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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.

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  $  \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 [[#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====
<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====
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

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

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 [[#References|[a2]]], Chapt. 11.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. van de Ven, "Compact complex surfaces" , Springer (1984) {{MR|}} {{ZBL|0718.14023}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Iitaka, "Algebraic geometry, an introduction to birational geometry of algebraic varieties" , Springer (1982) pp. Chapt. 10 {{MR|}} {{ZBL|0491.14006}} </TD></TR></table>