# 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 ) to the case when in the linear system $| m K _ {V} |$ the canonical class $K _ {V}$ is replaced by an arbitrary divisor $D$.

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