Kodaira dimension

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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 be a non-singular algebraic variety and let be a rational mapping defined by a linear system , where is the canonical class of . The Kodaira dimension of is defined as . Here, if for all , then it is assumed that . 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 . If is sufficiently large, then one has the estimate

where , are certain positive numbers. If , then there exists a surjective morphism of algebraic varieties such that: a) is birationally equivalent to ; b) ; and c) for some dense open set , all the fibres , , 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 the canonical class is replaced by an arbitrary divisor .


[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 be a compact connected complex manifold. Let be the canonical bundle on . There is a canonical pairing of sections

making into a commutative ring , called the canonical ring of . It can be proved to be of finite transcendence degree, . The Kodaira dimension of is now described as follows:

It is always true that , where is the algebraic dimension of , i.e. the transcendence degree of the field of meromorphic functions on . Let be the -th plurigenus of . Then one has: i) if and only if for all ; ii) if and only if or 1 for , but not always 0; iii) , with , if and only if has growth , i.e. if and only if there exists an integer and strictly positive constants , such that for large .

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


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