# Canonical ring

In mathematics, the **pluricanonical ring** of an algebraic variety *V* (which is non-singular), or of a complex manifold, is the graded ring

of sections of powers of the canonical bundle *K*. Its *n*th graded component (for ) is:

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

The 0th graded component is sections of the trivial bundle, and is one-dimensional as *V* is projective. The projective variety defined by this graded ring is called the **canonical model** of *V*, and the dimension of the canonical model is called the Kodaira dimension of *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.^{[1]}

## Properties[edit]

### Birational invariance[edit]

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

### Fundamental conjecture of birational geometry[edit]

A basic conjecture is that the pluricanonical ring is finitely generated. This is considered a major step in the Mori program. Caucher Birkar, Paolo Cascini, and Christopher D. Hacon et al. (2010) proved this conjecture.

## The plurigenera[edit]

The dimension

is the classically defined *n*-th * plurigenus* of

*V*. The pluricanonical divisor , via the corresponding linear system of divisors, gives a map to projective space , called the

*n*-canonical map.

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

## Notes[edit]

**^**Hartshorne (1975).*Algebraic Geometry, Arcata 1974*. p. 7.

## References[edit]

- Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James (2010), "Existence of minimal models for varieties of log general type",
*Journal of the American Mathematical Society*,**23**(2): 405–468, arXiv:math.AG/0610203, Bibcode:2010JAMS...23..405B, doi:10.1090/S0894-0347-09-00649-3, MR 2601039 - Griffiths, Phillip; Harris, Joe (1994),
*Principles of Algebraic Geometry*, Wiley Classics Library, Wiley Interscience, p. 573, ISBN 0-471-05059-8