Proceedings of the International Congress of Mathematicians
Berkeley, California, USA, 1986
Numerical Geometry of Algebraic Varieties
V. V. SHOKUROV
This report is devoted to the achievements and difficulties in the program
of constructing minimal models for algebraic varieties of any dimension. Two
essential results have been obtained within the framework of this program: the
contraction and the cone theorems. Of no less importance are the working out
of the terminology and the statement of two key conjectures: the flip and the
termination conjectures, whose solutions would lead to the completion of the
whole program. These conjectures have been proved in some special cases; for
example, they both hold for toric morphisms and the latter holds in dimension
< 4. The final results are in many cases due to many mathematicians from all
over the world, their names following the title of a result in the order of their
contribution to it.
1. Q-divisors. Let X be a normal projective algebraic variety over the
complex number field. A Q-Cartier divisor on X, or simply a Q-divisor on X,
is an element D G DìVQ X = DivX (8) Q, where DivX is the group of Cartier
divisors on X\ in other words, a Q-divisor is a linear combination of Cartier
divisors with coefficients in Q. The group DìVQ X also contains certain Weil
divisors of X, namely, such D that rD G DivX for some 0 ^ r G Z; for then
D = (l/r)(rD). A variety X is said to be Q-factorial if every Weil divisor of X
is Q-Cartier in this sense.
A canonical divisor Kx is a Weil divisor on X such that
where X r e g is the nonsingular locus of X. Since codim(X—X reg ) > 2, a canonical
divisor Kx is well defined and Kx = Kxreg under natural identification of Weil
divisors on X and on Xveg. In general a canonical divisor Kx is not a Q-divisor.
But when Kx is a Q-divisor a variety X is said to be Q- Gorenstein. A minimal
number r such that rKx G Div X is called the index of a Q-Gorenstein variety
X. These definitions do not depend on the choice of Kx because two canonical
divisors differ by a Cartier divisor. Obviously, every Q-factorial variety X is
Q-Gorenstein.
© 1987 International Congress of Mathematicians 1986
672
NUMERICAL GEOMETRY OF ALGEBRAIC VARIETIES
673
2. Intersections. Any morphism / : Y —» X of normal projective varieties
induces the inverse image map / * : D ì V Q X —• DìVQ Y. This gives a natural
intersection theory for Q-divisors. For example, we have a bilinear pairing
(.): D i v Q X x Z i X - + Q ,
where Z\X is the free Abelian group generated by curves on X.
A Q-divisor D G DìVQ X is said to be numerically effective or briefly nef if
(D.C) > 0 for every curve C Ç X. To each nef Q-divisor D we associate its
numerical dimension
v(D)=max{k\Dk£Q}i
where Dk ^ 0 means that DkC ^ 0 for some fc-cycle C on X. Obviously,
max{0, K(D)}
< v(D) <n = dimX,
where K(D) is the Iitaka ^-dimension of X.
Let D be a nef Q-divisor. Then the following conditions are equivalent:
2.1. u(D) = n;
2.2. Dn = DnX > 0;
2.3. K(D) = n.
A nef Q-divisor D satisfying any one of these conditions is said to be big.
3. Singularities. Throughout what follows we assume that X is Q-Gorenstein. Let / : Y —> X be a desingularization of X. Then for any canonical divisor
Kx there exists a canonical divisor Ky such that
KY=f*Kx + Y,aiE^
where ai G Q and the 25» vary over all prime divisors which are exceptional with
respect to / . The divisor Y^ai^i m ^ n e above equation does not depend on the
choice of canonical divisors Kx and Ky- The coefficient a» is referred to as the
discrepancy at Ei. A variety X is said to have only terminal (resp. canonical)
singularities if all discrepancies a» > 0 (resp. a» > 0). This notion does not
depend on the choice of a desingularization of X.
3 . 1 . THEOREM.
All canonical and terminal singularities are rational.
Note that when dimX = 2, X has only terminal singularities if and only if X
is nonsingular. Surface canonical singularities have many different descriptions
and are also known as du Val singularities.
4. Models. A variety X with only canonical singularities is said to be canonicaliî Kx is ample. A canonical variety, birationally equivalent to a given variety
Y, is a canonical model of Y. It follows from the invariance of the Kodaira dimension K(X) = K(KX) for birational transformations of nonsingular varieties
and from the definition of canonical singularities that a nonsingular variety X
has a canonical model X c a n only if K(X) = dimX, i.e., X is of general type.
674
V. V. SHOKUROV
4 . 1 . THEOREM (REID). A nonsingular variety X of general type has a
canonical model X c a n if and only if the canonical ring of X
R(X)=@H°(X,Ox(mKx))
m>0
is finitely generated. Moreover,
X c a n = Proj2î(X).
4 . 2 . COROLLARY. Any nonsingular variety X of general type has a unique
canonical model if it exists.
A variety X with only terminal singularities is said to be minimal if Kx is
nef. A minimal variety, birationally equivalent to a given variety Y, is a minimal
model of Y. A nonsingular algebraic surface X has a minimal model if and
only if K(X) > 0 and it is always unique. But when d i m X > 3 there may be
many minimal models of X and there is certainly no absolute minimal model in
general. In the 3-fold case, key counterexamples concern (—2)-curves and the
Kulikov modification or more generally Pagoda modifications in these curves.
However, there is hope that at least in dimension three any nonsingular variety
of general type has only finitely many minimal models.
A morphism h : X —> Z of normal projective varieties is said to be a relative
anticanonical variety if
4.3. h*0x = Oz and
4.4. — Kx is relatively ample for h, i.e., 0x(—rKx)\Y is an ample invertible
sheaf on any fibre Y = h~1(z), z G Z.
In addition, we assume that X has only canonical singularities and dim Y > 1
for all fibres Y. If Z is a point, an anticanonical variety X is said to be a Fano
variety.
A relative anticanonical variety h: X —> Z has the following properties:
4.5. Rlh*Ox = 0 for all i > 0 and, in particular, x(Ox) = x(Oz)4.6. (Rollar). Z has only rational singularities.
4.7. The general fibre Y = h~x (z), z G Z, is a Fano variety with only canonical
singularities. Moreover, Hl(Y, Oy) = 0 for all i > 0 and the group scheme Pic Y
is discrete and torsion free. If X has only terminal singularities, then so does Y.
Note that P 1 is a unique Fano variety in dimension one. So the general
fibre of a relative anticanonical variety with dimZ = dimX — 1 is P 1 , and
this variety is also known as conic bundle. Two-dimensional Fano varieties are
usually called Del Pezzo surfaces. Respectively, a relative anticanonical variety
with dimZ = d i m X — 2 is a Del Pezzo fibre space.
A relative anticanonical variety h: X —• Z with only terminal singularities
on X such that X is birationally equivalent to a given variety Y is called a
relative anticanonical model of Y. Obviously, Y has a relative anticanonical
model only if K(Y) = — oo. As a rule, a relative anticanonical model is not
unique. For example, 3-folds have three possible types of relative anticanonical
models: conic bundles over surfaces, Del Pezzo fibre spaces over curves and Fano
NUMERICAL GEOMETRY OF ALGEBRAIC VARIETIES
675
3-folds. P 3 has infinitely many relative anticanonical models of every type and
all of the types may be on the same birational transform. Indeed, P 1 x P 2 —• P 2
is a conic bundle, P 1 X P 2 —• P 1 is a Del Pezzo fibre space, and P 1 x P 2 is a
Fano 3-fold. But a nonsingular quartic 3-fold is a Fano 3-fold, and it is its only
relative anticanonical model known to date. Note that a nonsingular algebraic
surface X has a relative anticanonical model if and only if K(X) = — OO.
The following conjecture is a main driving force and at the same time is the
stumbling-block in the development of a modern algebraic geometry in higher
dimensions.
MODEL CONJECTURE. Every nonsingular projective (or complete) algebraic
variety has a minimal model or a relative anticanonical model.
In dimension two this conjecture is true due to the Enriques classification of
algebraic surfaces. Now we turn to successes and difficulties met in carrying out
the minimal model program.
5. Contractions. By a contraction we mean a surjective morphism ip: X —•
Z of normal projective varieties which satisfies property 4.3. Every such morphism is a morphism (p\mD\ ' X —• <p\mD\(X) Ç P-^, given by a complete linear
system \mD\ for some m > 0, where D = <p*H and H is an ample divisor on
% = ^Im-DlO^O- Note that the linear system \mD\ is free, i.e., \mD\ has no base
points, and hence the divisor D is semiample, i.e., \mD\ is free for some ra > 0.
Conversely, any semiample divisor D o n ! defines a contraction <p: X —• Z
such that mD ~ <p*H for some m > 0, where H is a (very) ample divisor on Z
and ~ means the linear equivalence of divisors. The following statement gives a
numerical criterion for semiampleness.
CONTRACTION THEOREM (KAWAMATA, REID, SHOKUROV) . Let D be
a Cartier divisor on a variety X with only canonical singularities such that
5.1. D is nef.
5.2. aD - Kx is nef and big for some a > 0. Then D is semiample.
Moreover, D is stably free, that is, for all ra > 0, the linear system \mD\ is free.
Equivalently there exists a contraction <p: X —• Z such that D ~ <p*H for an
ample divisor H G Div Z.
The contraction <p has the following properties:
5.3. Rl<p*0x = 0 for all i > 0 and, in particular, x{0x) = x(0z) (cf 4.5).
5.4. There exists a unique decomposition (p = hogof<pasa
composite of
contractions
X JU X~ ± Z,
where X~ is a normal projective variety with only canonical singularities, such
that g is a birational morphism with Kx = g*Kx-, and —Kx- is relatively
ample for h (cf. 4.4). Moreover, h is a relative anticanonical variety when cp is
not birational, i.e., dimZ < dimX.
676
V. V. SHOKUROV
The proof of the contraction theorem uses the vanishing and nonvanishing
theorems. Applying the contraction theorem to a divisor D = rKx on a minimal
variety X with big Kx we obtain
5.5. COROLLARY (KAWAMATA, BENVENUTE, SHOKUROV). If a nonsingular projective variety Y of general type has a minimal model X, then there
exists a canonical model Y can and a birational contraction <p: X —• y c a n such
that<p*KYc&n=Kx.
6. Kleiman-Mori cone. Two 1-cycles zi,z^ G Z\X are said to be numerically equivalent if, for any divisor D G DivX, (D.z\) = (D.Z2), and we then
write z\ = z%. Let N\X = (Z\Xf = ) ® R . This is a finite-dimensional real linear
space and p(X) = diniR, N±X is known as the Picard number of X.
Let V be a finite-dimensional real linear space. A convex subset S in V is
said to be polyhedral if
S =
{vGV\h(v)<0,...,fN(v)<0},
where f \ , . . . , f^ are linear forms on V. If V = W ® R, where W is a maximal
lattice in V and f±,..., fw are integral on W, then a subset S is said to be rational
polyhedral. A subset S Ç V is said to be locally polyhedral (resp. rational
polyhedral) if it is local like a polyhedral (resp. rational polyhedral) set in V.
By the Kleiman-Mori cone we mean the closure NE(X) of the convex cone
NE(X) generated by effective 1-cycles or, equivalently, by curves in N±X.
CONE
THEOREM(MORI,
KAWAMATA,
SHOKUROV,
REID,
KOLLäR).
Let X be a projective variety with only canonical singularities. Then the "halfcone" WE-(X) = {zG WE(X)\(Kx.z)
< 0} is locally rational polyhedral.
Note that
an arbitrary
A face F
An extremal
the cone theorem also holds for nonsingular projective varieties over
algebraically closed field [M].
of the cone NE(X) is said to be extremal if F - {0} Ç ~NË-(X).
ray is a one-dimensional face.
6 . 1 . COROLLARY (MORI). If Kx is not nef, then there exists a nontrivial
extremal face and, hence, an extremal ray.
7. E x t r e m a l c o n t r a c t i o n s . Let X be a projective variety with only terminal
singularities; for example, X may be nonsingular. If Kx is nef, then X is a
minimal variety. Otherwise, there is a nontrivial extremal face F of NE(X).
Then F uniquely determines a contraction (p = contjr : X —> Z, such that
7.1. For any curve C Ç X, we have <p(C) = pt. G Z -o» the numerical class
of C belongs to F. cont/r is called the contraction of the face F. Indeed, due to
the rationality in the cone theorem we have a nef divisor D G Div X such that
7.2. F = {z G NÊ(X)\(D.z) = 0}.
Moreover, D satisfies condition 5.2 in the contraction theorem and <p is determined by D. Obviously, <p does not depend on the choice of such D. Note also
NUMERICAL GEOMETRY OF ALGEBRAIC VARIETIES
677
that there exists a curve C Ç X contracted by <p because D is not ample in
the sense of Kleiman [KI]. If dimZ < dimX, then (p: X —» Z is a relative
anticanonical variety. Such extremal contractions are of fibre type.
For simplicity throughout what follows we assume in addition that X is Qfactorial and F = R is an extremal ray. Then ip = contß : X —» Z has the
following properties and classification:
7.3. There is an exact sequence
0 -» R[C] -> NtX - • NXZ -+ 0,
where C is a curve with the numerical class [C] G R. Hence p(Z) = p(X) — 1.
7.4. If dimZ < dimX, then (p is of fibre type.
7.5. If dimZ = dimX and there exists a prime Weil divisor E on X such
that dim ip(E) < dim E, then Z is again a Q-factorial variety with only terminal
singularities. Such a divisor E is unique and E = [j C, where the C vary over
all curves with [C] G R. This contraction (p is said to be of divisorial type, and
the divisor E is said to be the exceptional divisor associated to the ray R.
7.6. If dim Z = dimX and (p is not of divisorial type, then there exists a subvariety E C X such that 1 < dim E < dimX — 2 and <p\x-E is an isomorphism.
Such a contraction ip is said to be of flipping type, and a minimal subvariety E
with the above property is said to be the exceptional locus of (p.
The major difficulty in carrying out the minimal model program in dimension
> 3 arises when (p is of flipping type because in this case Z is not Q-factorial;
still worse, it is not Q-Goren&tein. If dimX = 2 or X is a nonsingular 3-fold
[M] we have no flipping extremal rays, i.e., rays with flipping contractions.
8. E x t r e m a l modifications. There is an entirely natural
FLIP CONJECTURE. Any extremal ray R of flipping type has an adjoint
diagram or a flip. This should be a commutative diagram
X »*-'*_> X+
£>=cont/j \
/ (p+
z
consisting of:
8.1. A normal projective Q-Gorenstein variety X+, which is said to be an
extremal transform of X in the ray R,
8.2. A rational map tr^ : X —• X+ which is an isomorphism except for loci
of codimension > 2. The map tr^ is an extremal modification in the ray R.
8.3. A contraction tp+ such that a canonical divisor Kx+ is relatively ample
for <p+. (Recall that — Kx is relatively ample for (p.)
8.4. PROPOSITION (on a platter on one's head).
An adjoint diagram
exists if and only if for some ample divisor H G Biv Z and any integer ra ^> 0
the adjoint-canonical ring
R(mip*H + Kx) = 0 H°(X, Ox(n(m<p*H + Kx)))
n>0
678
V. V. SHOKUROV
is finitely generated. Moreover, in this case
X+ = Proj R(m<p*H +
Kx).
A flip has the following properties:
8.5. If an extremal ray has a flip, then it is unique.
8.6. X+ is Q-factorial and has only terminal singularities.
8.7. Moreover, for any common desingularization
W
X
X+
we have a* > ai, with af > ai, if and only if <p o g(Ei) Ç E, where a« and af
are discrepancies of g and h at E{ respectively and E is the exceptional locus of
<P8.8. p(X+) = p(X).
The flip conjecture holds for flipping extremal rays with toric contractions, so
there is good reason to believe the flip conjecture to hold in general. In dimension
three we have
9. Some sufficient conditions for the existence of a flip. The best one
now is
KAWAMATA CONDITION. Let (p: X -> Z be a contraction for a flipping
extremal ray R and dimX = 3. Then a flip in R exists if the divisor Kx + D/2
is log-terminal for some effective reduced Weil divisor D G \ — 2Kx + <p*H\,
where H G Div Z.
Note that D is Q-Cartier. A divisor Kx + 22/2 is said to be log-terminal if
there is a desingularization / : Y —• X for which
(i) D' + 53 Ei is a normal crossing divisor, where D' is the strict transform of
D by / and £ Ei is the exceptional locus of / ;
(ii) 2(ai + 1) > ri for all i, where ai is the discrepancy at Ei and f*D =
D' + iZnEi.
It is known that (ii) does not depend on the choice of desingularization with
(i). So the Kawamata condition states that a flip exists if a general divisor
D G | — 2Kx + <P*H\ has mild singularities in the above sense.
Another condition concerns semistable extremal rays. A reduced Weil divisor
D C X is said to be simple if D is Q-Cartier and there is a desingularization
f:Y—*X
such that f*D = D'+^2i=1 Ei and it is a divisor with normal crossing,
1
where D is the strict transform of D by / and J2i=i Ei is the exceptional locus
of / . A minimal d for such desingularizations is the depth of D. An extremal ray
R is semistable when there exists a simple divisor D on X such that any curve
C Ç D provided the numerical class of C belongs to R.
9 . 1 . THEOREM (TSUNODA, SHOKUROV, MORI, KAWAMATA). If R is
a flipping semistable ray, then there exists a flip in R. Besides, if R is semistable
for D, then ìVR D is simple and the depth oftrji D is less than that of D.
NUMERICAL GEOMETRY OF ALGEBRAIC VARIETIES
679
We may apply the last theorem to a semistable family / : X —• C of surfaces
over a curve if the general fibre f~1(c),cG
C, is a minimal surface. For example,
in such a way a Kulikov relative model of a stable one-dimensional family of KS
surfaces is obtained after some easy modification [SI].
10. Termination. Now we turn to yet another difficulty in the minimal
model program. First note that p(Z) = p(X) — 1 for divisorial contractions of
extremal rays and p(X+) = p(X) for modifications in flipping extremal rays. So
there arises the problem as to how long we may produce modifications in flipping
extremal rays. The
TERMINATION CONJECTURE states that a sequence of modifications in flipping extremal rays must always terminate, i.e., there does not exist an infinite
sequence of such modifications.
This conjecture together with the flip conjecture implies that there exists a
finite chain of flipping modifications
Jf—>jr+—•
+X<+n) =Z
such that either
(i) Kz is nef, or
(ii) Z has a contraction of fibre type, or
(iii) Z has a contraction of divisorial type.
So they imply the model conjecture.
1 0 . 1 . THEOREM (SHOKUROV,
holds when dimX < 4.
10.2. COROLLARY.
dimension < 4.
KAWAMATA).
The termination conjecture
The flip conjecture implies the model conjecture in
11. Abundance. Let X be a projective normal variety and DìVQ X be a nef
Q-divisor on X. It is known that K(D) < v(D) (cf. §2). D is said to be abundant
if the equation K(D) = v(D) holds.
ABUNDANCE CONJECTURE. If X i s a minimal variety, then Kx is abundant.
We say that X is a good minimal variety if Kx is abundant.
1 1 . 1 . THEOREM (KAWAMATA). Let X be a good minimal variety. Then
the canonical divisor Kx is semiample. So there is a contraction
f:X—*Z
such that
(i) dim Z = v(Kx),
(ii) Kx\f-i(K) = 0 for all zGZ.
The abundance conjecture is true when v(Kx) = dimX, u(Kx)
when K(KX) = K(X) = dimX - 1. Moreover, we have
= 0 and
11.2. THEOREM (KAWAMATA). The model and the abundance conjectures for dimension < dim X — k imply that a minimal variety X is good when
K(X) > fc.
680
V. V. SHOKUROV
So if the model conjecture is true we have only to establish the following
statement in order to prove the abundance conjecture:
WEAK ABUNDANCE CONJECTURE. If X is a minimal variety and v(Kx) >
1, then K(X) > 1.
We have only the following result in dimension three.
11.3. THEOREM ( MlYAOKA). Let X be a minimal variety and dim X = 3.
Then K(X) > 0.
12. Bibliographical notes. A complete account of the current state in the
minimal model program and its applications can be found in [KMM].
The notion of a canonical and a terminal singularity and that of a canonical
model were introduced by Reid in [Rl]. Minimal models in dimension three
emerge and are constructed from canonical models in [R2]. Of the algebraic
3-folds, nonsingular Fano 3-folds seem to be the only class that has been studied
extensively in recent years (see [I, MM] and their references). In the singular
three-dimensional case we have only one essential result about Fano 3-folds,
which says that if X is a Gorenstein Fano 3-fold then a general divisor in the
anticanonical linear system | — Kx\ is a K3 surface, possibly with canonical
singularities [R3]. An important result on birational automorphisms of conic
bundles was obtained by Sarkisov [S].
The nonvanishing and the generalized vanishing theorem originally proved in
[S2, K I , V] are also given in [Koll, K2, K3]. The contraction theorem was
first proved only in dimension < 3, in the case where the nonvanishing theorem
follows immediately from the Riemann-Roch theorem [K4, B, K5, S3]. For
any dimension the contraction theorem was proved in [S2], where the author
improved the technique developed by Kawamata, Benveniste, and Reid [R3].
The first proof of the cone theorem for nonsingular varieties [M] has used
the ingenious method of modulo p reduction. Then in [K5, R3, S4] for dimension < 3 the cone theorem was derived from a combination of the contraction
theorem and the rationality theorem, which was proved by the same technique
as that used in the proof of the contraction theorem. Improving this technique
Kawamata and Kollär [K3, Kol2] have proved the theorem in the general case.
The flip conjecture was first stated and proved for toric morphisms in [R4].
General properties of flips can be found in [S2]. Sufficient conditions of semistable type (cf. Theorem 9.1) were first introduced by Tsunoda [T]. Other approaches were developed by Mori, Kawamata [K6], and by the author [S5].
Kawamata's and some other conditions are described in [K6].
Termination in dimension three was proved in [S2] and in dimension four in
[KMM].
The abundance conjecture and related results are discussed in [K2]. For the
last proof of the Miyoka theorem see [Mi].
Note added in proof. Now theflipand the model conjectures hold in dimension
3; see S. Mori, Flip theorem and the existence of minimal models for 3-folds,
Preprint.
NUMERICAL GEOMETRY OF ALGEBRAIC VARIETIES
681
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Y A R O S L A V L S T A T E P E D A G O G I C A L I N S T I T U T E , Y A R O S L A V L 150000, USSR