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A numerical invariant of an [[Algebraic variety|algebraic variety]], named after K. Kodaira who first pointed out the importance of this invariant in the theory of the classification of algebraic varieties.
 
A numerical invariant of an [[Algebraic variety|algebraic variety]], named after K. Kodaira who first pointed out the importance of this invariant in the theory of the classification of algebraic varieties.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k0556301.png" /> be a non-singular algebraic variety and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k0556302.png" /> be a [[Rational mapping|rational mapping]] defined by a [[Linear system|linear system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k0556303.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k0556304.png" /> is the [[Canonical class|canonical class]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k0556305.png" />. The Kodaira dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k0556306.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k0556307.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k0556308.png" />. Here, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k0556309.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563010.png" />, then it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563011.png" />. The Kodaira dimension is a birational invariant, that is, it does not depend on the representative in the birational equivalence class.
+
Let $  V $
 +
be a non-singular algebraic variety and let $  \Phi _ {m} : V \rightarrow \mathbf P ( n) $
 +
be a [[Rational mapping|rational mapping]] defined by a [[Linear system|linear system]] $  | m K _ {V} | $,  
 +
where $  K _ {V} $
 +
is the [[Canonical class|canonical class]] of $  V $.  
 +
The Kodaira dimension $  \kappa ( V) $
 +
of $  V $
 +
is defined as $  \max _ {m>} 1 \{  \mathop{\rm dim}  \Phi _ {m} ( V) \} $.  
 +
Here, if $  | m K _ {V} | = \emptyset $
 +
for all $  m \geq  1 $,  
 +
then it is assumed that $  \kappa ( V) = - \infty $.  
 +
The Kodaira dimension is a birational invariant, that is, it does not depend on the representative in the birational equivalence class.
  
Suppose that the ground field is the field of the complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563013.png" /> is sufficiently large, then one has the estimate
+
Suppose that the ground field is the field of the complex numbers $  \mathbf C $.  
 +
If $  m $
 +
is sufficiently large, then one has the estimate
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563014.png" /></td> </tr></table>
+
$$
 +
\alpha m ^ {\kappa ( V) }
 +
\leq    \mathop{\rm dim}  | m K _ {V} |  \leq  \
 +
\beta m ^ {\kappa ( V) } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563016.png" /> are certain positive numbers. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563017.png" />, then there exists a surjective morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563018.png" /> of algebraic varieties such that: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563019.png" /> is birationally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563020.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563021.png" />; and c) for some dense open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563022.png" />, all the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563024.png" />, are varieties of parabolic type (i.e. of Kodaira dimension zero).
+
where $  \alpha $,  
 +
$  \beta $
 +
are certain positive numbers. If $  \kappa ( V) > 0 $,  
 +
then there exists a surjective morphism $  f : V  ^ {*} \rightarrow W $
 +
of algebraic varieties such that: a) $  V  ^ {*} $
 +
is birationally equivalent to $  V $;  
 +
b) $  \kappa ( V) = \mathop{\rm dim}  W $;  
 +
and c) for some dense open set $  U \subset  W $,  
 +
all the fibres $  f ^ { - 1 } ( \omega ) $,  
 +
$  \omega \in U $,  
 +
are varieties of parabolic type (i.e. of Kodaira dimension zero).
  
There is a generalization of the notion of the Kodaira dimension (see [[#References|[2]]]) to the case when in the linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563025.png" /> the canonical class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563026.png" /> is replaced by an arbitrary [[Divisor|divisor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563027.png" />.
+
There is a generalization of the notion of the Kodaira dimension (see [[#References|[2]]]) to the case when in the linear system $  | m K _ {V} | $
 +
the canonical class $  K _ {V} $
 +
is replaced by an arbitrary [[Divisor|divisor]] $  D $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Algebraic surfaces" ''Proc. Steklov Inst. Math.'' , '''75''' (1967) ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) {{MR|1392959}} {{MR|1060325}} {{ZBL|0830.00008}} {{ZBL|0733.14015}} {{ZBL|0832.14026}} {{ZBL|0509.14036}} {{ZBL|0492.14024}} {{ZBL|0379.14006}} {{ZBL|0253.14006}} {{ZBL|0154.21001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> K. Ueno, "Classification theory of algebraic varieties and compact complex spaces" , Springer (1975) {{MR|0506253}} {{ZBL|0299.14007}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Iitaka, "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563028.png" />-dimensions of algebraic varieties" ''J. Math. Soc. Japan'' , '''23''' (1971) pp. 356–373 {{MR|285531}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Algebraic surfaces" ''Proc. Steklov Inst. Math.'' , '''75''' (1967) ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) {{MR|1392959}} {{MR|1060325}} {{ZBL|0830.00008}} {{ZBL|0733.14015}} {{ZBL|0832.14026}} {{ZBL|0509.14036}} {{ZBL|0492.14024}} {{ZBL|0379.14006}} {{ZBL|0253.14006}} {{ZBL|0154.21001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> K. Ueno, "Classification theory of algebraic varieties and compact complex spaces" , Springer (1975) {{MR|0506253}} {{ZBL|0299.14007}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Iitaka, "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563028.png" />-dimensions of algebraic varieties" ''J. Math. Soc. Japan'' , '''23''' (1971) pp. 356–373 {{MR|285531}} {{ZBL|}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563029.png" /> be a compact connected [[Complex manifold|complex manifold]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563030.png" /> be the canonical bundle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563031.png" />. There is a canonical pairing of sections
+
Let $  X $
 +
be a compact connected [[Complex manifold|complex manifold]]. Let $  {\mathcal K} _ {X} $
 +
be the canonical bundle on $  X $.  
 +
There is a canonical pairing of sections
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563032.png" /></td> </tr></table>
+
$$
 +
\Gamma ( X, {\mathcal K} _ {X} ^ {\otimes m } ) \otimes
 +
\Gamma ( X, {\mathcal K} _ {X} ^ {\otimes n } )  \rightarrow \
 +
\Gamma ( X, {\mathcal K} _ {X} ^ {m + n } )
 +
$$
  
making <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563033.png" /> into a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563034.png" />, called the canonical ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563035.png" />. It can be proved to be of finite transcendence degree, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563036.png" />. The Kodaira dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563037.png" /> is now described as follows:
+
making $  \mathbf C \oplus \oplus _ {m = 1 }  ^  \infty  \Gamma ( X, {\mathcal K} _ {X}  ^ {m} ) $
 +
into a commutative ring $  R ( X) $,  
 +
called the canonical ring of $  X $.  
 +
It can be proved to be of finite transcendence degree, $  \textrm{ tr  deg  } ( R ( X)) < \infty $.  
 +
The Kodaira dimension of $  X $
 +
is now described as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563038.png" /></td> </tr></table>
+
$$
 +
\kappa ( X)  = - \infty \  \textrm{ if }  R ( X) \simeq \mathbf C ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563039.png" /></td> </tr></table>
+
$$
 +
\kappa ( X)  = \textrm{ tr deg  } ( R ( X)) - 1 \  \textrm{ otherwise } .
 +
$$
  
It is always true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563041.png" /> is the algebraic dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563042.png" />, i.e. the transcendence degree of the field of meromorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563043.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563044.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563045.png" />-th plurigenus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563046.png" />. Then one has: i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563047.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563048.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563049.png" />; ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563050.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563051.png" /> or 1 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563052.png" />, but not always 0; iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563053.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563054.png" />, if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563055.png" /> has growth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563056.png" />, i.e. if and only if there exists an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563057.png" /> and strictly positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563059.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563060.png" /> for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055630/k05563061.png" />.
+
It is always true that $  \kappa ( X) \leq  a ( X) \leq  \mathop{\rm dim} ( X) $,  
 +
where $  a ( X) $
 +
is the algebraic dimension of $  X $,  
 +
i.e. the transcendence degree of the field of meromorphic functions on $  X $.  
 +
Let $  P _ {m} ( X) = h  ^ {0} ( {\mathcal K} _ {X} ^ {\otimes m } ) = \mathop{\rm dim}  H  ^ {0} ( {\mathcal K} _ {X} ^ {\otimes m } ) $
 +
be the $  m $-
 +
th plurigenus of $  X $.  
 +
Then one has: i) $  \kappa ( X) = - \infty $
 +
if and only if $  P _ {m} ( X) = 0 $
 +
for all $  m \geq  1 $;  
 +
ii) $  \kappa ( X) = 0 $
 +
if and only if $  P _ {m} ( X) = 0 $
 +
or 1 for $  m \geq  1 $,  
 +
but not always 0; iii) $  \kappa ( X) = k $,  
 +
with $  1 \leq  k \leq  \mathop{\rm dim} ( X) $,  
 +
if and only if $  P _ {m} ( X) $
 +
has growth $  m  ^ {k} $,  
 +
i.e. if and only if there exists an integer k $
 +
and strictly positive constants $  a $,  
 +
$  b $
 +
such that $  am  ^ {k} \leq  P _ {m} ( X) \leq  bm  ^ {k} $
 +
for large $  m $.
  
 
The Kodaira dimension is also called the canonical dimension. For the concept of the logarithmic Kodaira dimension see [[#References|[a2]]], Chapt. 11.
 
The Kodaira dimension is also called the canonical dimension. For the concept of the logarithmic Kodaira dimension see [[#References|[a2]]], Chapt. 11.

Latest revision as of 22:14, 5 June 2020


A numerical invariant of an algebraic variety, named after K. Kodaira who first pointed out the importance of this invariant in the theory of the classification of algebraic varieties.

Let $ V $ be a non-singular algebraic variety and let $ \Phi _ {m} : V \rightarrow \mathbf P ( n) $ be a rational mapping defined by a linear system $ | m K _ {V} | $, where $ K _ {V} $ is the canonical class of $ V $. The Kodaira dimension $ \kappa ( V) $ of $ V $ is defined as $ \max _ {m>} 1 \{ \mathop{\rm dim} \Phi _ {m} ( V) \} $. Here, if $ | m K _ {V} | = \emptyset $ for all $ m \geq 1 $, then it is assumed that $ \kappa ( V) = - \infty $. The Kodaira dimension is a birational invariant, that is, it does not depend on the representative in the birational equivalence class.

Suppose that the ground field is the field of the complex numbers $ \mathbf C $. If $ m $ is sufficiently large, then one has the estimate

$$ \alpha m ^ {\kappa ( V) } \leq \mathop{\rm dim} | m K _ {V} | \leq \ \beta m ^ {\kappa ( V) } , $$

where $ \alpha $, $ \beta $ are certain positive numbers. If $ \kappa ( V) > 0 $, then there exists a surjective morphism $ f : V ^ {*} \rightarrow W $ of algebraic varieties such that: a) $ V ^ {*} $ is birationally equivalent to $ V $; b) $ \kappa ( V) = \mathop{\rm dim} W $; and c) for some dense open set $ U \subset W $, all the fibres $ f ^ { - 1 } ( \omega ) $, $ \omega \in U $, are varieties of parabolic type (i.e. of Kodaira dimension zero).

There is a generalization of the notion of the Kodaira dimension (see [2]) to the case when in the linear system $ | m K _ {V} | $ the canonical class $ K _ {V} $ is replaced by an arbitrary divisor $ D $.

References

[1] I.R. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) MR1392959 MR1060325 Zbl 0830.00008 Zbl 0733.14015 Zbl 0832.14026 Zbl 0509.14036 Zbl 0492.14024 Zbl 0379.14006 Zbl 0253.14006 Zbl 0154.21001
[2] K. Ueno, "Classification theory of algebraic varieties and compact complex spaces" , Springer (1975) MR0506253 Zbl 0299.14007
[3] S. Iitaka, "On -dimensions of algebraic varieties" J. Math. Soc. Japan , 23 (1971) pp. 356–373 MR285531

Comments

Let $ X $ be a compact connected complex manifold. Let $ {\mathcal K} _ {X} $ be the canonical bundle on $ X $. There is a canonical pairing of sections

$$ \Gamma ( X, {\mathcal K} _ {X} ^ {\otimes m } ) \otimes \Gamma ( X, {\mathcal K} _ {X} ^ {\otimes n } ) \rightarrow \ \Gamma ( X, {\mathcal K} _ {X} ^ {m + n } ) $$

making $ \mathbf C \oplus \oplus _ {m = 1 } ^ \infty \Gamma ( X, {\mathcal K} _ {X} ^ {m} ) $ into a commutative ring $ R ( X) $, called the canonical ring of $ X $. It can be proved to be of finite transcendence degree, $ \textrm{ tr deg } ( R ( X)) < \infty $. The Kodaira dimension of $ X $ is now described as follows:

$$ \kappa ( X) = - \infty \ \textrm{ if } R ( X) \simeq \mathbf C , $$

$$ \kappa ( X) = \textrm{ tr deg } ( R ( X)) - 1 \ \textrm{ otherwise } . $$

It is always true that $ \kappa ( X) \leq a ( X) \leq \mathop{\rm dim} ( X) $, where $ a ( X) $ is the algebraic dimension of $ X $, i.e. the transcendence degree of the field of meromorphic functions on $ X $. Let $ P _ {m} ( X) = h ^ {0} ( {\mathcal K} _ {X} ^ {\otimes m } ) = \mathop{\rm dim} H ^ {0} ( {\mathcal K} _ {X} ^ {\otimes m } ) $ be the $ m $- th plurigenus of $ X $. Then one has: i) $ \kappa ( X) = - \infty $ if and only if $ P _ {m} ( X) = 0 $ for all $ m \geq 1 $; ii) $ \kappa ( X) = 0 $ if and only if $ P _ {m} ( X) = 0 $ or 1 for $ m \geq 1 $, but not always 0; iii) $ \kappa ( X) = k $, with $ 1 \leq k \leq \mathop{\rm dim} ( X) $, if and only if $ P _ {m} ( X) $ has growth $ m ^ {k} $, i.e. if and only if there exists an integer $ k $ and strictly positive constants $ a $, $ b $ such that $ am ^ {k} \leq P _ {m} ( X) \leq bm ^ {k} $ for large $ m $.

The Kodaira dimension is also called the canonical dimension. For the concept of the logarithmic Kodaira dimension see [a2], Chapt. 11.

References

[a1] A. van de Ven, "Compact complex surfaces" , Springer (1984) Zbl 0718.14023
[a2] S. Iitaka, "Algebraic geometry, an introduction to birational geometry of algebraic varieties" , Springer (1982) pp. Chapt. 10 Zbl 0491.14006
How to Cite This Entry:
Kodaira dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kodaira_dimension&oldid=23878
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article