# Kodaira dimension

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

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=47509
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article