In algebraic geometry, the Kodaira dimension κ(X) measures the size of the canonical model of a projective variety X. Soviet mathematician Igor Shafarevich in a seminar introduced an important numerical invariant of surfaces with the notation κ.
[1] Japanese mathematician Shigeru Iitaka extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension),[2] and later named it after Kunihiko Kodaira.
For an integer d, the dth tensor power of KX is again a line bundle.
For d ≥ 0, the vector space of global sections H0(X,KXd) has the remarkable property that it is a birational invariant of smooth projective varieties X.
For d ≥ 0, the dth plurigenus of X is defined as the dimension of the vector space of global sections of KXd: The plurigenera are important birational invariants of an algebraic variety.
In particular, the simplest way to prove that a variety is not rational (that is, not birational to projective space) is to show that some plurigenus Pd with d > 0 is not zero.
The Kodaira dimension gives a useful rough division of all algebraic varieties into several classes.
Smooth projective curves are discretely classified by genus, which can be any natural number g = 0, 1, ....
Abelian varieties (the compact complex tori that are projective) have Kodaira dimension zero.
Siu (2002) proved the invariance of plurigenera under deformations for all smooth complex projective varieties.
A fibration of normal projective varieties X → Y means a surjective morphism with connected fibers.
For example, a smooth hypersurface of degree d in the n-dimensional projective space is of general type if and only if
Nonetheless, there are some strong positive results about varieties of general type.
For example, Enrico Bombieri showed in 1973 that the d-canonical map of any complex surface of general type is birational for every
More generally, Christopher Hacon and James McKernan, Shigeharu Takayama, and Hajime Tsuji showed in 2006 that for every positive integer n, there is a constant
such that the d-canonical map of any complex n-dimensional variety of general type is birational when
There is a natural rational map X – → B; any morphism obtained from it by blowing up X and B is called the Iitaka fibration.
The minimal model and abundance conjectures would imply that the general fiber of the Iitaka fibration can be arranged to be a Calabi–Yau variety, which in particular has Kodaira dimension zero.
[5] In this sense, X is decomposed into a family of varieties of Kodaira dimension zero over a base (B, Δ) of general type.
Given the conjectures mentioned, the classification of algebraic varieties would largely reduce to the cases of Kodaira dimension
The minimal model and abundance conjectures would imply that every variety of Kodaira dimension
The Iitaka conjecture helped to inspire the development of minimal model theory in the 1970s and 1980s.
It is now known in many cases, and would follow in general from the minimal model and abundance conjectures.
Although the base space is not required to be algebraic, the assumption that all the fibers are isomorphic is very special.