, Ka¨hler–Einstein metrics

always exist by Yau and Aubin (cases  = 1, resp.

 = 0). However if X is Fano, there are obstructions,

and a Ka¨hler–Einstein met ric does not always exist.

An important consequence of the existence of a

Ka¨ hler–Einstein metric on a manifold X

with ample

canonical class is the Miyaoka–Yau inequality:

n2



2n þ 1

n2

In case of equality, X is covered by the

n-dimensional unit ball.

The same inequality holds in case K

= 0, and as a

consequence the Chern class c

(X) is in some sense

semipositive. If c

(X) = 0, then some finite unrami-

fied cover of X is a torus.

There is an interesting relation to stability. Recall

that a vector bundle E on a compact Ka¨ hler

manifold X

is semistable with respect to a given

Ka¨ hler form !, if for all proper coherent subsheaves

FE of rank-r the following inequality holds:

ðFÞ  !

n1



ðEÞ!

n1

In case of strict inequality, E is said to be stable.

The basic observation is now that the tangent

bundle of a manifold with a Ka¨ hler–Einstein metric

is semistable (with respect to the Ka¨hler–Einstein

metric). It is expected that Fano manifolds with

= 1 have (semi?-)stable tangent bundles, although

in certain situations they do not admit a Ka¨ hler–

Einstein metric.

Again the first two Chern classes of a semistable

vector bundle fulfill an inequality:

ðEÞ!

n2



r  1

ðEÞ!

n2

Equally important, semistable bundles with fixed

numerical data form moduli spaces, this being the

origin of the stability notion (Mumford). In this

context, the notion of an Hermite–Einstein bundle is

also important. Given a holomorphic vector bundle

E with a Her mitian metric h, there is a unique

connection F

on E compat ible both with h and the

complex structure. F

is a (1,1)-form with values in

End(E). Now suppose (X, ! )isKa¨hler and let F

the contraction of F

with !. Then (E, h) is said to

Hermite–Einstein on (X, !), if

F

¼ id

with some constant  and id: E !E the identity.

Notice that (X, !)isKa¨hler–Einstein if (T

, h)is

Hermite–Einstein over (X, !) with h the Ka¨hler

metric with Ka¨ hler form !. It is not so difficult to

see that Hermite–Einstein bundles are semistable

(with respect to the underlying Ka¨hler form) and

actually are directs sum of stable Hermite–Einstein

bundles. Conversely, a very deep theorem of

Uhlenbeck–Yau says that every stable vector bundle

on a compact Ka¨hler manifold is Hermite–Einstein.

This is known as the Kobayashi–Hitchin correspon-

dence; see Lu¨ bke and Teleman (1995).

556 Several Complex Variables: Compact Manifolds

Topology, Invariants and Cohomology

Besides the Kodaira dimension there are other

important invariants of compact complex manifolds.

Of course there are topological invariants such as the

Betti number b

(X) = dim H

(X, R) or the fundamen-

tal group 

(X). The fundamental group has been

studied intensively in the last decade. A central

question asks which groups can occur as fundamental

groups of compact Ka¨hler manifolds; another pro-

blem is the so-called Shafarevitch conjecture which

says that the universal cover of a compact Ka¨hler

manifold should be holomorphically convex. We

refer to Campana (1996) and Kolla´r (1995).

The plurigenera,

ðXÞ¼dim h

ðmK

are also extremely important. Here, Siu recently

proved that P

(X) is constant in families of

projective manifolds. Other important invariants are

(X,(

)

m

). For example, it is conjectured that if

ðX; 



m

Þ¼0

for all positive m, then X is rationally connected.

Tensor powers of the cotangent bundle somehow

capture more of the structure of X than the Kodaria

dimension but they are more difficult to treat. The

relevance of the dimensions

X; 



of holomorphic forms is easier to understand. More

generally one has the Hodge numbers

p;q

ðXÞ¼dim H

X; 



For compact Ka¨ hler manifolds, the Hodge decom-

position states

ðX; CÞ¼

pþq¼r

p;q

ðXÞ

Furthermore, Hodge duality,

p;q

ðXÞ¼H

q;p

ðXÞ

holds. These results form a cornerstone for the

geometry of compact Ka¨hler mani folds and the

starting point of Hodge theory. Hodge theory is,

for example, extremely important in the study of

families of manifolds and moduli.

Concerning the topology of projective (Ka¨ hler)

manifolds, the following two questions are very

basic.



Which invariants are topological (or diffeo-

morphic) invariants?



What are the projective or Ka¨hler structures on a

given compact topological manifold?

Concerning the first, Hodge decomposition

implies that the irr egularity h

(

) is actually a

topological invariant. However it is unknown

whether the number of holomorphic 2-forms is a

topological invariant of Ka¨ hler 3-folds. Both ques-

tions have been intensively studied in dimension 2.

However, in higher dimensions almost nothing is

known. For example, it is not known whether there

is projective manifold of general type of even

dimension which is homeomorphic to a quadric,

that is, a hypersurface of degree 2 in projective

space.

Other important tools in the study of projective/

Ka¨ hler manifolds are listed below.



Cohomological methods: Riemann–Roch theorem

and holomorphic Morse inequalities; vanishing

theorems (Kodaira, Kawamata–Viehweg, etc.);

Serre duality. References: Demailly (2000),

Demailly and Lazarsfeld, Fulton (1984), Grauert

et al. (1994), Lazarsfeld (2004).



methods: extension theorems, singular metrics,

multiplier ideal s, etc. Reference: Demailly and

Lazarsfeld (2001), Lazarsfeld (2004).



Theory of currents. Referen ce: Demailly 2000.



Cycle space and Douady space, resp. Chow

scheme and Hilbert scheme. Reference: Fulton

1984, Grauert 1994, Kolla´r 1996.

We restrict our remarks on just one of these

topics, vanishing theorems. The classical Kodaira–

Nakano vanishing theorem says that if X is a

compact manifold of dimension n with a positive

(ample) line bundle L, then

ðX; L  

Þ¼0

for p þ q > n. This is usually proved via harmonic

theory, that is, by representing the cohomology

space by harmonic (p, q)-forms with values in L

and by computing integrals of these forms. For

many purposes, for example, for Mori theory, it is

important to generalize this to a line bundle which

have some positivity properties but which are not

ample. This works only for p = n, however this is

the most important part of the Kodaira–Nakano

vanishing. The Kawamata–Viehweg vanishing theo-

rem in its most basic version says that given a nef

and big line bundl e L, then Kodaira vanishing still

holds:

ðX; L  K

Þ¼0

for q  1. But actually it is not necessary to assume

L nef, in fact the following is true. Let

D ¼

Several Complex Variables: Compact Manifolds 557

be an effective Q-divisor, that is, all a

are positive

rational numbers. Let ha

i be the fractional part of a

and suppose that the Q-divisor

has normal

crossings. Let da

e be the roundup of a

and put

L =

.IfD is big and nef, then

ðX; L  K

Þ¼0

for q  1. Of course L itself need not be nef! This

generalization is technically very important and yields

substantial freedom for birational manipulations. We

refer to Kawamata et al. (1987) and Lazarsfeld (2004).

Even this is not the end of the story: the Kawamata–

Viehweg theorem is embedded in the broader context

of the Nadel vanishing theorem where multiplier ideal

sheaves come into the play. See Demailly and

Lazarsfeld and Lazarsfeld (2004).

Homogeneous Manifolds

In this section we consider vector fields and

holomorphic group actions on compact (Ka¨hler)

manifolds. Our main reference is Huckleberry

(1990) with further literature given there.

We denote by Aut(X) the group of holomorphic

automorphisms of the compact manifold X (well

known to be a complex Lie group), and by

G := Aut

(X) the connected component containing

the identity. The tangent space at any point of

Aut

(X) can naturally be identified with H

(X, T

the (finite-dimensional) space of holomorphic

vector fields on X. In fact, by integration, a

vector field determines a one-parameter group of

automorphisms.

One says that X is homogeneous if G acts

transitively on X. Therefore, one can write

X ¼ G=H

where H is the isotropy subgroup of any point

2 X, that is, the subgroup of automorphisms

fixed x

. Conversely one can take a complex Lie

group G and a closed subgroup H and form the

quotient G/H whichisagainacomplexmanifold

and in fact homogeneous (of course not necessarily

compact).

Going back to a compact manifold X, the

condition to be homogenenous can be rephrased by

saying that the tangent bundle is generated by

global sections, that is, if x 2 X and e 2 T

X, x

, then

there exists v 2 H

(X, T

) such that v(x) = e. The

easiest case is when T

is trivial. If X is Ka¨ hler, this

is exactly the case when X is torus, X = C

= with

 ’ Z

a lattice, but without the Ka¨hler assump-

tion there are many more examples (the so-called

parallelizable manifolds).

More generally, let us consider the case that the

compact Ka¨ hler manifold X admits a vector field v

without zeros, but X is not required to be homo-

geneous. Then a theorem of Lieberman says that

there is a finite unramified cover f : X !X and a

splitting

X ’ F  T

with T a torus, such that f



(v) is the pullback of a

vector field on T. On the other hand, if v has a zero,

then a classical theorem of Rosenlicht says that X is

covered by rational curves, that is, X is uniruled. In

particular (X) = 1. Notice also that a manifold

of general type can never carry a vector field, in

other words, the automorphism group is discrete,

even finite.

Coming back to compact homogeneous Ka¨ hler

manifolds, the first thing to study is the Albanese

map. The Borel–Remmert theorem says that

X ’ T  Q

where T is the Albanese torus. This is proved using a

maximal compact subgroup K  G and by some

averaging proces s over K. Moreover, Q is a rational

homogeneous manifold. The structure of Q is more

precisely the following. One can write Q = S=P with

S a semisimple Lie group and P  S parabolic,

which means that P contains a maximal connect ed

solvable subgroup (the so-called Borel subgroup).

The main ingredie nts of the proof are the Tits

fibration, the Levi–Malcev decomposition of a Lie

group into its radical and a semisimple group, and

the Borel fixed point theorem:

Theorem 6 Let G  GL

solvable subgroup and X  P

n1

be a G- stable

subvariety. Then G has a fixed point on X.

In the homogenenous Ka¨ hler case, the rationality of

Q is seen by exhibiting an open subset in Q which is

algebraically isomorphic to C

Now things come down to classify all rational

homogenenous manifold S/P which is of course

classical. Notice that all rational homogeneous

manifolds are Fano. One knows that a rational

homogeneous manifold with Betti number b

 2

can be fibered over another rational homogenenous

manifold with fibers rational homogeneous – this is

actually a fiber bundle. The case that b

= 1 can be

rephrased by saying that P is maximal parabolic.

This fiber bundle might not be trivial as shown by

the projectivized tangent bundle P(T

Compact Hermitian symmetric spaces form a

particularly interesting subclass of homogeneous

Ka¨ hler manifolds. A manifold equipped with a

558 Several Complex Variables: Compact Manifolds

Hermitian metric is called Hermitian symmetric, if for

every x 2 X there exists an involutive holomorphic

isometry fixing x. Mok has shown the remarkable fact

that the simply connected compact Hermitian sym-

metric spaces are exactly those simply connected

compact manifolds carrying a Ka¨hler metric with

semipositive holomorphic bisectional curvature. The

only manifold having a metric with positive holo-

morphic bisectional curvature is P

(Siu-Yau, Mori).

See also: Classical Groups and Homogeneous Spaces;

Einstein Manifolds; Mirror Symmetry: A Geometric

Survey; Moduli Spaces: An Introduction; Riemann

Surfaces; Several Complex Variables: Basic Geometric

Theory; Topological Sigma Models; Twistor Theory:

Some Applications [in Integrable Systems, Complex

Geometry and String Theory].

Further Reading

Beltrametti M and Sommese AJ (1995) The Adjunction Theory of

Complex Projective Varieties. Berlin: de Gruyter.

Boucksom S, Demailly JP, Paun M, and Peternell T (2004)

The pseudo-effective cone of a compact Ka¨hler manifold

and varieties of negative Kodaira dimension math.AG/0405285

Campana F (1996) Kodaira dimension and fundamental group of

compact Ka¨ hler manifolds. In: Andreatta M and Peternell T

(eds.) Higher Dimensional Complex Varieties, pp. 89–162.

Berlin: de Gruyter.

Demailly JP (2000) Complex analytic and algebraic geometry,

http://www-fourier.ujf-grenoble.fr/ demailly.

Demailly JP and Lazarsfeld R (eds.) Vanishing Theorems and

Effective Results in Algebraic Geometry, ICTP Lecture Notes,

vol. 6, Trieste.

Fulton W (1984) Intersection Theory. Heidelberg: Springer.

Griffiths Ph and Harris J (1978) Priniciples of Algebraic

Geometry. New York: Wiley.

Grauert H, Peternell Th, and Remmert R (1994) Several Complex

Variables VII. Encyclopedia of Mathematical Sciences, vol. 74.

Heidelberg: Springer.

Huckleberry A (1990) Actions of groups of holomorphic

transformations. In: Barth W and Narasimhan R (eds.) Several

Complex Variables VI, Encyclopedia of Mathematical Science,

vol. 69, pp. 143–196. Berlin: Springer.

Kawamata Y, Matsuda K, and Matsuki K (1987) Introduction to

the minimal model problem. Advance Studies in Pure

Mathematics 10: 283–360.

Kolla´r J (1995) Shafarevitch Maps and Automorphic Forms.

Princeton University Press.

Kolla´r J (1996) Rational Curves on Algebraic Varieties. Ergeb-

nisse der Mathematik und ihrer Grenzgebiete, vol. 32.

Heidelberg: Springer.

Lazarsfeld R (2004) Positivity in Algebraic Geometry I, II.

Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48

and 49. Heidelberg: Springer.

Lu¨ bke M and Teleman A (1995) The Kobayashi–Hitchin

Correspondence. Singapore: World Scientific.

Matsuki K (2002) Introduction to the Mori Program, Universi-

text. Heidelberg: Springer.

Mori S (1987) Classification of higher-dimensional varieties.

Proceedings of Symposia in Pure Mathematics 46: 269–331.

Parshin AN and Shafarevich IR (eds.) (1999) Algebraic Geometry

V – Fano Varieties, vol. 47, Encyclopedia of Mathematical

Sciences. Heidelberg: Springer.

Siu YT (1987) Lectures on Hermitian–Einstein Metrics for Stable

Bundles and Ka¨hler–Einstein Metrics. DMV Seminar, vol. 8.

Basel: Birkha¨user.

Ueno K (1975) Classification Theory of Compact Complex spaces.

Lecture Notes in Math., vol. 439. Heidelberg: Springer.

Viehweg E (1995) Quasi-Projective Moduli for Polarozed Mani-

folds. Ergebnisse der Mathematik und ihrer Grenzgebiete,

vol. 30. Heidelberg: Springer.

Shock Wave Refinement of the Friedman–Robertson–

Walker Metric

B Temple, University of California at Davis, Davis,

CA, USA

J A Smoller, University of Michigan, Ann Arbor, MI, USA

Introduction

In the standard model of cosmology, the expanding

universe of galaxies is described by a Friedman–

Robertson–Walker (FRW) metric, which in spherical

coordinates has a line element given by (Blau and Guth

1987, Weinberg 1972)

¼dt

þR

ðtÞ



1kr

þr

½d

þsin

d





½1

In this model, which accounts for things on the

largest length scale, the universe is approximated by a

space of uniform density and pressure at each fixed

time, and the expansion rate is determined by the

cosmological scale factor R(t ) that evolves according

to the Einstein equations. Astronomical observations

show that the galaxies are uniform on a scale of

about one billion light years, and the expansion is

critical – that is, k= 0in[1] – and so, according to

[1], on the largest scale, the universe is infinite flat

Euclidian space R

at each fixed time. Matching the

Hubble constant to its observed values, and invoking

the Einstein equations, the FRW model implies that

the entire infinite universe R

emerged all at once

from a singularity (R=0), some 14 billion years ago,

and this event is referred to as the big bang.

Shock Wave Refinement of the Friedman–Robertson–Walker Metric 559

In this article, which summarizes the work of the

authors in Smoller and Temple (1995, 2003),we

describe a two-parameter family of exact solutions

of the Einstein equations that refine the FRW met ric

by a spherical shock wave cutoff. In these exact

solutions, the expanding FRW metri c is reduced to a

region of finite extent and finite total mass at each

fixed time, and this FRW region is bounded by an

entropy-satisfying shock wave that emerges from the

origin (the center of the explosion), at the instant of

the big bang, t = 0. The shock wave, which marks

the leading edge of the FRW expansion, propag ates

outward into a larger ambient spacetime from time

t = 0 onward. Thus, in this refinement of the FRW

metric, the big bang that set the galaxies in motion

is an explosion of finite mass that looks more like a

classical shock wave explosion than does the big

bang of the standard model. (The fact that the entire

infinite space R

emerges at the instant of the big

bang, is, loosely speaking, a consequence of the

Copernican principl e, the principle that the Earth is

not in a special place in the universe on the largest

scale of things. With a shock wave present, the

Copernican principle is violated, in the sense that

the Earth then has a special position relative to the

shock wave. But, of course, in these shock wave

refinements of the FRW metric, there is a spacetime

on the other side of the shock wave, beyond the

galaxies, and so the scale of uniformity of the FRW

metric, the scale on which the density of the galaxies

is uniform, is no longer the largest length scale.)

In order to construct a mathematically simple

family of shock wave refinements of the FRW met ric

that meet the Einstein equations exactly, we assume

k = 0 (critical expansion), and we restrict to the case

that the sound speed in the fluid on the FRW side of

the shock wave is constant. That is, we assume an

FRW equation of state p = , where , the square

of the sound speed

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

@p=@

, is constant, 0 < c

At  = c

=3, this catches the important equation of

state p = (c

=3) which is correct at the earliest stage

of big bang physics ( Weinberg 1972). Also, as 

ranges from 0 to c

, we obtain qualitatively correct

approximations to general equations of state.

Taking c = 1 (w e use the convention that c = 1, and

Newton’s consta nt G= 1 when convenient), the

family of solutions is then determined by two

parameters, 0 < 1 and r



 0. The second

parameter, r



, is the FRW radial coordinate r of

the shock in the limit t ! 0, the instant of the

big bang. (Since, when k = 0, the FRW metric is

invariant under the rescaling r ! r and R ! 

1

we fix the radial coordinate r by fixing the scale

factor  with the condition that R(t

) = 1 for some

time t

, say present time.) The FRW radial

coordinate r is singular with respect to radial

arclength

r = rR at the big bang R = 0, so setting



> 0 does not place the shock wave away from the

origin at time t = 0. The distance from the FRW

center to the shock wave tends to zero in the limit

t ! 0 even when r



> 0. In the limit r



!1,we

recover from the family of solutions the usual

(infinite) FRW metric with equation of state p =  –

that is, we recover the standard FRW metric in the

limit that the shock wave is infinitely far out. In this

sense our family of exact solutions of the Einstein

equations considered here represents a two-parameter

refinement of the standard FRW metric.

The exact solutions for the case r



= 0 were first

constructed in Smoller and Temple (1995) (see also

the notes by Smoller and Temple (1999)), and are

qualitatively different from the solutions when r



> 0,

which were constructed later in Smoller and

Temple (2003). The difference is that, when r



= 0,

the shock wave lies closer than one Hubble length

from the center of the FRW spacetime throughout

its motion (Smoller and Temple 2000), but when



> 0, the shock wave emerges at the big bang at a

distance beyond one Hubble length. (The Hubble

length depends on time, and tends to zero as t ! 0.)

We show in Smoller and Temple (2003) that one

Hubble length, equal to c=H,whereH =

R=R,isa

critical length scale in a k = 0 FRW metric because

the total mass inside one Hubble length has a

Schwarzschild radius equal exactly to one Hubble

length. (Since c=H is a good estimate for the age of

the universe, it follows that the Hubble length c=H

is approximately the distance of light travel starting

at the big bang up until the present time. In this

sense, the Hubble length is a rough estimate for the

distance to the further most objects visible in the

universe.) That is, one Hubble length marks precisely

thedistanceatwhichtheSchwarzschildradius

 2M

of the mass M inside a radial shock wave at distance

r from the FRW center, crosses from inside (

to outside (

r) the shock wave. If the shock wave

is at a distance closer than one Hubble length from

the FRW center, then 2M <

r and we say that the

solution lies outside the black hole, but if the shock

wave is at a distance greater than one Hubble

length, then 2M >

r at the shock, and we say that

the solution lies ‘‘inside’’ the black hole. Since M

increases like

, it follows that 2M <

r for

sufficiently small, and 2M >

r for

r sufficiently

large, so there must be a critical radius at which

2M =

r, and we show in what follows (see also

Smoller and Temple (2003)) that when k = 0, this

critical radius is exactly the Hubble length. When

the parameter r



= 0, the family of solutions for 0 <

  1 starts at the big bang, and evolves thereafter

560 Shock Wave Refinement of the Friedman–Robertson–Walker Metric

‘‘outside’’ the black hole, satisfying 2M=

r < 1 every-

where from t = 0 onward. But, when r



> 0, the

shock wave is further out than one Hubble length

at the instant of the big bang, and the solution

begins with 2M=

r > 1 at the shock wave. From this

time onward, the spacetime expands until even-

tually the Hubble length catch es up to the shock

wave at 2M=

r = 1, and then passes the shock wave,

making 2M=

r < 1 thereafter. Thus, when r



> 0,

the whole spacetime begins inside the black hole

(with 2M=

r > 1 for s uffi cient ly large

r), but

eventually evolves to a solution outside the black

hole. The time when

r = 2M actually marks the

event horizon of a white hole (the time reversal of

a black hole) in the ambient spacetime beyond the

shock wave. We show that, when r



> 0, the ti me

when the Hubble length c atches up to the shock

wave comes after the time when the shock w ave

comes into view at the FRW center, and when

2M =

r (assuming t is so large t hat we can neglect

the pressure from this time onward), the whole

solution emerges fr om the white hole as a finite

ball of mass expanding into empty s pace, satisfying

2M=

r < 1 everywhere thereafter. In fact, when r



> 0,

the zero pressure Oppenheimer–Snyder solution

outside the black hole gives the large-time asymp-

totics of the solution (Oppenheimer and Snyder

1939, Smoller and Temple 1988, 2004 and the

comments after Theorems 6–8 below).

The exact solutions in the case r



= 0 give a

general-relativistic version of an explosion into a

static, singular, isothermal sphere of gas, qualita-

tively similar to the corresponding clas sical explo-

sion outside the black hole (Smoller and Temple

1995). The main difference physically between the

cases r



> 0andr



= 0 is that, when r



> 0 (the case

when the shock wave emerges from the big bang at a

distance beyond one Hubble length), a large region

of uniform expansion is created behind the shock

wave at the instant of the big bang. Thus, when r



> 0,

lightlike information about the shock wave

propagates inward from the wave, rather than

outward from the center, as is the case when r



= 0

and the shock lies inside one Hubble length. (One

can imagine that when r



> 0, the shock wave can

get out through a great deal of matter early on when

everything is dense and compressed, and still not

violate the speed of light bound. Thus, when r



> 0,

the shock wave ‘‘thermalizes,’’ or more accurately

‘‘makes uniform,’’ a large region at the center, early

on in the explosion.) It follows that, when r



> 0,

an observer positioned in the FRW spacetime inside

the shock wave will see exactly what the standard

model of cosmology predicts, up until the time when

the shock wave comes into view in the far field. In

this sense, the case r



> 0 gives a black hole

cosmology that refines the standard FRW model of

cosmology to the case of finite mass. One of the

surprising differences between the case r



= 0 and the

case r



> 0 is that, when r



> 0, the important

equation of state p = /3 comes out of the analysis as

special at the big bang. When r



> 0, the shock

wave emerges at the instant of the big bang at a

finite nonzero speed (the speed of light) only for the

special value  = 1/3. In this case, the equation of

state on both sides of the shock wave tends to the

correct relation p = /3 as t ! 0, and the shock

wave decelerates to subluminous speed for all

positive times thereafter (see Smoller and Temple

(2003) and Theorem 8 below).

In all cases 0 < 1, r



 0, the spacetime

metric that lies beyond the shock wave is taken to

be a metric of Tolmann–Oppenheimer–Volkoff

(TOV) form (Oppenheimar and Volkoff 1939):

¼Bð



rÞd



þA

1



rÞd



½d

þsin

d

½2

The metric [2] is in standard Schwarzschild coordi-

nates (diagonal with radial coordi nate equal to the

area of the spheres of symmetry), and the metric

components depend only on the radial coordinate

Barred coordinates are used to distinguish TOV

coordinates from unbarred FRW coordinates for

shock matching. The mass function M(

r) enters as a

metric component through the relation

A ¼ 1 

2Mð



rÞ



½3

The TOV metric [2] has a very different character

depending on whether A > 0orA < 0; that is,

depending on whether the solution lies outside the

black hole or inside the black hole. In the case A > 0,

r is a spacelike coordinate, and the TOV metric

describes a static fluid sphere in general relativity.

(When A > 0, for example, the metric [2] is the

starting point for the stability limits of Buchdahl

and Chandresekhar for stars (Weinberg 1972,

Smoller and Temple 1997, 1998).) W hen A < 0,

is the t imelike coordinate,and [2] is a dynamical metric

that evolves in time. The exact shock wave solutions are

obtained by taking

r = R(t)r to match the spheres of

symmetry, and then matching the metrics [1] and [2] at

an interface

r =

r(t) across which the metrics are

Lipschitz continuous. This can be done in general.

In order for the interface to be a physically mean-

ingful shock surface, we use the result in Theorem 4

below (see Smoller and Temple (1994))thatasingle

additional conservation constraint is sufficient to rule

out -function sources at the shock (the Einstein

equations G = T aresecondorderinthemetric,and

Shock Wave Refinement of the Friedman–Robertson–Walker Metric 561