Pluricanonical ring
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In mathematics, the pluricanonical ring of an algebraic variety V (which is non-singular), or of a complex manifold, is a graded commutative ring that is made up of the sections of powers of the canonical bundle K. More precisely, it is the graded ring R such that for n ≥ 0 we have
- Rn = H0(V, Kn),
that is, the space of sections of the n-th tensor product Kn of the line bundle K (by definition the determinant bundle of the holomorphic cotangent bundle, in the case of a complex variety).
[edit] The plurigenera
The dimension
- h0(V, Kn)
is classically called the n-th plurigenus of V. By definition K is an ample line bundle if and only if the corresponding linear system of divisors, for n sufficiently large, gives an embedding of V in a projective space (of dimension one less than the n-th plurigenus). In general one has a n-canonical mapping to a projective space.
[edit] Kodaira dimension
The size of R and rate of growth of the Rn, which are of finite dimension for a complete variety, are basic invariants of V, and help to give the outline for the classification of algebraic varieties.
In the Kodaira dimension, named for Kunihiko Kodaira, of V is defined as one less than the transcendence degree of R, i.e.
- t − 1
where t is the number of algebraically independent generators one can find. Kodaira dimensions can take any value from 0 to the dimension of V.
Conventionally when R is the zero ring, which happens for example with V the projective line, the Kodaira dimension is set as −1. In contrast, for an elliptic curve K is a trivial bundle, meaning the plurigenera are all 1, and the Kodaira dimension is 0. The Kodaira dimension is the dimension of the image of the n-canonical mapping for n large enough.
[edit] General type
A variety of general type V is one with an n-canonical embedding as a projective variety; equivalently Kodaira dimension equal to its dimension. For example, complete non-singular algebraic curves are of general type if and only if they have genus ≥ 2. In contrast, for an elliptic curve K is a trivial bundle, meaning the plurigenera are all 1, and the Kodaira dimension is 0. The same is true for any abelian variety.
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.
