Interpretations of the Kodaira dimension

The following integers are equal if they are non-negative. A good reference is Lazarsfeld (2004), Theorem 2.1.33.

• The dimension of the Proj construction ${\displaystyle \operatorname {Proj} R(K_{X})}$, a projective variety called the canonical model of X depending only on the birational equivalence class of X. (This is defined only if the canonical ring ${\displaystyle R=R(K_{X})}$is finitely generated, which is true in characteristic zero and conjectured in general.)
• The dimension of the image of the d-canonical mapping for all positive multiples d of some positive integer ${\displaystyle d_{0}}$.
• The transcendence degree of the fraction field of R, minus one; i.e. ${\displaystyle t-1}$, where t is the number of algebraically independent generators one can find.
• The rate of growth of the plurigenera: that is, the smallest number κ such that ${\displaystyle P_{d}/d^{\kappa }}$ is bounded. In Big O notation, it is the minimal κ such that ${\displaystyle P_{d}=O(d^{\kappa })}$.

When one of these numbers is undefined or negative, then all of them are. In this case, the Kodaira dimension is said to be negative or to be ${\displaystyle -\infty }$. Some historical references define it to be −1, but then the formula ${\displaystyle \kappa (X\times Y)=\kappa (X)+\kappa (Y)}$ does not always hold, and the statement of the Iitaka conjecture becomes more complicated. For example, the Kodaira dimension of ${\displaystyle \mathbf {P} ^{1}\times X}$ is ${\displaystyle -\infty }$ for all varieties X.

Application

The Kodaira dimension gives a useful rough division of all algebraic varieties into several classes.

Varieties with low Kodaira dimension can be considered special, while varieties of maximal Kodaira dimension are said to be of general type.

Geometrically, there is a very rough correspondence between Kodaira dimension and curvature: negative Kodaira dimension corresponds to positive curvature, zero Kodaira dimension corresponds to flatness, and maximum Kodaira dimension (general type) corresponds to negative curvature.

The specialness of varieties of low Kodaira dimension is analogous to the specialness of Riemannian manifolds of positive curvature (and general type corresponds to the genericity of non-positive curvature); see classical theorems, especially on Pinched sectional curvature and Positive curvature.

These statements are made more precise below.

Dimension 1

Smooth projective curves are discretely classified by genus, which can be any natural number g = 0, 1, ....

Here "discretely classified" means that for a given genus, there is an irreducible moduli space of curves of that genus.

The Kodaira dimension of a curve X is:

• κ = ${\displaystyle -\infty }$: genus 0 (the projective line P¹): K_X is not effective, P_d = 0 for all d > 0.
• κ = 0: genus 1 (elliptic curves): K_X is a trivial bundle, P_d = 1 for all d ≥ 0.
• κ = 1: genus g ≥ 2: K_X is ample, P_d = (2d − 1)(g − 1) for all d ≥ 2.

Compare with the Uniformization theorem for surfaces (real surfaces, since a complex curve has real dimension 2): Kodaira dimension ${\displaystyle -\infty }$ corresponds to positive curvature, Kodaira dimension 0 corresponds to flatness, Kodaira dimension 1 corresponds to negative curvature. Note that most algebraic curves are of general type: in the moduli space of curves, two connected components correspond to curves not of general type, while all the other components correspond to curves of general type. Further, the space of curves of genus 0 is a point, the space of curves of genus 1 has (complex) dimension 1, and the space of curves of genus g ≥ 2 has dimension 3g − 3.

the classification table of algebraic curves
Kodaira dimension 　κ(C)
	genus of C : g(C)	structure
${\displaystyle 1}$	${\displaystyle \geq 2}$	curve of general type
${\displaystyle 0}$	${\displaystyle 1}$	elliptic curve
${\displaystyle -\infty }$	${\displaystyle 0}$	the projective line ${\displaystyle \mathbb {P} ^{1}}$

Dimension 2

The Enriques–Kodaira classification classifies algebraic surfaces: coarsely by Kodaira dimension, then in more detail within a given Kodaira dimension. To give some simple examples: the product P¹ × X has Kodaira dimension ${\displaystyle -\infty }$ for any curve X; the product of two curves of genus 1 (an abelian surface) has Kodaira dimension 0; the product of a curve of genus 1 with a curve of genus at least 2 (an elliptic surface) has Kodaira dimension 1; and the product of two curves of genus at least 2 has Kodaira dimension 2 and hence is of general type.

the classification table of algebraic surfaces
Kodaira dimension 　κ(C)
	geometric genus pg	irregularity q	structure
${\displaystyle 2}$			surface of general type
${\displaystyle 1}$			elliptic surface
${\displaystyle 0}$	${\displaystyle 1}$	${\displaystyle 2}$	abelian surface
	${\displaystyle 0}$	${\displaystyle 1}$	hyperelliptic surface
	${\displaystyle 1}$	${\displaystyle 0}$	K3 surface
	${\displaystyle 0}$	${\displaystyle 0}$	Enriques surface
${\displaystyle -\infty }$	${\displaystyle 0}$	${\displaystyle \geq 1}$	ruled surface
	${\displaystyle 0}$	${\displaystyle 0}$	rational surface

For a surface X of general type, the image of the d-canonical map is birational to X if d ≥ 5.

Any dimension

Rational varieties (varieties birational to projective space) have Kodaira dimension ${\displaystyle -\infty }$. Abelian varieties (the compact complex tori that are projective) have Kodaira dimension zero. More generally, Calabi–Yau manifolds (in dimension 1, elliptic curves; in dimension 2, abelian surfaces, K3 surfaces, and quotients of those varieties by finite groups) have Kodaira dimension zero (corresponding to admitting Ricci flat metrics).

Any variety in characteristic zero that is covered by rational curves (nonconstant maps from P¹), called a uniruled variety, has Kodaira dimension −∞. Conversely, the main conjectures of minimal model theory (notably the abundance conjecture) would imply that every variety of Kodaira dimension −∞ is uniruled. This converse is known for varieties of dimension at most 3.

Siu (2002) proved the invariance of plurigenera under deformations for all smooth complex projective varieties. In particular, the Kodaira dimension does not change when the complex structure of the manifold is changed continuously.

the classification table of algebraic three-folds
Kodaira dimension 　κ(C)
	geometric genus 　pg	irregularity q	examples
${\displaystyle 3}$			three-fold of general type
${\displaystyle 2}$			fibration over a surface with general fiber an elliptic curve
${\displaystyle 1}$			fibration over a curve with general fiber a surface with κ = 0
${\displaystyle 0}$	${\displaystyle 1}$	${\displaystyle 3}$	abelian variety
	${\displaystyle 0}$	${\displaystyle 2}$	fiber bundle over an abelian surface whose fibers are elliptic curves
	${\displaystyle 0}$　or ${\displaystyle 1}$	${\displaystyle 1}$	fiber bundle over an elliptic curve whose fibers are surfaces with κ = 0
	${\displaystyle 0}$　or ${\displaystyle 1}$	${\displaystyle 0}$	Calabi–Yau 3-fold
${\displaystyle -\infty }$	${\displaystyle 0}$	${\displaystyle \geq 1}$	uniruled 3-folds
	${\displaystyle 0}$	${\displaystyle 0}$	rational 3-folds, Fano 3-folds, and others

A fibration of normal projective varieties X → Y means a surjective morphism with connected fibers.

For a 3-fold X of general type, the image of the d-canonical map is birational to X if d ≥ 61.^[4]