Table Of ContentNORTH-HOLLAND
MATHEMATICS STUDIES 73
Recent Progress of
Algebraic Geometry
in Japan
Edited by
MASAYOSHINAGATA
Kyoto University
1983
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FOREWORD
This volume contains survey reports on developments of
algebraic geometry in Japan during the 1970's.
Topics and writers were selected after consultation with
S. Iitaka, M. Maruyama and K. Veno.
The editor would like to express his thanks to these three
friends and to the writers for their cooperation.
M. Nagata
Recent Progressof Algebraic Geometry in Japan
North-Holland Mathematics Studies 73
Birational Geometry and Logarithmic Forms
Shigeru IITAKA
Introduction. In this report, we shall give perspective view on
the recent progress of birational geometry on non-complete algebraic
varieties.
Birational geometry for complete varieties is treated by Ueno in
this volume. Here we shall study birational invariant properties for
non-complete algebraic varieties. But since any given algebraic vari-
ety has a projective model, i.e., there exists a projective variety
birationally equivalent to the given variety, it is enough to consider
complete varieties for the study of birational invariant properties.
Such a study is classical birational geometry, which has powerful tools
such as regular forms, theory of genera, especially Kodaira dimension.
By these, structure of complete algebraic varieties has been studied,
in detail.
Fortunately, for any non-complete algebraic variety, the spaces
of logarithmic forms and the logarithmic genera are defined, which are
counterparts (or generalizations) of the spaces of regular forms and
the genera for a complete algebraic variety. By using these, we can
develop a certain kind of birational geometry, which is called proper
birational geometry.
SHIGERU IITAKA
In proper birational geometry, one uses strictly rational maps,
proper birational maps, and logarithmic Kodaira dimension, respectively,
in place of rational maps, birational maps, and Kodaira dimension.
Since proper birational maps between normal affine varieties turn
out to be isomorphisms, proper birational geometry includes affine
geometry, i.e., the study of affine (in particular, normalJ varieties,
as well as classical birational geometry. Thus, for instance, in
affine geometry one can develop the theory of classification of affine
surfaces which is an analog of the theory of birational classification
of surfaces by Italian geometers. We exhibit an example.
Let {w € C[x,y]} be reduced polynomials, which are classified
into four classes as follo\\ls:
class W in terms of appropriate variables Aut C[x,y, l/lP]
C
I w= ex , or = n(x - J.} <Xl-dimensional
m £
W= x y + 1 + alx +.••+ a£x (a£ I 0, finite
£>0)
m
II l/J = x y + 1 I-dimensional
w= xy 2-dimensional
*
C -polynomials, that is, there exists
l/
II 2 '¥ € C[x,y] such that
at most
{p € C2., '¥(p) - Al/Jd(p) = 0, l/J(p) I O}"'C* I-dimensional
for a general A and some d ~ 0
III pOlynomial of hyperbolic type finite
2
Birational Geometry and Logarithmic Forms
In this way, theory of algebras (of polynomials) and theory of
geometry of complete varieties are unified. Such a unification is
Algebraic Geometry in the original sense.
The contents are as follows:
§l. Strictly rational maps and birational equivalence.
§2. Non-singular models and a-manifolds.
§3. Logarithmic forms.
§4. Logarithmic genera.
§5. D-dimension and logarithmic Kodaira dimension.
§6. Fundamental theorems.
§7. Fundamental conjectures.
§8. Quasi-abelian varieties and quasi-Albanese maps.
§9. Algebraic surfaces.
§lO. Analytic theory of Kodaira dimension.
§U. Algebraic varieties of hyperbolic type.
§12. Examples.
§I. Strictly rationai maps and proper birationa1 equivalence.
Let f: VI ~ V be a rational map of algebraic varieties VI
2
and V If the points of indeterminacy of f are eliminated by a
2.
proper birational morphism ~: Z ~ VI' i.e., f.~ is a morphism,
then f i~ said to be a strictly rational map.
Example 1. When V is a complete algebraic variety, every
2
rational map f VI ~ V turns out to be a strictly rational map.
2
Lemma 1. Let f VI ~ V2 be a strictly rational map with
3
SHIGERU IITAKA
the assumption that VI is normal. Then codim(V - dom(f)) ~ 2
l
where dom(f) is the set of all points at which f is defined.
Lemma 2. with the same notation as above, let p E VI and
assume by definition) is discrete. Then p E
dom(f)
Moreover, a strictly rational map f: VI ~ V is said to be a
2
proper rational map, if there is a proper birational morphism ~ Z ~
VI such that g = f'~ is also a proper morphism.
Lemma 3.
(resp.proper) rational maps such that a composition hofl can be
defined, then f20fl : VI ~ V3 is also a strictly rational map. If
f : V~ V' is a strictly (resp.proper) birational map and if f-l : V' ~
V is also strictly birational, then f is a proper birational map.
Thus one defines an equivalence relation VI ~ Vz between alge-
braic varieties by the existence of a proper birational map: VI ~ V .
2
Lemma 4. Let VI and Vz be affine normal varieties. Then
a strictly rational map ~ : VI ~ Vz is a proper birational map if
and only if ~ is an isomorphism.
In particular, let R be an integral domain finitely generated
over k and by R' denote the integral closure of R. Then one obtains
in which P Bir(V) is the group consisting of all proper birational
maps of V into itself.
Thus, the study of proper birational equivalence classes, that
may be called proper birational geometry, includes that of normal
affine varieties.
4
Birational Geometry and Logarithmic Forms
§2. Non-singular models and a-manifolds.
In what follows, all varieties considered are defined over C.
For any algebraic variety V, there exists a proper birational morphism
* *
~ : V + V such that V is a non-singular algebraic variety. The
*
pair (V,~) is said to be a non-singular model of V. But this
is not sufficient for the detailed study of non-complete algebraic
varieties.
Given an algebraic variety V, there exists a complete algebraic
variety V which contains V as its open subset V. V is said to
be a completion of V The existence of V has been shown by Nagata.
-*
Using Hironaka's Main Theorem I, one obtains a non-singular model (V ,
p) of V • Furthermore, one may assume that the algebraic boundary
D* =-V* - V* has only normal crossings each
*
component of which is non-singular. Such a divisor D is said to
have simple normal crossings.
A reduced divisor D on a complete normal variety V is by
definition an effective divisor with each multiplicity I When V
is non-singular, D has normal crossings at p € D if D is locally
0v
defined by WE ~V,p ,such that W=sl"'sr € C[[sl'" ·'sn]] ,~ p
Here, (sl,.."sn) is a system of formal regular parameters at p .
Furthermore, D is said to have simple normal crossings at p, if
D is loca~ly defined by W€ 0V-,p ,such that where
(Zl",.,zn) is a system of regular parameters at p .
V
The pair of a complete non-singular algebraic variety and a
divisor D with nermal crossings on V is said to be a a-manifold.
One calls V the interior of (V, D) Occasionally, V is said to be
5
SHIGERU IITAKA
a completion of V with ordinary (smooth, if D has simple normal
crossings) boundary D.
*
A a-manifold (-Vi<, D) constructed previously for a given
-*
algebraic variety V will play the same role as (V, p) does for
*
complete varieties. One says that (-Vi<, D) is a a-manifold of V
Byusing a-manifolds one will develop the theory of logarithmic forms
and logarithmic genera, which is the central part of proper birational
geometry.
§3. Logarithmic forms.
Let (V, D) be a a-manifold whose interior is 'V. By nl(*D)
one denotes the sheaf of germs of rational I-forms whose poles are
I
contained in D P. Deligne has introduced the sheaf n (log D) on
V as follows: First assume D has only simple normal crossings.
(i) nl(log D) is an Oy-Module and is a subsheaf of- nl(*D) ,
(ii) for a point p E V, choose a system (zl"",zn) of local
coordinates around p such that zl",zr = 0 defines D. Then
r dz. n
W E QI(log D)P~ W j=Ll aJ. _Jz.J_ + i=Lr+l b1.dz1.
where the a. and the b. are regular functions around p Then
J 1
dz.
L 0- _J + L 0 -dz '" On
V,p Zj V,p i V
I
Thus Q (log D) is a locally free sheaf of rank n
When D has normal crossings, choose a system of formal regular
parameters (~l""'~n) at p such that D is defined by ~
1:;1" '~r .
6
Birational Geometry and Logarithmic Forms
nl
1hen define, for til € (*0)p ,
1 dt;.
W E: n (log 0) ¢=:} W = La. -;- + L8.dt;.
P J"j 1 1
o
where the a. and the 8. belong to
J 1 V,p
One checks easily
W E: nl(log 0) ~ 1/JUJ and 1/!dw are regular forms around p .
p
Even when 0 has only normal crossings, one can prove that nlClog 0)
is a locally free sheaf of rank n.
For a positive integer i define
which is called the sheaf of germs of logarithmic i-forms of V with
i
logarithmic pole along 0 . In short one says that n Clog 0) is
the sheaf of germs of logaritmnic i-forms of V. The spaces HOCV,
i
n Clog D)) were studied precisely by P. Deligne, who has shown that
these depend only on V and every element of these is d-closed.
In proper birational geometry, more complicated sheaves are used.
Let o· GLCn,C) + GLeE) be a polynomial representation, E being a
C-vector space with finite dimension. Then, a locally free sheaf
nlClog 0)0 is constructed, whose rank is dim E The space HOCV,
nllog 0)0) is the space of logarithmic a-forms on V. The useful
representation is the tensor product of m times tensor products of
i
i-th skew symmetric representations, which we denote by M= Cml, ...,m .
n)
Let f: W+ V be a morphism between complete non-singular varieties.
Let 0 and O be divisors with normal crossings on V and W
1 2
respectively, such that f-1CD O Then by pulling back of
1) C 2,
rational forms, one obtains a linear map
7
