In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the Iitaka dimension of a
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
''L'' on an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
''X'' is the dimension of the image of the
rational map
In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible.
Definition
Formal d ...
to
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
determined by ''L''. This is 1 less than the dimension of the
section ring
This is a glossary of algebraic geometry.
See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry.
...
of ''L''
:
The Iitaka dimension of ''L'' is always less than or equal to the dimension of ''X''. If ''L'' is not effective, then its Iitaka dimension is usually defined to be
or simply said to be negative (some early references define it to be −1). The Iitaka dimension of ''L'' is sometimes called L-dimension, while the dimension of a divisor D is called D-dimension. The Iitaka dimension was introduced by .
Big line bundles
A
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
is big if it is of maximal Iitaka dimension, that is, if its Iitaka dimension is equal to the dimension of the underlying variety. Bigness is a
birational
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational f ...
invariant: If is a birational morphism of varieties, and if ''L'' is a big line bundle on ''X'', then ''f''
*''L'' is a big line bundle on ''Y''.
All
ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...
s are big.
Big line bundles need not determine birational isomorphisms of ''X'' with its image. For example, if ''C'' is a
hyperelliptic curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form
y^2 + h(x)y = f(x)
where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dist ...
(such as a curve of genus two), then its
canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers, it ...
is big, but the rational map it determines is not a birational isomorphism. Instead, it is a two-to-one cover of the
canonical curve In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers, ...
of ''C'', which is a
rational normal curve In mathematics, the rational normal curve is a smooth, rational curve of degree in projective n-space . It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For it is the ...
.
Kodaira dimension
The Iitaka dimension of the canonical bundle of a
smooth variety In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smo ...
is called its
Kodaira dimension In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''.
Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
.
Iitaka conjecture
![Fig001(new03)](https://upload.wikimedia.org/wikipedia/commons/4/45/Fig001%28new03%29.png)
Consider on complex algebraic varieties in the following.
Let K be the
canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers, it ...
on M. The dimension of H
0(M,K
m), holomorphic sections of K
m, is denoted by P
m(M), called m-genus. Let
::
then N(M) becomes to be all of the positive integer with non-zero m-genus. When N(M) is not empty, for
m-pluricanonical map is defined as the map
:
where
are the bases of H
0(M,K
m). Then the image of
,
is defined as the submanifold of
.
For certain
let
be the m-pluricanonical map where W is the complex manifold embedded into projective space P
N.
In the case of surfaces with κ(M)=1 the above W is replaced by a curve C, which is an elliptic curve (κ(C)=0). We want to extend this fact to the general dimension and obtain the analytic fiber structure depicted in the upper right figure.
![Fig002(new001)](https://upload.wikimedia.org/wikipedia/commons/9/98/Fig002%28new001%29.png)
Given a birational map
, m-pluricanonical map brings the commutative diagram depicted in the left figure, which means that
, i.e. m-pluricanonical genus is birationally invariant.
![Fig003](https://upload.wikimedia.org/wikipedia/commons/8/8a/Fig003.png)
It is shown by Iitaka that given n-dimensional compact complex manifold ''M'' with its Kodaira dimension κ(M) satisfying 1 ≤ κ(M) ≤ n-1, there are enough large ''m''
1,''m''
2 such that
and
are birationally equivalent, which means there are the birational map
. Namely, the diagram depicted in the right figure is commutative.
Furthermore, one can select
that is birational with
and
that is birational with both
and
such that
::
is birational map, the fibers of
are simply connected and the general fibers of
::
have Kodaira dimension 0.
The above fiber structure is called the Iitaka fiber space. In the case of the surface S (''n'' = 2 = dim(S)), W
* is the algebraic curve, the fiber structure is of dimension 1, and then the general fibers have the Kodaira dimension 0 i.e. elliptic curve. Therefore, S is the elliptic surface. These fact can be generalized to the general ''n''. Therefore The study of the higher-dimensional birational geometry decompose to the part of κ=-∞,0,n and the fiber space whose fibers is of κ=0.
The following additional formula by Iitaka, called Iitaka conjecture, is important for the classification of algebraic varieties or compact complex manifolds.
This conjecture has been only partly solved, for example in the case of
Moishezon manifold In mathematics, a Moishezon manifold is a compact complex manifold such that the field of meromorphic functions on each component has transcendence degree equal the complex dimension of the component:
:\dim_\mathbfM=a(M)=\operatorname_\mathbf\ma ...
s. The classification theory might been said to be the effort to solve the Iitaka conjecture and lead another theorems that the three-dimensional variety V is
abelian if and only if κ(V)=0 and q(V)=3 and its generalization so on. The
minimal model program
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its or ...
might be led from this conjecture.
References
*
*
*{{Citation , last1=Ueno , first1 = Kenji , mr=0506253 , title=Classification theory of algebraic varieties and compact complex spaces , year=1975 , series=Lecture Notes in Mathematics , volume=439 , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
Birational geometry
Vector bundles