Pluricanonical ring

In mathematics, the pluricanonical ring of an algebraic variety "V" (which is non-singular), or of a complex manifold, is the graded ring $R(V,K)=R(V,K\_V)$ of sections of powers of the canonical bundle "K".Its "n"th graded component (for $ngeq\; 0$) is:

:$R\_n\; :=\; H^0(V,\; K^n),$

that is, the space of sections of the "n"-th tensor product "K"^"n" of the canonical bundle "K".

The 0th graded component $R\_0$ is sections of the trivial bundle, and is canonically the ring of regular functions; thus "R" is an algebra over the regular functions on "V".

One can define an analogous ring for any line bundle "L" over "V"; the analogous dimension is called the Iitaka dimension. A line bundle is called big if the Iitaka dimension equals the dimension of the variety.

Birational invariance

The canonical ring and therefore likewise the Kodaira dimension is a birationalinvariant: Any birational map between smooth compact complex manifolds inducesan isomorphism between the respective canonical rings. As a consequence one candefine the Kodaira dimension of a singular space as the Kodaira dimension of adesingularization. Due to the birational invariance this is well defined, i.e., independent of the choice of the desingularization.

Fundamental conjecture of birational geometry

A basic conjecture is that the pluricanonical ring is finitely generated. This is considered a major step in the Mori program.

Announcements were made in 2007 claiming proofs [ [http://arxiv.org/abs/math.AG/0610203 Proof by Caucher Birkar, Paolo Cascini, Christopher D. Hacon, James McKernan] ] [ [http://arxiv.org/abs/math.AG/0610740 Proof by Yum-Tong Siu (analytic)] ] .

The plurigenera

The dimension

:$P\_n\; =\; h^0(V,\; K^n)\; =\; operatorname\{dim\}\; H^0(V,\; K^n)$

is the classically-defined "n"-th "plurigenus" of "V".The pluricanonical divisor $K^n$, via the corresponding linear system of divisors, gives a map to projective space $mathbf\{P\}(H^0(V,\; K^n))\; =\; mathbf\{P\}^\{P\_n\; -\; 1\}$,called the "n"-canonical map.

Kodaira dimension

The size of "R" is a basic invariant of "V", and is called the Kodaira dimension.

Definition

The Kodaira dimension, named for Kunihiko Kodaira, of "V" is defined as any of the following:

* The dimension of the Proj construction $operatorname\{Proj\},\; R$ (this variety is called the canonical model of "V").

* The dimension of the image of the "n"-canonical mapping for "n" large enough.

* The transcendence degree of "R", minus one, i.e. "t" − 1, where "t" is the number of algebraically independent generators one can find.

* The rate of growth of the plurigenera: it is the smallest number κ such that $P\_n/n^kappa,$ is bounded. In Big O notation, it is the minimal κ such that $P\_n\; =\; O(n^kappa)$.

Conventionally, when "R" is trivial ($R=R\_0$ = underlying field (constant functions); the plurigenera are all zero (other than $P\_0=1$); the pluricanonical divisors are not effective), which happens for example when "V" is rational, one takes $kappa=-1$ (in agreement with the transcendence degree definition), or sometimes $-infty$ (which is a conventional dimension of the empty set, as it preserves additivity under multiplication).

Kodaira dimensions can take any value from −1 to the dimension of "V".

Application

The Kodaira dimension is a relatively coarse invariant, and helps to give the outline for the classification of algebraic varieties: it is coarse in that there are generally several distinct families of varieties with a given Kodaira dimension.

Varieties with low Kodaira dimension are special, while varieties of maximal Kodaira dimension are (suggestively) called general type.

Geometrically, there is a 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 corresponds 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".

The above statements are made more precise below.

Dimension 1

Complex non-singular algebraic curves are discretely classified by genus, which can be any natural number $g=0,1,ldots$.

By "discretely classified" we mean that for a given genus, there is a connected, irreducible moduli space of curves of that genus.

The Kodaira dimension of an algebraic curve is:
* $kappa=-1$: genus 0 (projective line): "K" is not effective, $P\_n\; =\; 0$
* $kappa=0$: genus 1 (elliptic curves): "K" is a trivial bundle, $P\_n\; =\; 1$
* $kappa=1$: genus 2 or more: "K" is ample

Compare with the Uniformization theorem for surfaces (real surfaces, which are the analogue in differential geometry of algebraic curves): Kodaira dimension -1 corresponds to positive curvature, Kodaira dimension 0 corresponds to flat, Kodaira dimension 1 corresponds to negative curvature. Note that the generic surface is of general type: in the moduli space of surfaces, 2 components have Kodaira dimension below 1, while all other components have Kodaira dimension 1. Further, the component corresponding to genus 0 is a point, to genus 1 is 1-dimensional, and to genus $ggeq\; 2$ is $(3g-3)$-dimensional.

Dimension 2

The Enriques-Kodaira classification classifies surfaces: coarsely by Kodaira dimension, then in more detail within a given dimension.

General dimension

Rational varieties have negative Kodaira dimension (corresponding to positive curvature). Abelian varieties and Calabi-Yau manifolds (in dimension 1, elliptic curves; in dimension 2, complex tori and K3 surfaces) have Kodaira dimension zero (corresponding to admitting flat metrics and Ricci flat metrics, respectively).

General type

A variety of general type "V" is one of maximal Kodaira dimension (Kodaira dimension equal to its dimension)::$kappa(V)\; =\; operatorname\{dim\}\; V$

Equivalently, "K" is a big line bundle; equivalently, the "n"-canonical map is generically injective for "n" sufficiently large.

For example, a variety with ample canonical bundle is of general type.

In some sense varieties of general type are generic, hence the term(discrete invariants of varieties of general type vary in more dimensions, and moduli space of varieties of general type have more dimensions; this is made more precise for curves and surfaces). A smooth hypersurface of degree "d" in the "n"-dimensional projective space is of general type if and only if "d" is greater than "n+1". In this sense most smoothhypersurfaces in the complex projective space are of general type.

Varieties of general type are poorly understood, even for surfaces.For instance, in the Enriques-Kodaira classification, the surfaces of general type are a category, but are not described in much more detail. It is not even known what Chern numbers can be realized.

Examples

For genus $ggeq\; 22$, the Deligne-Mumford moduli space of curves $mathcal\{M\}\_g$ is of general type. [ [http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-727Spring-2006/EF58E7E1-1981-419A-B6F8-2E4607C35F9B/0/generaltype.pdf The Kodaira Dimension of the Moduli Space of Curves] ]

Application to classification

An important idea is that "Kodaira dimensions add" in fibrations. This motivates a classification programme for algebraic varieties, in which it is sought to represent "V" as a fibration over a variety of general type, with typical fiber of Kodaira dimension 0. This is quite a natural idea, given that the application of the Proj construction to the pluricanonical ring should produce a projective variety in which the sections of powers of "K" 'capture' as much as they can about "V".

Notes

References

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